Security Constrained Optimal Power Flow in a Mixed AC-DC Grid · Security Constrained Optimal Power...

eeh power systemslaboratory

Stephen Raptis

Security Constrained Optimal Power Flowin a Mixed AC-DC Grid

Master ThesisPSL 1305

EEH – Power Systems LaboratorySwiss Federal Institute of Technology (ETH) Zurich

Expert: Prof. Dr. Goran AnderssonSupervisors: Emil Iggland, Roger Wiget

Zurich, August 21, 2013

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Abstract

In recent years high voltage direct current transmission systems have becomepopular due to the economical and technical advantages they feature forsome applications. A proposal that seems very promising to be able to meetfuture demands of the world’s power systems is that of a mixed AC-DCgrid that is more heavily meshed. This mixed grid poses new challengesand aspects that have to be thoroughly addressed before it can be realized.Among the most critical issues in electric power transmission is the securityand reliability of the system. A good practice for power system operation isto ensure that the system is N-1 secure. The efficient solution of the security-constrained optimal power flow in AC grids is a problem with sufficientbackground research. On the other hand the solution of the SC-OPF in amixed grid is a problem with inadequate insight into it. This thesis presentsa formulation and solution of the SC-OPF in a linear fashion, and is anextension of the linear optimal power flow problem described in [1]. Themethod is applied to a test case and the results obtained are displayed andanalysed.

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Acknowledgments

I thank my supervisors Emil and Roger for the time and effort they put inthe project, and for the valuable input I had from them the past 7 months.

Zurich, August 21, 2013Stephen Raptis.

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Contents

List of Figures x

List of Tables xi

List of Acronyms xiii

List of Symbols xv

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Structure of Report . . . . . . . . . . . . . . . . . . . . . . . . 2

2 OPF and Security Constrained OPF 3

2.1 Basic OPF Concept . . . . . . . . . . . . . . . . . . . . . . . 32.2 Security Analysis and Principles . . . . . . . . . . . . . . . . 4

2.2.1 Remedial Actions . . . . . . . . . . . . . . . . . . . . . 42.2.2 Preventive Measures and Control . . . . . . . . . . . . 52.2.3 Corrective Measures and Control . . . . . . . . . . . . 5

2.3 SC-OPF Formulation . . . . . . . . . . . . . . . . . . . . . . . 52.3.1 Preventive Approach . . . . . . . . . . . . . . . . . . . 52.3.2 Preventive-Corrective Approach . . . . . . . . . . . . 62.3.3 Power System Security States . . . . . . . . . . . . . . 72.3.4 Trade-off between Generation Costs and Security . . . 8

3 System Modelling 11

3.1 Linear Optimal Power Flow . . . . . . . . . . . . . . . . . . . 113.2 Calculation of Active Power Flows in the AC Grid . . . . . . 113.3 Linear Optimal Power Flow in the Mixed Grid . . . . . . . . 12

3.3.1 Calculation of Active Power Flows in the DC Grid . . 123.3.2 Power Balance in the Mixed Grid . . . . . . . . . . . . 133.3.3 Matrix Formulation of Optimization Problem . . . . . 133.3.4 Optimization Vector ξ . . . . . . . . . . . . . . . . . . 133.3.5 Objective Function . . . . . . . . . . . . . . . . . . . . 143.3.6 Equality Constraints . . . . . . . . . . . . . . . . . . . 15

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viii CONTENTS

3.3.7 Inequality Constraints . . . . . . . . . . . . . . . . . . 17

4 Security Constrained OPF in the Mixed Grid 19

4.1 Method I: Preventive . . . . . . . . . . . . . . . . . . . . . . . 194.1.1 Line Outage Distribution Factors in the AC Grid . . . 204.1.2 Line Outage Distribution Factors in the DC Grid . . . 204.1.3 Generalized Generation Distribution Factors . . . . . 254.1.4 Line Constraints . . . . . . . . . . . . . . . . . . . . . 264.1.5 Formulation . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Method II: Preventive-Corrective . . . . . . . . . . . . . . . . 294.2.1 Objective Function . . . . . . . . . . . . . . . . . . . . 304.2.2 Post-Contingency Terminal Control . . . . . . . . . . 324.2.3 Line Outages in the AC Grid . . . . . . . . . . . . . . 334.2.4 Line Outages in the DC Grid . . . . . . . . . . . . . . 354.2.5 Terminal Station Outages . . . . . . . . . . . . . . . . 374.2.6 Generator Outages . . . . . . . . . . . . . . . . . . . . 384.2.7 Formulation of the Complete Problem . . . . . . . . . 40

5 Results-Case Studies 43

5.1 System Description - Case Study I . . . . . . . . . . . . . . . 435.2 Method I: Preventive . . . . . . . . . . . . . . . . . . . . . . . 44

5.2.1 Generation Profiles . . . . . . . . . . . . . . . . . . . . 445.2.2 Power Flow Distribution in the Mixed Grid . . . . . . 45

5.3 Method II: Preventive-Corrective . . . . . . . . . . . . . . . . 475.3.1 Costs Comparison . . . . . . . . . . . . . . . . . . . . 485.3.2 Interaction between AC and DC Grid . . . . . . . . . 525.3.3 Terminal Control Sensitivity Analysis . . . . . . . . . 545.3.4 DC Grid Capacity and System Performance . . . . . . 60

5.4 Solving Times . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.5 Case Study II . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6 Conclusion and Discussion 67

6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Appendices 69

A Numerical Example 69

B Example with Matrix Formulations 75

Bibliography 79

List of Figures

2.1 Security-State Diagram . . . . . . . . . . . . . . . . . . . . . 9

4.1 Line Before Outage . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Line After Outage . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Simulation of Line Outage Through Bus Injections . . . . . . 24

5.1 Combined AC and DC grid [1] . . . . . . . . . . . . . . . . . 445.2 Generation Profile for Preventive SC-OPF . . . . . . . . . . . 455.3 Distribution of Line Flows in the AC Grid . . . . . . . . . . . 465.4 Distribution of Line Flows in the DC Grid (Preventive) . . . 465.5 Terminal Station Power Transfers (Preventive) . . . . . . . . 475.6 Generation Profile for Prev-Corr SC-OPF for 100% Load Case

1: AC-DC line outages Case 2: AC-DC line and generatoroutages Case 3: AC-DC line and terminal outages Case 4:AC-DC line, terminal and generator outages . . . . . . . . . . 48

5.7 Generation Profile for Prev-Corr SC-OPF for 160% Load Case1: AC-DC line outages Case 2: AC-DC line and generatoroutages Case 3: AC-DC line and terminal outages Case 4:AC-DC line, terminal and generator outages . . . . . . . . . . 49

5.8 Operational Costs for 100% Load Case 1: AC-DC line outagesCase 2: AC-DC line and generator outages Case 3: AC-DCline and terminal outages Case 4: AC-DC line, terminal andgenerator outages . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.9 Operational Costs for 160% Load Case 1: AC-DC line outagesCase 2: AC-DC line and generator outages Case 3: AC-DCline and terminal outages Case 4: AC-DC line, terminal andgenerator outages . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.10 Linking of Preventive and Corrective SC-OPF . . . . . . . . . 525.11 Line Flows in the AC Grid for 100% Load Case 1: AC-DC

line outages Case 2: AC-DC line and generator outages Case3: AC-DC line and terminal outages Case 4: AC-DC line,terminal and generator outages . . . . . . . . . . . . . . . . . 53

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x LIST OF FIGURES

5.12 Line Flows in the DC Grid for 100% Load Case 1: AC-DCline outages Case 2: AC-DC line and generator outages Case3: AC-DC line and terminal outages Case 4: AC-DC line,terminal and generator outages . . . . . . . . . . . . . . . . . 54

5.13 Terminal Station Power Transfers for 100% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outagesCase 3: AC-DC line and terminal outages Case 4: AC-DCline, terminal and generator outages . . . . . . . . . . . . . . 55

5.14 Line Flows in the AC Grid for 160% Load Case 1: AC-DCline outages Case 2: AC-DC line and generator outages Case3: AC-DC line and terminal outages Case 4: AC-DC line,terminal and generator outages . . . . . . . . . . . . . . . . . 55

5.15 Line Flows in the DC Grid for 160% Load Case 1: AC-DCline outages Case 2: AC-DC line and generator outages Case3: AC-DC line and terminal outages Case 4: AC-DC line,terminal and generator outages . . . . . . . . . . . . . . . . . 56

5.16 Terminal Station Power Transfers for 160% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outagesCase 3: AC-DC line and terminal outages Case 4: AC-DCline, terminal and generator outages . . . . . . . . . . . . . . 56

5.17 Effects of Terminal Control on Costs: Case 1 . . . . . . . . . 575.18 Effects of Terminal Control on Costs: Case 2 . . . . . . . . . 585.19 Effects of Terminal Control on Costs: Case 4 . . . . . . . . . 595.20 Effects of Terminal Control for ΠAC = ΠDC : Case 2 . . . . . 595.21 Comparison of Absolute Costs for 2 levels of ΠAC ,ΠDC : Case 2 605.22 Effects of Terminal Control for ΠAC = ΠDC : Case 4 . . . . . 615.23 Dependency of Operational Costs from Capacities for Base

OPF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.24 Dependency of Operational Costs from Capacities for Case 2 625.25 Projection of figures 5.23 and 5.24 for comparison of depen-

dency on terminal and DC line capacities . . . . . . . . . . . 635.26 Calculation times for all cases . . . . . . . . . . . . . . . . . . 645.27 IEEE RTS-96 with interconnected DC grid . . . . . . . . . . 645.28 Effects of Terminal Control for Case 4 . . . . . . . . . . . . . 66

A.1 Small Test Grid . . . . . . . . . . . . . . . . . . . . . . . . . 70

B.1 3 Bus Test Grid . . . . . . . . . . . . . . . . . . . . . . . . . 76

List of Tables

3.1 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1 Nomenclature for Preventive-Corrective Method . . . . . . . 31

5.1 Cost of Security for the Cases studied for 100% Load . . . . . 505.2 Cost of Security for the Cases studied for 160% Load . . . . . 515.3 Cost of Security and Solving Times for Test Case of Figure 5.27 65

A.1 Generator Data . . . . . . . . . . . . . . . . . . . . . . . . . . 69A.2 AC Line Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 69A.3 DC Line Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 71A.4 Terminal Station Data . . . . . . . . . . . . . . . . . . . . . . 71A.5 Active Power Demand . . . . . . . . . . . . . . . . . . . . . . 71A.6 Decision Variable ξ for Base OPF and Case 1 . . . . . . . . . 72A.7 Line Flows for Base OPF and Case 1 SC-OPF . . . . . . . . 72A.8 Decision Variable ξ for Case 3 and Case 4 . . . . . . . . . . . 73A.9 Line Flows for Case 3 and Case 4 SC-OPF . . . . . . . . . . . 73

B.1 Nomenclature for Numerical Example . . . . . . . . . . . . . 75

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xii LIST OF TABLES

List of Acronyms

OPF optimal power flow.

DCOPF linearized optimal power flow.

SC-OPF security constrained optimal power flow.

LODF line outage distribution factor.

GGDF generalized generation distribution factor.

HVAC high voltage alternating current.

HVDC high voltage direct current.

VSC voltage source converter.

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xiv LIST OF ACRONYMS

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xvi LIST OF SYMBOLS

List of Symbols

AAC line adjacency matrix of AC grid.

ADC line adjacency matrix of DC grid.

AACprev matrix of AC line outage sensitivities for preventive method.

ADCprev matrix of DC line outage sensitivities for preventive method.

BAC AC network admittance matrix.

BDC DC network admittance matrix.

C length of vector ξ.

D number of DC lines.

D pairs of DC buses connected with a line.

EAC AC line adjacency matrix for generator outages of preventive method.

EI matrix of generator outage sensitivities for preventive method.

Fmaxkm limit on power flowing on line km.

G matrix of quadratic cost entries.

H generator allocation matrix.

I number of generators.

K number of AC nodes.

L number of terminals.

M number of DC nodes.

N number of AC lines.

N pairs of AC buses connected with a line.

PG generator active power output.

PmaxG generator maximum power output.

PminG generator minimum power output.

PL active power demand.

PT terminal station power transfer.

PmaxT,i limit on terminal power transfer.

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Pbase base power for per unit representation.

Pkm power flowing on line km.

QC generator quadratic costs diagonal matrix.

Rkm resistance of DC line connecting bus k to bus m.

S terminal allocation matrix for DC buses.

T terminal allocation matrix for AC buses.

U AC bus voltage magnitude.

V DC bus voltage.

WAC DC voltage difference penalty matrix.

WDC AC voltage angle difference penalty matrix.

X reactance matrix of AC network.

Y number of inequality constraints.

Z impedance matrix of DC network.

c set of single-element contingencies that are accounted for.

f objective function.

g set of equality constraints

h set of inequality constraints.

nij number of lines connecting bus i to j.

ri,m percentage of power of generator m, taken on by generator i.

u vector of control variables.

∆umaxc amount of corrective control actions that can be implemented.

xkm reactance of AC line connecting bus k to bus m.

z vector of state dependent variables.

ΛI identity matrix of size I.

ΛL identity matrix of size L.

ΠAC penalty for AC angle differences.

ΠDC penalty for DC voltage differences.

ΠTer penalty for changes in terminal power transfer.

Φiq matrix of post-contingency generator bounds.

Φeq matrix of post-contingency generator outputs.

Ψ distribution vector for generator outages.

α linear coefficient of generator cost.

β quadratic coefficient of generator cost.

xviii LIST OF SYMBOLS

γ total amount of contingencies accounted for.

δ AC bus voltage angle.

λ vector of linear cost coefficients.

ξ decision variable.

Chapter 1

Introduction

1.1 Motivation

Power systems nowadays operate in more strained conditions than were ex-pected during their planning stage [2]. The development of distributed gen-eration from renewable sources together with the introduction of electricitymarkets have resulted in uncertainty regarding operating conditions. Steadyincrease in load has to be met with appropriate upgrades of the generationand transmission systems.

A scheme that looks promising in dealing with these challenges is thehigh voltage direct current (HVDC) transmission system. HVDC links areideal for transferring large amounts of power over long distances and forconnecting renewable power infeeds to the grid. Recent advances in thevoltage source converter VSC-HVDC systems also offer potential high levelsof power controllability. Therefore HVDC technology will play a significantrole both in future planning and operation of power systems. As the numberof these point to point HVDC connections increases it becomes more sensibleto connect them directly and not through the vaster AC grid. This kind ofscheme could introduce additional flexibility to power systems and facilitatepower exchanges and trading between systems. These concepts give rise tothe DC grid. Though there are no technology gaps in small HVDC projects,large inter-regional grids still lack adequate research regarding, among oth-ers, power flow control and system security.

The secure and efficient operation of such a grid poses many difficulties.The system operator must ensure that the system is operating under thesatisfactory limits. A solution to this problem is known as the Security-Constrained Optimal Power Flow (SC-OPF) and is a problem that is con-sidered to have substantial research behind it for AC grids. On the otherhand, the solution of the SC-OPF in the DC grid is a problem that lackssufficient research. The goal of this thesis is to provide a means by which thesecurity-constrained optimal power flow can be applied to a mixed AC-DC

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2 CHAPTER 1. INTRODUCTION

network.

1.2 Structure of Report

In chapter 2 the formulation and basic concept of the optimal power flow isoutlined. Fundamental theory regarding security analysis is then presentedfollowed by extended formulations of the OPF problem to encompass secu-rity constraints.

In Chapter 3 the modelling of the AC and DC grid is presented. Theformulas for power flow calculations are derived preceded by an analyticaldescription of the linear OPF model for the mixed grid [1].

In chapter 4 two methods for the implementation of the SC-OPF in amixed AC-DC are developed and presented.

Chapter 5 contains all results from simulations performed on a mixedgrid test case.

Finally chapter 6 is reserved for conclusions and evaluations. A discus-sion is made about future improvements and possible implementations thatcan be augmented to the existing models.

Chapter 2

OPF and Security

Constrained OPF

In this chapter the basic concepts of the optimal power flow and security-constrained optimal power flow are outlined.

2.1 Basic OPF Concept

Power system optimization is a field that has progressed a lot in the pastcentury. Early in the 20th century optimal power flow (OPF) was a prob-lem engineers had to deal with using their experience and judgement. Withgradual advances in mathematical optimization and computational tools,OPF is now a problem that is solved several times a day in control stations.Even though the problem has been addressed for over 50 years, the complex-ities that arise in power system operation and control always create roomfor further improvement and new implementations.

OPF refers to the class of problems that find the optimal solution to anobjective function subject to the power flow constraints and other opera-tional constraints. It is generally a problem of “best” generation dispatch.The formulation of the OPF in compact notation [3] is the following:

Minimize f(z, u) (2.1)

subject to g(z, u) = 0 (2.2)

h(z, u) ≤ 0 (2.3)

The objective of function f(z, u) of equation (2.1) is usually the mini-mization of generation costs or of system active power losses, though differentobjectives can be specified depending on the application.

Vectors u and z consist of the control and state dependent variables re-spectively. Control variables u are all quantities that can be modified to

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4 CHAPTER 2. OPF AND SECURITY CONSTRAINED OPF

satisfy the power balance under consideration of the system limits. Theset of state dependent variables z contains all the variables that depend onthe state the system is in. Equation (2.2) represents the set of equality con-straints. Equality constraints express the power flow equations as well as thepower balance between generation-losses-load and have to be uncondition-ally satisfied. The inequality constraints of equation (2.3) depict networkoperating limits and limits on control variables. Such limitations are for in-stance the thermal capacities of transmission devices or the maximum activepower output of a generator.

2.2 Security Analysis and Principles

Apart from operating in an economically optimal way, it is crucial for apower system to operate with respect to certain criteria that ensure securityand reliability. The first concept mainly refers to the operation under thesatisfactory limits to avoid damage of equipment. A reliable system opera-tion is the situation where all loads are supplied without disruptions.

For these criteria to be met the system operator has to perform securityanalysis in order to know how robust the system is with respect to variouspossible contingencies. The fundamental goal of security analysis is to ensurea steady supply of power to all the loads of a system without disruptions.In other words it is performed to guarantee that an unexpected outage of anelement of the system will not lead to an uncontrollable cascading failure.The formulation of the OPF problem presented in section 2.1 does not takeinto account contingency events. According to ENTSO-E [4] “a contingencyis defined as the trip of one single of several network elements that cannotbe predicted in advance. A scheduled outage is not a contingency”. Theoptimization problem that takes into consideration contingency scenarios iscalled the Security Constrained-Optimal Power Flow (SC-OPF). The SC-OPF is an augmentation of the OPF through the inclusion of additionalconstraints that guarantee system security in the event of a contingency.

2.2.1 Remedial Actions

Remedial actions are the means applied by a transmission system operatorto ensure security of the power transmission grid. They refer to changes inthe settings of controllable quantities of the system that achieve this goal.For the scope of the project the following types of remedial actions are goingto be of concern:

• Preventive remedial actions

• Corrective remedial actions

2.3. SC-OPF FORMULATION 5

2.2.2 Preventive Measures and Control

Preventive control is a measure taken to provide for a need that might occurin the case a contingency happens, due to uncertainty of being able to han-dle the contingency constraints once they have occurred [4]. In other wordsit is an action that is made while still in normal operation. The aim is toplace the power system in a state, that will not suffer violations of operatinglimits following the occurrence of a contingency.

The preventive approach is a conservative one, since the system is posi-tioned in a state that has higher operational costs, to guarantee it is secureagainst the effects of a contingency that might or might not occur. It isnot concerned with possible actions that could be implemented in real-timeoperation after the contingency has occurred.

2.2.3 Corrective Measures and Control

Corrective control describes actions needed to clear violations of physicallimits after a contingency has occurred. Corrective control measures do nottake into account pre-contingency conditions and settings which means thatpost-contingency controls must move in order to satisfy post-contingencyconstraints. In the corrective-secure state the system might be suffering fromconstraint violations, thus an immediate concern of the corrective approachis the time a corrective action requires. For the corrective actions that areimplemented in the following chapters, it is assumed that the time requiredfor such actions do not endanger the system. In other words, under nocircumstances will the system enter a state of non-correctable emergencybecause a corrective action takes too long to have effect.

2.3 SC-OPF Formulation

The concept of the SC-OPF is to augment the initial OPF problem withadditional constraints that relate to contingency states or to the effects anoutage of an element would have on the system. Two approaches that leadto different formulations are presented in the following sections.

2.3.1 Preventive Approach

The preventive SC-OPF is the problem described by the following equations[5]:

Minimize f(z0, u0) (2.4)

subject to gc(zc, u0) = 0 c=0,1,2,...,γ (2.5)

hc(zc, u0) ≤ 0 c=0,1,2,...,γ (2.6)


where subscript “0” represents the pre-contingency(base-case) state beingoptimized, and subscript “c” (c > 0) represents the post-contingency statesfor the γ contingency cases that are selected.

Therefore this formulation addresses pre-contingency but also the set “c”of post-contingency constraints. The fact that the set of equations (2.5) and(2.6) depend on controls u0 instead of uc, for all post-contingency states,makes this formulation a preventive SC-OPF. Control levels are restrictedto their pre-contingency condition settings even in the post-contingency sit-uations. The set of post-contingency state variables is represented by zc.

Each set of contingency-related equality constraints is like the set ofequality constraints of the base OPF, only it corresponds to the systemwith one element removed. The sets of inequality constraints of contingencycases are like the equivalent constraints of the base OPF, except that thesystem has one element less and the limits for line flows might be different.

2.3.2 Preventive-Corrective Approach

The preventive-corrective SC-OPF can be stated in the following sets ofequations [5]:

Minimize f(z0, u0) (2.7)

subject to g0(z0, u0) = 0 (2.8)

h0(z0, u0) ≤ h0max (2.9)

gc(zc, uc) = 0 c=1,2,...,γ (2.10)

hc(zc, uc) ≤ hcmax c=1,2,...,γ (2.11)

|u0 − uc| ≤ ∆ucmax c=1,2,...,γ (2.12)

where subscript “0” represents the pre-contingency state and subscript “c”(c > 0) represents the post-contingency states for the γ contingency casesthat are addressed. The term corrective is derived from the fact that post-contingency controls uc are allowed to move in order to satisfy(correct) post-contingency constraints. The problem is preventive-corrective because theamount of corrective control actions that can be expended is not unlimited.It is bound by an amount ∆uc

max and also by the pre-contingency controlsetting u0 as in equation (2.12). If the amount of expendable corrective ac-tions do not suffice to satisfy all post-contingency constraints, the algorithmresorts to preventive rescheduling of pre-contingency controls u0.

The vector of maximum allowed adjustments ∆ucmax is given by the

following equation:

∆ucmax = Tc ·

ducdt

(2.13)


Tc is the time available for corrective actions after a contingency has occurredand duc

dt is the rate of response of controls to a contingency. It is thereforeevident that corrective control is heavily concerned with the amount of timean action requires to be implemented.

In the case studies that are considered, the quantity ∆ucmax is going

to be chosen as an input for sensitivity analysis purposes. It is the sys-tem operator’s responsibility to know what amount of corrective actions isexpendable or to develop mechanisms to increase this amount.

2.3.3 Power System Security States

In the following chapter two methods for solving the SC-OPF are goingto be formulated. The first method is fully preventive while the second ispreventive-corrective. To clarify the fundamental goals of each method andlink the concepts associated with them to real operating situations, figure2.1 is presented. Several operating conditions are depicted, and have certainsecurity-related states attributed to them.

The goal when applying a preventive method is to constantly maintainthe system in the “Secure” state. In this state it is guaranteed that thesystem will suffer no violations of inequality constraints or loss of load inthe event of a contingency. It is the best level that can be achieved froma security perspective. Under certain circumstances the system can movefrom the “Secure” to the “Alert” state. An increase in power demand forinstance can trigger this transition. In the “Alert” state the system is stillnot suffering from any constraint violations, but a contingency would lead tooperating constraints being violated. These violations would not be able tobe corrected without cutting power supply to some loads of the system. Toreturn to the “Secure” state the system operator must implement preventivecontrol actions.

The preventive-corrective approach aims to keep the system in the “Cor-rectively Secure” state. In this state it is guaranteed that any possible vi-olations caused by a contingency can be quickly cleared through correctivecontrol actions without any further consequences. The system operator per-forms security analysis in order to develop appropriate control strategiesthat can be flexibly applied under a contingent scenario. If the availableamount of corrective actions is not enough to clear all post-contingency vio-lations, the system moves to the “Alert” state. Preventive measures are thenrequired to ensure the extra level of security that corrective control cannotprovide.

In the correctively secure state a contingency will trigger a transitionto the correctively secure state. Would-be violations are rapidly clearedthrough implementation of corrective actions. To provide security againstfurther contingencies though, preventive measures are needed or the systemrisks suffering a non-correctable emergency.


Cases where the system is in a restorative mode after loss of load hasbeen suffered are out of the scope of this work and are not discussed.

2.3.4 Trade-off between Generation Costs and Security

The operating costs of a SC-OPF dispatch are always larger or equal tothose of a normal OPF. To achieve a level of security the operating point ofa power system moves away from the most profitable point. This happensbecause the dispatching that is defined by a SC-OPF is more distributedamong the participating generators of the system. Cheap generating unitshave to reduce their output to avoid congestion of lines of the system. Thedifference is compensated by more expensive generators and therefore lightlyloaded lines of the base OPF are utilized more in the SC-OPF. Enhancedsecurity comes at the expense of additional operating costs. If P0, Pp and Pc

are used to denote the standard OPF, preventive and preventive-correctiveSC-OPF respectively, then the relationships for system costs are:

Cost(P0) ≤ Cost(Pc) ≤ Cost(Pp)

The difference in cost between the standard OPF and the SC-OPF has beentermed “Cost of Security” [6] and expresses the additional costs required toguarantee system security. It is an index of the “price to pay” to achieve acertain level of security and will be used later to quantify and evaluate thecosts induced by the SC-OPF.


All load supplied. No

operating limits violated. In

the event of a contingency

there will be no violations.

Secure

All load supplied. No

operating limits violated. Any

violations caused by a

contingency can be corrected

by appropriate control action

Correctively

Secure

All load supplied. No operating

limits violated. In the event of a

contingency there will be

violations of inequality

constraints that can’t be

corrected without loss of load

Alert

All load supplied, any

violation is quickly cleared

through corrective actions.

Correctively

Alert Operating limits violated. The

violations cannot be corrected

without loss of load.

Non-Correctable

Emergency

Preventive Method

Transition due to reduction in

reserve margins or high

probability of disturbance

Transition through

implementation of preventive

actions

Transition due to a

contingency event

Preventive-Corrective

MethodTransition due to reduction

in reserve margins or high

probability of disturbance

Transition through

implementation of

preventive actions

Transition through

implementation of

preventive and corrective

actions

Transition due to a

contingency event

un

inte

ntio

na

l

de

libe

rate

un

inte

ntio

na

ld

elib

era

te

Figure 2.1: Security-State Diagram


Chapter 3

System Modelling

This chapter introduces the linear optimal power flow problem along withthe theoretical background that supports this modelling. Its application on amixed AC-DC grid is then presented.

3.1 Linear Optimal Power Flow

The OPF problem stated in section 2.1 is a non-linear, non-convex problem.As the size of the problem increases the OPF calculation requires substan-tial computational effort. This fact often makes the OPF unattractive forcertain applications and implementations. A good alternative in these casesis to use the so called DCOPF modelling. The DCOPF model is based onapproximations that lead to a linear version of the OPF problem. The ap-proximations and simplified power flow equations that are derived can befound in [7].

The DCOPF model features many advantages [8]. The solutions ob-tained from it are reliable and the complexity of the problem significantlyreduced compared to the full OPF. The solvers of such problems are verypowerful and fast, while they guarantee convergence to the global optimum.Active power flows that are calculated are considerably accurate and themodel is especially powerful when it comes to contingency analysis.

The following sections explain the fundamental concepts that lead to alinear formulation of the OPF.

3.2 Calculation of Active Power Flows in the AC

Grid

Power flowing on an AC line is given by the following set of non-linearequations [9]:

Pkm = U2kGkm − UkUmGkmcos(δk − δm)− UkUmBkmsin(δk − δm)] (3.1)

11

12 CHAPTER 3. SYSTEM MODELLING

Qkm = −U2k (Bkm+Bsh

km)+UkUmBkmcos(δk− δm)−UkUmGkmsin(δk− δm)](3.2)

Uk, Um stands for the voltage magnitude at bus k,m, δ for the voltage angle,andGkm, Bkm for the shunt conductance and susceptance respectively. Withthe approximations made in [7] the linear formula for power flow calculationin the AC grid is derived. These assumptions are the following:

• All voltage magnitudes are assumed to be equal to the base voltage,i.e Un = 1 p.u ∀ n ∈ 1, 2, ...K.

• All branch resistances are neglected (RkmAC = 0). It follows thatGkm = 0.

• No reactive power quantities are considered (Qkm = 0).

• Bus voltage angles are assumed very small to make the formula |sin(α)| ≈α valid.

With these assumptions the steady-state power flowing from bus k to m isgiven by the linear equation:

PACkm ≈

δk − δmxkm

(3.3)

xkm being the reactance of the line connecting bus k and m and δk, δm thebus voltage angles in radians.

3.3 Linear Optimal Power Flow in the Mixed Grid

The linearized OPF that includes DC grids was conceived and formulated in[1]. The work carried out in the present thesis is based on this formulationand extends it to construct a problem that incorporates security assessments.For consistency purposes this work [1] is thoroughly documented in sections3.3.1 - 3.3.7.

3.3.1 Calculation of Active Power Flows in the DC Grid

The active power flowing on a DC line can be calculated through equation(3.4):

PDCkm =

Vk · (Vk − Vm)

Rkm(3.4)

which is non-linear. To incorporate power flow equations in the DC gridto the linearized OPF problem a suitable approximation is in order. If itis presumed that all voltages at DC nodes are close to their nominal valueof 1 p.u, then equation (3.4) can be approximated by the following linearequation:

PDCkm ≈

(Vk − Vm)

Rkm(3.5)

3.3. LINEAR OPTIMAL POWER FLOW IN THE MIXED GRID 13

Vk, Vm are the voltages of DC nodes k,m and Rkm is the resistance of theDC line connecting them.

3.3.2 Power Balance in the Mixed Grid

As is the case for the AC grid, active power losses on the DC grid areassumed to be zero. Hence the power balance equations for AC and DCnodes are:

PLk =∑

PGk −N∑

n

Pkm −∑

PTk for all AC nodes (3.6)

PLk =∑

PGk −M∑

n

Pkm +∑

PTk for all DC nodes (3.7)

PLk, PGk and PTk represent the load, generation infeed and terminalpower transfer at node k. The sings of terms PTk for the AC and DC nodescan be assigned arbitrarily, as long as they have opposite signs to one anotherso that the power balance holds.

3.3.3 Matrix Formulation of Optimization Problem

The DC optimal power flow problem can be constructed with the use ofmatrices in the following form:

Minimize f(ξ) =1

2· ξT ·G · ξ + λ · ξ (3.8)

subject to

Ceq · ξ = beq (3.9)

Ciq · ξ ≤ biq (3.10)

Problem (3.8) - (3.9) constitutes a linearly constrained optimizationproblem with quadratic objective function and can be solved as a quadraticprogramming problem.

3.3.4 Optimization Vector ξ

Vector ξ in the mixed DCOPF contains variables of both grids:

ξ = [PG1 PG2 ... PGI δ2 δ3 ... δK

PT1 PT2 ... PTL VDC2 VDC3 ... VDCM ]T[C×1]


where the subscripts introduced comply to the nomenclature of table 3.1.The set of control variables consists of all active power generations PG andterminal power transfers PT , whereas the set of state dependent variablescontains all AC voltage angles δ and all DC voltages VDC . To solve the powerflow equations only the relative difference in angles and relative differencein DC voltages are needed, therefore a slack bus in both the AC and DCgrid can be assigned. In the formulations presented AC bus 1 and DC bus1 are chosen as slack buses for the AC and DC grid. This way the angle ofthe AC slack bus and the Voltage of the DC slack bus is removed from ξ.

Table 3.1: Nomenclature

K Number of AC nodes

N Number of AC lines

M Number of DC nodes

D Number of DC lines

I Number of generators

L Number of terminals

C I+K+L+M-2 Length of vector ξ

Y 2(I+N+L+D) Amount of inequality constraints

3.3.5 Objective Function

The objective of the proposed algorithm is the minimization of generationcosts. Therefore the most important component of the objective functionis the quadratic and linear cost of all generators. Penalties for AC angledifferences and DC voltages are introduced for stability purposes and forassessing the AC-DC interaction.

I∑

i=1

[α ·PGi+β ·P 2Gi]+ΠAC ·

∑

km∈N

[δk−δm]2+ΠDC ·∑

km∈D

[VDCk−VDCm]2

(3.11)

N= Pairs of AC buses connected with a branch.D= Pairs of DC buses connected with a branch.α = linear coefficient of generator cost.β = quadratic coefficient of generator cost.


Matrix G of equation (3.8) and its components are constructed as:

G =

QC 0 0 0

0 WAC 0 0

0 0 0[L×L] 0

0 0 0 WDC

[C×C]

QC =

2β1 0 0 0

0 2β2 0 0

0 0. . . 0

0 0 0 2βI

[I×I]

WAC =

∑

k 6=2

Jk2 −J23 · · · −J2K

−J32∑

k 6=3

Jk3 · · · −J3K

......

. . ....

−JK2 −JK3 · · ·∑

k 6=K

JkK

[K−1×K−1]

WDC =

∑

k 6=2

Ik2 −I23 · · · −I2M

−I32∑

k 6=3

Ik3 · · · −I3M

......

. . ....

−IM2 −IM3 · · ·∑

k 6=M

IkM

[M−1×M−1]

regarding the entries of matrix WAC :

Jkm =

1 if there is a line connecting bus k and m

0 otherwise

The entries of WDC are derived in a similar way. The diagonal entry ofmatrix G that consists of zeros means there is no application of costs forterminal operation. Vector λ of equation (3.8) contains the generator linearcoefficients:

λ =[

α1 α2 ... αI 0[1×(C−I)]

]

(3.12)

3.3.6 Equality Constraints

In the mixed grid the equality constraints are practically the equations thatexpress the power balance at AC and DC nodes. Ceq of equation (3.9) con-tains the appropriate matrices to represent equality constraints and are pre-sented below. Matrix H defines the generated in-feed at AC nodes throughthe allocation of generators at AC nodes:

H[K×I] (3.13)


Hij =

1 if generator j is connected at node i

0 otherwise(3.14)

The power flowing on AC lines can be expressed in matrix form:

PAC1

PAC2...

= [BAC ] ·

δ1

δ2...

(3.15)

The admittance matrix BAC is constructed in the following way:

BAC =

−B12 −B13 · · · −B1K∑

k 6=2

Bk2 −B23 · · · −B2K

.... . . · · ·

......

.... . .

...

−BK2 −BK3 · · ·∑

k 6=K

BkK

[K×(K−1)]

(3.16)

Bkm =

1xkm

if AC nodes k,m have a line connecting them

0 otherwise(3.17)

The first column of the admittance matrix is removed due to the assignmentof an AC slack bus, in this case AC bus 1 is chosen as the slack bus. Thepower flow equations in (3.5) for the DC grid take on the following form:

PDC1

PDC2...

= [BDC ] ·

V1

V2

...

(3.18)

The equivalent matrix for the DC grid is constructed similarly to result inBDC[M×(M−1)], where the entries are:

Bkm =

1Rkm

if DC nodes k,m have a line connecting them

0 otherwise(3.19)

The next matrices to be formulated indicate the interconnection of theAC-DC grid through the allocation of the terminals at buses.

T[K×L] : Allocation of terminal stations at AC buses (3.20)


where

Tpq =

−1 if terminal q is connected at AC bus p

0 otherwise

S[M×L] : Allocation of terminal stations at DC buses (3.21)

Spq =

1 if terminal q is connected at DC bus p

0 otherwise

Matrix T allocates terminals to AC nodes and matrix S allocates them toDC nodes. For the power balance to hold, elements of T and S have oppositesigned elements. The power flowing on one side of the terminal must equalthe power flowing on the other side since no losses are considered. With allmatrices defined the equality matrix Ceq is constructed in the following wayand represents equations (3.6) and (3.7):

Ceq =

H −BAC T 0

0 0 S −BDC

[(K+M)×C]

(3.22)

Constant vector beq of equation (3.9) contains the loads connected at thebuses of the mixed grid:

beq = [PL,1PL,2...PL,(K+M)]T[(K+M)×1] (3.23)

The first row of Ceq represents the power balance at all AC nodes:

∑

PGi −∑

km∈N

BACkm [δk − δm]−

∑

PTk =∑

PLi (3.24)

while the second row represents the power balance at all DC nodes:

−∑

km∈D

BDCkm [Vk − Vm] +

∑

PTk =∑

PLi (3.25)

3.3.7 Inequality Constraints

The inequality constraints for the system modelling that is implemented arestated below:

|PACkm | ≤ Fmax

AC ∀ AC lines (3.26)

|PDCkm | ≤ Fmax

DC ∀ DC lines (3.27)

|PT,i| ≤ FmaxT,i ∀ terminals i (3.28)

PminG,i ≤ PG,i ≤ Pmax

G,i ∀ generators i (3.29)


To obtain these inequality constraints the following matrices are con-structed:

AAC =

J12 · · · J1K

J22 · · · J2K

.... . .

...

JN2 · · · JNK

[N×(K−1)]

(3.30)

Jij =

BAC if line i starts at node j

−BAC if line i ends at node j

0 otherwise

(3.31)

The analogous matrix for the DC grid is constructed with the samephilosophy leading to ADC[D×(M−1)]. Then Ciq is formulated as follows:

Ciq =

ΛI 0 0 0

−ΛI 0 0 0

0 AAC 0 0

0 −AAC 0 0

0 0 ΛL 0

0 0 −ΛL 0

0 0 0 ADC

0 0 0 −ADC

[Y×C]

(3.32)

ΛI and ΛL are identity matrices. The first two rows of the matrix corre-spond to the generating limits of units. The 3rd and 4th rows represent thelimits for AC line flows and the next two rows bound the terminal powertransfers. The last two rows correspond to the limits for DC line flows.

Vector biq contains all upper and lower bounds as shown in the start ofsection 3.3.7, and is constructed by the following components:

bmaxg =[Pmax

G1 PmaxG2 ... Pmax

GI ]T[I×1] (3.33)

bming =[Pmin

G1 PminG2 ... Pmin

GI ]T[I×1] (3.34)

bl,AC =[Pl1 Pl2 ... PlN ]T[N×1] (3.35)

bl,DC =[Pl1 Pl2 ... PlO]T[D×1] (3.36)

bterm =[Pt1 Pt2 ... PtL]T[L×1] (3.37)

biq = [bmaxg bmin

g bl,AC − bl,AC

bterm bterm bl,DC − bl,DC ]T[Y×1]

Chapter 4

Security Constrained OPF in

the Mixed Grid

In this chapter the formulation of 2 methods for the application of SC-OPFin the mixed grid are presented, along with the methodology that is used tosupport the modellings implemented. The term SC-OPF will be used to referto the linearized model of the security constrained optimal power flow.

4.1 Method I: Preventive

The preventive approach to the security constrained optimal power flow isbased on the notion that the effects of a possible contingency should be an-ticipated and dealt with while in the normal state of the system. To achievethis the operator of a power system is required to do contingency analysisand take the necessary measures to ensure system security.

One way to proceed is to run a power flow simulation for every contin-gency scenario considered and assess the results. Any violations monitoredare registered and dealt with appropriately. This approach can prove ineffi-cient and computationally forbidding when dealing with large systems.

The use of a DCOPF model offers a vantage point when it comes tocontingency analysis, because the effect of a contingency can be approxi-mately quantified in a pre-determined way. Constraints for line and gen-erator outages can be included in a single optimization problem as linearsensitivities, providing a straightforward approach and notably reducing thecomputational effort required. The preventive method that is developedand presented takes into account all AC and DC line contingencies as wellas generator contingencies.

19

20CHAPTER 4. SECURITYCONSTRAINEDOPF IN THEMIXED GRID

4.1.1 Line Outage Distribution Factors in the AC Grid

When an AC line outage occurs there is a redistribution of the power initiallyflowing on that line. The percentage of this power taken up by each of theremaining lines is calculated through the line outage distribution factors(LODF) [10]. The LODFij,km denotes the fraction of the power initiallyflowing on line k −m, that is flowing on line i− j after line r − s has beenoutaged. It is calculated as:

LODFACij,km =

nkm · xkmnij · xij

·(Xik −Xim −Xjk +Xjm)

[nrs · xkm − (Xkk +Xmm − 2Xkm)](4.1)

xij : reactance of line i− j

Xij : entry in the ith row and jth column of the bus reactance matrix X

nij : number of lines connecting bus i and j

The admittance matrix BAC is singular and can’t be inverted. By re-moving the row and column that corresponds to the slack bus the reducedmatrix Bred

AC [K − 1 ×K − 1] can be inverted. The resulting matrix is thenaugmented by a row and column of zeros to result in reactance matrix X.

BredAC = BAC(2 : K, 2 : K) (4.2)

X = inv(BredAC) (4.3)

With the use of the LODFs the total power flowing on line i − j afterthe outage of line k −m has occurred can be calculated as:

Pij,1 = Pij,0 + LODFACij,km · Pkm,0 (4.4)

where subscripts 0, 1 stand for pre and post-contingency conditions.

4.1.2 Line Outage Distribution Factors in the DC Grid

Equation (3.18) can be written in the following way if multiplied with[BDC ]

−1:V = [Z] ·P (4.5)

where V stands for the vector of DC voltage entries, P stands for thepower injections in the DC grid and Z is the impedance matrix derived fromBDC . Matrix BDC is singular because it refers to a set of linearly dependentequations and hence is not invertible. By declaring a reference value ofvoltage at a chosen DC slack bus, the row and column corresponding tothat bus are removed from BDC . Inversion of the reduced matrix is then

4.1. METHOD I: PREVENTIVE 21

performed while a row and column of zeros are added to result in matrix Z

of equation (4.5). Thus if one assigns bus 1 as the slack bus matrix Z is:

Z =

0 | · · · · · · 0K

− | − − −... |... | B−1

DC

0K |

(4.6)

To calculate changes in voltages for a given set of changes in bus powerinjections the next equation can be used

∆V = [Z] ·∆P (4.7)

Suppose that bus i has a 1 p.u increase in power injection that is equallycompensated by a 1 p.u decrease in the reference bus. Then the ∆V valuesare equal to the derivative of the voltages with respect to a change in powerinjection in bus k. The sensitivity factors for this case are

ω =dfkmdPi

=d

dPi

[

Vk − Vm

Rkm

]

=1

Rkm

[

dVk

dPi−dVm

dPi

]

=1

Rkm(Zki−Zmi) (4.8)

where:

dfkm : change in flow on line km

dPi : incremental change in power injection at bus i

Zki =dVk

dPi: element k-i of matrix Z

Zmi =dVm

dPi: element m-i of matrix Z

Rkm : resistance of line km

Figure 4.1 shows line k − m under normal conditions while figure 4.2shows the same line once it has been dropped. In a situation where linek − m is out the circuit breakers are open as in figure 4.2. In this case nocurrent is flowing through the breakers and line k − m is totally isolatedfrom the rest of the system.

A line outage can be modelled by adding a pair of injections at thesystem, one at each bus corresponding to the line that is out, as seen infigure 4.3. The breakers are closed but injections ∆Pk and ∆Pm are addedto bus k and m respectively. If ∆Pk = P 1

km and ∆Pm = −P 1km then no

power is flowing through the breakers despite the fact that they are closed.Therefore for the rest of the network it is as though line k −m is outaged.


Bus k

Bus m

Pkm

Before

Outage

Figure 4.1: Line Before Outage

Now equation (4.7) can be used to obtain the effects of the injections at busk,m.

∆Vk = Zkk∆Pk +Zkm∆Pm (4.9)

∆Vm = Zmk∆Pk + Zmm∆Pm (4.10)

The following notations are now defined:

Vk, Vm, Pkm : to be the values before the outage

∆Vk,∆Vm,∆Pkm : to be the incremental changes

V 1k , V

1m, P 1

km : to be the values after the outage

So for the case where line km is out it follows that

P 1km = ∆Pk = −∆Pm (4.11)


Bus k

Bus m

After

Outage

Figure 4.2: Line After Outage

and

P 1km =

V 1k − V 1

m

Rkm(4.12)

Equations (4.9) and (4.10) are written

∆Vk = (Zkk −Zkm)∆Pk (4.13)

∆Vm = (Zmm −Zmk)∆Pk (4.14)

and with use ofV 1k = Vk +∆Vk (4.15)

V 1m = Vm +∆δm (4.16)

P 1km =

Vk − Vm

Rkm+

∆Vk −∆Vm

Rkm(4.17)


Bus k

Bus m

Simulation of

Outage

P1km

ΔPm

ΔPk

Figure 4.3: Simulation of Line Outage Through Bus Injections

which can also be written like

P 1km = Pkm + (Zkk +Zmm − 2Zkm)∆Pk (4.18)

P 1km can be replaced by ∆Pk according to equation (4.11)

∆Pk =

[

1

1− (Zkk+Zmm−2Zkm)Rkm

Pkm

]

(4.19)

If a sensitivity factor θ is defined as the ratio of change in voltage any-where in the system towards the initial flow Pkm flowing over a line

θi,km =∆Vi

Pkm(4.20)

then changes in bus voltage is:

∆Vi = Zik∆Pk +Zim∆Pm (4.21)


Using the relationships for ∆Pk and ∆Pm from equation (4.19) the sensitiv-ity factor is:

θi,km =(Zik −Zim)Rkm

Rkm − (Zkk +Zmm − 2Zkm)(4.22)

By definition the line outage distribution factor is described by the followingequation:

LODFij,km =∆Fij

F 0km

(4.23)

LODFij,km : line outage distribution factor when monitoring line i− j after

the outage of line k −m

∆Fij : change in flow on line i− j

F 0km : original flow on line k −m before the outage

expanding the expression gives:

LODFij,km =∆Fij

F 0km

=

∆Vi−∆Vj

Rij

F 0km

=1

Rij

(

∆Vi

Pkm−

∆Vj

Pkm

)

=1

Rij(θi,km − θj,km) (4.24)

and by applying equation 4.20 the final equation is reached:

LODFDCij,km =

Rkm

Rij·

(Zik −Zim −Zjk +Zjm)

Rkm − (Zkk +Zmm − 2Zkm)](4.25)

By applying equation (4.25) the power flowing on line i−j due to the outageof line k −m is:

Pij,1 = Pij,0 + LODFDCij,km · Pkm,0 (4.26)

4.1.3 Generalized Generation Distribution Factors

The redistribution of line flows during a generator contingency can be cal-culated with the use of the Generalized Generation Distribution Factors(GGDFs) [11], [12]. GGDFm

ij describes the fraction of generation of unit mbefore it goes out, that flows on line i− j after the outage has occurred:

GGDFmij =

1

xij·Eij · [BAC ]

−1 ·Ψm,i (4.27)

Eij is a vector of size [1×K] that has value 1 at the column that correspondsto the starting node of the line and value −1 at the column that corresponds


to the ending node of the line. Ψm is a vector of size [K × 1]:

Ψm,i =

−1 if outaged generator m was connected at bus iPi

max∑

j 6=m

PJmax if there is a generator connected at bus i

0 otherwise

(4.28)This modelling is based on the assumption that lost power of unit m is

compensated by the rest of the units in proportion to their nominal maxi-mum capabilities.

4.1.4 Line Constraints

With the implementations described in sections 4.1.1 - 4.1.3 the inequalityconstraints of (3.26)-(3.29)

∣

∣

∣

∣

∣

δk − δmxkm

∣

∣

∣

∣

∣

≤FmaxAC,km ∀ AC lines k −m

∣

∣

∣

∣

∣

VDC,k − VDC,m

Rkm

∣

∣

∣

∣

∣

≤FmaxDC,km ∀ DC lines k −m

|PT,i| ≤PmaxT,i ∀ terminals i

PminG,i ≤ PG,i ≤Pmax

G,i ∀ generators i ∈ I

are augmented by the following inequality constraints:∣

∣

∣

∣

∣

δk − δmxkm

+ LODFACkm,ij ·

δi − δj

xij

∣

∣

∣

∣

∣

≤ FmaxAC,km

∀ monitored AC lines k −m and outaged AC lines i− j (4.29)

∣

∣

∣

∣

∣

VDC,k − VDC,m

Rkm+ LODFDC

km,ij ·VDC,i − VDC,j

Rij

∣

∣

∣

∣

∣

≤ FmaxDC,km

∀ monitored DC lines k −m and outaged DC lines i− j (4.30)

∣

∣

∣

∣

∣

δk − δmxkm

+GGDF skm · PGs

∣

∣

∣

∣

∣

≤ FmaxAC,km

∀ monitored AC lines k −m and outaged generators s (4.31)

Every AC line induces one constraint regarding its normal limit and N−1constraints to serve as preventive measures for the outage of the line. Eachgenerator contingency accounted for induces N constraints for flows on AClines. Every monitored DC line induces one constraint regarding normallimits and D− 1 constraints for DC line outages. N, I and D are defined intable 3.1.


4.1.5 Formulation

To obtain a formulation of the problem as in section 3.3.3, elements Ciq andbiq have to be modified. Ceq, beq and G remain the same due to the factthat the preventive method does not take into account control and statevariables for contingency cases nor does it solve the equivalent power flowsfor the modelling that is used. The formulation of the set of equations statedin (4.31) in matrix form is achieved with the use of two matrices. The firstone is the matrix of generator outage sensitivities:

EI =[

EI,1 EI,2 ... EI,N

]T

[(N ·I)×I](4.32)

EI,1 =

GGDFGen1br1 0 · · · 0

0 GGDFGen2br1 0 0

......

. . ....

0 0 · · · GGDFGenIbr1

[I×I]

(4.33)

Matrices EI,2,...,EI,N are constructed similarly and refer to the branchwith index i = 1, 2, ..., N . The second matrix needed to formulate equations(4.31) is the line adjacency matrix for generator outages and is as follows:

EAC =[

EAC,1 EAC,2 · · · EAC,N

]

[(N ·I)×(K−1)](4.34)

EAC,1 =

J12 · · · J1K

J12 · · · J1K

......

...

J12 · · · J1K

[I×(K−1)]

(4.35)

Entries of matrix EAC,i are the entries of matrix AAC in equation (3.30)corresponding to the ith line, repeated in every row. Thus EI,1 and EAC,1

together quantify the effects of all generator outages to the power flowingon AC branch 1. For each monitored line the appropriate row of AAC isrepeated over I number of times (once for every generator outage considered)to produce the desired result.

Regarding the AC line contingencies and the inequality constraints ofequations (4.29) the following matrix is introduced:

AACprev =[

AACprev1 AACprev2 · · · AACprevN

]T

[N2×(K−1)](4.36)

Each element of AACprev quantifies the effects of all AC line outages to themonitored line that corresponds to the subscript of this element. Matrix


AACprev1 is going to be defined as the sum of two matrices, the rest of theelements are constructed with the same methodology:

AACprev1[N×(K−1)] = Pp1 +Qp1 (4.37)

Pp1 =

P12 · · · P1K

.... . .

...

PN2 · · · PNK

[N×(K−1)]

(4.38)

Pij =

BAC,br,1 if monitored branch 1 starts at node j, for every row i

−BAC,br,1 if monitored line 1 ends at node j, for every row i

0 otherwise

(4.39)

Qp1 =

Q12 · · · Q1K

.... . .

...

QN2 · · · QNK

[N×(K−1)]

(4.40)

Qij =

BAC,br,i · LODFbr1,bri if outaged branch i starts at node j

−BAC,br,i · LODFbr1,bri if outaged branch i ends at node j

0 if i is equal with index of monitored branch 1

0 otherwise

(4.41)Each element of AACprev therefore contains one constraint for normal

operating conditions and N− 1 constraints for line contingency related sit-uations.

In an equivalent way the formulation of ADCprev can be obtained and willnot be presented here. The final inequality matrix then has the following

4.2. METHOD II: PREVENTIVE-CORRECTIVE 29

form:

Ciq,prev =

ΛI 0 0 0

−ΛI 0 0 0

EI EAC 0 0

−EI −EAC 0 0

0 AACprev 0 0

0 −AACprev 0 0

0 0 ΛL 0

0 0 −ΛL 0

0 0 0 ADCprev

0 0 0 −ADCprev

[Yprev×C]

(4.42)

where Yprev = 2 · [I + (N · I) +N2 + L+D2].Constant vector biq,prev is as follows:

biq,prev = [bmaxg bmin

g bl,AC,G − bl,AC,G

bl,AC,P − bl,AC,P bterm − bterm bl,DC,P − bl,DC,P ]T[Yprev×1]

Vectors bl,AC,G and bl,AC,P contain the short-term limits for AC lines andvector bl,DC,P contains the short-term limits for DC lines. With Ciq,prev andbiq,prev defined the preventive SC-OPF is formulated as in section 3.3.3.

4.2 Method II: Preventive-Corrective

Methodology and Implementations

Contrary to the preventive method of section 4.1 this method explicitlyaccounts for post-contingency equality and inequality constraints. Thereforea decision variable ξc for every contingency related state is included in the


total decision variable:

ξcorr =

ξ0

ξ1

...

ξγ

=

P 0G

δ0

P 0T

V 0DC

δ1

P 1T

V 1DC...

δγ

P γT

V γDC

(4.43)

The controls for the base-case are the generator power outputs PG andthe terminal power transfers PT . In the contingency states only the termi-nal power transfers are included in the control variables. This means thatpost-contingency corrective control is limited to changes in terminal powertransfers. For both pre and post-contingency states the state variable con-tains the AC bus angles and the DC bus voltages.

The sections to come explain analytically the formulation of the quadraticprogramming problem which includes the following contingency scenarios:

• AC Line Outages

• DC Line Outages

• Terminal Outages

• Generator Outages

The reason behind the choice to include these sets of contingencies issimple. They are the most significant for the given system modelling. Seeingas all outages of elements of the above sets are accounted for, the length ofthe decision variable and the total amount of inequality constraints can becomputed. These quantities are presented in table 4.1 and they are linkedto elements C,Y,D,I,N,L from table 3.1.

4.2.1 Objective Function

For the preventive-corrective method the objective function presented in sec-tion 3.3.5 is extended by adding one more quadratic term. This term is the


Table 4.1: Nomenclature for Preventive-Corrective Method

Ccorr C+(C-I)(I+N+D+L) Length of vector ξcorr

YAC N(Y-2I) Inequality constraints invoked by AC line outages

YDC D(Y-2I) Inequality constraints invoked by DC line outages

YTer L(Y-2I) Inequality constraints invoked by terminal outages

YGen I· Y Inequality constraints invoked by generator outages

YControl 2L(I+N+D+L) Inequality constraints for terminal control

Ycorr Y+YAC + YDC + YTer

+YGen + YControl Total number of inequality constraints

terminal difference penalty cost of the form [P 0T − P c

T ]2 and is introduced

for all assessed contingency scenarios. The rationale behind this implemen-tation lies in the fact that it gives additional room for sensitivity analysis.Furthermore it is beneficial when it comes to evaluating the interaction be-tween the AC and DC grid and it also enhances system stability. A directassignment of costs for terminal power transfers is not performed, though animplementation of this sort could be envisioned. Thus the objective functionnow takes on the following form:

I∑

s=1

[α ·PGs+β ·P 2Gs]+ΠAC ·

∑

km∈N

[δk − δm]2+ΠDC ·∑

km∈D

[VDCk−VDCm]2

+ΠTer ·

γ∑

c=1

[P 0T − P c

T ]2 (4.44)

N= Pairs of AC buses connected with a branchD= Pairs of DC buses connected with a branchΠTer is the penalty for terminal power differences.

To include these terms in the objective function the matrix G is constructed


as:

G =

QC 0 · · · · · · · · · · · · · · · · · · · · · 0

0 WAC 0 · · · · · · · · · · · · · · · · · · 0

0 0 c · ΛL 0 0 −ΛL 0 0 −ΛL 0

0 · · · · · · WDC 0 · · · · · · · · · · · · 0

0 · · · · · · · · · · · · · · · · · · · · · · · · 0

0 · · · −ΛL 0 0 ΛL 0 0 0 0

0 · · · · · · · · · · · · · · · · · · · · · · · · 0

0 · · · · · · · · · · · · · · · · · · · · · · · · 0

0 · · · −ΛL 0 0 0 0 0 ΛL 0

0 · · · · · · · · · · · · · · · · · · · · · · · · 0

(4.45)

For clarity regarding the implementation, a case where two contingencies areconsidered is depicted to portray the basics of the formulation. A display offurther contingency states would be rather trivial. Here ΛL is the identitymatrix of size L. In this way the terms [P 0

T − P 1T ]

2 + [P 0T − P 2

T ]2 are derived

from equation (3.8). Beware that all contingency state terminal flows arerelated to the pre-contingency terminal flows P 0

T . This means that the pre-contingency terms P 0

T occur squared as many times as the total number ofoutages accounted for. This explains why the appropriate entry of matrixG in (4.45) is c · ΛL(c = 2 for the example), where c is the number ofcontingencies regarded.

Vector λ in this case is augmented by an appropriate amount of zeros.The only linear coefficients in the objective function are the ones belongingto the generator costs:

λ =[

α1 α2 ... αI 0[1×(Ccorr−I)]

]

(4.46)

4.2.2 Post-Contingency Terminal Control

The HVDC scheme displays significant advantages when it comes to con-trollability. This fact urges for a direct implementation of post-contingencycorrective control of the terminal power transfers. In the proposed SC-OPFthis control takes on the form

|P 0T − P c

T | = ∆P cT ≤ ∆P c

Tmax (4.47)

which bounds the allowed change in terminal power transfer in post contin-gency states.

In the event of a contingency, the algorithm first attempts to satisfy any


post-contingency constraint violations through post-contingency correctiveactions ∆P c

T of the terminal flows. If these actions do not suffice to clear allviolations then pre-contingency control levels P 0

G, P0T must move to satisfy

these constraints.Even though there is an application of costs for the ∆P c

T actions, thecosts incurred by such actions are significantly less than the ones that wouldarise in order to achieve the same goal through preventive measures. Gen-eration costs of the units supplying the system dominate the value of theobjective function, and as such the change in the dispatching of generationwill lead to a more expensive system operation as a trade-off between secu-rity and optimality.

To implement control over the absolute value of the difference in ter-minal power, two constraints are needed for each terminal variable. Thecombination of :

P 0T − P c

T ≤ ∆P cTmax

P kT − P 0

T ≤ ∆P cTmax

(4.48)

achieves this result. In matrix form this is formulated below :

TerCon · ξ ≤ ∆P cTmax (4.49)

0 · · · · · · · · · · · · · · · 0... · · · · · · · · · · · · · · ·

...... 0 ΛL 0 0 −ΛL 0... · · · · · · · · · · · · · · ·

...... · · · · · · · · · · · · · · ·

...... · · · −ΛL 0 0 ΛL 0

0 · · · · · · · · · · · · · · · 0

·

P 0G

δ0

P 0T

V 0DC

δ1

P 1T

V 1DC

≤

0

0

∆P 1Tmax

0

0

∆P 1Tmax

0

(4.50)

Again here IL is the identity matrix of size L, whereas ∆P cTmax is the maxi-

mum amount of change in the power transferred by terminals in contingencysituations.

4.2.3 Line Outages in the AC Grid

Note: for purposes of space efficiency lower bounds for generator limits, neg-ative limits for terminal flows and AC and DC line flows are not depictedin the inequality constraints to follow. They need to be included in the in-equality matrices for the correct formulation of inequality constraints as inequation (3.32).


As stated earlier, constraints for post-contingency line flows are explicitlyaccounted for through a new set of equality and inequality constraints. Theinequality matrix CiqAC contains all the entries that formulate the post-contingency inequality constraints for all AC line outages:

CiqAC[YAC×N(C−I)]

Inequality constraints regarding the contingency states where an AC line isout can be represented as follows:

CiqAC[YAC×N(C−I)] · ξAC[N(C−I)×1] ≤ biqAC[YAC×1]

is depicted in the following way:

C1iqAC 0

... 0

0 C2iqAC

... 0

0 0. . . 0

0 0 · · · CNiqAC

ξ1AC

ξ2AC...

ξNAC

≤ biqAC (4.51)

where for a chosen indexing of AC lines, 1, ..., N refers to the outage ofAC line number 1,..., outage of AC line number N. Vector ξcAC is the decisionvariable that corresponds to the state where AC line c is out. Vector biqAC

is of length N(C − I) and contains the post-contingency ratings for AC andDC lines, as well as limits for terminal flows.

To make it easier to comprehend, the formulation of CiqAC for a singleline outage is presented. The pre-contingency state is also shown in or-der distinguish which elements change in the post-contingency state. Theaugmented matrix to include additional outages is constructed in a similarmanner.

C0,1iqAC =

ΛI 0 0 0 · · · · · · · · ·

0 A0AC 0 0 · · · · · · · · ·

0 0 ΛL 0 · · · · · · · · ·

0 0 0 A0DC 0 · · · · · ·

0 · · · · · · 0 A1AC 0 0

0 · · · · · · 0 0 ΛL 0

0 · · · · · · · · · 0 0 A0DC

[2(Y−I)×(2C−I)]

(4.52)

For the contingency state the Line Adjacency matrix A1AC has all entries

equal to zero in the row that corresponds to the outaged line. The Line


Adjacency matrix for the DC Grid stays the same, since it is obvious thatan outage in the AC grid has no effect on the topology of the DC grid. Thesame applies for the terminal stations, where for a line outage matrix ILremains unaltered.

Regarding the equality constraints the matrix that contains all entriesfor each contingency scenario is:

CeqAC[N(K+M)×N(C−I)] (4.53)

CeqAC[N(K+M)×N(C−I)] · ξAC[N(C−I)×1] = beqAC[N(C−I)×1] (4.54)

is depicted in this way:

C1eqAC 0

... 0

0 C2eqAC

...

0 0. . . 0

0 0 · · · CNeqAC

ξ1AC

ξ2AC...

ξNAC

= beqAC (4.55)

C0,1eqAC =

H0 −B0AC T 0 · · · · · · 0

0 0 S −B0DC 0 · · · 0

H0 0 · · · 0 −B1AC T 0

0 · · · · · · · · · 0 S −B0DC

[2(K+M)×(2C−I)]

(4.56)In the Equality matrix the entry that changes is the AC bus Admittancematrix BAC , to simulate this change in the AC grid topology. Note that thegeneration profile H remains the same for pre and post-contingency statesto model the pre-deterministic response that is assumed for the generators.

4.2.4 Line Outages in the DC Grid

The formulation of constraints for DC line contingency scenarios is as follows:

CiqDC[YDC×D(C−I)] (4.57)

CiqDC[YDC×D(C−I)] · ξDC[D(C−I)×1] ≤ biqDC[YDC×1]

is depicted in the following way:

C1iqDC 0

... 0

0 C2iqDC

... 0

0 0. . . 0

0 0 · · · CDiqDC

ξ1DC

ξ2DC...

ξDDC

≤ biqDC (4.58)


and for a chosen indexing of DC lines, 1, ...,D refer to the outage of DC linenumber 1,..., outage of AC line number D.

C0,1iqDC =

ΛI 0 0 0 · · · · · · · · ·

0 A0AC 0 0 · · · · · · · · ·

0 0 ΛL 0 · · · · · · · · ·

0 0 0 A0DC 0 · · · · · ·

0 · · · · · · 0 A0AC 0 0

0 · · · · · · 0 0 ΛL 0

0 · · · · · · · · · 0 0 A1DC

[2(Y−I)×(2C−I)]

(4.59)

The line adjacency matrix for the contingency state A1DC has all entries

equal to zero in the row that corresponds to the outaged line. The LineAdjacency matrix for the AC Grid remains the same.

CeqDC[D(K+M)×D(C−I)] (4.60)

CeqDC[D(K+M)×D(C−I)] · ξDC[D(C−I)×1] = beqDC[D(K+M)×1] (4.61)

is depicted in this way:

C1eqDC 0

... 0

0 C2eqDC

...

0 0. . . 0

0 0 · · · CDeqDC

ξ1DC

ξ2DC...

ξDDC

= beqDC (4.62)

C0,1eqDC =

H0 −B0AC T 0 · · · · · · 0

0 0 S −B0DC 0 · · · 0

H0 0 · · · 0 −B0AC T 0

0 · · · · · · · · · 0 S −B1DC

[2(K+M)×(2C−I)]

(4.63)

Here it is the DC grid that changes topology, resulting in a different ad-mittance matrix B1

DC .


4.2.5 Terminal Station Outages

Terminal stations of an AC-DC network are very important for power flowstudies and security assessments since they define the interaction betweenthe two grids. Therefore it is essential to include terminal outages in con-tingency analysis. A terminal contingency is the situation where due toan unexpected disruption the power flowing through the terminal is zero,PT = 0.

The implementation used to simulate a terminal outage is to change theappropriate entries of vector biq so that the following inequality constraintsare formulated:

P cT ≤ 0

−P cT ≤ 0

The second of these inequalities is equivalent with: P cT ≥ 0. For the 2

inequality constraints to hold the power flow P cT which refers to the outage

of terminal c has to be 0. Therefore this implementation achieves the desiredresult of simulating a terminal station contingency. Another way that seemsreasonable to simulate an outage of this sort is to change the appropriateelements of matrices S, T of equations (3.20) and (3.21). However this is notadvised because an implementation like this leaves a control variable free tochange and affect the optimization process.

CiqTer[YTer×L(C−I)] (4.64)

CiqTer[YTer×L(C−I)] · ξTer[L(C−I)×1] ≤ biqTer[L(C−I)×1]

C1iqTer 0

... 0

0 C2iqTer

... 0

0 0. . . 0

0 0 · · · CLiqTer

ξ1Ter

ξ2Ter...

ξLTer

≤ biqTer (4.65)

C0,1iqTer =

ΛI 0 0 0 · · · · · · · · ·

0 A0AC 0 0 · · · · · · · · ·

0 0 ΛL 0 · · · · · · · · ·

0 0 0 A0DC 0 · · · · · ·

0 · · · · · · 0 A0AC 0 0

0 · · · · · · 0 0 ΛL 0

0 · · · · · · · · · 0 0 A0DC

[2(Y−I)×(2C−I)]

(4.66)


From matrix CiqTer it is clear that there is absolutely no change betweenelements of pre and post-contingency states. An outage of a terminal stationdoes not affect the topology of the AC or DC grid and the methodology usednecessitates no alterations for matrices T and S.

The matrix that represents equality constraints also displays no changebetween entries for pre and post-contingency states:

C1eqTer 0

... 0

0 C2eqTer

...

0 0. . . 0

0 0 · · · CLeqTer

ξ1Ter

ξ2Ter...

ξLTer

= beqTer (4.67)

C0,1eqTer =

H0 −B0AC T 0 · · · · · · 0

0 0 S −B0DC 0 · · · 0

H0 0 · · · 0 −B0AC T 0

0 · · · · · · · · · 0 S −B0DC

[2(K+M)×(2C−I)]

(4.68)

4.2.6 Generator Outages

The model that is applied for the control scheme of the generating units,allows for a useful approximation regarding the responses of these units to agenerator contingency. When a generating unit is outaged, other generatorsare required to compensate for the lost power. The modelling that is goingto be used for this situation is as follows: In the event of a generator outage,automatic control of all participating units takes effect to ensure the lostgeneration is compensated by these units. The portion of this additionalpower that each unit must supply is proportionate to the unit’s nominalmaximum power [11]. In equation form this translates to:

PGi,1 = PGi,0 + PGm · ri,m (4.69)

where

ri,m =PmaxGi

∑

j 6=m

PmaxGj

(4.70)

and PGi,0, PGi,1 is the power of unit i before and after the contingency andPGm the power of outaged unit m. Hence the generator power outputs arenot included as controls in the post-contingency states, which is already evi-dent from equation (4.43), since their response is defined in a pre-determinedway.


CiqGen[YGen×I(C−I)] (4.71)

CiqGen[YGen×I(C−I)] · ξGen[I(C−I)×1] ≤ biqGen[YGen×1]

C1iqGen 0

... 0

0 C2iqGen

... 0

0 0. . . 0

0 0 · · · CIiqGen

ξ1Gen

ξ2Gen...

ξIGen

≤ biqGen (4.72)

C0,1iqGen =

Λ0I 0 0 0 · · · · · · · · ·

0 A0AC 0 0 · · · · · · · · ·

0 0 ΛL 0 · · · · · · · · ·

0 0 0 A0DC 0 · · · 0

Φ1iq 0 · · · · · · · · · · · · 0

0 · · · · · · 0 A0AC 0 0

0 · · · · · · 0 0 IL 0

0 · · · · · · · · · 0 0 A0DC

[2(Y−I)×(2C−I)]

(4.73)The difference between base and post-contingency case is in matrices ΛI

and Φiq, where the latter matrix is introduced to define the pre-determinedoutputs of generators. In the post-contingency state, these online generatorsare producing increased power compared to what they were producing inthe pre-contingency state. These power generations though still have to bebounded to the same pre-contingency capacities that each unit has. Thusa generator contingency invokes I − 1 number of constraints regarding thecapacities of remaining generating units and are expressed in matrix GLI .

Φ1iq =

0 0 · · · · · · 0

r2,1 1 0 · · · 0

r3,1 0 1 0 0...

......

......

rI,1 0 · · · · · · 1

[I×I]

(4.74)


Φ2iq =

1 r1,2 0 · · · 0

0 0 · · · · · · 0

0 r3,2 1 0 0...

......

......

0 rI,2 0 0 1

[I×I]

(4.75)

The above matrices show the build-up of matrix Φiq for the outage of gen-erators with indices 1 and 2, the equivalent matrices for the rest of thegenerator contingencies are constructed accordingly. The desired result thatthese entries produce is the following inequality:

Φiq · P0G ≤ P 0max

G (4.76)

which enforces constraints to generating units according to each one’s limit.The formulation of the equality constraints for a single generator outage isdepicted below:

C0,1eqGen =

H −B0AC T 0 · · · · · · 0

0 0 S −B0DC 0 · · · 0

Φ1eq 0 · · · 0 −B0

AC T 0

0 · · · · · · · · · 0 S −B0DC

[2(K+M)×(2C−I)]

(4.77)The difference lies between elements H and Φeq[K×I]. H denotes the gener-ator allocation matrix stated in 3.13. Here matrix Φ1

eq[K×I] is defined :

Φ1eq,ij =

Φ1iq,ij if there is a generator connected at bus i

0 otherwise(4.78)

Matrix Φeq assigns the in-feeds of the remaining units to the buses they areconnected.

4.2.7 Formulation of the Complete Problem

The individual matrix formulations presented in the previous sections canbe combined with each other to construct a single optimization problem.Depending on the types of contingencies included in the cases that are envi-sioned, suitable concatenations can produce the desired matrices G,Ceq, Ciq

and vectors beq, biq, λ of equations (3.8) − (3.9).The case where all types of contingencies are incorporated in the opti-

mization problem is as follows :

Ciq,total · ξcorr ≤ biq,total


CiqBase 0 · · · · · · 0

0 CiqAC 0 · · · 0

0 · · · CiqDC 0 0

0 · · · · · · CiqTer 0

0 · · · · · · · · · CiqGen

0 TerCon,AC 0 · · · 0

0 · · · TerCon,DC 0 0

0 · · · · · · TerCon,Ter 0

0 · · · · · · · · · TerCon,Gen

·

ξ0

ξAC

ξDC

ξTer

ξGen

≤

biq0

biqAC

biqDC

biqTer

biqGen

(4.79)The variables ξAC ,ξDC ,ξTer,ξGen are used to indicate the N AC line out-

age, D DC line outage, L terminal station outage and I generator outagestates respectively. Every single contingency is explicitly expressed throughits own decision variable leading to the following relation for the total deci-sion variable:

ξcorr = ξ0 + ξAC + ξDC + ξTer + ξGen

Ceq,total · ξcorr = beq,total

CeqBase 0 · · · · · · 0

0 CeqAC 0 · · · 0

0 · · · CeqDC 0 0

0 · · · · · · CeqTer 0

0 · · · · · · · · · CeqGen

·

ξ0

ξAC

ξDC

ξTer

ξGen

=

beq0

beqAC

beqDC

beqTer

beqGen

(4.80)

The biq constant inequality vector changes at the entries regarding con-tingency states. For instance if one is considering short term ratings for ACand DC lines, then the corresponding limits of the inequality vector containincreased values. The beq constant equality vector is a repetition of the orig-inal base-case vector for all cases since no loss of load scenarios are included.

The large number of contingencies that are being addressed result inmatrices that are very sparsely populated.


Chapter 5

Results-Case Studies

In this chapter the methods that have been described in chapter 4 are appliedon test cases. Simulations for sensitivity analysis then follow accompaniedby comments and explanations regarding the results. The last section isreserved for conclusions and a discussion about future improvements andimplementations.

In the following sections the base quantity for power is Sbase = 100 MWsince only active power quantities are considered, therefore 1 p.u refers to100 MW. All simulations regarding the results that follow were done usingthe quadprog solver of the MATLAB optimization toolbox.

5.1 System Description - Case Study I

The test case that is studied is a hybrid grid that was envisioned in [1], andis shown in figure 5.1. It is the IEEE 14 bus test case with an interconnected5-bus DC grid. The technical characteristics of the AC grid can be found inthe equivalent test case of [13], while for the DC grid all lines have resistanceequal to Rkm = 2.78Ω and terminals have capacities equal to 1 p.u. Thetwo grids are interconnected at four points, where the terminals are situated.Bus 25 of the DC grid is not connected to a terminal and is a pure DC bus.

System Characteristics

The power system is supplied by five generators that are connected at ACbuses. Generator 1 has the smallest cost coefficients making it the cheapestoption for power supply. In the range of power demand of the system gener-ator 2 follows as the next cheapest option, while generators 3,4 and 5 havethe same higher operating costs. The highest power demands are situatedat buses 3 and 4, therefore the general flow of power has direction and originfrom the bottom left part of figure 5.1 towards the bottom right.

43

44 CHAPTER 5. RESULTS-CASE STUDIES

4

5

6

7

8

9

10

1112

13 14

1

32

VS

C

VS

C

VS

CV

SC

25

AC Line

Generator

Voltage source converter

DC Line

Transformer

VSC

Figure 5.1: Combined AC and DC grid [1]

5.2 Method I: Preventive

The first method to be assessed is the preventive SC-OPF. The model pro-vides an optimal power flow solution that is secure against any single con-tingency event of an AC line, DC line or a generator. The effects of everycontingency from the sets of these elements are considered, regardless ofwhether they result in constraint violations or are not severe at all. Simu-lations for the base OPF and the preventive SC-OPF for case 14 result inoperating costs of 7643.63 $/h and 7944.54 $/h respectively.

5.2.1 Generation Profiles

The target of the objective function is the minimization of generator costs.The optimization depends on the quadratic and linear cost coefficients ofthe generators, their physical limits and from the level of demand. Figure5.2 shows the generation profiles of the base OPF and SC-OPF. In the base-case generator 1 undertakes most of the generation as the cheapest unit withthe largest capacity, while generator 2 also participates with a much smalleroutput. In the SC-OPF though, the security constraints require for a more


1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

Base OPFPreventive SC−OPFGenerator Limit

Generator

Pow

erOutput[p.u]

Figure 5.2: Generation Profile for Preventive SC-OPF

distributed power dispatch as can be seen from the figure. Generator 1 nowproduces 70 MW less power while units 3,4 and 5 which are shut down inthe base OPF now participate in the power generation.

5.2.2 Power Flow Distribution in the Mixed Grid

Figure 5.3 displays the distribution of flows on AC lines for the OPF and SC-OPF. In the second case the power flowing on the AC grid is more distributedover all the lines. Lines that carry power of cheap generating units are lessloaded so that they can operate safely under a contingent situation. Anexample that displays this behaviour is the set of power flows on lines 1− 2and 1− 5 which are connected to the bus with the cheapest generating unit.In the SC-OPF the power flowing on these lines drops from 0.82 and 0.39p.u for the base OPF case to 0.6721 and 0.3279 p.u. This way both lines canoperate within their limits if the other line is outaged. Due to the topologyof the grid if either of these lines is out, the other compensates and transfersthe initial power that was flowing on the outaged line. Notice that the sumof the power flowing on lines 1−2 and 1−5 is 1 p.u in the SC-OPF. Thereforein the event one of the two lines is outaged, the other line will have a powerflow of 1 p.u which is the limit of all AC lines of the system. The algorithmfinds the optimum solution under security constraints given the fact that thepaths that transfer power from cheap generating units are congested duringthe outage of a single element of the system.

The distribution of power flows in the DC grid is shown in figure 5.4The topology of the DC grid and the fact that post-contingency terminalcontrol is not performed in the preventive method, intensely handicap the


1−2 1−5 2−3 2−4 2−5 3−4 4−5 4−7 4−9 5−6 6−11 6−12 6−13 7−8 7−9 9−10 9−1410−1112−1313−14

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Base OPFPreventive SC−OPFLine Limit

AC Line from bus k to m

Pow

erFlow

[p.u]

Figure 5.3: Distribution of Line Flows in the AC Grid

1−25 3−25 3−9 9−13 13−1 9−25 13−25

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Base OPFPreventive SC−OPFLine Limit

DC Line from bus k to m

Pow

erFlow

[p.u]

Figure 5.4: Distribution of Line Flows in the DC Grid (Preventive)

total power flowing on the DC grid in the SC-OPF case. The total injectionof power into the DC grid drops from 1 p.u to 0.5 p.u to ensure secureoperation in the case of unexpected contingencies. Terminal 1 that injectspower in the DC grid has to halve its transfer rate to make sure that neitherof DC lines 1− 25 or 31− 1 suffer from overloads if one of them is outaged.


The results can be seen in figure 5.5.

1 3 9 13

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Base OPFPreventive SC−OPFTerminal Limit

Terminal connected at Bus

Pow

erFlow

[p.u]

Figure 5.5: Terminal Station Power Transfers (Preventive)

5.3 Method II: Preventive-Corrective

In the following sections several cases are envisioned regarding the type ofcontingencies that are included in the preventive-corrective SC-OPF prob-lem. These are :

• Case 1: SC-OPF that accounts for all AC and DC line outages.

• Case 2: SC-OPF that accounts for AC and DC line outages, andgenerator outages.

• Case 3: SC-OPF that accounts for AC and DC line outages, as wellas outages of terminal stations.

• Case 4: SC-OPF that accounts for AC and DC line outages, terminaloutages and generator outages.

Generation Profiles

The dispatching of power generation among the participating units for thecases that are considered is depicted in figure 5.6. In all cases generator 1is required to reduce its power output. Generator 2 has a small increasein the power it is supplying, and generators 3,4 and 5 also participate in


the supply of the loads. Compared to the generation profile of figure 5.2however, the shift in generation levels between base OPF and SC-OPF casesis not so severe. In the preventive method there has to be an enforcementof preventive measures to place the system in a secure state only throughchanges in the generation dispatch. The corrective method provides moreflexibility due to corrective actions that can be implemented in a possiblecontingent situation. This means that the generation dispatch is not requiredto change as much as in the preventive method.

Notice how in case 4 of the corrective SC-OPF, where all contingencyscenarios are included, generator 1 is providing the system with 169.48 MWin pre-contingency steady state. In the preventive case generator 1 has anoutput of 150 MW, and in this case constraints for terminal outages are notaccounted for. So the corrective method that includes security constraintsfor a greater set of contingencies offers a cheaper solution (7828.5 $/h) thanthe preventive method (7944.5 $/h).

1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

Base OPFCase 1Case 2Case 3Case 4Generator Limit

Generator

Pow

erOutput[p.u]

Figure 5.6: Generation Profile for Prev-Corr SC-OPF for 100% LoadCase 1: AC-DC line outagesCase 2: AC-DC line and generator outagesCase 3: AC-DC line and terminal outagesCase 4: AC-DC line, terminal and generator outages

5.3.1 Costs Comparison

The next indicative quantity to be addressed is the operational cost of thesystem in each of the cases described. What is clear from figure 5.8 is thatthe inclusion security constraints for line outages leads to a more expensivesystem operation. A further augmentation to include terminal contingency


1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

Base OPFCase 1Case 2Case 3Case 4Generator Limit

Generator

Pow

erOutput[p.u]

Figure 5.7: Generation Profile for Prev-Corr SC-OPF for 160% LoadCase 1: AC-DC line outagesCase 2: AC-DC line and generator outagesCase 3: AC-DC line and terminal outagesCase 4: AC-DC line, terminal and generator outages

scenarios further increases operational costs. In this case the total generat-ing capacity is rather large compared to the demand and that is why theaugmentation of security constraints for the generator outages does not giverise to extra costs. Thus the most severe of generator outages is not in factcausing any post-contingency violations which means that no re-dispatchis required. An increase in the total load of the system by 60% resultsin the operational costs of figure 5.9. The notation 100% Load refers tothe original loading of the IEEE system which is 259 MW. The figure de-picts the effects an increased demand has on the system costs. Unlike 5.8,here the augmentation of constraints for system security during generatoroutages has a notable influence on the total costs. Under heavier loadingsituations the generators have large power outputs, which makes the outageof a unit a more severe and stressful phenomenon. Hence the preventivemeasures needed in these circumstances result in an even more conservativeand costly dispatch of generation.

It is also interesting to point out that while Case 2 differs from Case 1 ina small amount, Case 4 differs from Case 3 much more. This highlights thedependency of system performance from terminal operation. The inclusionof terminal contingencies affects the amount of corrective actions that canbe taken during a disruption as well as the actual power that is fed andextracted from the DC grid. Tables 5.1 and 5.2 list the cost of security for


7500

7550

7600

7650

7700

7750

7800

7850

7900

Base OPF(Case 0)

Lines(Case 1)

Lines+Gen(Case 2)

Lines+Terminals(Case 3)

All Contingencies(Case 4)

$/h

Figure 5.8: Operational Costs for 100% LoadCase 1: AC-DC line outagesCase 2: AC-DC line and generator outagesCase 3: AC-DC line and terminal outagesCase 4: AC-DC line, terminal and generator outages

the various cases and two loading levels. It can be an indicative value forpower system planning and development.

Table 5.1: Cost of Security for the Cases studied for 100% Load

Scenario Type of Contingencies in-cluded

Operational Cost[$/h]

Cost of Se-curity [$/h]

Case 0 No Security 7643.63 -

Case 1 Lines 7683.48 39.85

Case 2 Lines,Generators 7683.48 39.85

Case 3 Lines,Terminals 7828.51 184.88

Case 4 Lines,Generators,Terminals 7828.51 184.88

It is interesting to compare Case 2 of the corrective SC-OPF methodto the preventive method since the set of contingencies accounted for is thesame. Figure 5.10 shows the cost of the corrective approach with varyinglimits of expendable terminal corrective control. That is, the points on x-


1.38

1.385

1.39

1.395

1.4

1.405

1.41

1.415

1.42

1.425

1.43

1.435x 10

4

Base OPF(Case 0)

Lines(Case 1)

Lines+Gen(Case 2)



$/h

Figure 5.9: Operational Costs for 160% LoadCase 1: AC-DC line outagesCase 2: AC-DC line and generator outagesCase 3: AC-DC line and terminal outagesCase 4: AC-DC line, terminal and generator outages

Table 5.2: Cost of Security for the Cases studied for 160% Load

Scenario Type of Contingencies in-cluded

Operational Cost[$/h]

Cost of Se-curity [$/h]

Case 0 No Security 13915.11 -

Case 1 Lines 13998.8 83.69

Case 2 Lines,Generators 14016.95 101.84

Case 3 Lines,Terminals 14205.34 290.23

Case 4 Lines,Generators,Terminals 14289.94 374.83

axis depict the percentage of allowed change in terminal power transfersin post-contingency states. The maximum (100%) permissible correctivecontrol is here equal to the the capacity of the terminal stations, which is 1p.u. The cost of the preventive approach is plotted on the same graph andis a horizontal line since no corrective actions are allowed in the preventive


method. As the amount of allowed corrective actions is reduced the cost ofthe corrective SC-OPF rises and as this amount approaches 0 the solutionof the corrective method converges to the solution of the preventive method.This is to be expected because with no corrective actions allowed the secondmethod is practically the same problem as described by the first method.The difference then lies in the modelling and formulations of the problems.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17650

7700

7750

7800

7850

7900

7950

Case 2 CorrectiveCase 2 Preventive

∆PT /PmaxT

$/h

Figure 5.10: Linking of Preventive and Corrective SC-OPF

5.3.2 Interaction between AC and DC Grid

Figure 5.11 shows the line flows in the AC grid. In the security constrainedcases the power flows are more distributed over the lines. Lines that wouldusually be congested, or loaded very close to the their capacity, are lessheavily loaded to avoid any violations in the event of a contingency. Thedifference is compensated by other lines which in the normal OPF would belightly loaded.

Be that as it may, the ability to make corrective actions through theterminals means that lines that are connected close to cheap generatingunits can in fact be loaded close to their limits, because in the event of acontingency the terminals can alleviate possible over-loadings. So with thelimits of the AC lines at 1 p.u and given the fact that generator 1 is thecheapest unit, one would expect lines 1 − 2 and 1 − 5 to be more heavilyloaded. The reason they are not is owed to the fact that the DC grid is moreintensely loaded as can be seen in figure 5.12, where DC lines 1 − 25 and13− 1 are both at their limit of 0.5 p.u.

However observe that at case 1 and 2, lines 1−2 and 1−5 are delivering0.75 and 0.37 p.u power. If no corrective actions were implemented during anoutage of either of the two lines, the line operating would be forced to deliver


1.12 p.u power, which would be unacceptable. Through the application ofcorrective actions 12 MW will be shifted to the DC grid in the event ofa contingency of line 1 − 2 or 1 − 5. The equivalent holds for the DCgrid, where lines 1 − 25 and 13 − 1 are loaded at 0.44 p.u. An outageof one of the two lines will result in congestion of the operating line (0.5p.u), and the additional power will be fed back into the AC grid to avoidconstraint violations. Therefore the implementation of corrective actionsmakes it possible for generator 1, the cheapest unit, to produce significantlymore power compared to a fully preventive dispatch.

Another observation is that the cases which include generator outages,i.e cases 3 and 5 do not display significant differences compared to cases 2and 4 respectively. This is justified by the fact that the total load of thesystem is rather small compared to the total capacity of the generators, sothe augmentation of extra security for generator outages to cases 2 and 4does not lead to a more expensive dispatch.

1−2 1−5 2−3 2−4 2−5 3−4 4−5 4−7 4−9 5−6 6−11 6−12 6−13 7−8 7−9 9−10 9−14 10−1112−1313−14

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Base OPFCase 1Case 2Case 3Case 4Line Limit


Pow

erFlow

[p.u]

Figure 5.11: Line Flows in the AC Grid for 100% LoadCase 1: AC-DC line outagesCase 2: AC-DC line and generator outagesCase 3: AC-DC line and terminal outagesCase 4: AC-DC line, terminal and generator outages

The flows in the DC grid shown in figure 5.12 present a slightly differentpicture than the distribution of flows in the AC grid. Here the security con-strained cases have reduced line flows in almost all of the lines. With thegiven topology the total power flowing in the DC grid for the constrainedcases is actually less, so the redistribution results in the majority of linesbeing less loaded. One can also observe this fact in figure 5.13 where it isevident that the terminals are feeding the DC grid with less power in the se-curity constrained cases. However in comparison to the preventive method


1−25 3−25 3−9 9−13 13−1 9−25 13−25

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Base OPF Case1Case 2Case 3Case 4Line Limit


Pow

erFlow

[p.u]

Figure 5.12: Line Flows in the DC Grid for 100% LoadCase 1: AC-DC line outagesCase 2: AC-DC line and generator outagesCase 3: AC-DC line and terminal outagesCase 4: AC-DC line, terminal and generator outages

and the results of figure 5.4 the amount of power flowing in the DC gridis substantially larger. Terminal corrective control can deal with clearingmany of the violations that occur when a DC line is outaged. It follows thatthe DC grid can in fact be more heavily loaded with the corrective approachthan with the preventive one.

Terminal connected at AC bus 1 is operating as a rectifier in all simula-tions and is feeding power into the DC grid since generator 1 is the largestand cheapest unit. The rest of the terminals are operating as inverters, feed-ing the power flowing in the DC grid back into the AC grid to supply theloads.

For a loading level of 1.6 times the original one the power flows can beseen in figures 5.14, 5.15 and 5.16

The lines in the AC grid are transferring more power compared to thecase in figure 5.11. The DC lines however have similar distribution andloading levels to the case of figure 5.12 because the DC grid was alreadystressed to its limit even for the original conditions. Therefore the AC gridtakes up the slack to deliver the additional power.

5.3.3 Terminal Control Sensitivity Analysis

To evaluate the effects of terminal corrective control, simulations are runfor varying loading levels and different security constrained cases. For adesignated loading level and case, the amount of corrective control that can


1 3 9 13

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Base OPFCase 1Case 2Case 3Case 4Terminal Transfer Limit


Pow

erFlow

[p.u]

Figure 5.13: Terminal Station Power Transfers for 100% LoadCase 1: AC-DC line outagesCase 2: AC-DC line and generator outagesCase 3: AC-DC line and terminal outagesCase 4: AC-DC line, terminal and generator outages

1−2 1−5 2−3 2−4 2−5 3−4 4−5 4−7 4−9 5−6 6−11 6−12 6−13 7−8 7−9 9−10 9−14 10−1112−1313−14

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1



Pow

erFlow

[p.u]

Figure 5.14: Line Flows in the AC Grid for 160% LoadCase 1: AC-DC line outagesCase 2: AC-DC line and generator outagesCase 3: AC-DC line and terminal outagesCase 4: AC-DC line, terminal and generator outages

be expended affects the behaviour of the system. Figure 5.17 shows theeffects of corrective control on the system costs for case 1. The maximumamount of allowed change in power transfers Pmax

T is here considered to be


1−25 3−25 3−9 9−13 13−1 9−25 13−25

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5



Pow

erFlow

[p.u]

Figure 5.15: Line Flows in the DC Grid for 160% LoadCase 1: AC-DC line outagesCase 2: AC-DC line and generator outagesCase 3: AC-DC line and terminal outagesCase 4: AC-DC line, terminal and generator outages

1 3 9 13

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Base OPFCase 1Case 2Case 3Case 4Terminal Transfer Limit


Pow

erFlow

[p.u]

Figure 5.16: Terminal Station Power Transfers for 160% LoadCase 1: AC-DC line outagesCase 2: AC-DC line and generator outagesCase 3: AC-DC line and terminal outagesCase 4: AC-DC line, terminal and generator outages

equal to the capacity of the terminals. The simulations were performed fora set of values for ∆PT , or equivalently ∆uc if the notation of section 2.3.2is used. How these values reflect on the parameters of equation 2.13 is out


of the scope of the current work.When the amount of corrective actions that can be expended is limited,

the generation costs rise. This can be seen in the first part of the graph,where tighter limits on the allowed change in terminal power transfers resultin higher costs. The costs are normalized to the costs the system has whenthe terminals are assigned 100% flexibility. Therefore the normalized costsof the y-axis are described by the relation:

Normalized Cost =Cost(∆PT = x)

Cost(PmaxT = x)

(5.1)

The steepness of the curve depends on the loading of the system. As the loadincreases, the steepness of the curve decreases. This happens because thesystem is closer to congestion, meaning that the flexibility of the terminalshas a reduced impact since all lines are already closer to their limits. What is

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.995

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

100% Load120% Load140% Load160% Load

∆PT /PmaxT

Norm

alizedCost

Figure 5.17: Effects of Terminal Control on Costs: Case 1

interesting is the fact that after a certain point in the graph, approximatelywhen ∆PT /P

maxT = 0.25 the increase in flexibility does not result in reduced

costs. After this point the terminal corrective actions can’t further decreasethe cost of operation. This is due to the fact that for these simulations thepenalization of the AC grid is much more severe than the penalization of theDC grid. Thus the DC grid is congested because it is the cheapest optionand this creates an upper bound, after which any increase in the amount ofcorrective control can’t further optimize the solution.


In figure 5.18 the same simulations as in figure 5.17 are performed for case2 that includes line outages and terminal outages. In this case the amountof corrective controls allowed have a greater impact on system costs. This isevident from the increased normalized costs and by the fact that the regionwherein the amount of corrective actions allowed affect the costs is largerthan in case 1.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09


∆PT /PmaxT

Norm

alizedCost


To display the effects a congested DC grid has on terminal control andflexibility, simulations for ΠAC = ΠDC are carried out. Figure 5.20 refersto case 2 with an application of equal penalties for AC angle differencesand DC voltage differences. A first remark to be made is that comparedto figure 5.18 the relative costs are lower when the terminals are firmlyconstrained to their pre-contingency settings. This is to be expected seeingas the power flowing in the system is more distributed between the two gridsand thus reduced loading of the DC grid doesn’t affect operating costs asmuch. However, unlike figure 5.18 the system costs continue to decrease inproportion to increased expendable control of the terminals, almost until themaximum allowed value of ∆PT /P

maxT . This behaviour is observed because

there is more room for terminal corrective actions when the DC grid is notcongested. This means that the system in this case can benefit from severechanges in post-contingency controls.It follows that one or more violationsin post-contingency states of the system can be entirely cleared throughcorrective control of terminal flows instead of using preventive generationdispatching to achieve the security level desired.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

100% Load120% Load140% Load

∆PT /PmaxT

Norm

alizedCost


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07


∆PT /PmaxT

Norm

alizedCost

Figure 5.20: Effects of Terminal Control for ΠAC = ΠDC : Case 2


A comparison of absolute costs invoked by these two levels of penaltycosts is shown in figure 5.21 for 100% load. Figures 5.19 and 5.22 portraysimulations for case 4 where all contingencies are accounted for. The resultsare similar to the simulations for case 2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17800

7900

8000

8100

8200

8300

8400

8500

ΠAC=30,ΠDC=0.1

ΠAC=ΠDC=10

∆PT /PmaxT

Absolute

Cost($/h)

Figure 5.21: Comparison of Absolute Costs for 2 levels of ΠAC ,ΠDC : Case2

5.3.4 DC Grid Capacity and System Performance

The amount of power the DC lines and the terminals are able to transmitundoubtedly influence the overall performance of the system. Seeing thatthe DC networks are still at an early stage, some insight in the way technicalspecifications such as capacities affect costs is really useful.

The first simulation considers a large amount of combinations for thelimits Pmax

T and FmaxDC and is performed for the non-secure base OPF. The

result is the surface plot of figure 5.23. Along the x-axis the quantity thatchanges is the terminal capacity, while along the y-axis it is the DC linecapacity. Each pair of these values induce a cost that is depicted on z-axis.Very close to the axes the cost of operation rises dramatically as was ex-pected. For a fixed value of terminal capacity the steepness of the curvesrunning along y-axis depict the sensitivity to DC line capacities. The sameapplies for curves running along x-axis for sensitivity to terminal capacities.The steepness of the former is greater, though the curves related to the DC

5.4. SOLVING TIMES 61

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

100 % Load120 % Load140 % Load

∆PT /PmaxT

Norm

alizedCost

Figure 5.22: Effects of Terminal Control for ΠAC = ΠDC : Case 4

line capacities even out horizontally much earlier than the curves related toterminal capacities. Thus operating costs depend more on terminal capaci-ties, which is logical given the fact that power flowing through the terminalsis a prerequisite to power flowing on DC lines. In figure 5.24 the same anal-ysis is done as in figure 5.23, only for Case 2. The two surface plots aresimilar, though in Case 2 the costs continue to decrease for higher valuesof capacities. In figure 5.23 there is an area wherein increase in capacitiesdoes not further minimize costs, leading to a horizontal area, but with theinclusion of security measures the system can further benefit from increasedcapacities. This is also evident from the fact that the difference betweenhighest and lowest cost is twice as big in the security constrained case thanin the base-case.

5.4 Solving Times

A significant advantage of using linear models is the fact that the solversavailable obtain the solution very fast. The solution for the base OPF isobtained in 7 − 10 milliseconds whereas the full non-linear OPF for thesame problem requires approximately 2.5 seconds to reach a solution [1].The solving times for all cases that are studied are seen in figure 5.26


00.2

0.40.6

0.81

00.2

0.40.6

0.81

7600

7650

7700

7750

7800

7850

7900

7950

Absolute

Cost($/h)

Terminal Capacity (p.u)DC Line Capacity (p.u)

Figure 5.23: Dependency of Operational Costs from Capacities for BaseOPF

00.2

0.40.6

0.81

00.2

0.40.6

0.81

7800

7900

8000

8100

8200

8300

8400

8500

Absolute

Cost($/h)

Terminal Capacity (p.u)DC Line Capacity (p.u)

Figure 5.24: Dependency of Operational Costs from Capacities for Case 2

5.5 Case Study II

The next test case to be studied is a larger grid compared to the one studiedin sections 5.1-5.4. It is the RTS96 test case [14] with an added DC grid.

5.5. CASE STUDY II 63

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17600

7650

7700

7750

7800

7850

7900

7950

7650

7700

7750

7800

7850

7900

(a) Base-Case

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17800

7900

8000

8100

8200

8300

8400

8500

7900

8000

8100

8200

8300

8400

(b) Case 2

Figure 5.25: Projection of figures 5.23 and 5.24 for comparison of depen-dency on terminal and DC line capacities


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Base OPF(Case 0) Preventive

Lines+Gen(Case 1)

Lines+Gen(Case 2)



Tim

e[sec]

Figure 5.26: Calculation times for all cases

The DC grid consists of eight DC buses, six of which are interconnected tothe AC grid through a terminal, and sixteen DC lines. The system is seenin figure 5.27.

118121

122

117

116119 120

123

113

114

112111

115

124

103

104

105

110109 106

108

107102101

218221

222

217

216219 220

223

213

214

212211

215

24

203

204

205

210209 206

208

207202201

318

321322

317

316319 320

323

313

314

312311

315

324

303

304

305

310309 306

308

307302301

325

2

3

46

5

1

87

Figure 5.27: IEEE RTS-96 with interconnected DC grid

5.5. CASE STUDY II 65

The costs induced by the cases described in the beginning of section 5.3are gathered in table 5.3, along with the time required for the solver toobtain the solutions.

Table 5.3: Cost of Security and Solving Times for Test Case of Figure 5.27

Scenario Type of Contingen-cies included

OperationalCost [$/h]

Cost of Secu-rity [$/h]

SolvingTime[sec]

Case 0 No Security 123394 - 0.078

Case 1 Lines 126198 2804 9.6

Case 2 Lines,Generators 128140 4746 30.8

Case 3 Lines,Terminals 126224 2830 9.8

Case 4 All Contingencies 128164 4770 31.1

Figure 5.28 depicts the effects of post-contingency terminal control on theoperating costs. Contrary to the first case which was presented earlier, theamount of corrective control allowed has no noticeable effect on operationalcosts for the system under consideration. Several cheap generating units aredistributed over the entire system and the buses where they are placed havemany interconnecting lines with adjacent buses. This means that the systemis robust. The topology of the system and its technical characteristics mightcontribute to the fact that operational costs are independent of terminalcorrective control. It is probable that a different positioning of the terminalscould lead to a situation where operational costs depend on the amount ofpost-contingency corrective actions.

The reduced flexibility of the system is highlighted by another interestingobservation, derived by comparing tables 5.1 and 5.3. If the cost of includingsecurity measures for all contingencies (case 4) is expressed in relation to thecost of the base-OPF, then for the IEEE 14-bus system:

∆Cost14(%) =Costcase4 − Costbase

Costbase=

7828.51 − 7643.63

7643.63= 2.4% (5.2)

Equivalently for the RTS-96 system:

∆CostRTS−96(%) =Costcase4 − Costbase

Costbase=

128164 − 123394

123394= 3.86%

(5.3)It is proportionately more expensive to provide security for the RTS-96

system than for the IEEE 14-bus system.Solving times in this case increase a lot. But Cases 2 and 4 that require


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

100 % Load115 % Load125 % Load

∆PT /PmaxT

Norm

alizedCost

Figure 5.28: Effects of Terminal Control for Case 4

approximately 30 seconds to reach a solution are accounting for more than200 contingencies, so the solution is obtained fast for the size of the prob-lem. Effective contingency screening could significantly reduce computa-tional time, since most of the non-binding contingencies would be neglected.

Chapter 6

Conclusion and Discussion

This chapters serves as a summary of key points and gives an outlook ofpossible future research challenges.

6.1 Conclusion

In the current work two methods of applying the security constrained opti-mal power flow in a mixed AC-DC grid were presented.

The preventive SC-OPF is a tool that could prove to be very useful inthe future of power system operation. A robust formulation and solutionto this problem is a state of the art challenge since an effective methodcould significantly enhance the stability and reliability of power systems.The additional operating costs that are induced by a preventive dispatchare insignificant compared to the prospect of a guaranteed supply. Extremefailures like major blackouts can be avoided, along with the severe economicconsequences these situations impel.

The preventive-corrective method generally poses as a cheaper alterna-tive to the fully preventive method. Corrective actions implemented afterthe occurrence of a contingency are usually attributed “zero costs” becausethey only have to be implemented rarely. However these actions have tobe performed based on a set of viable control strategies developed by thesystem operator. If these strategies prove to be poor, preventive measuresneeded to ensure security could be more expensive than the equivalent mea-sures of a preventive method. The importance of post-contingency terminalcontrol with respect to operating costs was analytically described in theprevious chapter. Flexible changes in the power exchanged between the ACand DC grid are suitable to clear a substantial amount of post-contingencyconstraint violations. These corrective actions provide a cheaper solutionthan a preventive MW dispatch of generation.

For such control schemes to exist, the appropriate strategies have to be

67

68 CHAPTER 6. CONCLUSION AND DISCUSSION

planned and the physics behind such real time operations of VSC terminalshave to be studied extensively.

It can be argued that with the development of liberalized power marketsthe level of security of power systems has been weakened [2]. Especially withthe introduction of intra day markets, there is substantial pressure not tointerfere with the market. Under these circumstances the use of correctivecontrol is favoured over that of preventive rescheduling.

6.2 Future Work

A future extension of the proposed methods would be a formulation of theSC-OPF that encompasses non-linear effects. A non linear OPF problemformulation for the mixed grid is shown in [15]. Further development toinclude security constraints for this more precise problem seems promising.

Appendix A

Numerical Example

In this section a numerical example of the preventive-corrective SC-OPF inthe mixed grid is presented. The test case is a small custom system that isshown in figure A.1. All system quantities are presented in the tables thatfollow:

Table A.1: Generator Data

Gen No At AC Bus Pmax [p.u] Pmin [p.u] Linear Cost Coefficient Quadratic Cost Coefficient

1 1 1.6 0 18 0.0585295

2 3 1.2 0.25 20 0.25

3 4 1.7 0.15 40 0.12

Table A.2: AC Line Data

Line No From AC Bus To AC Bus Reactance [p.u] Rating [p.u]

1 1 2 0.08947 0.5

2 1 3 0.21502 0.45

3 1 4 0.17291 0.4

4 2 3 0.15879 0.45

5 3 4 0.22388 0.45

Table A.6 contains the decision variable ξ for the base OPF and the pre-contingency settings ξ0 of Case 1 that includes security constraints for ACand DC line outages. The distribution of power flows is registered in tableA.7.

The results for Cases 3 and 4 can be found in tables A.8 and A.9

69

70 APPENDIX A. NUMERICAL EXAMPLE

4

1

2

3

G1

G2

G3

T1

T2

T3

VSC

VSC

2

3

1

VSC

PL3

PL2

PL1

Figure A.1: Small Test Grid

71

Table A.3: DC Line Data

Line No From DC Bus To DC Bus Resistance [Ω] Rating [p.u]

1 1 2 1.3908 0.35

2 1 3 3.2208 0.4

3 1 4 2.3424 0.4

4 4 5 4.3920 0.4

5 3 4 3.27936 0.35

6 2 5 1.37616 0.5

Table A.4: Terminal Station Data

Terminal No AC Bus DC Bus Rating [p.u]

1 1 1 0.9

2 4 2 0.7

3 2 5 0.9

Table A.5: Active Power Demand

PL1[p.u] PL2[p.u] PL3[p.u] PL4[p.u]

0.2 0.8 0.65 0.5


Table A.6: Decision Variable ξ for Base OPF and Case 1

Base OPF Case 1 SC-OPF

Cost of Operation 6205.55 $/h Cost of Operation 6225.3 $/h

Generation Profile

G1 1.6

Generation Profile

G1 1.5438

G2 0.4 G2 0.4477

G3 0.15 G3 0.1585

AC Bus Angles

δ1 0

AC Bus Angles

δ1 0

δ2 -0.0382 δ2 -0.03803

δ3 -0.04178 δ3 -0.04130

δ4 -0.0343 δ4 -0.0425

Terminal Transfers

PT1 0.5797

Terminal Transfers

PT1 0.4934

PT2 -0.3948 PT2 -0.3955

PT3 -0.1849 PT3 -0.0979

DC Bus Voltages

V1 0

DC Bus Voltages

V1 0

V2 -0.5563 V2 -0.5138

V3 -0.5789 V3 -0.3994

Table A.7: Line Flows for Base OPF and Case 1 SC-OPF

Base OPF Case 1 SC-OPF

Power Flow Power Flow

AC Line

1-2 0.4274

AC Line

1-2 0.425

1-3 0.1943 1-3 0.192

1-4 0.1984 1-4 0.2462

2-3 0.0223 2-3 0.0206

3-4 -0.0333 3-4 0.0057


DC Line

1-2 0.4

DC Line

1-2 0.3694

1-3 0.1797 1-3 0.124

2-3 0.005 2-3 -0.026

73

Table A.8: Decision Variable ξ for Case 3 and Case 4

Case 3 SC-OPF Case 4 SC-OPF

Cost of Operation 6981 $/h Cost of Operation 7023.8 $/h

Generation Profile

G1 1.126

Generation Profile

G1 1.0409

G2 0.8353 G2 0.7692

G3 0.1881 G3 0.3397

AC Bus Angles

δ1 0

AC Bus Angles

δ1 0

δ2 -0.0357 δ2 -0.0337

δ3 -0.0091 δ3 -0.0093

δ4 -0.0227 δ4 -0.0115

Terminal Transfers

PT1 0.353

Terminal Transfers

PT1 0.3534

PT2 -0.2336 PT2 -0.2698

PT3 -0.1193 PT3 -0.0836

DC Bus Voltages

V1 0

DC Bus Voltages

V1 0

V2 -0.334 V2 -0.3588

V3 -0.363 V3 -0.3072

Table A.9: Line Flows for Case 3 and Case 4 SC-OPF

Case 3 SC-OPF Case 4 SC-OPF


AC Line

1-2 0.3991

AC Line

1-2 0.3769

1-3 0.0426 1-3 0.0436

1-4 0.1317 1-4 0.0669

2-3 -0.1671 2-3 -0.1532

3-4 0.0607 3-4 0.0097


DC Line

1-2 0.2402

DC Line

1-2 0.258

1-3 0.1127 1-3 0.095

2-3 0.0065 2-3 -0.0117


Appendix B

Example with Matrix

Formulations

To get a clear idea of the way the various matrices are formulated, a simple3 AC bus 3 DC bus system of figure B.1 is studied. All three AC buses areinterconnected with a DC bus through a terminal station. The matrices thatare needed to formulate the fully preventive SC-OPF problem are going to beexplicitly defined. The dimensions of the matrices to follow are according tothe nomenclature of table 3.1: The equality constraints, which only account

Table B.1: Nomenclature for Numerical Example

K=3 Number of AC nodes

N=3 Number of AC lines

M=3 Number of DC nodes

D=3 Number of DC lines

I=2 Number of generators

L=3 Number of terminals

C=9 Length of vector ξ

Yprev=58 Amount of inequality constraints

for the pre-contingency state as documented in section 4.1.5, are constructedas follows:

H =

1 0

0 1

, T =

−1 0 0

0 −1 0

0 0 −1

, S =

1 0 0

0 1 0

0 0 1

75

76 APPENDIX B. EXAMPLE WITH MATRIX FORMULATIONS

1

2

3

G1

G2

T1

T2

T3

VSC

VSC

2

3

1

VSC

PL2

PL1

Figure B.1: 3 Bus Test Grid

BAC =

−11.1769 −4.6507

17.4746 −6.2976

−6.2976 10.9484

, BDC =

−0.719 −0.3104

0.9467 −0.2276

−0.2276 0.5381

77

beq =[

0.2 0.8 0 0 0 0]T

(B.1)

ΛI =

1 0

0 1

, ΛL =

1 0 0

0 1 0

0 0 1

EI =

−0.4641 0

0 0.4641

−0.5359 0

0 0.5359

−0.4641 0

0 0.4641

, EAC =

−11.1769 0

−11.1769 0

0 −4.6507

0 −4.6507

6.2976 −6.2976

6.2976 −6.2976

AACprev =

−11.177 0

−11.177 −4.65

−17.474 6.297

−11.177 −4.65

0 −4.65

6.297 −10.948

17.474 −6.297

6.297 −10.948

6.297 −6.297

, ADCprev =

−0.719 0

−0.719 −0.31

−0.946 0.227

−0.719 −0.31

0 −0.31

0.227 −0.538

0.946 −0.227

0.227 −0.538

0.227 −0.227

biq,prev = [bmaxg bmin

g bl,AC,G − bl,AC,G

bl,AC,P − bl,AC,P bterm − bterm bl,DC,P − bl,DC,P ]T[Yprev×1]

bmaxg =

1.2

1.1

, bming =

0

0.3

, bl,AC,G =

bl,AC

bl,AC

...

bl,AC

[(N×I)×1]

, bl,AC,P =

bl,AC

bl,AC

...

bl,AC

[N2×1]

78 APPENDIX B. EXAMPLE WITH MATRIX FORMULATIONS

bl,DC,P =

bl,DC

bl,DC

...

bl,DC

[D2×1]

, bl,AC =

0.4

0.3

0.4

, bl,DC =

0.3

0.3

0.35

QC =

0.117059 0

0 0.5

,WAC =

120− 60

−60120

,WDC =

0.2− 0.1

−0.10.2

With the matrices presented the preventive SC-OPF problem can beformulated as in section 4.1.5.

Bibliography

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[11] P.N. Biskas and A.G. Bakirtzis. A decentralized solution to the securityconstrained dc-opf problem of multi-area power systems. In Power Tech,2005 IEEE Russia, pages 1–7, 2005.

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