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On Frequency Control Schemes in Power Systems with Large Amounts of Wind Power

CAMILLE HAMON

Licentiate Thesis Stockholm, Sweden 2012

TRITA-EE 2012:061 ISSN 1653-5146 ISBN 978-91-7501-585-9

KTH School of Electrical Engineering SE-100 44 Stockholm SWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie licentiatexamen i elektriska energisystem tisdagen den 11 december 2012 klockan 10.00 i E2, Lindstedsvägen 3, Kungl Tekniska högskolan, Stockholm. © Camille Hamon, December 2012 Tryck: Universitetsservice US AB

iii Abstract In recent years, large investments have been made in wind power, and this trend is expected to continue in the coming decades. Integrating more wind power in the production mix offers great opportunities for the society, such as reducing greenhouse gas emissions and the dependence on foreign fuel. Large wind power penetration does, however, require changes in the way power systems are planned and operated. The power transfers across the electrical grid are determined by the load and the production. A secure operation of power systems requires that these power transfers stay within certain limits. Frequency control schemes are crucial for ensuring the balance between the electric demand and the production. They enable system operators to re-dispatch the production (for example via the activation of balancing bids) during real-time operations to follow the load variations. With wind power, these frequency control schemes must not only meet the variations of the load but also those of the wind. An optimal use of the frequency control reserves would allow system operators to operate the system in the most cost effective and secure manner, that is, using the cheapest available resources while taking into account the stability limits of the system and the uncertainty. With no wind power, the load is the main source of uncertainty, and it can be forecasted accurately. This enables system operators to dispatch the generation in the most cost-effective way to meet the load while keeping the system within its stability limits. Adding wind power to power systems, on the other hand, introduces a new source of uncertainty on the production side, which is more difficult to forecast. The tools used today for computing the stability limits and operating the system do not consider the whole range of possible future load and wind power production levels, but only pick a few likely values in this range. In this work, we propose a new approach which accounts for the whole uncertainty in the load and wind power, and gives the optimal re-dispatch which ensures a given level of system security given this uncertainty. The approach is a so-called Stochastic Optimal Power Flow (S-OPF) formulation, developed in the scope of this project for the optimal activation of balancing bids. It is a nonlinear optimization problem with one probabilitistic constraint ensuring a certain level of system security – computed as the probability that the system stays within its stability limits – and whose objective function is the minimization of the generation re-dispatch. Compared to what is done today, the S-OPF formulation enables system operators to consider the uncertainty when making decisions. An approximation of the proposed S-OPF formulation is developed to render the problem tractable. In particular, the stability boundary, defined as the set of stability limits, is approximated by second-order approximations. The accuracy of these second-order approximations are analyzed in the IEEE 9 bus system by computing the distance between the actual boundary and its approximation. The S-OPF problem is then solved in the IEEE 39 bus system using the approximated stability boundaries. Monte Carlo simulations are run in order to assess the accuracy of the approximation and check whether the optimal solution of the approximation does ensure the specified level of system security.

Acknowledgments I would like to begin by thanking Lennart Söder and Mikael Amelin for creating the project, and giving me the opportunity to work on it. Lennart Söder has been my main supervisor, and has given much appreciated and helpful feedback throughout my work. The financial support from Vindforsk is gratefully acknowledged. The reference group is also to be thanked for the feedback on this project. In particular, Elin Broström has provided useful information on the Swedish power system. I am grateful to all colleagues in the EPS department for the nice working atmosphere and many fikas. Thanks to that, it is a pleasure to come to work everyday, and I am less inclined to jobba hemma. I am indebted to Magnus for his constructive supervision which has included useful discussions, several passes to thoroughly proofread this thesis and hopeless tries to make me pronounce the Swedish “y” and “u” correctly. Special thanks to Richard for nice discussions and sharing his best fika plans, Pia for the nice time sharing the office, Valentin and Topp for all the support, Angela, Yalin and Tu for the fun and laughs, and Yuwa for all the warmth and kindness.

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Contents

Acknowledgments

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Contents

vi

List of Figures

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List of Tables

xii

1

Introduction 1.1 Background . . . . . . . . . . . 1.2 Aims and scope of this thesis 1.3 An introductory example . . . 1.4 Thesis outline . . . . . . . . . 1.5 Contributions . . . . . . . . . 1.6 List of publications . . . . . .

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I

Background

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Technical background 2.1 Wind turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Electricity markets . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Frequency control schemes . . . . . . . . . . . . . . . . . . . 2.4 Challenges for the operation of frequency control schemes 2.5 Generation re-dispatch and operation of tertiary control . . 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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17 17 20 22 29 33 41

Mathematical foundations 3.1 Newton’s method . . . . . . . . . . . . . . . 3.2 Elements of differential geometry . . . . . . 3.3 Gram-Schmidt orthonormalization . . . . . 3.4 Cumulants and Cornish-Fisher expansion .

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CONTENTS 4

Elements of bifurcation theory 4.1 Dynamical systems . . . . . . . . . . . . . . . . . . . . . 4.2 Equilibria and topological classification of equilibria . 4.3 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Topological normal forms of saddle-node bifurcations 4.5 Topological normal forms for Hopf bifurcations . . . . 4.6 Center manifold theory . . . . . . . . . . . . . . . . . . . 4.7 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . .

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II Approximations of the stability boundary 5

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Stability in power systems 5.1 Power system models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Voltage instability, small-signal stability and bifurcation theory . . . 5.3 Stability boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Normal to the stability boundary . . . . . . . . . . . . . . . . . . . . . 5.5 Iterative method to get the closest point on the stability boundary . 5.6 Summary and challenges with larger amounts of variable resources

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71 . 71 . 75 . 81 . 93 . 100 . 100

Polynomial approximations of the stability boundary 6.1 Review of existing approximations of the stability boundary 6.2 Second-order approximations of the stability boundary . . 6.3 Weingarten maps . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Finding the approximation point on each type of surface . . 6.5 Contingencies and corrective actions . . . . . . . . . . . . . 6.6 Considering margins . . . . . . . . . . . . . . . . . . . . . . . 6.7 Note on the parameters . . . . . . . . . . . . . . . . . . . . . . 6.8 The importance function . . . . . . . . . . . . . . . . . . . . . 6.9 Comparison with the iterative method . . . . . . . . . . . . . 6.10 Distance to the second-order approximations . . . . . . . . 6.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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103 103 104 106 111 118 119 119 120 121 121 123

Second-order approximations: case study in the IEEE 9 bus system 7.1 Setup of the case study . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Objectives with the case study . . . . . . . . . . . . . . . . . . . . 7.3 Case 1: Illustration of the method . . . . . . . . . . . . . . . . . . 7.4 Case 2: Accuracy of the second-order approximations . . . . . . 7.5 Computational issues . . . . . . . . . . . . . . . . . . . . . . . . .

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125 125 127 129 130 132

III Stochastic optimal power flows 8

59 59 60 61 63 64 65 68

Stochastic optimal power flow

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135 137

viii

CONTENTS 8.1 8.2 8.3 8.4 8.5

9

Stochastic optimal power flow for generation re-dispatch . Usage of the S-OPF formulation within the operating period Approximation of the constraint . . . . . . . . . . . . . . . . Solving the S-OPF problem . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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137 140 144 156 157

Application of stochastic optimal power flow in the IEEE 39 bus system 9.1 Setup of the case study . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Objectives with the case study . . . . . . . . . . . . . . . . . . . . . . . 9.3 Case 1: correct pre-distribution . . . . . . . . . . . . . . . . . . . . . . 9.4 Case 2: incorrect pre-distribution . . . . . . . . . . . . . . . . . . . . . 9.5 Computational time . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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161 161 162 164 166 167

10 Conclusion and future work 169 10.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 10.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

IV Appendices

175

A Power System Data 177 A.1 Reference Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 A.2 IEEE 9 bus system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 A.3 IEEE 39 bus system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 B Derivatives of A and Φ B.1 Implicit function z = Φ(λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 First-order derivatives of A and Φ . . . . . . . . . . . . . . . . . . . . . . . . B.3 Second derivative of A and Φ . . . . . . . . . . . . . . . . . . . . . . . . . .

183 183 183 186

Bibliography

189

List of Figures

1.1 1.2 1.3 1.4 1.5

1.6 1.7 1.8 2.1 2.2 2.3 2.4 2.5 2.6

2.7 2.8 2.9

Forecasted load and planned production: they are equal on average but not on a continuous basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The different phases in power system planning and operation. Time frame considered in this thesis: Operating period (one hour or less). . . . . . . . . . Cumulative installed wind power capacities in 2010 (dark gray) and 2011 (additional capacity compared to 2010 in light gray), numbers from [46]. . . . . Global cumulative installed wind power capacities for the period 2001-2016, numbers from [46]. Years 2012-2016 are forecasts. . . . . . . . . . . . . . . . . Share of each renewable energy source for reaching the 34% target for the share of electricity consumption from renewable sources, according to the submitted NREAPs, from [44]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Share of net electricity production coming from wind power in Sweden [%], from [40]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference power system from [59]. . . . . . . . . . . . . . . . . . . . . . . . . . The problem considered in this thesis. . . . . . . . . . . . . . . . . . . . . . . . An example of power curve for a 1500 kW wind turbine. . . . . . . . . . . . . . The different time frames for power system operation and planning. The operating period lies in the scope of this thesis. . . . . . . . . . . . . . . . . . . Influence of load forecast errors on frequency control schemes: the production is not planned optimally. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Load, wind power (WP) production and net load on Gotland, 16 March 2009, from Gotlands Energi AB (GEAB). . . . . . . . . . . . . . . . . . . . . . . . . . . Generators are driven by turbines, and supply electric power to the loads. . . Requirements for the automatic active reserves in Nordel: Frequency controlled normal operation reserve (solid line) and Frequency controlled disturbance reserve (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . The different layers of frequency control schemes, inspired by [104]. . . . . . Illustration of the σ-method with the probability distributions of load and net load (NL) variations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Increase in reserve requirements due to wind power, from [53]. . . . . . . . . ix

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6 6 9 11 19 21 23 24 24

27 29 31 32

x

List of Figures

2.10 Normalized standard deviation of wind power variations approximated as a function of the mean distance between the wind turbines, from [100] (Publication II). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.11 The Swedish power system, the four bidding areas and the three bottlenecks from [60]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.12 Computation of transmission limits across one bottleneck, from [93]. . . . . 38 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4.1 4.2 4.3 4.4

SMIB system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Newton’s method applied to the power flow problem. . . . . . . . . . . . . . . Problem of projecting A onto the sphere in a given direction. . . . . . . . . . Convergence of Newton’s method. The first three iterations are shown [blue diamond]. The last three iterations are very close to P (A). . . . . . . . . . . . Gram-Schmidt process: taking away the projection of v 2 onto b 1 . . . . . . . . The unit sphere S2 and its normal n at P = (0.41, 0.82, 0.4). . . . . . . . . . . . The unit sphere S2 and its tangent plane at P = (0.41, 0.82, 0.4) . . . . . . . . . The two possible generic co-dimension one bifurcations. . . . . . . . . . . . . Bifurcation diagram and some trajectories for the dynamical system in (4.6). A saddle-node bifurcation occurs at β = 0, from [64]. . . . . . . . . . . . . . . . Subcritical Hopf bifurcation, from [63]. . . . . . . . . . . . . . . . . . . . . . . . Supercritical Hopf bifurcation, from [63]. . . . . . . . . . . . . . . . . . . . . .

45 46 47 49 54 56 56 62 64 65 66

5.1 5.2

P-V curves with SNB and SLL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Difference between a harmless breaking point and a SLL (harmful breaking point). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.3 Stability boundary in parameter space, and restricted to the load space for a given value u 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.4 Stability boundary in the IEEE 9 bus system in a three dimensional load space, made of different smooth parts: Hopf (dark and light blue), SLL (green) and line thermal limits (red, orange and yellow). . . . . . . . . . . . . . . . . . . . . 87 5.5 The tangential intersection of a SNB and a SLL surface [88]. . . . . . . . . . . 91 5.6 A harmless breaking point, a SLL (harmful breaking point) and a tangential intersection point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.7 Pre- and post-contingency stability boundaries. Colors as the same as in Figure 5.4. The post-contingency stability boundary is the innermost one. . . . 93 5.8 Limited knowledge of the stability boundary. . . . . . . . . . . . . . . . . . . . 94 5.9 Two dimensional stability boundary of a fictitious power system, and forecasted load (system without wind power) and net load (with wind power) increase paths. P 1 and P 2 are the net loads at buses 1 and 2, respectively. . . 101 5.10 Illustration of the need for computing more stability limits. . . . . . . . . . . 102 6.1 6.2 6.3

One predictor-correction step towards the most important point. . . . . . . . 114 Search after the most important point on a set of corner points . . . . . . . . 116 The tangential intersection of a SNB and a SLL surface [88]. . . . . . . . . . . 117

List of Figures

xi

6.4 6.5

Computation of the distance to second-order approximations. . . . . . . . . 122 Overview of the method to compute second-order approximations of all boundary surfaces of interest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.1

The stability boundary of the pre-contingency IEEE 9 bus system, consisting of six different smooth parts (corresponding to the six different colors). . . . Searches to the most important points on these smooth parts (black lines) from different start points (black circles). All searches on the same smooth part converge to the same closest point (white circles). . . . . . . . . . . . . . The stability boundary and the second-order approximations of the light blue part (in gray). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spherical coordinates defined from λbase , adapted from [7]. . . . . . . . . . . Absolute values of the distances between the approximations and the real surface. The smooth parts are colored according to Figure 7.2. . . . . . . . . .

7.2

7.3 7.4 7.5

8.1 8.2 8.3 8.4

127

129 130 132 133

8.5 8.6 8.7 8.8

The two phases in solving the S-OPF problem. . . . . . . . . . . . . . . . . . . Approach 1: monitoring and acting; ∆t is at most a few seconds. . . . . . . . Approach 2: repeatedly acting; ∆t is a few minutes. . . . . . . . . . . . . . . . An approximation of a stability boundary consisting of two smooth parts Σia1 and Σia2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometrical interpretation of V1 and V2 . . . . . . . . . . . . . . . . . . . . . . . Case of three smooth parts intersecting. . . . . . . . . . . . . . . . . . . . . . . A parameter space with two SLL surfaces and one SNB surface. . . . . . . . . Flowchart of the methodology to solve the S-OPF problem. . . . . . . . . . . .

141 143 143

A.1 A.2 A.3

Reference power system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 IEEE 9 bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 IEEE 39 bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

147 149 151 154 158

List of Tables

2.1

The four main wind turbine designs. . . . . . . . . . . . . . . . . . . . . . . . .

18

5.1 5.2 5.3

Description of the parameters in (5.4) and (5.5). . . . . . . . . . . . . . . . . . Different types of stability limits. . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical expressions for the normals to different smooth parts. In all formulas, the normal is of unit length and points towards instability. . . . . . . .

73 82 99

6.1

Analytical expressions for the derivatives dN . . . . . . . . . . . . . . . . . . . . 112

7.1 7.2

Power transfer limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 The different stability limits in the IEEE 9 bus system. . . . . . . . . . . . . . . 128

9.1 9.2 9.3

The variation of the optimal solution when varying α. . . . . . . . . . . . . . . 164 Optimal solutions when using u c = 0. . . . . . . . . . . . . . . . . . . . . . . . . 165 The variation of the optimal solution when varying α. . . . . . . . . . . . . . . 166

A.1 A.2 A.3

Data for the reference power system of Figure A.1. . . . . . . . . . . . . . . . . 178 Data for the IEEE 9 bus system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Data for the IEEE 39 bus system . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

xii

Chapter 1

Introduction 1.1 Background 1.1.1 Power systems Power systems aim at continuously delivering electricity to end consumers in a secure manner. The secure and stable operation of power systems ensure that the socioeconomic costs are minimized, since system instability can lead to costly blackouts. Fulfilling this goal requires two phases: a planning phase to prepare power systems to meet the expected consumption – the forecasted electrical load – at a certain time, and an operation phase to react to unplanned events and continue delivering the electricity securely in real-time. The time frame for the operation phase is the operating period, whose length vary for different power systems from 5 minutes up to one hour [112, 68]. The planning phase The planning phase spans the period ahead of the operating period. It is usually organized around, but not limited to, a wholesale electricity market where market participants, the so-called balance responsible players, submit production and consumption bids for future operating periods. During the planning phase, production plans are determined in order to meet the forecasted load. These plans are based on offers submitted by the balance responsible players and transmission limits set by system operators. This is called market clearing. The market is usually cleared the day before the actual operating periods. After the market has been cleared, and before the beginning of each operating period, the market players can continue to trade on intra-day markets. This intra-day trading is used to adjust to new information such as updated forecasts. During the operating period, the balance responsible players are then responsible for supplying electricity according to their offers if they have been accepted. They are financially penalized for deviations from their offers, which are usually energy based (e.g. given in MWh per hour). This means that the balance responsible players commit themselves to produce a certain amount of power which covers the load on average during operating 1

2

CHAPTER 1. INTRODUCTION

periods, i.e. they are responsible for the energy balance between the planned production and the forecasted consumption. This is shown in Figure 1.1, where the planned production is equal to the forecasted load on average over the operating period, but imbalances arise on a continuous basis (shaded area).

Load

Planned production Time Operating period (one hour or less) Figure 1.1: Forecasted load and planned production: they are equal on average but not on a continuous basis.

The operating phase and frequency control schemes During the operating period, the deviations between the actual production and the actual load are taken care of by frequency control schemes which activate frequency control reserves to maintain the balance between production and consumption. This is the responsibility of the system operator. Frequency control reserves are power reserves kept in participating power plants. Some frequency control reserves are continuously controlled so as to quickly respond to changes in the system, while some others correspond to discrete actions taken by the system operator who can ask power producers to manually increase or decrease the production levels of some of their power plants. The former are controlled by the so-called primary and secondary frequency control schemes, while the latter are controlled by the so-called tertiary frequency control schemes. This thesis deals with the optimal operation of frequency control reserves with large amounts of wind power. “Optimal” means that the frequency control reserves should be deployed at a cost as low as possible while ensuring a secure operation of the power system in the sense that the system must be operated within certain stability limits. For example, transmission limits defining the maximum allowable power transfers across certain transmission corridors 1 are computed by system operators to this purpose. The 1. Transmission corridors are transmission lines or sets of transmission lines. They can also be called flowgates [31].

3

1.1. BACKGROUND

new challenges arising with the expected large amounts of wind power will be described in subsequent sections. The time frame of interest will be the time frame spanned by the operating periods (one hour or less). The operation of frequency control reserves has an impact on the planning phase, because power systems are planned assuming that the real-time operations are handled in a certain way. Hence, although not directly dealing with the planning phase, the work developed here will influence it. In power systems with very large penetration of wind power, the question arises of whether wind power should also be used for frequency control purposes 2 . Methods developed for the operation of frequency control reserves can be used to answer this question by finding the optimal way of using the available reserves, be it from conventional generators 3 or from wind power plants. The different phases described above are depicted in Figure 1.2, where the operating period has been shaded to emphasize that this is the focus of this thesis. The dotted line between “Operating period” and the two planning phases shows the influence that the way in which the system is operated has on decisions taken in the planning phase.

Seasonal, weekly and day-ahead planning

Intra-day planning

Operating period

Time Market clearing Figure 1.2: The different phases in power system planning and operation. Time frame considered in this thesis: Operating period (one hour or less).

1.1.2 Two substantial changes faced by power systems In the past decades, electric power systems have witnessed two important changes. First, in many countries, electricity supply has been deregulated. National vertical utilities have been subject to competition, and electricity markets have been created, the goal being to increase competition on the supply side. Operation of the transmission and distribution grids are considered to be natural monopolies, and have therefore remained so, while the supply side was restructured [112]. 2. Note that it is already required in some countries and implemented in modern wind farm controllers [61] 3. Conventional generators can be controlled so as to deliver a certain amount of power up to their rated capacity. Thermal, nuclear, and to some extent hydro power plants are examples of conventional generators. The term is usually used in contrast with wind turbines, solar power plants, and other variable generation resources which are limited by the natural variations of the wind or the sun.

4


Second, concerns have been raised about environmental issues and dependence on foreign countries for supplying the fuel necessary to run conventional power plants [46]. Both of these concerns have led governments to set up goals for increasing the share of renewable energy sources in the electricity production mix. Among renewable energy sources, hydro power has already been developed to a large extent. Wind power, solar power, tide and wave power, and geothermal power are other renewable energy sources in which investments have not been as large as in hydro power. Many of these energy sources are variable, which means that unlike fuel-based energy sources (such as coal, gas and nuclear), the output of power plants using renewable energy sources is partly defined by the natural variations of the sources themselves (for instance variations in wind speed and incident solar radiation). Among these variable resources, wind power is regarded as the most technologically mature for large-scale electric power production as of today [56]. Large amounts of wind power have been installed, and will continue to be installed in the coming future. In 2011, 21 countries installed more than 1 GW wind power [46] each. Figure 1.3 shows the cumulative installed capacities in different parts of the world for years 2010 and 2011. Year 2010 witnessed a substantial growth in the Asian market with large investments in China and India. Large amounts of wind power were also installed in Europe and North America. The global installed wind power capacity has increased steadily, and is expected to continue doing so as shown in Figure 1.4 where the global cumulative installed capacities during the past decade as well as forecasts from the Global Wind Energy Council (GWEC) up to 2016 are plotted. Europe Asia North America Pacific Region Latin America 2010 2011

Africa and Middle East 0

20

40 60 Installed capacity [GW]

80

100

Figure 1.3: Cumulative installed wind power capacities in 2010 (dark gray) and 2011 (additional capacity compared to 2010 in light gray), numbers from [46]. In Europe, national overall targets for the share of energy from renewable resources

5

Cumulative installed capacity [GW]

1.1. BACKGROUND

500

Historical data Market forecasts

400 300 200 100 0 2002 2004 2006 2008 2010 2012 2014 2016

Figure 1.4: Global cumulative installed wind power capacities for the period 2001-2016, numbers from [46]. Years 2012-2016 are forecasts.

in gross final consumption in 2020 have been set through the Directive 2009/28/EC on Renewable Energy [43]. The overall European target is to reach 20% of the overall share of energy from renewable sources in 2020. Article 4 of the directive requires member states to submit so-called National Renewable Energy Action Plans (NREAPs) in which each country sets up roadmaps for reaching their individual targets. These action plans can be consulted in [21]. As far as the electricity sector is concerned, putting these plans together implies that 34% of EU’s total electric energy consumption will come from renewable energy sources in 2020 [44]. Figure 1.5 shows the contribution of each renewable source to the 34% target according to the NREAPs. Wind power is expected to play the most important role. Figure 1.6 shows the share of Sweden’s net electricity production coming from wind power in the period 2003–2011. We see that wind power has increased steadily. For Sweden, the European directive set the target for the share of energy from renewable sources in gross final consumption of energy in 2020 to 49%. Sweden’s NREAP describes how the Swedish government plans to reach the target [92]. In the electricity sector, 62% of the gross final consumption of electricity is expected to come from renewable energy sources. In addition, the Swedish Parliament specified a target for a planning framework of 30 TWh of wind power in 2020, including 10 TWh offshore wind power [78]. As a comparison, wind power plants generated 6.1 TWh of electricity in Sweden in 2011 [40]. Since wind power is expected to be the variable resource with the largest share in the electricity mix, challenges arising because of large amounts of wind power will be

6


Wind power

14

Hydro power

10.5 6.7

Biomass 2.4

Solar PV 1

Others

0 2 4 6 8 10 12 14 Share of total electricity consumption in 2020 [%]

Wind power share of electricity production [%]

Figure 1.5: Share of each renewable energy source for reaching the 34% target for the share of electricity consumption from renewable sources, according to the submitted NREAPs, from [44].

4 3.5 3 2.5 2 1.5 1 0.5

07 20 08 20 09 20 10 20 11

06

20

05

20

04

20

20

20

03

0

Figure 1.6: Share of net electricity production coming from wind power in Sweden [%], from [40].

studied in this thesis. Most of the challenges posed by wind power to existing electric power systems also apply for other variable resources.

1.1. BACKGROUND

7

1.1.3 Expected challenges Deregulation The two substantial changes constituted by deregulation and large amounts of wind power bring about new challenges for the operation of power systems. Deregulation has already been studied extensively, since issues around it appeared earlier than today’s challenges with large amounts of wind power. As early as 1989, the electricity supply industry was unbundled in England and Wales [47], shortly followed by the Nordic countries in the early nineties [112]. Early work looking at impacts of deregulation includes [20]. The two main issues identified and relevant to this thesis are that first, the dispatch order is changed as more actors enter the market, which modifies the power flows in the system; and second, the power plants are no longer controllable by system operators, who must then develop other methods to optimally use the controllable resources in the system. Note that concerning the latter point, we are observing a return to controllability of power plants by system operators as more wind power is integrated into power systems, as can be seen for example in certain markets in the U.S. and Spain where most of the wind generators are controllable by system operators [2, 55]. Large amounts of wind power The impacts due to large amounts of wind power, on the other hand, have begun to be studied later, since the installed amounts of wind power did not contribute to a significant share of the total generation mix until recently. In the past few years, however, much work has been put into identifying these impacts in the context of power system operation and planning [10, 50, 55, 58, 66, 97]. Together with the new challenges associated with the integration of wind power comes the need for new tools designed to meet these challenges. The new tools needed depend on which aspect of power system operation is of interest. This, in turn, determines which time horizon is considered. This thesis focuses on the operation of frequency control schemes, and, thus, on the operating period (no longer than one hour). As described in Section 1.1.1, these frequency control schemes activate power reserves in order to maintain the balance between production and consumption and to continue operating the system in a secure and stable manner. The power reserves used by frequency control schemes are located in different areas of the power system. Because transmission capacities are limited between the areas, the cheapest reserves cannot always be activated [82]. In addition, as will be seen in this thesis, injections of power reserve at different locations in the electrical grid will have different effects on the system security. Therefore, optimally maintaining the realtime balance between production and consumption with frequency control schemes requires tools which activate power reserves in the right locations to maximize system security and minimize the costs. In particular, the location of the primary and secondary control reserves is important, because they are automatically activated and, if badly located, their activation can increase the loading in critical transmission corridors, eventually leading the system to instability [85].

8


Traditionally, deviations between production and consumption could be due to load forecast errors, disturbances such as the loss of a generating unit, or the fact that balance responsible players are only responsible for delivering energy over the operating period. With large amounts of wind power in power systems, however, wind variations and wind forecast errors will entail larger needs for balancing power [52], thus increasing the need for robust tools to optimally handle frequency control schemes.

1.2 Aims and scope of this thesis This thesis focuses on the new challenges put on the operation of frequency control schemes by large amounts of wind power, considering the stability limits of the power system. Power systems with small amounts of wind power are planned and operated to meet the expected load, which can be forecasted with a good accuracy [12]. Hence, it has been possible to operate power systems in a deterministic manner, since the load patterns are well known and thus the uncertainty in such power systems is low. Large amounts of wind power will introduce a larger uncertainty in operation and planning [50]. Therefore, in the literature, a shift from the deterministic framework in which the system is operated today towards a probabilistic (or stochastic) framework has been advocated [66, 97, 116]. This stochastic framework is deemed more adapted to deal with the larger uncertainty due to wind power. It is well expressed by Mark Lauby et al. in [66] in the following manner (the boldface is for emphasizing)

“

Traditionally, many power system tools and techniques have been based on deterministic approaches in which a small number of single snapshots of the system (e.g. peak load) are used for analysis and application, in planning and/or operational time frames. These deterministic approaches have been driven to some extent by a need to limit computational burdens. With advances in computational capabilities, however, this is no longer a significant issue. In addition, changes in the structure of the power industry - in particular, introduction of competitive markets and the associated changes in the generation merit order - have made deterministic, single-snapshot analysis less meaningful in representing the underlying system. Furthermore, and most relevant, the increase in penetration of variable generation further undermines the value of deterministic snapshot analysis. To give a specific example: while the critical operating points of bulk power systems have traditionally been known, with the advent of wind and solar generation, these points are difficult to find, and require analysis of many more points. Hence, there is a need to study multiple scenarios, a process that is largely deterministic but could become computationally intractable. Probabilistic techniques are developed by studying the underlying distribution of scenarios rather than a specific set of points that make up the distribution.

”

9

1.3. AN INTRODUCTORY EXAMPLE

Following these recommendations, this thesis aims at addressing the new challenges due to wind power by describing in detail the stochastic framework advocated in the emphasized sentences in the quote above, and proposing tools for operating frequency control schemes that fall in this framework. Within frequency control schemes, this thesis will focus on tertiary control, that is, system operators’ decisions to change the output power of participating units within the operating period. The other frequency control schemes – primary and secondary frequency control – are assumed to be operated as they are today. The thesis does not address issues relating to power system planning, i.e. we assume a certain location and setup of primary and secondary resources, which need not be optimal. Optimizing these parameters is left as future work. We now give an introductory example that illustrates the scope of this thesis.

1.3 An introductory example We consider the power system in Appendix A.1 that is reproduced in Figure 1.7. The system has three generators and one load.

Pg2 2

Pg1

1

5

6

7

4

Load B 3

Pg3 Figure 1.7: Reference power system from [59]. We assume in the following that both generators 1 and 3 participate in frequency control schemes (i.e. can be used to maintain the balance between production and consumption in real time) but that for technical reasons, it is cheaper to produce power in generator 1 than in generator 3. As discussed in Section 1.1.1, frequency control reserves are either continuously controlled – primary or secondary control – or activated by discrete actions taken by the system operator – tertiary control. Here, we assume that generator 1 is the only generator participating in primary frequency control while reserves can be activated discretely by changing generator 3’s generation level, i.e. generator 3 participates in tertiary frequency control. We do not consider secondary frequency control in this example. Generator 2 generates a certain amount of power according to its production plan, but cannot be controlled to participate in frequency control schemes. When the load

10


varies 4 , the system operator can either let generator 1 continuously adjust its production level to follow these variations, or decide to activate reserves in generator 3 to relieve part of the production from generator 1. As will be seen in Section 5, power systems become unstable if the electrical grid becomes too loaded. A limit, called stability limit, exists on the amount of power which can be transferred from generator 1 to the load through the grid. For now, we just assume that such a limit exists, and the reader is referred to Section 5 for further detail about stability limits. Generator 3 is closer to the load than generator 1. Consequently, for a certain load, if generator 3’s production is increased by the system operator, generator 1’s production is decreased, the electrical grid is relieved from part of its loading, and the system comes farther away from the stability limit. For that reason, the larger the production in generator 3 the larger the maximum load that can be covered before reaching the stability limit. When maintaining the balance between production and consumption in real time, the system operator must therefore make a trade-off between, on the one hand, letting the automatically activated and cheap power reserves from generator 1 deal with the load variations and, on the other hand, maintaining an acceptable level of system security by activating more expensive reserves in generator 3 (to reduce the power transfer from generator 1). Hence, the system operator must optimally set the generation level in generator 3 in order for the power system to be able to supply as much load as possible before reaching the stability limit. As discussed before, the load can be forecasted in an accurate manner so that the system operator can support his decision with the load forecast, and be confident that the future load will be close to the forecasted one. This is the deterministic approach widely used today, and introduced in the quote above: the system is operated and planned based on a small number of possible loading situations – the “single snapshots” in the quote – determined from the load forecasts. It has been possible to operate power systems in this way because the load patterns are well known, making them possible to be forecasted with a good accuracy. Consider now the same system with generator 2 being a wind farm. While the system operator could assume a certain production level for generator 2 before (the level set in this generator’s production plan), the wind farm’s production will vary within the operating period due to the natural wind variations. Generators 1 and 3 must now make up for both the wind and the load variations. When the system operator decides upon the optimal production level of generation 3, it must therefore take into account the uncertainty coming from both the wind and the load. Wind power forecast errors are larger than those of the load [50, 97]. With large amounts of wind power, the future operating conditions (future wind power production and load) will thus be more difficult to forecast. Using a small number of “snapshots” may thus be insufficient to run the system in a secure manner, because the operating conditions not considered in these snapshots are disregarded although they are likely to occur in reality. 4. Note that the load can both increase and decrease, but the challenging case as far as system stability is concerned, is the case of load increase as will be seen in Chapter 5. Load variations in this example refer thus mainly to load increase.

11

1.4. THESIS OUTLINE

What was advocated above was to develop tools which consider the entire probability distribution of the load and wind power instead of only a small number of possible operating conditions (which are discrete points on these distributions). The aim of this thesis is to propose such tools.

1.4 Thesis outline Figure 1.8 summarizes the above discussion: throughout the operating period, the system operator continuously monitors the power system (step 1 in the figure), and takes counteractions such as the activation of tertiary frequency control reserves if necessary (step 2 in the figure). Both these steps require some knowledge, and hence the computation, of the stability limits to assess whether the system is in a secure state and, if it is not, take the optimal decisions which will make it secure again. The time iteration indicates that these steps are repeated during the whole operating period. t = 0: beginning of the operating period t = t + ∆t

No

1. System too close to or beyond the stability limits?

Computation of stability limits.

Yes 2. Take actions to come farther away from the stability limits, e.g. activation of balancing bids. Figure 1.8: The problem considered in this thesis.

The outline of the thesis is as follows – Chapter 2 gives the technical background related to the wind power impacts on power systems. Electricity markets and frequency control schemes are described. The aforementioned issues will be explained in more detail. The reader familiar with these impacts and the associated challenges can safely omit this chapter without missing any of the contributions of this thesis. – Chapter 3 gives mathematical foundations necessary to understand the work carried out in this thesis. Useful examples are given to understand how the mathe-

12


– –

– –

–

– –

matic methods presented in this chapter work. The topics addressed in this chapter are : – applications of Newton’s method to a set of relevant problems, – differential geometry, which will be used to study and approximate the stability boundary of power systems, – the Gram-Schmidt orthonormalization procedure and its application to tangent planes, – the Cornish-Fisher expansion. Chapter 4 presents some fundamentals in bifurcation theory. This is important in order to understand concepts in voltage stability. Chapter 5 aims at defining what is meant throughout this thesis by stability boundary. Stability limits due to voltage stability, small-signal stability and operational limits are discussed. The chapter ends on a discussion about new challenges associated with the stability boundary due to large amounts of wind power. The discussion around Figure 1.8 showed the importance of the stability limits in the scope of this thesis. Chapter 6 builds on the discussion in the end of Chapter 5, and proposes a secondorder approximation of the stability boundary. In Chapter 7 the second-order approximations developed in Chapter 6 are computed in the IEEE 9 bus system, and the accuracy of the approximations is assessed. Chapter 8 proposes a new formulation of a stochastic optimal power flow (SOPF) to find the least-cost generation redispatch. This can be applied for operating frequency control schemes (step 2 in Figure 1.8). A method to solve this new formulation is given. This method uses the second-order approximations presented in Chapter 6. Chapter 9 applies the method of Chapter 8 for solving SOPF to the IEEE 39 bus system. Chapter 10 concludes, and future research areas are proposed.

1.5 Contributions This thesis is the first phase in a project aiming at researching frequency control schemes in power systems with large amounts of wind power. This first phase has consisted in developing a general method which can be used to optimally operate tertiary control reserves under uncertainty while ensuring a specified level of system security. The method solves an optimization problem for generation re-dispatch. System security is ensured by taking into account the power system’s loadability limits. Uncertainty is accounted for by considering the entire probability distribution of the uncertain system parameters such as load and wind power production. Applications of the developed method to the specific case of power systems with large amounts of wind power have been left as future work for the second phase of the project. Future work is discussed in Chapter 10.

1.6. LIST OF PUBLICATIONS

13

The main scientific contributions of the thesis are the following: 1. Second-order approximations of the stability boundary taking voltage stability, small-signal stability and operational limits into account. This is presented in Chapter 6. These second-order approximations build on the work by Magnus Perninge in [83, 85, 88, 89]. The second-order approximations of the stability boundary can be used to take optimal decisions so as to operate the system in a secure manner (see next contribution). 2. A formulation of a stochastic optimal power flow (SOPF) for generation redispatch, and a method to solve this optimization problem. The method uses the second-order approximations developed in this thesis. This is presented in Chapter 8. Stochastic optimal power flows can be used to operate the frequency control schemes within the stochastic framework advocated in Section 1.2. 3. Application of the stochastic optimal power flow to the IEEE 39 bus system. This is presented in Chapter 9.

1.6 List of publications The following publications have been written in the scope of this thesis: Publication I Hamon, C.; Söder, L.; "Review paper on wind power impact on operation of reserves," Energy Market (EEM), 2011 8th International Conference on the European, pp.895-903, 25-27 May 2011. C. Hamon carried out the work and wrote the paper under the supervision of L. Söder. Publication II Söder, L.; Abildgaard, H.; Estanqueiro, A; Hamon, C; Holttinen, H; Lannoye, E; Lázaro, E.G.; O’Malley, M.; Zimmermann, U; "Experience and challenges with short term balancing in European systems with large share of wind power," IEEE Transactions on Sustainable Energy, vol. 3, no. 4, pp. 853–861, Oct. 2012. C. Hamon analyzed the Swedish and German data under the supervision of L. Söder. Publication III Hamon, C.; Perninge, M.; Söder, L.; "Stochastic Optimal Power Flow Problem with Stability Constraints; Part I: Approximating the Stability Boundary," Accepted for publication in IEEE Transactions on Power Systems. C. Hamon carried out the work and wrote the paper under the supervision of M. Perninge and L. Söder. Publication IV Perninge, M.; Hamon, C.; "Stochastic Optimal Power Flow Problem with Stability Constraints; Part II: The Optimization Problem," Accepted for publication in IEEE Transactions on Power Systems. M. Perninge developed the theory and carried out the work using the results from Publication III.

Part I

Background

15

Chapter 2

Technical background

Contents 2.1

Wind turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.2

Electricity markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.3

Frequency control schemes . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.4

Challenges for the operation of frequency control schemes . . . . . .

29

2.5

Generation re-dispatch and operation of tertiary control . . . . . . .

33

2.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

This chapter gives background about wind power, electricity markets and frequency control schemes. New challenges associated with large amounts of wind power for power system operation and planning are discussed. A state-of-the-art of methods for generation re-dispatch is given. Especially relevant in the scope of this thesis are the methods which account for uncertainties in power systems.

2.1 Wind turbines 2.1.1 Wind turbine designs Wind turbines harvest the kinetic energy stored in the wind, and convert it into electric power which is fed to the electrical grid. Different designs exist, but the following two components are encountered in all wind turbines: – Rotor blades: they interact with the incoming wind, extract power from it, and transfer it as mechanical power to the shaft of the wind turbine, which connects the blades and the generator. Some designs use gearboxes in order to transfer power from the slowly rotating blades to the fast rotating generator. – Generators: the generators receive mechanical power from the rotor blades, and transforms it into electric power. 17

18

CHAPTER 2. TECHNICAL BACKGROUND

There are four main design alternatives for wind turbines [1, p.56] whose main characteristics are gathered in Table 2.1. Control systems in variable-speed wind turbines can change the rotational speed of the generators, and, hence, of the blades, while this cannot be done in fixed-speed wind turbines. Table 2.1: The four main wind turbine designs. Name

Fixed or variable speed

Type of generator

Power electronics

Type A

Fixed speed

No

Type B Type C

Limited variable speed Variable speed

Type D

Variable speed

Squirrel cage induction generator Wound rotor induction generator Doubly-fed induction generator Synchronous generator

No On the rotor side only Full-scale converters

In the end of the nineties, type A was the most common type of wind turbine. Today, the variable speed wind turbines, types C and D, have become the most common [1, p.65]. Variable speed wind turbines have many advantages over fixed speed ones: the energy extraction is more optimal, and the components’ lifetime is longer because wear and tear are reduced. From the power system point of view, fixed and variable speed wind turbines behave differently. On the one hand, variable speed wind turbines are more flexible in operation thanks to their power electronics. On the other hand, these power electronics decouple the generators from the grid, which as will be seen in Section 2.4 gives rise to some issues. These designs are different from that used by conventional generators where synchronous generators are driven by turbines (such as gas or steam turbines) and directly connected to the grid (see Section 2.3.2 and Figure 2.5).

2.1.2 Wind power production The power delivered by wind turbines depends on the wind speed and the efficiency of the machine. Each wind turbine model has a power curve that defines how much power is produced for each wind speed. An example of power curve is given in Figure 2.1. The maximum power that a wind turbine can deliver is called the rated power. The cut-in wind speed is the speed at which a wind turbine starts producing power. The rated wind speed is the speed at which the wind turbine starts producing at rated power. For wind speeds higher than the rated wind speed, power control is used to limit the amount of power extracted from the wind, which is necessary since the wind turbine is

19

2.1. WIND TURBINES

Power [kW]

1,500

1,000

500

0

Rated wind speed

Cut-in wind speed

1

3

5

7

9

11 13 15 17 Wind speed [m/s]

Cut-out wind speed

19

21

23

25

Figure 2.1: An example of power curve for a 1500 kW wind turbine.

designed to handle at maximum its rated power. The cut-out wind speed is the wind speed at which the wind turbine is shut down to avoid damage.

2.1.3 Wind farms and connection to power systems Wind farms are clusters of wind turbines which share the same connection point to the electric grid. Today’s wind farms have central controllers which steer the individual wind turbines. Inside a wind farm, the wind turbines see different wind speeds not only because they are standing at different locations, but also because wind turbines can shade others standing behind them. This is called the shadow effect [71, Section 9.4.2.5]. The aggregated power curve of the wind farm can therefore not be obtained as the sum of the power curves of the individual wind turbines. If a weather front with high winds passes through a wind farm, it can happen that most, if not all, wind turbines will be shut down because their cut-out wind speed will be exceeded. From the power system perspective, this is equivalent to losing a generation unit. When the weather front has passed, the wind turbines will start producing again, thus creating a large injection of power into the grid. Wind variations will create fluctuations in the output power of wind farms. If wind farms or individual wind turbines are installed over a small geographic area, their power fluctuations will be highly correlated, whereas this correlation becomes weak if wind turbines are spread over a wide geographic area. This smoothing effect reduces the power fluctuations seen by the rest of the power system [1, Chapter 3]. As will be seen in Chapter 5, an important aspect when operating power systems in a stable way is that of voltage stability. Voltage stability is strongly coupled with reactive power support. For wind turbines, reactive power capabilities depend on the design (see Table 2.1) of the wind turbines, and on the control systems of the power electronics. This

20


issue is not in the scope of the thesis, however, and the reader is referred to, for example, [1, 38].

2.2 Electricity markets Power systems are operated to bring electricity to the end consumers. This electricity is traded on electricity markets by market players, or balance responsible players, who can be producers, consumers or retailers. Another actor is the system operator, who is responsible for maintaining an adequate level of security of supply. In the following, an overview of electricity markets is given. Two examples of market will be used for illustration: Nordel It gathers the transmission system operators of Denmark, Finland, Iceland, Norway and Sweden [42]. UCTE Union for the Coordination of the Transmission of Electricity. It represents 29 transmission system operators of continental Europe [41]. Since 2009, both are part of ENTSO-E, European Network of Transmission System Operators for Electricity, but the sets of rules which apply in Nordel and UCTE are still different. Different time periods are relevant for electricity trading and the actual supply of electricity to the consumers [94], see also Figure 2.2: Long-term financial markets Producers, consumers and retailers trade with each other mainly to hedge against future price risks. The transactions are not reported to system operators. Operating reserve planning Primary control reserves have been presented in Section 1.1.1 and will be further discussed in Section 2.3.3. In Sweden, the system operator purchases these reserves on two dedicated markets, one for reserves for the day after, and one for two days after [103]. Balance responsible players submit bids to the market to offer primary control reserves. Day-ahead markets Balance responsible players trade either bilaterally or by supplying bids to power pools for each operating period during the next day. Operating periods are typically one hour, such as in Nordel, but can also be shorter. Australia, for instance, has 5-minute operating periods [68]. The bids supplied by the market players are used to dispatch the generation for each operating period during the next day: this is called market clearing. By supplying offers on the day-ahead markets, the players commit to fulfilling them if their offers are accepted. If they do not, they will pay financial compensations. Intra-day markets After having submitted their offers to the day-ahead market and before the actual operating periods, market players can still trade on intra-day markets. The intra-day market trade allows the players to take into account new information, such as better forecasts or unavailability of power plants, before the actual operation period.

21

2.2. ELECTRICITY MARKETS

Balancing markets Ancillary services such as load following or balancing can be procured on other markets than those used for production planning. In Nordel, the balance responsible players can submit regulating bids to the balancing market up to 45 minutes before the operating period [81]. These bids are activated by the system operators if necessary during the operating period to maintain balance between production and consumption. Operating periods The balance responsible players whose offers have been accepted have the responsibility to supply the offered energy over the operating period. However, within the operating period, maintaining the balance between production and consumption is the responsibility of the system operator. To this purpose, power reserves are controlled by frequency control schemes. This thesis is concerned with issues related to the operating period. Post-delivery markets In post-delivery markets, imbalances that occurred during operating periods are financially settled. Market clearing: Production plans decided. Seasonal and weekly planning

Day-ahead planning

End of planning.

Intra-day planning

Operating period

Trading for adjustments (updated forecasts, units’ availability)

Maintaining continuous balance between production and consumption.

Time Production offers based on load and wind day-ahead forecasts.

Figure 2.2: The different time frames for power system operation and planning. The operating period lies in the scope of this thesis.

In short, the balance responsible players, on the one hand, have the responsibility of maintaining balance between production and consumption on average over the operating periods by following their production plans 1 , see Figure 1.1. System operators, on 1. On average means that the planned production covers the load on an energy basis, but not on a power basis.

22


the other hand, are responsible for maintaining the real-time balance between production and consumption within the operating periods, and use frequency control schemes to this purpose.

2.3 Frequency control schemes As discussed above, frequency control schemes are used to maintain the balance between production and consumption within the operating periods. Production is scheduled ahead of the operating period to meet the expected load. The latter is estimated with forecasts. Forecasts are also used to estimate how much wind power plants can produce. The offers submitted by the market participants depend on these forecasts. During the actual operating period, deviations between the actual load and the planned production occur resulting in imbalances between production and consumption. As will be seen, these deviations result in a change in frequency, which is undesirable for a secure and reliable operation because power systems are designed to work at a nominal frequency (e.g. 50 Hz in Europe and 60 Hz in the U.S.). Hence, the frequency should be kept within certain limits, and power reserves are assigned to meet these deviations. These reserves are activated by the frequency control schemes, which are usually divided in different layers, where each layer has a specific role and acts within a certain time frame. These layers can be classified into inertial response, primary, secondary and tertiary control. Inertial response is not strictly speaking included in the frequency control schemes, but it is an important mechanism which is tightly related to primary, secondary and tertiary control. In addition to the different layers, another distinction can be made between spinning and non-spinning reserves. Spinning reserves are the power reserves from already connected generators, while non-spinning reserves are the power reserves available from generators which must be started up.

2.3.1 Real-time imbalances between production and consumption and net load Kirchoff’s laws dictate that, physically, the electric power delivered by generators is equal to the electric power consumed by the loads (including losses in the electrical grid). Hence, these two quantities are always in balance. The meaning of “imbalance between production and consumption” will be defined when describing inertial response. Two main sources of imbalances between production and consumption exist: 1. As explained in Section 2.2, the production plans made ahead of the operating period are on an energy basis over the operating period. Deviations between the planned production and the actual load can therefore arise within the operating period on a power basis, as illustrated in Figure 1.1. 2. The second source of imbalance comes from forecast errors and unexpected events. Wind power producers submit their production offers based on wind forecasts,

23

2.3. FREQUENCY CONTROL SCHEMES

Actual load

Market clearing: production planned according to forecasts Intra-day trading for adjustments

Production plan Forecasted load

End of dayahead planning

0

T Operating period (one hour or less)

Figure 2.3: Influence of load forecast errors on frequency control schemes: the production is not planned optimally.

and the production is planned to meet the load based on load forecasts. Therefore, due to errors in the wind or load forecasts, there will be situations with deficit or surplus of production within the operating period. For example, if the wind is weaker than forecasted, wind power plants will produce less than planned. Furthermore, unexpected events such as outages in generators or lines also cause deviations from the plans. Trading on the intra-day market, which ends before the start of the operating hour, allows the players to take into account new information, such as updated power plant statuses or new load and wind forecasts. This helps reduce the deviations from the day-ahead plans. The remaining deviations have to be met by other production resources which are activated by the frequency control schemes. The influence of load forecast errors is illustrated in Figure 2.3: the actual load (thick line) is larger than the forecasted one (dashed line). Due to the deviations described above, the production plan (horizontal line) is not optimally adapted to the actual load. The striped and dotted areas are the deviation between the production plan and forecasted or actual load, respectively. The difference between these two deviations correspond to the additional use of frequency control schemes due to forecast errors. The effect of wind forecast errors is similar. In the scope of this thesis, the additional reserve requirements put on load frequency schemes due to large amounts of wind power are of interest. To study these additional requirements, the net load is defined as the load minus the wind power production; that is, it is the load to be covered by the rest of the production fleet (not wind power). Figure 2.4 shows the load, wind power production and net load in Gotland, Sweden on 16 March 2009. By studying the net load forecast errors, the additional reserve requirements can be estimated as will be seen in Section 2.4.2.

24

CHAPTER 2. TECHNICAL BACKGROUND 140

Load / Production [MW]

120 100 80 60 Load WP production Net load

40 20

0

5

10 15 Hour of the day

20

Figure 2.4: Load, wind power (WP) production and net load on Gotland, 16 March 2009, from Gotlands Energi AB (GEAB).

2.3.2 Inertial response Most of the generators in power systems are driven by turbines. The turbines deliver mechanical power to the generators, that transform it into electrical power supplied to the loads through the electrical network, as depicted in Figure 2.5, where P m is the mechanical power delivered by the turbine, converted into electric power P e by the generator, and supplied to the load.

Pe

Pm Turbine

Generator

Electrical grid

Load

Figure 2.5: Generators are driven by turbines, and supply electric power to the loads. According to Kirchoff’s laws, the electric power produced by the generators is always equal to the power consumed by the load (including losses in the electrical grid). When an imbalance occurs between the load and the output power of the turbines, it is compensated for by using the kinetic energy stored in the rotating masses of the generators synchronously connected to the grid. The generators that are not synchronously con-

25


nected to the power system (such as generators in most modern wind turbines) will not participate in the inertial response, unless their control system has been intentionally designed to this purpose. As for the synchronous generators directly connected to the grid, the well known swing equation for synchronous generator i reads Mi

d∆ωi = P mi − P ei − P Di , dt

(2.1)

where ∆ωi is the speed deviation from synchronous speed, M i is the inertia coefficient, P mi and P ei are as defined above, and P Di is the damping power [69]. The term on the right-hand side is called the accelerating power P ai = P mi − P ei − P Di . When there is balance between production and consumption, the system is at steady state, and the accelerating power is zero. When an imbalance occurs, for example following changes or the loss of a generator, the remaining generators keep supplying the new load according to Kirchoff’s laws but the mechanical power delivered by the turbine does not change 2 . Hence, the accelerating power becomes nonzero, and the speeds of the synchronously connected generators deviate from synchronous speed. Since the speeds of all synchronous generators are tightly coupled together and to the system frequency, they will experience almost the same speed variations. This will result in a change in system frequency, ∆ f = 2π∆ωi , ∀i . In the case of a load increase, for example, the accelerating power becomes negative so that the generators’ speed, and thus the frequency, decreases. The same happens when a generating unit is lost. It is here important to make the following remark, which has consequences on power system operations with large amounts of wind power. Remark 2.1 (Inertia and rate of change of frequency, from [69]) Since the synchronous generators experience almost the same speed deviations, the following holds ∀i , j ,

d∆ω d∆ωi d∆ω j = = ⇐⇒ ∀i , j , dt dt dt

P mi − P ei − P Di P m j − P e j − P D j = , (2.2) Mi Mj

where ∆ω is the common deviation in speed experienced by the generators. Assuming a total load change of ∆P (or the loss of a generator producing ∆P ), this total change is spread over all participating generators so that µ ¶ X © ª X d∆ω ∆P = (P mi − P ei − P Di ) = By (2.2) = Mi , (2.3) dt i i so that

d∆ f ∆P = 2π P , dt i Mi

(2.4)

from which we see that the larger the total inertia from synchronous generators the smaller the rate of change in frequency. 2. That is, it does not change immediately. Turbines are usually equipped with governing systems which change their load reference set point, which defines the delivered mechanical power as will be seen when describing primary control.

26


This remark is important in power systems with large amounts of wind power. Wind turbines use different generator technologies as seen in Section 2.1.1: Fixed-speed wind turbines with induction machines These wind turbines have induction generators directly connected to the grid. Induction machines contribute less to system inertia than synchronous machines [65]. Variable-speed wind turbine Variable-speed wind turbines use either doubly-fed induction generators (DFIG) whose stator is directly connected to the grid and rotor connected through power electronics, or synchronous generators that are completely decoupled from the grid by power electronics. For both these types, it is shown in [65] that the rate of change of frequency will be larger if wind turbines replace conventional generators. This effect is more pronounced for variable-speed wind turbines, and most modern wind turbines have designs which allow variable speed operation. Concerns have been raised about this issue, since from the system stability standpoint, this rate of change should be as low as possible. This issue is all the more challenging in small power systems such as Ireland where the study in [65] was made. Research has been made to mitigate this effect by adding control capabilities to wind turbines to emulate inertial response [24, 65, 77].

2.3.3 Primary control As seen above, disturbances such as load variations or losses of generation units are translated in changes in frequency. The inertial response of the generators slows down the rate at which the frequency changes, all the more so as the total inertia is large. Without any additional action, the frequency would keep on changing. In order to stabilize the frequency at a new value, primary control is used. The governing systems of the turbines driving the generators change the turbines’ output power depending on the frequency deviation until either the balance between turbines’ output power and the load is restored or the primary control reserves are depleted. The required change in power output as response to changes in frequency are defined in grid codes, and vary in different power systems. They are usually two types of primary control reserves: the first one is used for normal operations, that is, for small frequency deviations, while the second one is used when frequency deviations are larger, usually due to a large disturbance in the system such as the loss of a generating unit. Example 2.1 (Nordel) Primary control is termed “automatic active reserve” in Nordel. Figure 2.6 shows the requirements for the automatic active reserve as defined in [80]. The frequency controlled normal operation reserve must be at least 600 MW and fully activated for frequency deviations of ±0.1 Hz. The frequency controlled disturbance reserve must be at least 1000 MW and fully activated at 49.5 Hz. Note that the latter starts being activated when the frequency falls under 49.9 Hz.

27

1600

600

49.5

49.6 49.7 49.8 Frequency [Hz]

Activated reserve [MW]


49.9 50

Figure 2.6: Requirements for the automatic active reserves in Nordel: Frequency controlled normal operation reserve (solid line) and Frequency controlled disturbance reserve (dashed line).

The droop ρ can be defined as the ratio between the relative change in production and the relative change in frequency following this change in production, that is ρ=

∆ f / fn , ∆P m /P mn

(2.5)

where the subscript n denotes the nominal value, and ∆ f and ∆P m are the changes in frequency and in mechanical power output due to primary control. From the frequency stability standpoint, it is desirable to have a small droop because a disturbance would then lead to a smaller frequency deviation. Primary control reserves must be activated within a time frame predefined in grid codes. For Nordel, this is two to three minutes for the frequency controlled normal operation reserve, and 30 seconds for the frequency controlled disturbance reserve [80]. In UCTE, 50% of the primary control reserves must be activated within 15 seconds, and 100% within 30 seconds [104]. Primary control restores the power balance between production and consumption, but does not restore the frequency at its nominal value. The location of the primary control reserves is usually determined in the planning phase. For example, in Sweden, the balance responsible players willing to participate in primary control can submit bids to markets dedicated to primary control reserve for one day after and two days after the current day [103].

2.3.4 Secondary control Secondary control reserves are automatically controlled reserves aimed at restoring the frequency to its nominal value and at relieving the primary control reserves to make them available when a new disturbance occurs.

28


In many countries, automatic generation control (AGC) is used for secondary control purposes. AGC is usually used for more than restoring the frequency to its nominal value [69]. In continental Europe, for example, AGC is used to maintain the tie line interchanges at their contracted values. Hence, it measures two errors: the frequency deviation from its nominal value and the power interchange deviations from their contracted values. The sum of these two errors 3 is called “area control error” (ACE). The ACE is then processed by a central controller, usually a proportional-integral or purely integral controller, which computes the required change in production to bring these errors to zero. The change in production is then distributed among the participating generators, as a change in the load reference set point of their turbine governor [69]. The distribution of the reserves among participating generators is done in the planning phase. Secondary control starts acting after primary control has been activated. In UCTE, secondary control must be fully activate within 15 minutes [104]. In Nordel, secondary control is not used as of today, but will be introduced under the name Load Frequency Control (LFC) in January 2013 [37]. The corresponding reserves are termed Frequency Restoration Reserves. The introduction of secondary control in the Nordic system was deemed necessary because the frequency quality has kept deteriorating over the past decade. Compared to what is done in UCTE, it will only be used to restore the frequency to 50 Hz because the Nordic system is handled commonly by the system operators, and, hence, is treated as one control area.

2.3.5 Tertiary control Tertiary control refers to the manual activation of power reserves by the system operator. In Nordel, tertiary control is the main tool to maintain the power balance as of today, since secondary control has not yet been introduced. Tertiary control is procured on the balancing market, where the power producers can submit regulating bids. If necessary, the system operators in the Nordic system can activate regulating bids, which are chosen depending on their price and location to avoid overloading transmission lines [94]. Balancing bids must be activated within 15 minutes [102, Section 4.4]. In UCTE, tertiary control is used to relieve and support secondary control reserves [104]. Note that the term “secondary control” can be used to refer to tertiary control in Nordel (manual activation of balancing bids) [102], although it does not work as the secondary control in UCTE (AGC). The term tertiary control will be used throughout this work to refer to the manual activation of balancing bids. This thesis deals with tertiary control. In Section 8, a new method for redispatching generation is proposed.

2.3.6 Summary Frequency control schemes consist of different layers. In response to an event such as a load change or the loss of a generation unit, the frequency will change. The inertial 3. The frequency deviation is pre-multiplied with the frequency bias factor to make its magnitude consistent with the power interchange deviation [69, Section 9.1.3].

2.4. CHALLENGES FOR THE OPERATION OF FREQUENCY CONTROL SCHEMES Frequency ¯ ¯ deviation ¯∆ f ¯

29

Primary control Secondary control

Tertiary control

Sec.

Minutes

Hours

Time

Figure 2.7: The different layers of frequency control schemes, inspired by [104].

response of the synchronously connected generators (or of the non synchronously connected generators equipped with a dedicated control loop) will limit the rate of change of frequency. Then, the reserves dedicated to primary control will be automatically activated within a few seconds (and fully activated within less than two to three minutes) in order to stabilize the frequency at a new value, which results in a steady-state frequency deviation from the nominal value. The secondary control reserves will automatically react to this steady-state frequency deviation, and be activated in order to bring back the frequency to zero and refill the primary control reserves. In some systems such as the Nordic system, secondary control has not been implemented yet. In other systems, secondary control also controls the generation to restore the tie-line interchanges to their contracted value. Finally, the tertiary control will act in order to relieve the secondary control reserves. Figure 2.7, inspired by [104], depicts the situation. Note that the time scales for the different layers are the ones defined by UCTE. In Nordel, tertiary control acts on the same time scale as secondary control in UCTE (within 15 minutes). The inertial response is strictly speaking not part of the frequency control schemes, but its role is important in the study of frequency stability.

2.4 Challenges for the operation of frequency control schemes As explained in Section 2.3.1, two main issues related to the operation of frequency control schemes arise when integrating large amounts of wind power: forecast errors in the net load increase, and the larger net load variability.

30


2.4.1 Forecast errors The load can be forecasted with a good accuracy [12]. Systems with small amounts of wind power can thus be planned efficiently because the expected operating point during operating periods of the next day (the expected load) can be determined with small forecast errors. Thus, the deviations from the production plans that the frequency control schemes must deal with are in general small compared to the same system with large amounts of wind power. Indeed, in this case, the wind forecast errors must also be met by the frequency control schemes.

2.4.2 Increased variability in net load Principles Assume that the forecasts were perfect. Wind power, because it has very low marginal cost, is usually dispatched before conventional resources, which are used to cover the rest of the load, i.e. the net load. As seen in Section 2.3, the variability in net load is met by power reserves activated by frequency control schemes. The increased use of power reserves due to wind power can be assessed by comparing the probability distribution of the load variations to that of the net load variations. The so called σ-method can be used for this purpose. A certain level of the variations which must be covered by the power reserves is chosen (for example 97% of the probability distribution of the variations). The difference between the power reserves needed to cover this level of the net load variations and this level of the load variations can then be used as an estimation of the additional reserve requirements due to wind power. The σ-method was originally proposed in [99], and later explained in detail in [54, 75]. An illustrative example for the σ-method is now given. Example 2.2 (The σ-method) Assume that the load and net load variations are normally distributed N (0, σL ) and N (0, σN L ), respectively. Assume that the system operator seeks that 95% of these variations are met by frequency control schemes 4 . For normal distributions centered on zero, 95% of the probability mass is located in the interval ±2 times the standard deviation. Figure 2.8 depicts the situation. It is expected that with wind power, the net load variations increase, and, hence, that the standard deviation of these variations is larger than that of the load [50, 97], so that σN L > σL . The areas corresponding to 95% of the variations have been shaded in the figure. In this example, the amount of power reserves necessary to meet the system operator’s requirements is then 2σL when only the load is considered and 2σN L when the wind is also considered. Hence, the additional reserve requirement due to wind power is 2(σN L − σL ). The time horizon of the considered variations is important since the larger the time horizon the larger the variations can be. The time horizon of interest depends on which 4. The following numbers are chosen for the sake of the example. In practice, three times the standard deviation is often taken as a confidence level [53, 75].

2.4. CHALLENGES FOR THE OPERATION OF FREQUENCY CONTROL SCHEMES

Load variations NL variations

0.4

Frequency [%]

31

0.3

0.2

0.1

0

−2σN L

−2σL

0

2σL

2σN L

Variations [MW]

Figure 2.8: Illustration of the σ-method with the probability distributions of load and net load (NL) variations.

kind of reserves are considered (primary, secondary or tertiary) and on how fast the reserve capacity can be started up. For example, it is not consistent to consider the fourhour variabilities of wind power to assess reserve requirements if the reserve capacity can be started up within one hour [53]. As seen in Section 2.3, primary control must be fully activated within a few minutes and secondary control in UCTE and tertiary control in Nordel within 15 minutes. Therefore, intra-hour variations are relevant to assess the impact of wind power on reserve requirements. The assessment of reserve requirements belongs to the phase of operating reserve planning in Section 2.2. This method can also be used to study the distributions of the load and net load forecast errors. Results In [53] results from studies using this method are gathered, and reproduced here in Figure 2.9. The studies do not all use the method on the same sort of data (some use the distribution of forecast errors, some the wind power variations), nor the same time horizon, but it is clear that more wind power increases the reserve requirements. Another observation from [53] is that the additional requirements decrease in larger balancing areas since large geographical areas have a smoothing effect on the overall wind power variations. It can be observed in Figure 2.9 that, for example, the additional requirements, normalized by the installed capacity, for the whole Nordic system are lower than that for

32


Figure 2.9: Increase in reserve requirements due to wind power, from [53].

only Finland. This is also observed in [100] where the standard deviations of short-term wind power variations from different countries and areas within these countries were studied. Using a least-squares fit, the following formulas are proposed for the standard deviations normalized by the installed capacity of 15- and 30-minute wind power variations as a function of distance d in kilometers:

σ15 = 4.67 exp(−0.0106d ), σ30 = 5.85 exp(−0.009d ).

(2.6) (2.7)

These are depicted in Figure 2.10. It appears clearly that the standard deviation of the overall wind power variations decreases as the distance between wind power plants increases and that the longer the considered time horizon the larger the variations can be. The smoothing effect due to geographical spreading is based on the assumption that the grid is strong enough to transfer power across the power system. Note that in [100], the curve fitting has been made for average distances up to 300 km. The smoothing effect for larger distances may not follow (2.6) and (2.7). In particular, it cannot be concluded from (2.6) and (2.7) that the normalized standard deviation of wind variations tend to zero as the average distance between wind turbines tends to infinity. Wind power is thus expected to require larger amounts of power reserves. Making use of wind resources spread over large geographical areas mitigate this effect to a certain extent, assuming that the grid is strong enough to transfer power from these areas. Also, the amount of power reserves is not as important as how fast reserve power can be activated (start-up times and ramping rates). This is because when wind power produces electricity, reserve capacity increases in the power plants that it replaces [52].

Standard deviation as % of installed wind power capacity

2.5. GENERATION RE-DISPATCH AND OPERATION OF TERTIARY CONTROL

6

33

15-minute variations 30-minute variations

4

2

0

0

50

100

150 200 Distance [km]

250

300

Figure 2.10: Normalized standard deviation of wind power variations approximated as a function of the mean distance between the wind turbines, from [100] (Publication II).

2.5 Generation re-dispatch and operation of tertiary control The role of the different layers of frequency control schemes have been presented in Section 2.3. In this section, the operation of these frequency control schemes is described. Special attention is given to tertiary control since this is the scope of this thesis. As mentioned in Section 2.3.5, tertiary control corresponds to a re-dispatch of the generation in order to restore primary and secondary control reserves, and to bring back the frequency to 50 Hz. Below, we review operation tools for generation dispatching.

2.5.1 Optimal power flows The problem of generation dispatching has two components: an objective function to minimize (such as the production cost) and some constraints which the solution must satisfy. These constraints include equality constraints (such as AC or DC power flow equations) ensuring that the solution corresponds to a physical equilibrium point of the system 5 , and inequality constraints representing operational limits (such as minimum and maximum generation capacity, maximum active power transfers on certain transmission lines or lower and upper bounds for bus voltages). Hence, the problem of generation dispatching can be set up as an optimization problem with the power flow equations as constraints. This type of problems are called optimal power flows, and are 5. The power flow equations ensure the power balance at each bus in the system.

34


formulated as [113] min C (x, λ, u) u

s.t.

f (x, λ, u) = 0,

h(x, λ, u) ≤ 0,

(2.8a) (2.8b) (2.8c)

where x ∈ Rn x , λ ∈ Rm and u ∈ Rnu are the state variables (such as voltage magnitudes and angles), the parameters and the control variables, respectively. The function C : Rn x ×Rm ×Rnu → R is the objective function to be minimized, f : Rn x ×Rm ×Rnu → Rn x represents the power flow equations ensuring that the solution corresponds to a possible equilibrium, and h : Rn x × Rm × Rnu → Rnl contains n l operational constraints 6 . Optimal power flows were studied as early as 1962 by Carpentier in [19]. In the following, the system operator’s perspective is taken. In the context of tertiary control, the control variables will be the output power of the participating generators and the objective function the overall operating costs associated with the generation redispatch. The parameters are quantities that are considered given when the problem is solved, and can be, for example, the active and reactive power loads. Note that some consumers can also participate in frequency control schemes by accepting to reduce or increase their consumption on request of the system operator. In this case, these loads are included in the control variables and not in the parameters. The optimal solution will give the feasible optimal redispatch of the generation which minimizes the chosen objective function. In the following sections, different variants of the optimal power flow problem are examined.

2.5.2 Security-constrained optimal power flows Security-constrained optimal power flows (SCOPF) are optimal power flows whose solutions are feasible for a set of selected contingencies [4]. The optimal solution ensures that the system satisfies the so-called N − k criterion, where k is the number of simultaneous contingencies that the system must be able to survive. Examples of contingencies are tripping of major transmission lines or loss of large generation units. The security-constrained optimal power flows are either preventive [4] or corrective [76]. In the preventive case, the optimal setting of the control variables remains unchanged in the post-contingency systems, and ensures that the system survives the contingencies. In the corrective case, the optimal setting of the control variables can be different in the post-contingency systems. The corrective SCOPF takes into account the fact that even though the constraints are violated with the setting of control variables for the pre-contingency system, actions can be taken after contingencies occur to eliminate the constraint violations, if the actions can be completed fast enough. An example of 6. A large part of the power in the Swedish power system comes from hydro power plants which are subject to hydrological constraints and court decisions. These constraints are included either in f or h depending of their form (equalities of inequalities).


35

such actions is generation re-dispatching, which can be done if the generators respond quickly to re-dispatching orders. Let n c be the number of contingencies. The preventive SCOPF can be formulated as follows: min C (x 0 , λ0 , u 0 )

(2.9a)

u

s.t.

f i (x i , λi , u 0 ) = 0,

i = 1, . . . , n c ,

h i (x i , λi , u 0 ) ≤ 0,

i = 1, . . . , n c ,

(2.9b) (2.9c)

where, compared to the optimal power flow formulation in 2.8, the equality and inequality constraints must hold for each contingency i = 1, . . . , n c . The pre-contingency system corresponds to i = 0. The objective function is to minimize the operating costs in the pre-contingency system. Automatic actions such as primary and secondary control and transformers’ actions to maintain voltages are included in the preventive SCOPF. The optimal preventive actions are given by the optimal solution u 0∗ . The corrective SCOPF includes other actions that are taken by the system operator after contingencies happen. Manual generation re-scheduling is an example of a postcontingency corrective action [14]. Corrective SCOPF can be formulated as follows: min C (x 0 , λ0 , u 0 )

(2.10a)

u

s.t.

f i (x i , λi , u i ) = 0,

h i (x i , λi , u i ) ≤ 0,

i = 1, . . . , n c ,

i = 1, . . . , n c ,

ku i − u 0 k ≤ ∆u imax ,

i = 1, . . . , n c ,

(2.10b) (2.10c) (2.10d)

where the two differences with preventive SCOPF are that, first, there is one optimal setting u i∗ of the control variables for each contingency and, second, that coupling constraints (2.10d) are added between the base case control actions and each of the post-contingency control actions. These constraints account for the fact that the postcontingency control actions must be taken quickly enough in order to ensure system stability and, thus, the amount of post-contingency control actions ku k − u 0 k is limited by the rate of change of the control actions (e.g. generators have maximum ramp rates which limit post-contingency generation re-scheduling). In [14] these coupling constraints are taken as ¶ µ du ku i − u 0 k ≤ Ti , i = 1, . . . , n c , (2.11) dt max

where Ti is the time horizon allowed for post-contingency control actions to restore feasibility after contingency i , and (du/dt )max is the maximum rate of change of control actions. The objective function for corrective SCOPF is the same as for preventive SCOPF. Additional constraints to ensure that the post-contingency state is feasible in the short term, i.e. before the system operator has time to take corrective control actions, can be added [13]: f is (x i , λi , u 0 ) = 0,

h is (x i , λi , u 0 ) ≤ 0,

i = 1, . . . , n c ,

i = 1, . . . , n c ,

(2.12) (2.13)

36


where f is and h is , of same dimensions as f and h before, are the short-term post-contingency equality and inequality constraints, and we see that u 0 appears in the constraints since the corrective actions u k have not yet been taken. Since the post-contingency control actions can be taken in the case of corrective SCOPF, the requirements on the base-case control actions u 0 are less restrictive than in the case of preventive SCOPF. Hence, the optimal value of the objective function for corrective SCOPF is never higher than that of preventive SCOPF [16]. Since solving preventive and corrective SCOPF is computationally demanding, a need to reduce the complexity of the problem has arisen. A possible solution is to filter out non binding contingencies (the contingencies for which the binding constraints are the same in the post-contingency system as in the pre-contingency system, entailing that no corrective actions are necessary) [14, 15, 17].

2.5.3 Situation in Sweden The way in which the Swedish power system is operated is described in detail in [93]. A summary of this reference is given here. The Swedish power system is operated mainly with respect to voltage stability. It is characterized by large power transfers from the North, where large generation capacities exist, to the South, where most of the load centers are. As the power transfer across the grid becomes larger – either because of an increase in the load or because a contingency has occurred and put additional stress to the grid – the bus voltages decrease [106]. Beyond a certain loadability limit, a voltage collapse will occur, characterized by falling voltages, possibly resulting in blackouts if no corrective actions are taken. This phenomenon will be studied in detail in Section 5. The Swedish system is divided in four bidding areas. Each bidding area has its own area price for electricity. If there is no line congestion in the electrical grid, the four price areas have the same price; otherwise, they are different. The three cuts separating the areas are called bottlenecks, and correspond to critical corridors for power transmission. The Swedish power system, its bidding areas and three bottlenecks are shown in Figure 2.11. A voltage security assessment is performed every 15 minutes by the so-called SPICA system to compute, for each of the three bottlenecks, loadability limits – corresponding to transmission limits – beyond which voltage instability arises. The power system must be able to satisfy the N − 1 criterion, that is, to remain stable after one of some selected severe contingencies has occurred. Therefore, as for SCOPF, the voltage security assessment takes contingencies into account. Each bottleneck corresponds to a job in SPICA, and each job contains contingencies. For each job, and each contingency in the job, the power transfer across the bottleneck from the production area to the load area is increased until the loadability limit is found. The way the power is increased is defined by the system operator, but just one way of increasing this power transfer is studied. Consider Figure 2.12 where the transmission limit from the production area to the load area is to be calculated. The power transfer across the bottleneck is increased by changing the production in a certain way at certain buses (corresponding to ∆PG a , . . . ,


37

Bidding area 1

Bidding area 2 Bottlenecks

Bidding area 3

Bidding area 4 Figure 2.11: The Swedish power system, the four bidding areas and the three bottlenecks from [60].

∆PG e in the figure) to meet a certain load increase at certain buses (corresponding to ∆P L a , . . . , ∆P L d in the figure). When the loadability limits are found, an operational margin (called Transmission Reliability Margin or TRM [83]) is subtracted from it to get the transmission limits. The system operator monitors the transfer across the bottlenecks, and can decide to re-dispatch the generation (i.e. to activate regulating bids via tertiary control) if the transfers come close to the limit computed by the SPICA system. Balancing bids can also be activated to maintain the frequency within an appropriate range. This aspect of tertiary control is handled commonly by transmission operators of the whole Nordic system. The fact that computations of transmission limits only consider one way of increasing the power transfer is a shortcoming of the current approach, because the loadability limits depend on this increasing path. With more wind power in power systems, the way in which the power transfers change will become more unpredictable, thus impairing the validity of the current approach.

38


Figure 2.12: Computation of transmission limits across one bottleneck, from [93].

2.5.4 Optimal power flow and uncertainty A secure dispatch as given by SCOPF is secure only for the considered contingencies and operating conditions. Operating conditions refer here to λ in the formulations above, which contains parameters not controllable by the system operators. Wind power and load are two examples of such parameters. When solving a SCOPF, λ is given a value which, according to the system operator, reflects the operating conditions for which the study is done. Hence, if SCOPF is used for re-dispatching generation within the next fifteen minutes (as for tertiary control), the parameters can be set to their expected value in the next fifteen minutes. These expected values can be obtained by forecasts. In the Swedish case, a load increase pattern expected to happen within the next fifteen minutes can be used to find the loadability limits of each bottleneck. Two shortcomings are associated with such approaches: 1. it does not consider the probabilities of the contingencies to happen; 2. it considers only a small amount of operating conditions. The first point means that the optimal solution from SCOPF is feasible in the sense that no constraints are violated for any contingency, irrespective of the possibly low probability with which these contingencies happen.


39

The second point means that the operating conditions not used for solving SCOPF are disregarded, even though their probability of occurrence is not zero. While the former shortcoming entails that the system is operated in a too conservative way, the latter shortcoming implies that the variations in parameter are disregarded. It was seen in Section 2.4 that with large amounts of wind power, the net load forecast errors will increase in the sense that the variance of these forecast errors will increase. Hence, the uncertainty faced by the system operator will increase, and new tools must be developed in order to account for this uncertainty [97]. Today, system operators hedge against risks associated with uncertainty usually by having some operational margins as it was done in Sweden’s case. Considering uncertainties directly when computing the optimal decisions would allow a more flexible and efficient use of the system resources. Several approaches have recently been proposed to meet the two aforementioned shortcomings of SCOPF. Expected-security-cost optimal power flow In [22, 23], Condren et al. developed an expected-security-cost optimal power flow (ESCOPF). It is a corrective security-constrained optimal power flow in the sense that the optimal setting of post-contingency control variables is allowed to be different from that of the pre-contingency control variables. The difference with the classical corrective SCOPF formulation presented above is that ESCOPF includes the probabilities of the studied contingencies and the costs of the corrective actions in the objective function, which is defined as min π0C 0 (x 0 , λ0 , u 0 ) +

u 0 ,...,u nc

nc X

πi C k (x i , λi , u 0 , u i ),

(2.14)

i =1

Pn c i where πi is the probability of occurrence for contingency i , π0 = 1− i =1 π is the probability associated with the pre-contingency system, C 0 is the cost in the pre-contingency system 7 , and C i is the cost associated with the post-contingency corrective actions for contingency i . Note that the post-contingency costs also depend on the pre-contingency setting of the control variables because these costs are functions of the difference u i −u 0 (measuring, for example, how much generation re-dispatch is required from the precontingency setting). Hence, ESCOPF addresses the first shortcoming mentioned above by considering the probability associated with all contingencies and the cost of the post-contingency control actions. However, it does not address the second shortcoming. Probabilistic optimal power flows In probabilistic optimal power flows (P-OPF), the parameters λ are modeled by their probability density function (PDF), and we seek at obtaining the PDF of all variables in 7. In the original formulations [23], the total social welfare was maximized.

40


the problem [95]. Probabilistic optimal power flows are usually solved in the following way. First, the optimal settings of the control variables are obtained by solving a classical SCOPF or OPF with, for example, the expected value of the uncertain parameters. Then, the system is linearized around this optimal solution in order to express the variables as linear functions of the parameters. This allows the computation of cumulants or moments of the other variables from those of the parameters. Finally, using the cumulants of moments, an approximation of the PDF is calculated, for example using the Gram-Charlier expansion [95, 109, 116, 117]. By considering the PDF of the parameters, P-OPF takes into account the uncertainty which the system is subjected to, thus addressing the second shortcoming mentioned above. Stochastic optimal power flows Stochastic optimal power flows (S-OPF) address both shortcomings. While P-OPF computes the PDF of the optimal setting of the control variables in order to assess the effect of uncertainty on this optimal solution, S-OPF includes the uncertainty in the optimization problem itself. When considering the entire distribution functions of the parameters, the constraints must be changed from being deterministic to being probabilistic because the probability of violating the deterministic constraints is almost surely nonzero. Work on S-OPF includes [31, 96, 98, 115, 116]. A general formulation of a SOPF problem is min E [C (x, λ, u)]

(2.15)

u

s.t. P [h i (x, λ, u) ≤ 0] ≥ 1 − αi ,

i = 1, . . . , n h ,

(2.16)

where αi > 0 are small and n h is the number of probabilistic constraints. In [31], a S-OPF was formulated to maximize the power transfer over a set of buses with the constraint that the probability that the transfers across some bottlenecks violate their limit is kept low. As seen in Section 2.5.3, similar constraints (although deterministic) are taken into account by the Swedish system operator. Stochastic control The methods above, while addressing the shortcomings of SCOPF, give an optimal generation re-dispatch for one point in time only. However, since the system operator is responsible for maintaining the balance between production and consumption within the operating period, it seeks at optimizing power system operation not only at one point in time but throughout the whole operating period. Hence, not only the cost of the decisions must be taken into account but also the expected costs of taking these decisions for the rest of the hour. A promising approach based on stochastic control which addresses this issue is proposed in [83]. In a first stage, the expected costs of system failure for different values of the control variables are estimated for a list of selected contingencies using Monte Carlo

2.6. SUMMARY

41

simulations. Then, just before the operating period, stochastic processes are chosen to model the uncertain parameters such as load variation and wind power production. Finally, stochastic control is used to obtain optimal dispatch strategies for the whole operating period. The author indicates that more research is needed to include continuous secondary control and study how the method behaves in large power systems.

2.6 Summary Important background to this thesis was given in this section. In particular, it was seen that large amounts of wind power will entail larger needs for reserve requirements and larger uncertainty in the system. The deterministic tools used today by system operators to operate the system cannot account for this uncertainty. New tools for operating power systems under uncertainty were presented. Particularly interesting is the S-OPF approach. Because the uncertainty faced by the system operator is accounted for in the optimization problem itself, the obtained setting of control variables is optimal for this uncertainty, which makes S-OPF a promising tool for operating power systems with large amounts of wind power. In Section 8, a new formulation of S-OPF is proposed for generation re-dispatch which can be used by system operators to activate regulating bids.

Chapter 3

Mathematical foundations Contents 3.1 3.2 3.3 3.4

Newton’s method . . . . . . . . . . . . . . . Elements of differential geometry . . . . . Gram-Schmidt orthonormalization . . . . Cumulants and Cornish-Fisher expansion

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

43 48 54 56

In this chapter, we present some mathematical instruments that will be used throughout this thesis. The theory is illustrated as much as possible with simple examples related to the work presented in this thesis, and we refer to the sections in which these instruments will be used.

3.1 Newton’s method Newton’s method can be used to find local zeros of a function.

3.1.1 Theory In many applications, we seek a solution to a general system of equations F (x) = 0,

(3.1)

F (x 0 + ∆x) ' F (x 0 ) + J (x 0 )∆x,

(3.2)

¡ ¢T where F : x ∈ Rn 7→ F (x) = f 1 (x) . . . f n (x) ∈ Rn is differentiable. We call x ∗ the solution to (3.1). Such a system of equations can be solved by Newton’s method. Newton’s method is an iterative method that from a starting point x 0 ∈ Rn at which F (x 0 ) 6= 0, looks for a solution to (3.1) by approximating F by its first-order Taylor expansion and solving (3.1) for this approximation. The first-order Taylor expansion of F around x 0 is

43

44

CHAPTER 3. MATHEMATICAL FOUNDATIONS

where J (x 0 ) is the Jacobian matrix of F at x 0 :  ∂ f1

 J (x 0 ) =  

∂x 1 (x 0 )

∂ f1 ∂x 2 (x 0 )

∂ fn ∂x 1 (x 0 )

∂ fn ∂x 2 (x 0 )

.. .

.. .

... .. . ...

 ∂ f1 ∂x n (x 0 ) .. .

∂ fn ∂x n (x 0 )

 . 

(3.3)

By approximating F to its first-order approximation in (3.2), we can solve (3.1) for ∆x, if the Jacobian in x 0 is nonsingular: F (x 0 ) + J (x 0 )∆x = 0

⇐⇒ ∆x = −J (x 0 )−1 F (x 0 ).

(3.4)

From this, we get a new approximation value x 1 = x 0 + ∆x. The process can be repeated to get a sequence of points x 0 , x 1 , . . . , x p ∈ Rn which, under certain conditions, converges to x ∗ . The numerical method stops when a defined criteria becomes small enough, for example if kF k∞ < ε for some small user-defined ε.

3.1.2 Convergence Two issues associated with any numerical method are whether and at which rate the method converges to x ∗ . This depends on the starting point x 0 (initial guess) and on the smoothness of the function F . The initial guess x 0 must lie in the basin of attraction of x ∗ for the method to converge. In some cases, Newton’s method can be numerically unstable even for points lying in the region of attraction [73]. However, if a good initial guess can be made, and if F is sufficiently smooth, Newton’s method converges quickly [101].

3.1.3 Applications to power systems In 1961, Newton’s method was proposed as a way to solve power flows 1 [108]. Power flow equations take the form of (3.1), and although there exist many other methods to solve power flows, Newton’s method (together with fast decoupled power flows) is the most used power solver [74]. Power flow equations appears in the optimal power flow formulations presented in Section 2.5. Throughout this thesis, Newton’s method will also be used to solve other problems of the form (3.1), see for example Chapter 6.

3.1.4 Examples Example 3.1 (Power flow) In this example, Newton’s method is applied to solve the power flow problem in a singlemachine infinite-bus system (SMIB), see Figure 3.1. 1. Power flows solve the problem of finding unknown quantities such as bus voltage magnitudes and angles by using the equations ensuring the power balance at each bus in the grid.

45

3.1. NEWTON’S METHOD V2

jX V1

P+jQ Figure 3.1: SMIB system

Solving the power flow in this example means finding the amplitude V2 and the phase angle θ2 of the load bus voltage V2 = V2 e j θ2 . The load consumes P + jQ. The apparent power transmitted from the generator to the load bus, through the line, is S = V2

µ

V1 − V2 jX

¶∗

,

(3.5)

where (·)∗ denotes the conjugate. The generator bus is an infinite bus 2 with V1 = 1 and θ1 = 0 so that V1 = V1 e j θ1 = 1. The power balance at the bus load gives P + jQ = V2

µ

V1 − V2 jX

¶∗

.

(3.6)

¡ ¢ Writing V2 = V2 cos θ2 + j sin θ2 , developing the power balance equation above, and equating real and imaginary parts, we get (

f 1 (V2 , θ2 ) = f 2 (V2 , θ2 ) =

which we can write as

1 2 sin θ2 + P = 0, XV ¡ 2 ¢ 1 X V2 − V2 cos θ2 +Q

(3.7)

= 0,

·

f 1 (V2 , θ2 ) F (V2 , θ2 ) = 0 where F (V2 , θ2 ) = 2 f (x 2 , θ2 )

¸

(3.8)

In this case, it is actually possible to solve the system of equations analytically, i.e. to get analytical expressions for both V2 and θ2 , by solving the above equation for sin θ2 and cos θ2 and using cos2 + sin2 = 1. The voltage V2 can then be expressed as [107] V2 =

s

1 −Q X + 2

r

1 − X 2P 2 − X P . 4

(3.9)

In higher dimensions, however, solving the problem analytically is not possible, and we can use Newton’s method instead. The latter can also be applied on this example, 2. An infinite bus is a bus at which the voltage is supposed to be constant.

46


since we seek at finding V2 and θ2 that satisfy F (V2 , θ2 ) = 0. To apply Newton’s method, we need the Jacobian of F : Ã ! f V12 (V2 , θ2 ) f θ1 (V2 , θ2 ) 2 J= 2 , (3.10) f V2 (V2 , θ2 ) f θ2 (V2 , θ2 ) 2

where all derivatives can be analytically obtained from (3.7) to get J=

µ

1 X

1 X

sin θ2 (2V2 − cos θ2 )

¶ 1 X V2 cos θ2 . 1 X V2 sin θ2

(3.11)

The iteration step (3.4) can then be used to find the solution by Newton’s method. Figure 3.2a shows the convergence of the error kF k∞ compared with the chosen threshold ε = 10−8 . Figure 3.2b shows the convergence of the voltage amplitude toward the exact value calculated from (3.9). Note that already after the third iteration, the voltage is very close to its exact value.

1

kF k∞ ε

V during PF Exact value of V

0.98 V [p.u.]

kF k∞

10−1

10−8

0.96 0.94

10−15

0

1 2 3 Number of iterations

(a) Convergence criterion of the power flow

4

0.92

0

1 2 3 Number of iterations

(b) Voltage amplitude during the power flow

Figure 3.2: Newton’s method applied to the power flow problem.

4

47

3.1. NEWTON’S METHOD

Example 3.2 (Projection onto a surface) In this example, we consider the unit sphere in R3 restricted to the positive octant of the space, described by

x 2 + y 2 + z 2 = 1,

x, y, z ≥ 0.

(3.12)

Also, we consider a point A ∈ R3 not on the surface, for example A = [0.8 1.6 0.4]T . The problem is to find the projection of this point onto the sphere along the line going through M = [0 0 0.4]T and A, i.e. along the line spanned by the vector u = A − M = [0.8 1.6 0]T . We can normalize u so that it becomes u = u/ kuk = [0.4472 0.8944 0]T . The situation is depicted in Figure 3.3. This example is useful to understand the work about the approximation of the stability boundary in Section 6.

1

0.5 M

A

0.5

1

1.5

2

1 Figure 3.3: Problem of projecting A onto the sphere in a given direction.

£ ¤T Let P (A) = x P y P z P be the sought projection. That the vector going from A to P (A) should be parallel to the vector u = [0.4472 0.8944 0]T can be seen as forcing it to be on the line spanned by this vector and going through A so that P (A) − A must be orthogonal to all vectors of the plane Π orthogonal to u (which is equivalent to P (A) − A being parallel to u). In order to analytically express this condition, we need a basis of this plane Π. Such a basis could be obtained by the Gram-Schmidt procedure (see Section 3.3 for more detail about the Gram-Schmidt procedure), starting with vector u. Here, however, since u is a unit vector with third coordinate 0, we get directly v = [−0.8944 0.4472 0]T and w = [0 0 1]T . So the condition of the projection being

48


parallel to u (and hence, orthogonal to v and w) can be written ( ( 〈P (A) − A | v〉 = 0, (P (A) − A) ⊥ v ⇐⇒ 〈P (A) − A | w〉 = 0, (P (A) − A) ⊥ w ( (x P − 0.8)v 1 + (y P − 1.6)v 2 + (z P − 0.4)v 3 = 0, ⇐⇒ (x P − 0.8)w 1 + (y P − 1.6)w 2 + (z P − 0.4)w 3 = 0, ( x P v 1 + y P v 2 − 0.8v 1 − 1.6v 2 = 0, ⇐⇒ z P − 0.4 = 0,

(3.13)

where v = [v 1 v 2 v 3 ]T = [−0.8944 0.4472 0]T and w = [w 1 w 2 w 3 ]T = [0 0 1]T . These two equations describe the condition that must be satisfied for the projection P (A) to lie on the line spanned by u and going through A, Now, the projection must also lie on the sphere so that its coordinates must also satisfy £

Hence, P (A) = x P

yP

¤T

x P2 + y P2 + z P2 = 1.

z P is completely characterized by  1   f (x P , y P , z P ) = x P v 1 + y P v 2 − 0.8v 1 − 1.6v 2 = 0, F (x P , y P , z P ) = 0 ⇐⇒ f 2 (x P , y P , z P ) = z P − 0.4 = 0,   3 f (x P , y P , z P ) = x P2 + y P2 + z P2 − 1 = 0.

(3.14)

(3.15)

This problem can thus be solved by Newton’s method. The Jacobian at (x, y, z) is   1   f x (x, y, z) f y1 (x, y, z) f z1 (x, y, z) v1 v2 0 0 1 (3.16) J =  f x2 (x, y, z) f y2 (x, y, z) f z2 (x, y, z) =  0 3 3 3 2x 2y 2z f x (x, y, z) f y (x, y, z) f z (x, y, z)

Six iterations are necessary to solve the problem to obtain kF k∞ < 10−8 . Figure 3.4 shows these iterations. We see that already after the third iteration, the current point is very close to P (A). The projection is P (A) = [0.41 0.82 0.4]T .

Remark 3.1 It is possible, in the example above, to solve (3.15) analytically to get the exact value of P (A). However, getting an analytical expression is not always possible (see for example Section 5.3 in which no parametrization of the stability boundary is known); in these cases, Newton’s method can be used.

3.2 Elements of differential geometry In this section, we will go through useful elements of differential geometry for the study of surfaces. This will be used in order to study the stability boundary presented in Section 5.3 and to approximate it. The interested reader is referred to, for example, [91] for further information.

49

3.2. ELEMENTS OF DIFFERENTIAL GEOMETRY

1 P (A) 0.5

0.2

0.4

0.8

0.6

1

1.4

1.2

1.8

1.6

2

A

(3)

(2) (1)

0.5 1

Iterations (i)

1.5 Figure 3.4: Convergence of Newton’s method. The first three iterations are shown [blue diamond]. The last three iterations are very close to P (A).

3.2.1 Surfaces In the following, we consider a space of dimension m, such as Rm . A surface Σ in Rm will be defined here as a level set of a real-valued smooth function f : Rm → R that is ¯ © ª Σ = λ = (λ1 , . . . , λm ) ∈ Rm ¯ f (λ) = 0 . (3.17)

Example 3.3 (Unit sphere) The unit sphere Sm−1 is the sphere in dimension m with center 0 and radius 1, and is defined by f (λ1 , . . . , λm ) = λ21 + . . . λ2m − 1 = 0.

(3.18)

3.2.2 Gradient, normal vectors, tangent planes, Gauss map Definition 3.1 (Gradient) The gradient of a surface Σ ∈ Rm defined by f (λ1 , . . . , λm ) = 0 is given by ·

∂f ∇f = ∂λ1

...

∂f ∂λm

¸T

.

(3.19) ♣

50


Theorem 3.1 Let λ ∈ Σ, then ∇ f (λ) is normal to Σ.

♦

The reader is referred to Example 3.4 to see how this applies to the unit sphere in R3 . A unit normal to Σ at λ can then be defined as ∇ f (λ) °. n Σ (λ) = ° °∇ f (λ)°

(3.20)

Definition 3.2 (Tangent hyperplane) Let λ ∈ Σ, and n Σ be the normal to Σ at λ. The tangent hyperplane Tλ Σ at λ is a m − 1 dimensional manifold containing all points orthogonal to n Σ : ¯ © ª Tλ Σ = y ∈ Rm ¯ (y − λ) ⊥ n Σ .

(3.21) ♣

It is possible to get a basis {u 1 , . . . , u m−1 } of Tλ Σ from the normal by using the GramSchmidt orthonormalization process, see Section 3.3 and in particular Example 3.4. Consider a surface Σ defined by the level set of f : λ ∈ Σ ⇐⇒ f (λ) = 0. Let λ0 = £ 0 ¤T λ1 . . . λ0m ∈ Σ. The first-order Taylor expansion of f around λ0 gives 0 = f (λ0 + ∆λ) ≈ f (λ0 ) +

n X

∆λi

i =1

∂f (λ0 ) ∂λi

n ª X ∂f (α0 ) = f (λ0 ) = 0 = ∆λi ∂λ i i =1

©

(3.22)

where ∆λ = [∆λ1 . . . ∆λn ]T is a small displacement on Σ from λ0 . Hence, in the first order, we have that ∆λ ⊥ ∇ f (λ0 ), that is, ∆λ belongs to the tangent hyperplane Tλ Σ. The tangent hyperplane Tλ Σ gives a first-order local approximation of the surface. In the following, we define C = [u 1 . . . u m−1 ]

(3.23)

to be the m × (m − 1) matrix containing the basis vectors. This matrix will often be used in the following. Definition 3.3 (Gauss map) The map N : Σ → Sm−1 which assigns the unit normal to Σ to each point on Σ is called the Gauss map: ∀x ∈ Σ, N (λ) = n Σ (λ) ∈ Sm−1 .

(3.24) ♣

51

3.2. ELEMENTS OF DIFFERENTIAL GEOMETRY

3.2.3 Curvatures and second-order approximations As seen above, the tangent hyperplane, and thus the Gauss map, carries the firstorder approximation of a surface Σ. Principal curvatures of a surface can also be determined, and carry second-order information about a surface. Intuitively, following the surface, the more the Gauss map (i.e. the normal) varies, the larger the curvature of the surface. For example, if the Gauss map is constant, the normal is identical all over the surface, which is then a plane and has zero curvature. Hence, we see that curvatures are related to the rate of change of the Gauss map. Weingarten map and second fundamental form Definition 3.4 (Weingarten map) Let N : Σ → Sm−1 be the Gauss map for Σ. The Weingarten map dNλ at λ ∈ Σ is defined as the map from the tangent hyperplane to Σ, Tλ Σ, to itself giving the derivative of the Gauss map: dNλ : Tλ Σ → Tλ Σ. ♣ Remark 3.2 (Interpretation of the Weingarten map) It maps the derivative of the Gauss map along a vector in the tangent hyperplane. Note that since the normal vector is a unit vector, its derivative is orthogonal to it, and, hence, lie in the tangent plane as seen in the function signature dNλ : Tλ Σ → Tλ Σ. The Weingarten map thus measures how the normal changes along vectors in the tangent hyperplane. Note that the definition only makes sense for changes along vectors in the tangent hyperplane because the normal does not exist for changes in the direction of the normal itself (since any change in this direction from a point on the surface makes this point leave the surface, and, thus, the normal is not defined anymore). The Weingarten map can be represented by a (m −1)×(m −1) symmetric real-valued matrix (remember that we consider a m-dimensional space) ∀x c ∈ Tλ Σ,

dNλ (x c ) = dNλ x c ,

(3.25)

where dNλ is sometimes called the curvature tensor. Let {u 1 , . . . , u m−1 } be an orthonormal basis of the tangent plane, and let C = [u 1 . . . u m−1 ] be the matrix whose columns j are the basis vectors. Let dNλ be the j -th column of the curvature tensor. By definition, this column is the derivative of the Gauss map along u j , the j -th basis vector. Hence, i,j

the i -th element in this column, dNλ corresponds to how the normal changes along the i -th basis vector u i for an infinitesimal change du ∈ R along u j . For such a change, the normal thus approximately (up to the first order) becomes j

N (λ + du · u j ) = N (λ) + duC dNλ .

(3.26)

Definition 3.5 (Second fundamental form) The second fundamental form of Σ at λ ∈ Σ, IIλ : Tλ Σ → R, is defined for a displacement x c in the tangent hyperplane as IIλ (x c ) = − 〈dNλ (x c ) | x c 〉 .

(3.27)

52


Note the minus sign in front of the scalar product.

♣

Using the curvature tensor, the second fundamental form can be expressed as, for x c ∈ Tλ Σ, IIλ (x c ) = −x cT dNλ x c ,

(3.28)

Let C be the matrix in (3.23) whose columns are the m − 1 vectors of a basis of the tangent plane Tλ Σ. Remember that Σ is the level set of the function f : Rm → R. Recall ° ° ° ° that the normal is ∇ f / ∇ f . The Weingarten map can be expressed as 1 ° C T H ( f )C x c , dNλ (x c ) = ° °∇ f (λ)°

(3.29)

where H ( f ) is the Hessian of f at λ. Second-order approximations

Before giving the expression of the second-order approximations, we give a geometric interpretation of the second fundamental form. Remark 3.3 (Interpretation of the second fundamental form) We consider the space R3 , a smooth surface Σ in R3 and a point λ ∈ Σ. Let Tλ Σ be the tangent plane of Σ at λ, and let {u 1 , u 2 } be a basis of Tλ Σ. Around λ, any point on Σ can be expressed locally in the coordinate system of the tangent ° °plane. Formally, let ε > 0. There exists a function σ : Tλ Σ → Σ, such that ∀ y ∈ Σ, ° y − λ° ≤ ε, there exists t ∈ Tλ Σ, with t = vu 1 + wu 2 , v, w ∈ R, such that y = σ(t ) = σ(v, w). In particular, λ = σ(0, 0). Note that the derivatives of σ at (0, 0) with respect to v and w, σv and σw , are in the tangent plane. A second-order Taylor expansion around (0, 0) gives σ(v, w) = σ(0, 0) + vσv + wσw +

¢ 1¡ 2 v σv v + 2v wσv w + w 2 σw w , 2

(3.30)

where σ and its first and second derivatives are evaluated at (0, 0). Let now consider the quantity 〈σ(v, w) − σ(0, 0) | N 〉, where N is the normal to Σ at λ. This quantity corresponds to the distance between the tangent plane and σ(v, w) as measured in the direction of the normal. Since both σv and σw belongs to the tangent plane, we have that ¢ 1¡ 2 v 〈σv v | N 〉 + 2v w 〈σv w | N 〉 + w 2 〈σw w | N 〉 2 ¢ 1¡ 2 = Lv + 2M v w + N w 2 2 · ¸· ¸ 1 L M v = [v w] , M N w 2

〈σ(v, w) − σ(0, 0) | N 〉 =

(3.31) (3.32) (3.33)

53

3.2. ELEMENTS OF DIFFERENTIAL GEOMETRY with L = 〈σv v | N 〉

(3.34)

N = 〈σw w | N 〉 .

(3.36)

M = 〈σv w | N 〉

(3.35)

It can be proven, see [91, Section 7.2], that in this case, the second fundamental form at λ is exactly, for any (v, w) in the tangent plane · ¸· ¸ L M v II(v, w) = [v w] . (3.37) M N w Hence, the geometric interpretation of the second fundamental form is that it measures, up to a factor 21 , the distance from the tangent plane to the second-order Taylor expansion of the surface Σ. An immediate application of the remark above is that the map Γλ : Tλ Σ → Rm defined, for x c ∈ Tλ Σ by

1 Γλ (x c ) = λ +C x c + IIλ (x c )n Σ (λ) (3.38) 2 where, as above, C is a matrix whose columns are the vectors of a basis of the tangent plane, is a second-order approximation of Σ locally around λ. We note that the structure of the second-order approximation is very similar to the geometric interpretation given by the above remark, where λ + C x c corresponds to a displacement x c on the tangent plane from λ, and the last term to the distance from the tangent plane to the secondorder approximation. Principal curvatures Recall the definition of the second fundamental form using the curvature tensor dNλ from (3.27). The principal curvatures are defined as follows

Definition 3.6 (Principal curvatures and principal directions) Let Σ be a smooth surface in Rm , and λ be a point on Σ. Let {u 1 , . . . , u m−1 } be an orthonormal basis of the tangent plane. Let dNλ be the (m − 1) × (m − 1) curvature tensor. The eigenvalues κ1 , . . . , κm−1 of dNλ are called the principal curvatures, and the corresponding eigenvectors d 1 , . . . , d m−1 are the principal directions. ♣ Since the matrix dNλ is symmetric, the eigenvectors (principal directions) can be chosen such that they form an orthonormal basis {d 1 , . . . , d m−1 }. Expressed in this basis, dNλ is diagonal with the principal curvatures on the diagonal. Hence, for a small displacement on the surface around λ, the normal becomes (up to the first order)   κ1 0 . . . . . . 0 0 κ 0 ... 0    T 2   C dλ. N (λ + dλ) = N (λ) +C  . (3.39) . . . . .. .. .. ..   ..  0

...

...

0

κm−1

54


v2

v2

b2 v1

v1

b1

〈v 2 | b 1 〉 b 1 (a) Original vectors v 1 and v 2 .

(b) Creation of the orthonormal basis (b 1 , b 2 ).

Figure 3.5: Gram-Schmidt process: taking away the projection of v 2 onto b 1 .

where C = [d 1 . . . d m−1 ] and C T dλ corresponds to the local coordinates of the displacement dλ, that is, the projection of dλ onto the tangent plane. We see from this expression that the larger a principal curvature κi is the more the normal will change along the corresponding principal direction d i , which is the intuitive definition of curvatures.

3.3 Gram-Schmidt orthonormalization Let us consider an inner product space of dimension n. In the following, we consider P Rn with the scalar product defined by 〈u | v〉 = ni=1 u i v i , where the u i and v i are the elements of each vector. The Gram-Schmidt orthonormalization process takes a set of n linearly independent vectors {v 1 , . . . , v n } (here the v i ’s are vectors), and transforms it into an orthonormal basis {b 1 , . . . , b n } of Rn . It works as follows: b1 =

1 v1, kv 1 k

b 2 = v 2 − 〈v 2 | b 1 〉 b 1 , ······

(3.40a) b2 =

1 b2 , kb 2 k

b n = v n − 〈v n | b 1 〉 b 1 − · · · 〈v n | b n−1 〉 b n−1 ,

(3.40b)

bn =

1 bn . kb n k

(3.40c)

¯ ® ¯ ® Each operation b j = v j − v j ¯ b 1 b 1 − ·©· · v j ¯ b j −1 ªb j −1 takes away from v j the projection of v j onto the space spanned by b 1 , . . . , b j −1 . This is illustrated in Figure 3.5 in R2 . Note that the normalization steps are optional; if they are omitted, we get an orthogonal basis. Often, we would like to build an orthonormal or orthogonal basis of Rn starting from one nonzero vector u. We saw that the Gram-Schmidt orthonormalization process re-

55

3.3. GRAM-SCHMIDT ORTHONORMALIZATION

quires to start with n linearly independent vectors. The idea is to start with u and (n −1) unit vectors of the canonical basis {e 1 , . . . , e n } where e j has its j -th element equal to 1 and all the others equal to zero. The n − 1 canonical vectors are chosen so that u does not lie in the space spanned by them. To do so, we have two cases 1. All elements of u are nonzero: any (n − 1) canonical vectors will work.

2. There exists i for which u i = 0: since u is a nonzero vector, there also exists k for which u k 6= 0. Then we choose not to take e k for one such k among the (n − 1) canonical vectors. Note that this can be extended to get an orthonormal basis starting from k given vectors, k = 1, . . . , n. Example 3.4 (Basis of the tangent plane) We consider again the sphere S2 used in Example 3.2. We recall that it is defined by F (x, y, z) = x 2 + y 2 + z 2 − 1 = 0,

x, y, z ≥ 0.

(3.41)

In this example, we study the problem of finding a basis of the tangent plane to S2 at any point (x, y, z) on S2 . The gradient of a surface parametrized by F (x 1 , . . . , x m ) = 0 at any point (x 1 , · · · , x m ) on this surface is normal to the surface at this point. The tangent plane is the orthogonal complement of the normal, i.e. the plane of all vectors orthogonal to the normal and going through (x 1 , · · · , x m ). Starting the Gram-Schmidt orthonormalization process £ ¤T with the gradient ∇F = F x1 , . . . , F xm , we get an orthonormal basis of Rm {b 1 , b 2 , . . . , b m } with b 1 = ∇F / k∇F k normal to the surface. The space spanned by {b 2 , . . . , b m } and containing (x 1 , · · · , x m ) is orthogonal to b 1 and therefore orthogonal to the normal to the surface; that is, it is the tangent plane to the surface. Applying this in our example, for any point (x, y, z) on the sphere, the gradient is     Fx x ∇F (x, y, z) = F y  = 2  y  . (3.42) Fz z £ ¤T Hence, a normal to the sphere at any point (x, y, z) is the vector n = x y z . Suppose that we look for the tangent plane at a point P = (0.41, 0.82, 0.4). Figure 3.6 shows the sphere and its normal at P . Now, we use the Gram-Schmidt orthonormalization starting with the normal n. As explained above, it results in a basis {n/ knk , u, v} (corresponding to b 1 , b 2 and b 3 in (3.40)) with (u, v) being a basis of the tangent plane. In our example, at P , we get u = [−0.59 0.57

− 0.57]T , T

v = [−0.7 0 0.72] .

(3.43) (3.44)

The tangent plane at P is therefore TP S2 = {au + bv | a ∈ R, b ∈ R}. This is illustrated in Figure 3.7 which shows the tangent plane to the sphere at P. We plotted −v instead of v for a better visualization.

56


n P

Figure 3.6: The unit sphere S2 and its normal n at P = (0.41, 0.82, 0.4). In Chapter 6, this approach will be used to get the tangent plane to the stability boundary.

n P u −v

Figure 3.7: The unit sphere S2 and its tangent plane at P = (0.41, 0.82, 0.4)

3.4 Cumulants and Cornish-Fisher expansion In this section, we define what the cumulants of a random variable are, and we present the Cornish-Fisher expansion. The Cornish-Fisher expansion is used in Chapter

3.4. CUMULANTS AND CORNISH-FISHER EXPANSION

57

8. In the following, we consider a random variable Z . Definition 3.7 (Cumulant generating function and cumulants) The cumulant generating function of Z is defined as: ∞ ¤¢ X ¡ £ tj κ j (Z ) , K (t ) = log E e t Z = j! j =1

(3.45)

£ ¤ where κ j (Z ), ∀ j ≥ 1 are the cumulants of Z , and E e t Z is the moment generating function. ♣ From the definition above, the cumulants can be computed as ∀ j ≥ 1, κ j (Z ) = K ( j ) (0),

(3.46)

where K ( j ) is the j -th derivative of the cumulant generating function. Theorem 3.2 (Cornish-Fisher) Let ¢ ¢ ¢ 1 ¡ 3 1 ¡ 3 1¡ 2 z − 1 κ3 (Z ) − z − 3z κ4 (Z ) + 4z − 7z κ23 (Z ) 6 24 36 + higher order terms.

k Z (z) =z −

(3.47)

Let U ∼ N (0, 1). Then, when the series converges, P [Z < z] = P [U < k Z (z)] .

(3.48) ♦

Assume in the following that the probability distribution of Z is not known but that its cumulants are known. £ ¤The Cornish-Fisher expansion in the theorem above can be used to compute P Z ≤ y [31]: P [Z ≤ z] = Ψ (k Z (z)) ,

(3.49)

where Ψ is the cumulative probability distribution of the standard normal distribution U ∼ N (0, 1). Note that, in practice, when Z is close to Gaussian, it is often enough to neglect the terms with cumulants of order higher than four [31]. This corresponds to neglecting the terms called “higher order terms” in the expression of k Z in (3.47). Another way of computing the cumulants up to the fourth order is: κ1 (Z ) = E [Z ] , £ ¤ κ2 (Z ) = E (Z − κ1 )2 , £ ¤ κ3 (Z ) = E (Z − κ1 )3 , £ ¤ κ4 (Z ) = E (Z − κ1 )4 − 3κ22 .

(3.50a) (3.50b) (3.50c) (3.50d)

Chapter 4

Elements of bifurcation theory

Contents 4.1

Dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.2

Equilibria and topological classification of equilibria . . . . . . . . . .

60

4.3

Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4.4

Topological normal forms of saddle-node bifurcations . . . . . . . . .

63

4.5

Topological normal forms for Hopf bifurcations . . . . . . . . . . . . .

64

4.6

Center manifold theory . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

4.7

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

In this chapter, we give some elements of bifurcation theory which are important for understanding power system stability issues. In the next chapter, we will review how it has been used in the context of power systems to determine stability limits beyond which the system goes unstable. Most of the theory is from [62].

4.1 Dynamical systems Bifurcation theory deals with the analysis of parameter-dependent dynamical systems. Here, we will consider continuous dynamical systems of the form x˙ = f (x t , λ),

(4.1)

where x t ∈ X are time-dependent state variables belonging to the state space X , λ ∈ Rm are parameters, t ∈ T with usually T = R, and f is a function from X ×Rm to X . The state space will often be Rn . In order to simplify the notations, the time dependency of the variables will be implicit, and we omit the t in x t . The behavior of the system for a particular value of λ is often of interest. In this case, the dependence of f on λ is often dropped. The function f is then considered as 59

60

CHAPTER 4. ELEMENTS OF BIFURCATION THEORY

a function from X to X and we write x˙ = f (x).

(4.2)

Definition 4.1 (Evolution operator) The evolution operator ϕt : X → X takes an initial state x 0 ∈ X at t = 0 and gives the corresponding state at time t : x t = ϕt x 0 . ♣ The state variable changes from its initial state according to (4.2), and the evolution operator is not always known explicitly. A formal definition of a dynamical system is © ª given by a triplet T, X , ϕt describing this system.

Definition 4.2 (Trajectory) A trajectory, or orbit starting at x 0 is a subset of the state space containing all states that can be reached from x 0 or from which x 0 can be reached: ¯ © ª Trajectory from x 0 = x ∈ X ¯ x = ϕt x 0 , ∀t ∈ T .

(4.3)

Note that the x belonging to the trajectory for which x = ϕt x 0 with t < 0 lie before x 0 in history, i.e., x 0 can be reached from x. ♣

4.2 Equilibria and topological classification of equilibria Definition 4.3 (Equilibrium) An equilibrium x 0 ∈ X is a fixed point of ϕt , i.e., ∀t , x 0 = ϕt x 0 . From (4.2), we also have f (x 0 ) = 0. ♣ An important notion is that of stability of an equilibrium. Definition 4.4 (Stability of an equilibrium) An equilibrium x 0 is called stable if 1. for any sufficient a neighborhood U containing x 0 there exists neighborhood V also containing x 0 such that ∀x ∈ V, ∀t > 0, ϕt x ∈ U ;

2. there exists a neighborhood U0 containing x 0 such that ∀x ∈ U0 , ϕt x → x 0 as t → +∞. ♣

Sufficient conditions for stability of continuous dynamical system were given by Lyapunov in [67]: Theorem 4.1 (Lyapunov’s sufficient conditions for stability) Consider the dynamical system given in (4.2) with x ∈ Rn and f smooth. Suppose that it has an equilibrium x 0 and denote by A the Jacobian matrix of f (x) evaluated at the equilibrium, i.e., A = f x (x 0 ). Then x 0 is stable if all eigenvalues µ1 , µ2 , . . . , µn of A satisfy Re(µ) < 0. ♦

4.3. BIFURCATIONS

61

In the following, let n 0 , n − and n + be the numbers of eigenvalues of A with negative, zero and positive real part, respectively, at an equilibrium x 0 . Generically, n 0 = 0 (the dynamic Jacobian A having an eigenvalue with zero real part is a “special” case), which leads to the following definition. Definition 4.5 (Hyperbolicity) An equilibrium for which n 0 = 0 is called hyperbolic. A hyperbolic equilibrium for which n − n + 6= 0, i.e. with eigenvalues on both sides of the imaginary axis is called a hyperbolic saddle. ♣ It is important to be able to compare different dynamical systems to each other. A classification is given by the notion of locally topologically equivalent systems. Definition 4.6 (Locally equivalent systems) © topologically ª A dynamical system T, Rn , ϕt is called locally © ª topologically equivalent near an equilibrium x 0 to a dynamical system T, Rn , ψt near an equilibrium y 0 if there exists a homeomorphism 1 h : Rn → Rn which is 1. defined in a small neighborhood U ⊂ Rn of x 0 ;

2. satisfies y 0 = h(x 0 );

3. maps trajectories of the first system in U onto trajectories of the second system in V = h(U ) ∈ Rn , preserving the direction of time. ♣ Note that this definition applies also for the same system at two different equilibria. In this case, the following theorem gives necessary and sufficient conditions for locally topologically equivalence, and can be used for the topological classification of hyperbolic equilibria of a given system. Theorem 4.2 (Theorem 2.2 in [62]) © ª Two hyperbolic equilibria x 0 and y 0 of a dynamical system T, Rn , ϕt are locally topologically equivalent if and only if these equilibria have the same number n − and n + of eigenvalues with Re(µ) < 0 and Re(µ) > 0, respectively. ♦ Thus, using this theorem, the equilibrium points can be classified according to their numbers n − and n + . This enables us to define what bifurcations are.

4.3 Bifurcations For each value of the parameters λ in (4.1), the dynamical system can have several equilibria. A bifurcation in the system occurs when one of its equilibria changes type as the parameters λ vary, where the type is understood as in theorem 4.2, i.e., as being defined by the numbers of eigenvalues with negative and positive real parts. The value λ∗ of the parameters at which the bifurcation occurs is called bifurcation point. 1. A homeomorphism is a continuous bijection whose inverse also is continuous.

62


Im

Im µ1

µ

µ∗1

∗

µ

Re

Re µ∗2 µ2

(a) Saddle-node bifurcation

(b) Hopf bifurcation

Figure 4.1: The two possible generic co-dimension one bifurcations.

In the following, we will study the generic bifurcations of co-dimension one, that is, we assume that λ is real valued. The function f in (4.1) is assumed smooth in both x and λ. One of the equilibria of f changes type if either a simple real eigenvalue becomes zero or if a pair of simple complex eigenvalues crosses the imaginary axis. With one parameter only, these two cases are the only possibilities for an equilibrium to change type. Definition 4.7 (Saddle-node bifurcation) A saddle-node bifurcation (SNB) is a bifurcation in which a simple real eigenvalue becomes zero. This is illustrated in Figure 4.1a. ♣ Definition 4.8 (Hopf bifurcation) A Hopf bifurcation is a bifurcation in which the real part of a pair of simple complex eigenvalues becomes zero. This is illustrated in Figure 4.1b. ♣ There exists one so-called topological normal form for each of these two types of bifurcation. Dynamical systems experiencing either of the aforementioned bifurcations can be transformed to the corresponding topological form by invertible coordinate and parameter changes. Therefore, these topological normal forms can be studied, and the conclusions drawn from this study will apply for any system experiencing saddle-node and Hopf bifurcations, respectively. Moreover, it is interesting to study the simplest of these bifurcations, that is, x ∈ R and x ∈ R2 for SNB and Hopf bifurcations, respectively. Later on, we will describe the center manifold theory, and see how results from the simplest cases can be applied in higher dimensions.

4.4. TOPOLOGICAL NORMAL FORMS OF SADDLE-NODE BIFURCATIONS

63

4.4 Topological normal forms of saddle-node bifurcations In this section, we consider the simplest dynamical systems where SNB can occur, i.e. dynamical systems of the form x˙ = f (x, λ) with x ∈ R and λ ∈ R. The theorem below gives the normal form of saddle-node bifurcations and the conditions for which a dynamical system can be transformed to this normal form. First, we need a definition of what is understood by generic systems in the context of SNB. Definition 4.9 (Generic systems for SNB) Suppose that a scalar one-parameter system x˙ = f (x, λ),

x ∈ R,

λ ∈ R,

(4.4)

with f smooth has at λ = 0 an equilibrium x = 0. This system is called generic if the following conditions are satisfied: f xx (0, 0) 6= 0, f λ (0, 0) 6= 0.

(4.5a) (4.5b)

The two conditions above are called the nondegeneracy and the transversality conditions, respectively. ♣ Now, we can state the theorem giving the topological normal forms of SNB. Theorem 4.3 (Topological normal form of SNB – Theorem 3.2 in [62]) Any generic scalar one-parameter system of the form (4.4) having at λ = 0 the equilibrium x = 0 with µ = f x (0, 0) = 0, is locally topologically equivalent near the origin to one of the following normal forms: y˙ = β ± y 2 .

(4.6) ♦

The theorem above means that by studying the system in (4.6), we can draw conclusions for all scalar one-parameter systems experiencing saddle-node bifurcations. For example, Figure 4.2 shows the bifurcation diagram 2 and some trajectories of the dynamical system in the topological normal form with the p sign “+”. Forpeach value of the parameter β < 0, two equilibrium points exist: y 01 = β and y 02 = − β. One is stable, and the other one unstable. At β = 0, the two equilibria coalesce. For β > 0, the system does not have any equilibrium. According to Theorem 4.3, the behavior in this example is generic for the dynamical systems fulfilling the sufficient conditions of the Theorem (any generic dynamical system having an SNB). 2. A bifurcation diagram shows the equilibria for different values of the parameters.

64


Figure 4.2: Bifurcation diagram and some trajectories for the dynamical system in (4.6). A saddle-node bifurcation occurs at β = 0, from [64].

4.5 Topological normal forms for Hopf bifurcations Similarly as for SNB, we study here the simplest systems where Hopf bifurcations can occur, i.e., dynamical systems of the form x˙ = f (x, λ),

x ∈ R2 ,

λ ∈ R.

(4.7)

For Hopf bifurcations, generic systems satisfy the conditions below. Definition 4.10 (Generic systems for Hopf bifurcations) Suppose that a two-dimensional system of the form (4.7) with smooth f has for all sufficiently small |λ| the equilibrium x=0 with eigenvalues µ1,2 (λ) = α(λ) ± i ω(λ),

(4.8)

where α(0) = 0, ω(0) = ω0 > 0. The system is called generic if the following conditions are satisfied: l 1 (0) 6= 0,

where l 1 is the first Lyapunov coefficient;

0

α (0) 6= 0.

(4.9a) (4.9b)

The two conditions are nondegeneracy and transversality conditions, respectively. We can now give the topological normal form for Hopf bifurcations.

♣

4.6. CENTER MANIFOLD THEORY

65

Theorem 4.4 (Topological normal form for Hopf bifurcations – Theorem 3.4 in [62]) Any generic two-dimension, one-parameter system of the form 4.7 having at λ = 0 the equilibrium x = 0 with eigenvalues µ1,2 (0) = ±i ω0 , ω0 > 0, is topologically equivalent near the origin to one of the following normal forms: · ¸ · ¸· ¸ · ¸ ¡ ¢ y1 y˙1 β −1 y 1 = ± y 12 + y 22 . (4.10) y˙2 1 β y2 y2 ♦ Depending on the sign (“+” or “−”) in the topological normal form for Hopf bifurcations, the Hopf bifurcations are either subcritical (“+” sign) or supercritical (“−” sign). Figure 4.3 shows the phase portraits of the dynamical system in the topological form as the parameter β varies for a subcritical Hopf bifurcation. The system has one equilibrium point at the origin. This equilibrium is stable for β < 0 but unstable for β > 0. An unstable limit cycle exists for β < 0.

Figure 4.3: Subcritical Hopf bifurcation, from [63]. Figure 4.4 shows phase portraits of the dynamical system in the topological form as the parameter β varies for a supercritical Hopf bifurcation. The system has one equilibrium point at the origin. As for the case of subcritical Hopf bifurcation, this equilibrium is stable for β < 0 but unstable for β > 0. A stable limit cycle exists for β > 0. According to Theorem 4.4, the behavior in this example is generic for the dynamical systems fulfilling the sufficient conditions of the Theorem (any generic dynamical system having a Hopf bifurcation).

4.6 Center manifold theory In the previous sections, we have seen how one-dimensional and two-dimensional systems behave when experiencing saddle-node and Hopf bifurcations, respectively. In order to extrapolate these results, we need to go through the center manifold theory that

66


Figure 4.4: Supercritical Hopf bifurcation, from [63].

explains the behavior of n-dimensional systems, n > 2, by the behavior of the systems restricted to one- and two-dimensional manifolds called center manifolds. Once again, we begin by considering dynamical systems without parameter of the form x˙ = f (x), x ∈ Rn ,

(4.11)

and assume that f is smooth and x = 0 is an equilibrium at which n 0 , n − and n + denote the numbers of eigenvalues of the Jacobian A = f x (0) with zero, negative and positive real parts, respectively. Furthermore, we consider the case of a nonhyperbolic equilibrium for which we have either a zero real eigenvalue, or a pair of conjugate eigenvalues with zero real parts (so that n 0 = 1 or n 0 = 2). The n 0 eigenvalues with zero real parts are called critical, and the critical space spanned by the corresponding eigenvectors is denoted T c . The essence of the center manifold theory is contained in the following two theorems. Theorem 4.5 (Center manifold theorem – Theorem 5.1 in [62]) There is a locally defined smooth n 0 -dimensional invariant manifold 3 W c (0) of (4.11), called the center manifold, that is tangent to T c at x 0 = 0. Moreover, there is a neighborhood U of x 0 , such that if ϕt x ∈ U for all t ≥ 0 (t ≤ 0), then ϕt x → W c (0) for t → +∞ (t → −∞). ♦ Theorem 4.6 (Reduction principle – Theorem 5.1 in [62]) There exists δ > 0, and a function H : Rn0 → Rn− +n+ such that for kvk < δ, the system (4.11) is locally topologically equivalent near the origin to the system ( v˙ = B v + g (v, H (v)), (4.12) u˙ = Cu. 3. An invariant manifold is a manifold which the trajectories of the system do not leave once they reach it.

67

4.6. CENTER MANIFOLD THEORY

for some n 0 ×n 0 and (n − +n + )×(n − +n + ) matrices B and C , where x has been expressed in the eigenbasis of the Jacobian, with the elements corresponding to the critical eigenvectors in v ∈ Rn0 and the rest in u ∈ Rn− +n+ . ♦ The function H in the theorem above parametrizes the center manifold close to the equilibrium x = 0 by the critical coordinates v: W c = {(v, u) | u = H (v) for kvk < δ}, i.e. it expresses the noncritical coordinates as a function of the critical ones. The matrices B and C have all their eigenvalues with zero and nonzero real parts, respectively, which means that the noncritical dynamics (corresponding to u) either grow or decay exponentially. If we now assume that the system is stable before bifurcation so that n + = 0 at the bifurcation, then all eigenvalues of the matrix C have negative real parts, which makes the dynamics associated with u decay exponentially fast, so that the dynamics of the whole system in (4.12) will be decided by the dynamics restricted to the center manifold. At a bifurcation in a one-parameter dynamical system, among the n eigenvalues of the system, either one single real eigenvalue becomes zero, or the real parts of a single pair of complex conjugate eigenvalues become zero. This means that n 0 = 1 or n 0 = 2, and the center manifold is of dimension one or two. Therefore, the study of the simplest dynamical systems for saddle-node and Hopf bifurcations in sections 4.4 and 4.5, respectively, allows us to also describe the dynamics on the center manifold of a oneparameter stable system that experiences a bifurcation. The above applied for dynamical systems without parameters. In order to apply the center manifold theory to parameter-dependent dynamical systems of the form x˙ = f (x, λ),

x ∈ Rn ,

λ ∈ R,

(4.13)

we add one equation for the dynamics of λ, and we get λ˙ = 0,

x˙ = f (x, λ).

(4.14)

If we assume that the original system had an equilibrium at x = 0 with n 0 eigenvalues with zero real parts, then the extended one has an equilibrium at (λ, x) = (0, 0) with (n 0 + 1) eigenvalues with zero real parts. Indeed the Jacobian of the extended system is J=

µ

0 f λ (0, 0)

¶ 0 . f x (0, 0)

(4.15)

Now, by theorem 4.5, there exists a center manifold W c ⊂ Rn+1 of dimension (n 0 + 1) for system (4.14) which is tangent to T c , the eigenspace associated with the n 0 eigenvalues of J . An interesting property of W c is that, because λ˙ = 0, it is made up of invariant manifolds Wλc = W c ∩{(λ, x) | λ = λ0 }. In particular, at λ = 0 where the bifurcation occurs, the 0 restriction W0c is a center manifold for the nonextended system (4.13) at x = 0.

68


4.7 Overview Important definitions and theorems for understanding the basics of bifurcation theory have been gathered in this chapter. Applications of the study to power system stability will be given in the next chapter, but most important were the following notions: – Initially stable parameter-dependent dynamical systems can lose stability in bifurcations when the parameters vary. – The bifurcation types of interest for us are saddle-node and Hopf bifurcations, described in sections 4.4 and 4.5, for one- and two-dimensional dynamical systems, respectively. – In one-parameter systems of dimension more than two, saddle-node and Hopf bifurcations are still the ones of interest, and the dynamics of the systems can be studied on a one- or two-dimensional so-called center manifold.

Part II

Approximations of the stability boundary

69

Chapter 5

Stability in power systems

Contents 5.1

Power system models . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.2

Voltage instability, small-signal stability and bifurcation theory . . .

75

5.3

Stability boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

5.4

Normal to the stability boundary . . . . . . . . . . . . . . . . . . . . .

93

5.5

Iterative method to get the closest point on the stability boundary . . 100

5.6

Summary and challenges with larger amounts of variable resources . 100

This chapter discusses aspects of power system stability – voltage and small-signal stability, operational limits – which define different types of stability limits. The set of these limits form the stability boundary, which is described in detail.

5.1 Power system models Power systems are usually modeled by a set of differential algebraic equations (DAE) of the form x˙ = f (x, y, λ), 0 = g (x, y, λ),

(5.1a) (5.1b)

where x ∈ Rn x are state variables, y ∈ Rn y are algebraic variables, and λ ∈ Rm are parameters. The smooth functions f : Rn x × Rn y × Rm → Rn x and g : Rn x × Rn y × Rm → Rn y are the differential and algebraic equations, respectively. Generators’ internal voltages and speeds are examples of state variables. The algebraic variables include bus voltages and angles. Loads’ active or reactive power consumption are typical examples of parameters. 71

72

CHAPTER 5. STABILITY IN POWER SYSTEMS

If the behavior of the system for a specific value of the parameters is of interest, the dependence on λ can be omitted: x˙ = f (x, y), 0 = g (x, y).

Equilibrium of the system are given by ( 0 = f (x, y, λ), 0 = g (x, y, λ).

⇐⇒ F (z, λ) = 0,

(5.2a) (5.2b)

(5.3)

£ ¤T where we sometimes substitute z ∈ Rn z , n z = n x + n y , for x T y T and F : Rn z × Rm → Rn z for the whole set of equations at equilibrium. Switching devices need special attention for modeling. Automatic voltage regulators (AVR) switching between voltage and overexcitation control are examples of switching devices. Switching devices can be modeled by different equations, depending on the state of the device, which means that the sets of equations f and g change as well. Furthermore, the type of one variable (state or algebraic variable) can change depending on the state of the device, and this must be dealt with carefully in the system modeling. We now give a small example of the forms that f and g can take. Example 5.1 (One-axis model with AVR) A power system with synchronous generators modeled by their one-axis model and equipped with AVR can be modeled as:  δ˙i = ωi , ∀i ∈ G\{slack}   Ã !  0   E qi Vi  1 ω˙ =   i Mi P mi − X 0 sin (δi − θi ) − D i ωi , ∀i ∈ G\{slack} di ¶ µ f i (x, y) = (5.4) 0  X d i −x d i 0 xd i 0  1 ˙  E + V cos − θ , ∀i ∈ G E = E − (δ )  0 0 0 i i i f i  qi T X d i qi Xdi  d 0i  ¡ ¢   ˙0 1 E f i = T −E f i + K Ai (Vref-i − Vi ) , ∀i ∈ G a ei  0  0 = −E f i + E lim , i ∈ Gb  fi  ¡ ¡ ¢ ¡ ¢¢ Pnb 0 = P − P + g (x, y) = (5.5) Lk Gk Vk j =1 V j G k j cos θk j + B k j sin θk j , ∀k ∈ B,  ¡ ¡ ¢ ¡ ¢¢  Pnb  0 = Q Lk −QGk + Vk j =1 V j G k j sin θk j − B k j cos θk j , ∀k ∈ B

where G is the set of all generators, B the set of all buses, G a the set of generators under AVR control and G b the set of generators under overexcitation control. The other parameters are described in Table 5.1. The last two sets of equations in g are the power flow equations ensuring the power balance at each bus in the system. Also: iT h 0 0 0 x = δ1 , . . . , δng , ω1 , . . . , ωng , E q1 , . . . , E qn g , E f k,k∈G a , (5.6) iT h 0 y = E f k,k∈G , θ1 , . . . , θnb ,V1 , . . . ,Vnb . (5.7) b

73

5.1. POWER SYSTEM MODELS

Table 5.1: Description of the parameters in (5.4) and (5.5). Parameters

Description Generators i , i ∈ G

δi ωi Mi P mi 0 E qi Xdi 0 Xdi Di 0 Td 0i Ef i E lim fi K Ai Tei Vref-i

Generator’s rotor angle Generator’s speed Inertia coefficient Mechanical power Electromagnetic force (EMF) behind transient reactance Direct-axis synchronous reactance Direct-axis transient reactance Damping coefficient Open-circuit transient time constant Excitation EMF AVR i , i ∈ G Limit of the exciter Gain of the exciter Time constant of the exciter Terminal voltage reference Loads

PL QL

Active power of load Reactive power of load Electric grid

V θ Gk j Bk j θk j

Bus voltage amplitude Bus voltage angle Conductance of line between buses k and j Susceptance of line between buses k and j θk − θ j

74


The last equation in f i and the first in g model the behavior of the voltage regulator under voltage control and overexcitation control, respectively. If one voltage regulator reaches its limit E lim , the corresponding field voltage E f i becomes algebraic. f i

Let f a,i be the equation of the field voltage under voltage control (last equation in (5.4)) and f b,i the equation of the field voltage under overexcitation control (first equation in (5.5)). In [107], a power system model suitable for voltage stability studies in steady-state with AVR was introduced. It is made of the following equations at equilibrium:

f

a,i

ψ(z, λ) = 0,

(z) · f

b,i

(5.8a)

(z) = 0,

i = 1, . . . , n s ,

(5.8b)

f b,i (z) ≥ 0,

i = 1, . . . , n s ,

(5.8d)

f

a,i

(z) ≥ 0,

i = 1, . . . , n s ,

(5.8c)

where n s is the number of switching devices, and ψ : Rn z +m → Rn z +m−n s all equations in F (z, λ) except those representing the switching devices. Equation (5.8b) ensures that at least one of f a,i and f b,i is zero. The system of equations in (5.1a) and (5.1b) can be linearized around an equilibrium point (x 0 , y 0 , λ0 ) to get d∆x/dt = f x (x 0 , y 0 , λ0 )∆x + f y (x 0 , y 0 , λ0 )∆y,

0 = g x (x 0 , y 0 , λ0 )∆x + g y (x 0 , y 0 , λ0 )∆y.

(5.9a) (5.9b)

or · ¸ · d∆x/dt f x (x 0 , y 0 , λ0 ) = 0 g x (x 0 , y 0 , λ0 )

f y (x 0 , y 0 , λ0 ) g y (x 0 , y 0 , λ0 )

¸·

¸ · ¸ ∆x ∆x = J (x 0 , y 0 , z 0 ) , ∆y ∆y

(5.10)

where J (x 0 , y 0 , λ0 ) =

·

f x (x 0 , y 0 , λ0 ) g x (x 0 , y 0 , λ0 )

f y (x 0 , y 0 , λ0 ) g y (x 0 , y 0 , λ0 )

¸

(5.11)

is the system Jacobian. When the matrix g y (x 0 , y 0 , λ0 ) is nonsingular, which is the general case, the system model can be reduced to ³ ´ d∆x/dt = f x (x 0 , y 0 , λ0 ) − f y (x 0 , y 0 , λ0 )g y−1 (x 0 , y 0 , λ0 )g x (x 0 , y 0 , λ0 ) ∆x (5.12) = A(x 0 , y 0 , λ0 )∆x, where A(x 0 , y 0 , λ0 ) is the dynamic Jacobian. The bifurcation theory studied in Chapter 4 studies parameter-dependent dynamical systems of the form x˙ = f (x, λ).

(5.13)

5.2. VOLTAGE INSTABILITY, SMALL-SIGNAL STABILITY AND BIFURCATION THEORY 75 We see that the system in (5.12) is of this form with ∆x substituted for x and f (x, λ) = A(x 0 , y 0 , λ0 )∆x. Hence, the Jacobian f x is A(x 0 , y 0 , λ0 ) in this case, and the study of the eigenvalues of A at an equilibrium point gives information about the stability of the linearized system according to Theorem 4.1. The so-called small-signal stability property of the original system (5.1a) and (5.1b) is studied by studying the stability of the linearized system (5.12).

5.2 Voltage instability, small-signal stability and bifurcation theory In this section, a short literature review of important work carried out in the field of voltage stability and small-signal stability is given. It is seen how bifurcation theory was used to explain voltage collapse and oscillatory instability. The most important results are summarized after the literature review. Before going through the literature study, we would like to make the following remark. Remark 5.1 We see that the equations 0 = g (x, y, λ) were not present in typical dynamical systems studied in Chapter 4, which were represented by equations of the form x˙ = f (x, λ),

(5.14)

that is, the algebraic equations were not part of the studied models for dynamic systems. For the two types of generic codimension one bifurcation studied in the previous chapter (saddle-node and Hopf bifurcations) we make the following remarks: – Saddle-node bifurcations (SNB): we will see below that in general, it is enough to study the Jacobian of the equations (5.3) modeling the system in steady-state. This Jacobian matrix is the system Jacobian in (5.11). – Hopf bifurcations (HB): Hopf bifurcations are used to study small-signal stability, in which case the model (5.12) for the linearized system is used.

5.2.1 Literature review In 1989, Ian Dobson and Hsiao-Dong Chiang, using the center manifold theory, proposed a model that accounts for the qualitative behavior of power systems during voltage collapse [30]: initial slow decrease in voltage magnitudes, followed by a much faster decrease. This model predicts that the system, when reaching a saddle-node bifurcation point, will become unstable and follow the trajectory defined by the unstable part of the center manifold. This was explained in Section 4.6. Formally, consider a power system including load dynamics modeled by x˙ = f (x, λ),

x ∈ Rn ,

λ ∈ Rm.

(5.15)

76


In [30], it was studied how such a system can lose stability at a saddle-node bifurcation point. Suppose that x 0 is a stable equilibrium, and that at λ∗ a saddle-node bifurcation occurs, which means that one of the real eigenvalues of the Jacobian f x becomes zero. Let µ be this eigenvalue and v be a right eigenvector for µ so that f x v = µv = 0. All other eigenvalues have negative real parts. Let T c be the critical space as defined in Section 4.6, i.e. the space spanned by the eigenvector v. According to the reduction principle, explained in Theorem 4.6 and the discussion that followed, it is enough, locally around (x 0 , λ∗ ), to study the dynamics of the system on a center manifold W c of the system. Furthermore, after bifurcation, the system trajectory will leave the equilibrium point on the center manifold, which is tangent to v at x 0 according to the Center Manifold Theorem (4.5). Hence, v defines the direction in which the system will leave the equilibrium point into instability. The authors conclude that after leaving this saddle node, the system states move slowly along some trajectory on the center manifold (because of the zero eigenvalue), and then accelerate along this same trajectory away from the equilibrium. Thus, the authors remarked, the bifurcation theory, and in particular the center manifold theory, allows us to study power systems in a way that fits with the qualitative behavior of voltage collapses as described above (first slow decrease of voltages followed by a faster decrease). In [105], Thierry Van Cutsem presented a method to compute reactive power margins to voltage collapse. The novelty of the paper lies in the consideration of the reactive power limits of the generators. The method solves an optimization problem, checks then if the reactive power productions are under the limits, enforces the limits for a certain group of generators that have exceeded their limits, and re-solve the optimization problem until no more reactive power limit is violated. In [26], Ian Dobson discussed the use of the right and left eigenvectors corresponding to the zero eigenvalue at a saddle-node bifurcation point. These two vectors give information about the geometry of the stability boundary. From [30], we know that the trajectory of voltage collapse is defined by the center manifold. The right eigenvector is tangent to the center manifold (see Section 4.6). Hence, the author explains in [26], the right eigenvector gives the initial direction of voltage collapse. Near the bifurcation, it approximates the direction from the stable equilibrium to the unstable one. It is also shown how the left eigenvector can be used to calculate the normal to the stability boundary (the set of saddle-node bifurcation points) in parameter space. If the surface is parametrized by the loads, the normal can be used in load shedding because it is the direction in which the distance to the stability boundary increases the fastest. In [32], Ian Dobson and Liming Lu used the normal calculated from the left eigenvector to compute optimal control strategies with other parameters than the load. It is also shown that it is enough to use the static equations in order to find saddle-node bifurcation points. More precisely, if the dynamic model of a power system, including the load dynamics, is of the form z˙ = h(z, λ),

(5.16)

we can study the system modeled by the static equations (5.3) instead. This is very useful in practice, the authors observe, because the dynamics of the load are not well

5.2. VOLTAGE INSTABILITY, SMALL-SIGNAL STABILITY AND BIFURCATION THEORY 77 known and thus hard to model accurately. This observation allows us to disregard these dynamics for the computation of saddle-node bifurcations, and to use the system of equations (5.3) even if it is not of the form usually encountered in bifurcation studies, see Remark 5.1. Furthermore, a direct method to compute the bifurcation point in a given direction of load increase is given. In [25], Ian Dobson studied Hopf bifurcations in power systems. They correspond to oscillatory instabilities. An iterative method for finding the closest Hopf or saddlenode bifurcation point to the operating point is given. The method computes successive bifurcation points by following the normal to the bifurcation surface at the bifurcation point in the previous step. Convergence conditions on the curvatures of the bifurcation surface are given. It is proposed that the distance to this closest bifurcation point can be used as an index of voltage collapse, and its sensitivities to parameters are given. The method will be presented in detail in Section 5.5. In [29], Ian Dobson, Fernando L. Alvarado and Christopher L. DeMarco gave the first-order sensitivity of the loading margin to Hopf bifurcations. It uses a formula for the normal to the Hopf bifurcation surface presented in [25]. Comparisons on which parameters influence the loading margins the most are made with saddle-node bifurcations. It is observed that the settings of generators’ voltage regulators do not influence SNB but do influence Hopf bifurcations. In [33], Ian Dobson and Liming Lu studied the influence of reactive power limits on bifurcation points. It is shown that, in general power systems, instabilities can occur suddenly when generators reach their reactive power limits (this corresponds to what we will call later switching loadability limits – SLL). Transcritical bifurcation theory is used to explain this phenomenon, and shows that there are two possible consequences: either the system converges to a new stable equilibrium or the system collapses. In [34], Ian Dobson and Liming Lu gave a direct method for finding the closest saddle-node bifurcation point to the operating point. Thus, the method is meant for solving the same problem as the iterative method given in [25]. The problem is set up as a system of equations which is solved by Newton’s method. The direct method requires checking conditions on the curvature of the surface to ensure that convergence to a local minimum has occurred. Compared to the iterative method, it is faster (due to the quadratic convergence of Newton’s method), but less robust (because Newton’s method is sensitive to initial guesses), and more computationally cumbersome because it requires the computation of the curvatures. In [27], Ian Dobson extended the direct method presented in [34] to the case of Hopf bifurcations. Moreover, formulas for the curvatures of the saddle-node bifurcation surface are given. The curvatures are required in the direct method to check that the method reaches a local minimum. Together, the normal and the curvatures give information about the local geometry of the bifurcation surface. In [6], the iterative method for finding the worst case load power margin presented in [25] is used on larger systems (up to 173 buses), where it converges in a few iterations. As already stated in [27, 34], the iterative method is preferred because it does not require the computation of the curvatures of the bifurcation surface in order to ensure convergence to a local minimum. This paper also uses Monte Carlo simulation to

78


generate random directions to initialize the iterative method. This allows the method to find multiple local minimum load power margins on different hypersurfaces which make up the overall stability boundary. The stability boundary and its different parts will be studied in detail in Section 5.3.2. In [48], Scott Greene, Ian Dobson and Fernando L. Alvarado gave formulas for the first- and second-order sensitivities of the loading margin to the saddle-node bifurcation points. These sensitivities can be used to take decisions to increase the loading margin, and to compare the effect of varying different control parameters. Linear and quadratic estimates of the loading margin are computed from these sensitivities. Case studies on the IEEE 118 bus system shows a good fit of the linear estimate for many parameters. In cases where the linear estimate fails to approximate the loading margin, the quadratic estimate can be used. A shortcoming of the method is that it is assumed that the reactive power limits of all generators stay the same when estimating the new loading margin under parameter variations. This does not have a noticeable impact in the case study, but the authors point out that care must be taken in other systems. In [111], Costas D. Vournas, Michael E. Karystianos and Nicholas G. Maratos formulated an optimization problem whose solutions are loadability limits. The optimization problem is formulated such that both saddle-node bifurcation points and loadability limits corresponding to enforcement of reactive power limits can be found. It is shown how the Lagrangian multipliers can be used to derive the normal to the loadability surface – the set of loadability limits – irrespective of whether the limit corresponds to a saddle-node bifurcation or a reactive power limit.

5.2.2 Bifurcation diagrams and P-V curves Saddle-node bifurcation points and switching loadability limits can be depicted with help of bifurcation diagrams giving the voltage at a chosen bus as a function of the system parameters λ. When the parameter is chosen to be the active load, we obtain the network P-V curves [107], depicting the variations of the voltage magnitude at a specific bus as the active load increases. Figure 5.1a shows a P-V curve at the nose of which the system encounters a SNB. When the loading P increases, the system equilibrium (P,V ) point moves to the right along the upper part of the curve until it reaches the nose. At the nose, the system loses stability, and the voltage V collapses along the lower part of the P-V curve 1 . Figure 5.1b shows a system which loses stability in a SLL before encountering a SNB. As in the previous picture, the voltage V decreases along the solid curve as the loading P increases. Before reaching the nose of the solid curve (an SNB point), a switching device (for example an AVR as in Example 5.1) changes state which makes the system’s equilibria follow a new P-V curve (the dashed one). However, the system reaches directly the lower part of the curve which is unstable 2 . Hence, the system loses stability 1. The reader is referred to, for example, the examples in [107, Section 7.3.5] to understand the reasons why the voltage collapses at an SNB point. 2. The reader is here referred to [107, Section 7.6.2] to see the effect of switching devices on system stability and the corresponding P-V curves.

5.2. VOLTAGE INSTABILITY, SMALL-SIGNAL STABILITY AND BIFURCATION THEORY 79 immediately when switching from the solid curve to the dashed one. V

V

SLL

SNB

P (a) P-V curve with a SNB.

P (b) P-V curve with an SLL.

Figure 5.1: P-V curves with SNB and SLL.

5.2.3 Summary The differential algebraic equations in (5.2a) and (5.1b) can be used to study voltage and small-signal stability. We went through three phenomena in which the system can lose stability: 1. A saddle-node bifurcation (SNB) can occur. This happens when one of the real eigenvalue of the system Jacobian of (5.3) (see also (5.11)) becomes zero. This leads to voltage collapse. 2. A Hopf bifurcation can occur. This happens when the real parts of one pair of complex conjugate eigenvalues of the dynamic Jacobian of the reduced system (see (5.12)) become zero. The system becomes small-signal unstable. 3. The system can become suddenly unstable when one of the generators hits its reactive power limits. This type of instability will be called switching loadability limit (SLL). It does not correspond to a bifurcation point. Rather, a single real negative eigenvalue of the dynamic Jacobian becomes suddenly positive at an SLL. SLLs are also referred to as limit induced bifurcations [74], or similarity-induced bifurcations [35]. A comprehensive reference which explains various aspects of voltage stability is [35], and the interested reader is referred to it for more detailed explanations of the phenomena described above. Remark 5.2 (Switching loadability limits and breaking points) A breaking point is a point at which one of the switching devices in the system switches between two states. An example of that is the excitation systems of the generators that can switch between AVR control and OXL control, see Example 5.1. When the excitation

80


V

V

Harmless breaking point SLL

P (a) Harmless breaking point.

P (b) Breaking point corresponding to a SLL.

Figure 5.2: Difference between a harmless breaking point and a SLL (harmful breaking point).

system goes from AVR control to OXL control, the system is at a breaking point, and the corresponding generator has reached its reactive power limit. In the power system model (5.8) of Example 5.1, this corresponds to the situation where there exists one i ∈ {1, . . . , n} such that f a,i = 0 and f b,i = 0. It can happen as discussed above that the system becomes unstable at such points if a SLL is encountered. All SLLs correspond to such breaking points. However, it is important to note that not all breaking points are SLL, so that a generator can reach its reactive power limit without the system becoming unstable. We can use P-V curves to illustrate this. Figure 5.2a shows a harmless breaking point (i.e. not corresponding to a SLL). After the breaking point, the system continues on a new P − V curve – the dashed one – but does not lose stability. In Figure 5.2b, the system encounters one breaking point corresponding to a SLL, and thus immediately loses stability. Remark 5.3 (Static equations for finding saddle-node bifurcation points) This remark is related to Remark 5.1 and, more specifically, to the use of the static equations (5.3) in studying SNBs. In [107], it was observed that the dynamic Jacobian A can be seen as the Schur complement of the system Jacobian J . Recall from (5.11) and (5.12) that · ¸ fx f y J= , (5.17) gx g y A = f x − f y g y−1 g x .

(5.18)

The determinant of a matrix is related to that of its Schur complement. In the case of the two Jacobians, we have det J = det g y det A.

(5.19)

5.3. STABILITY BOUNDARY

81

Hence, for all cases in which g y is nonsingular, having a zero eigenvalue in J is equivalent to having a zero eigenvalue in A. Therefore the static equations can be used to study saddle-node bifurcation points, and the results apply to the linearized system (5.12). Another relating result was proven by Dobson in [26, Section 4], [32, Section 2], and more comprehensively in [28]. Dobson showed that the static equations can also be used to find saddle-node bifurcations of more detailed dynamics models. As already observed, this is especially useful for neglecting the load dynamics, which are not well known.

5.3 Stability boundary 5.3.1 Stability limits In the previous section, we studied three types of instability that can arise in power systems at some bifurcation points. These instability phenomena typically occur when the system becomes too loaded and stressed [107]. The loading of the system when it reaches a bifurcation point (either a SNB or Hopf bifurcation point) or a SLL is termed stability limit. In the cases of SNB and SLL, it is also called loadability limit. For the operation of power systems, it is very useful to study these stability limits since system operators strive for keeping the system stable. The bifurcation points and SLLs are not the only stability limits. Other types of stability limits are operational limits. Such limits include voltage constraints at certain buses and thermal line limits. As the system loading increases, bus voltages will decrease. Acceptable bus voltage ranges are set by system operators, and it happens often that voltages leave these ranges before any bifurcation point or SLL is reached [18]. Larger system loading means also larger power transfers in the electrical network. It can happen in that case that the thermal limits of some lines are exceeded before the system loses stability at a bifurcation point or at an SLL. System operators want the system to be within these operational limits too, and it is therefore necessary to include them in the study of stability limits. To summarize, in the scope of this thesis, stability limits will be understood as being one of the types given in Table 5.2. Also given in the table are the consequences of crossing these limits. The types given in the category “Operational limits” are just examples, and more of these operational limits can be taken into account. Example 5.2 (Situation in Sweden - continuation) Recall the way the Swedish system is operated from Section 2.5.3. Svenska Kraftnät, the Swedish transmission operator, computes transmission limits for the three main bottlenecks in Figure 2.11 by using a model of the Swedish power system, simulating a contingency in it and virtually increasing power transfers across the bottlenecks in this model until the loadability limits, corresponding to the maximum possible power transfers, are hit. These loadability limits are stability limits. More exactly, Svenska Kraftnät considers the stability limits corresponding to voltage stability – SNB and SLL – in the computation of the loadability limits. It was seen in Section 2.5.3 that a margin is subtracted from the computed loadability limits to set the transmission limits, which define the maximum

82


Table 5.2: Different types of stability limits. Type

Name

Consequence

Bifurcation points

SNB Hopf bifurcation

Voltage collapse Oscillatory instability

Bus voltage constraints

No immediate consequences but grid codes require bus voltages to stay within specified ranges Can lead to instability if the violation is sustained

Operational limits Lines’ thermal limits Others

SLL

Immediate instability

allowed power transfers across the corridors. If the power transfers come close to these transmission limits, the system operator must act, for example by manually activating frequency control reserves (balancing bids in the case of Sweden). As explained in Section 1.1.1, this thesis deals with the optimal operation of frequency control reserves, “optimal” meaning in the most cost-effective and secure manner. The example above shows the importance of studying the stability limits, both to monitor whether the current state is secure and to take the optimal actions if necessary by ensuring that the system stays within certain stability limits.

5.3.2 The stability boundary The stability boundary is a codimension 1 surface in parameter space made of all stability limits in the system. The stability region or stability domain is the region bounded by the stability boundary in which the system is stable and within the selected operational limits. Let us consider the system equations again: x˙ = f (x, y, λ), 0 = g (x, y, λ),

(5.20a) (5.20b)

where x ∈ Rn x are the state variables, y ∈ Rn y are the algebraic variables, and λ ∈ Rm are the parameters. Parameters can be loads at different buses or areas but also other quantities such as control variables. Examples of control variables include tap ratios and generators’ power output. Since instabilities arise when the system becomes stressed, the parameters of interest include important active or reactive power loads in the system. £ ¤T Hence, it is useful to divide the vector λ of system parameters in λ = u T ζT where the vector ζ corresponds to the non controllable parameters such as the loads, and the vector u to the other parameters, such as control variables.

83


In the following the stability boundary will be denoted as Σ. For a given u the stability boundary can be represented in load space only, and is then composed of all stability limits in the system defined by this vector u. For example, in Figure 5.3a, the stability boundary is depicted in a three dimensional space (ζ1 , ζ2 , u) consisted of two loads ζ1 and ζ2 and one control variable u. In Figure 5.3b, the stability boundary is depicted in the two-dimensional load space, for a specific value u 0 of the control variable. It is the intersection of the stability boundary in the three-dimensional space with the plane u = u 0 as seen in Figure 5.3a. We will use λ in the following, and do not distinguish between load and other parameters, unless explicitly said otherwise.

ζ2

u

ζ2

ζ1

Plane u = u 0

(a) Stability boundary in parameter space. Intersection by the plane u = u 0 marked by the thick black curve.

ζ1 (b) Stability boundary in load space only, for u = u 0 , corresponding to the thick black curve in Figure 5.3a.

Figure 5.3: Stability boundary in parameter space, and restricted to the load space for a given value u 0 .

5.3.3 Finding the stability limits In the following, we extend the model for power systems in steady state introduced in Example 5.1 in the following way: ψ(z, λ) = 0,

f a,i (z) · f b,i (z) = 0, f

a,i

f

b,i

(z) ≥ 0, (z) ≥ 0,

h(z, λ) ≥ 0.

z ∈ Rz ,

λ ∈ Rm

(5.21a)

i = 1, . . . , n s ,

(5.21b)

i = 1, . . . , n s ,

(5.21d)

i = 1, . . . , n s ,

(5.21c) (5.21e)

As seen in the example, ψ(z, λ) contains all the steady-state equations from (5.3) but those modeling the n s switching devices. The three equations (5.21b), (5.21c) and (5.21d)

84


take switching devices such as AVRs into account. The last equation with h(z, λ) ∈ R p allows us to add p operational constraints to the model such as constraints on bus voltages or power transfers on the lines. Such power system models have been used before, for example in [110]. Here, we assume that the parameters are the loads, i.e. we do not consider other parameters than the loads in λ ∈ Rm . Several tools exist to find stability limits: direct methods, optimization methods, continuation methods and quasi steady-state (QSS) simulations with eigenvalue analysis. Most of these methods find the stability limit in one particular direction d ∈ Rm of load increase at a time, that is, they solve the following problem: max

{s : Equations (5.21) hold, and the system

x∈Rn ,s∈R+

is small-signal stable with λ = λp + sd },

(5.22)

where λp is the present operating point. Note that equations (5.21) take into account all kinds of limits discussed before except for stability limits corresponding to Hopf bifurcations (at which the system is small-signal unstable). Therefore, we add the small-signal stability condition explicitly in (5.22). Solving problem (5.22) gives a solution (x ∗ , s ∗ ) and the associated parameter value ∗ λ = λp + s ∗ d . The value of the parameter λ∗ is the stability limit in direction d . A commonly used concept is that of loading margin which is defined as the distance from ° ° the operating point to the stability limit in direction d , i.e. it is equal to s ∗ = °λ∗ − λp °. A brief discussion about different methods for finding stability limits is given below. More in-depth explanations about the pros and cons of each method are given in [35, Section 4.3.5] and [107, Section 9.3] Direct methods Direct methods are methods that solve a system of equations Ψ(z, λ, r ) = 0 describing the stability limits to be found. The additional vector r represents quantities necessary to set up the equations Ψ. Newton’s method can be used to get a solution to the system of equations. For example, in [32], the system of equations for finding a saddle-node bifurcation point (z ∗ , λ∗ ) is formulated as   f (z ∗ , λ∗ )    g (z , λ ) ∗ ∗ Ψ(z ∗ , λ∗ , w ∗ ) = 0 ⇐⇒  w J (z ∗ , λ∗ )  ∗    w∗u − 1

=0 =0 =0

(5.23)

= 0.

In addition to the system equations, the third equation is satisfied at a point at which the system Jacobian J has a zero eigenvalue, i.e. at which the system encounters a saddle-node bifurcation point. The fourth equation ensures that the solution is nontrivial, i.e. that the left eigenvector w ∗ of the solution is not zero. Any nonzero vector u can be used in this last equation. A formulation with the right eigenvector can also be used [11]. Choosing between using the left or right eigenvector formulation depends on the application. In the literature review, we


85

saw that the left eigenvector can be used to obtain the normal to the stability boundary, which is in turn used to help take optimal control actions [26]. Hence, in general, the formulation with the left eigenvector given above is preferred. Similar formulations exist for finding Hopf bifurcation points. For example, in [5], the following system of equations was proposed:   f (z ∗ , λ∗ ) =0      g (z ∗ , λ∗ ) =0     A(z , λ )v + −ωv ∗ ∗ 1∗ 2∗ = 0 Ψ(z ∗ , λ∗ , v 1∗ , v 2∗ , ω) = 0 ⇐⇒ (5.24)  A(z , λ )v + ωv =0  ∗ ∗ 2∗ 1∗     =0 kv 1∗ k − 1    kv 2∗ k − 1 =0

The vector v 1∗ + i v 2∗ in the solution will be an eigenvector corresponding to a purely imaginary eigenvalue µ = i ω, ω > 0 appearing at a Hopf bifurcation. A similar formulation is used in [27]. A shortcoming of the direct method is that it cannot be used to determine SLLs [35, Section 4.3.5.1]. Optimization methods The problem (5.22) of finding the stability limits can also be solved by optimization methods, [111]. It was originally proposed in [105]. However, Hopf bifurcations are usually not taken into account in these formulations. One advantage of the optimization methods is that the Lagrangian multipliers can be used to get the normal to the stability boundary. Another advantage is that the well-developed theory available for solving optimization problems can be used. Continuation methods Continuation methods are similar to running successive power flows, but overcome the problem of having a singular Jacobian at SNBs (in which case no power flow solution can be found by Newton’s method). Continuation methods consist of three steps: predictor, corrector, and parametrization [3], [107, Section 9.3.2], [74, Chapter 5]. From a given equilibrium point of (5.21), the tangent vector to the power system equations is computed in the predictor step to predict the next equilibrium point after a load increase by a predefined step length. The corrector step looks for the actual new equilibrium point from this predicted point, usually by maintaining either the loading or the critical voltage constant. The parametrization step allows for changing the parameter from loading to one of the state variables, typically a bus voltage. This allows in practice to have nonsingular Jacobians, even at SNBs. One advantage of the continuation methods is that it gives the solution paths to the stability limits. This can be used to get, for example, voltage profiles – the P-V curves presented in Section 5.2.2 – up to the stability limit. Another advantage is that all types of stability limits can be studied with continuation methods 3 . A disadvantage of using this method is that it is more computationally expensive than for example direct methods or the optimization method [35, Section 4.3.5.2], [107, Section 9.3.3]. 3. For example, Hopf bifurcations can be found by checking at each iteration whether a pair of complex conjugate eigenvalues of the dynamic Jacobian A has crossed the imaginary axis.

86


QSS simulations Quasi steady-state approximations are models of the form (5.20), which are obtained by assuming that the fast dynamics in the power system are infinitely fast and stable. These models can then be used to run time-domain simulations, and monitor when the system encounters a stability limit [107, Sections 5.4 and 6.4], [110, 18]. Using QSS simulations is much faster than using complete timedomain simulation [35]. As described above, stability limits are encountered when the loads in the system increase. The way the load changes are compensated for has a determining influence on the stability limits. When running continuation power flows for example, the load changes are compensated by the so-called slack bus (or buses if the model has a distributed slack bus), which responds automatically to meet the load changes. In terms of frequency control schemes, see Section 2.3, the slack buses correspond to the ones participating in the automatic frequency control schemes, such as primary and secondary frequency control. As was stated in Section 1.2, the setting for these frequency control schemes is assumed to be known. Different settings would give different stability limits, and thus different stability boundaries.

5.3.4 Smooth parts of the stability boundary In general, the stability boundary Σ will be made of several smooth parts Σi , corresponding to different types of stability limits. This is because different types of limit are encountered when solving problem (5.22) of finding stability limits for different load increase directions. Example 5.3 As an example, 10 000 continuation power flows were run in the IEEE 9 bus system described in Appendix A.2 to find enough points to represent the stability boundary accurately. The three loads of the power system were chosen as the parameters. The results can be seen in Figure 5.4 4 . The stability boundary consists of 6 smooth parts: three parts corresponding to line thermal limits (in red, orange and yellow), one corresponding to an SLL (green part) and two corresponding to Hopf bifurcations (light and dark blue). More detail about this example is given in Section 7.1. Let us consider again the steady-state equations describing the system: F (z, λ) = 0.

(5.25)

Comparing this to the model (5.21) taking into account the switching devices, the set of equations F includes all equations in ψ, and for each switching device i ∈ {1, . . . , n s }, the equations f a,i or f b,i that are active (equal to zero). Note that at a breaking point corresponding to switching device j , both f a, j and f b, j are included in F . Therefore, F (z, λ) = 0 is generically a set of n z equations, but for the specific case of a SLL, it has 4. The loads in the figure are expressed in per unit, that is, they are normalized by the base power, 100 MW in this case. Hence, a load of 2 p.u. is 200 MW.

87


n z + 1 equations. For all types of stability limit, equations (5.25) hold since all stability limits are still equilibria of the system. A stability limit will therefore be characterize by the equations F = 0, and by extra conditions that differ for the different types of stability limit. Hence, we can characterize each type of stability limit by a set of equations of the form Ψ(z, λ, r ) = 0,

(5.26)

where r ∈ Rt is a vector of t additional variables necessary to characterize the smooth part Σi corresponding to the considered type of stability limit. The set Ψ(z, λ, v) = 0 is a set of n z + t + 1 equations. Definition 5.1 A Corner Point (CP) of the stability boundary is a point λ in parameter space where two or more smooth parts of the stability boundary intersect. ♣

Figure 5.4: Stability boundary in the IEEE 9 bus system in a three dimensional load space, made of different smooth parts: Hopf (dark and light blue), SLL (green) and line thermal limits (red, orange and yellow).

88


Corner points are non-smooth parts of the stability boundary [87], and are of codimension two or higher. The co-dimension two corner points correspond to the cases where two different smooth parts intersect. In Figure 5.4 of Example 5.3, the corner points are all edges between smooth parts of different colors. Characterization of SNB For all stability limits (z, λ) corresponding to SNB, there exists a vector r ∈ Rn z such that the following holds [51]   F (z, λ) = 0, ΨSNB (z, λ, r ) = F z (z, λ)v = 0,   T v v − 1 = 0,

(5.27)

The two last equations ensure that F z is singular, and therefore that the point (z, λ) is a SNB. A similar formulation could be given with a left eigenvector. Note that these equations are very similar to the ones used in the direct method, see (5.23). The direct method uses the characterization of SNB points (5.27) to find a SNB point in one particular parameter increase direction. To check if one given point (z 0 , λ0 ) belongs to a smooth part corresponding to saddlenode bifurcations, we could also identify the eigenvalue µ of F z with the smallest magnitude, and check that µ = 0 so that the point satisfies ΨSNB (z, λ) = 2

( F (z, λ) = 0, µ = 0.

(5.28)

Characterization of SLL At an SLL bifurcation point, there exists one switching device i for which both f a,i (z) and f b,i (z) are zero and thus included in F , see Example 5.1. Therefore, F (z, λ) = 0 is a set of n z + 1 equations, and we define ΨSLL (z, λ) = F (z, λ).

(5.29)

The vector r is empty in this case. SLLs corresponding to different switching device belong to different smooth parts of the stability boundary. Characterization of Hopf bifurcations Hopf bifurcations are characterized by a pair of conjugate eigenvalues of the dy¡ ¢ namic Jacobian A crossing the imaginary axis, i.e. Re µ = 0, where µ is one of the

89


eigenvalues in this pair. Once again, the equations (5.23) used to find Hopf bifurcations with the direct method can be used. A point (z, λ) is a Hopf bifurcation point of the system if there exist vectors v 1 , v 2 ∈ Rn x and ω ∈ R+ such that   f (z, λ) = 0     g (z, λ) = 0     A(z, λ)v + −ωv = 0 1 2 HB Ψ (z, λ, v 1 , v 2 , ω) = 0 ⇐⇒  A(z, λ)v 2 + ωv 1 = 0      kv 1 k − 1 = 0    kv 2 k − 1 = 0

(5.30)

To check if one given point (z 0 , λ0 ) belongs to a smooth part corresponding to saddlenode bifurcations, we could also identify the eigenvalue µ of A with the largest real part, and check that Re(µ) = 0 so that the point satisfies  λ) = 0,  F (z, ¡ ¢ HB Ψ2 (z, λ) = Re µ = 0,   ¡ ¢ Im µ 6= 0.

(5.31)

Characterization of operational limits Operational limits set constraints on quantities such as bus voltages, transmitted powers or line currents. These quantities are functions of the state and the algebraic £ ¤T variables x and y, i.e. they are functions of z = x T y T , and we can express them as functions l i (z), i = {1, . . . , n ol } where n ol is the number of such quantities considered. i i Operational limits are then upper or lower bounds, l low and l up , for each quantity l i (z). A stability limit is then an operational limit if there exists i = {1, . . . , n ol } such that OL

Ψ

(z, λ) =

( F (z, λ) = 0, L i (z) = 0,

(5.32)

where i i − l i (z) or L i (z) = l i (z) − l low . L i (z) = l up

(5.33)

The vector r is empty in this case. Stability limits corresponding to different i belong to different smooth parts of the stability boundary. For one quantity i , stability limits corresponding to upper or lower bounds belong to different smooth parts of the stability boundary. In the power system i i model (5.21), equations in h(z, λ) ≥ 0 are either l up − l i (z) ≥ 0 or l i (z) − l low ≥ 0 for some i. Some examples of analytical expression of operation limits l i (z) are given below.

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Example 5.4 (Examples of operational limits) Bus voltages Bus voltages belong to y, the vector of algebraic variables. Hence, for each bus voltage Vk there exists j such that Vk = y j , with y j being the j -th component of y, so that the corresponding function l i is just l i (z) = y j .

Line currents Line currents can be computed from the bus voltages, the series impedances and the charging capacitances of lines. For example, for the simple case where a line between buses j and k does not have any charging capacitance, it can be modeled by a series impedance Z . The current on this line can then be calculated as ¯ ¯ ¯ ¯ ¯ Vk − V j ¯ ¯ y i k − y i j ¯ i ¯ ¯, ¯ ¯ = (5.34) I = l (z) = ¯ ¯ Z ¯ ¯ Z where the two algebraic variables y i k and y i j are the two voltage amplitudes Vk and V j .

Power transfers Considering again a simple line as the one described for the case of line currents, we can define the power transfer on this line as the mean value between the power transferred from the transmitting bus j and the power received at the receiving bus k. These powers can be expressed respectively as µ µ ¶ ¶ µ µ ¶ ¶ yi j − yik ∗ V j − Vk ∗ P f = Re V j = Re y i j , Z Z µ µ ¶∗ ¶ µ µ ¶∗ ¶ yi j − yik V j − Vk P t = Re Vk = Re y i k , Z Z

(5.35) (5.36)

so that the power transferred over the line can be computed by l i (z) =

µ³ ´ µ y i − y i ¶∗ ¶ 1 j k Re y i j + y i k . 2 Z

(5.37)

5.3.5 SNB-SLL intersections Corner points at the intersection between SNB and SLL surfaces need special attention. For any other type of intersection (such as SLL-SLL, Hopf-SLL, . . . ), the smooth parts intersect transversally. At SNB-SLL corner points, however, the intersection is tangential (except in very rare cases) [87, 88]. Section 8.3 will describe why special attention needs to be given to these tangential intersections. We adopt the notion from [87], and call points where the intersection is tangential, tangential intersection points (TIPs). This situation is depicted in Figure 5.5 together with an operating point λ0 in a two dimensional parameter space, and two possible directions d 1 and d 2 of parameter increase. The stability boundary in the two dimensional parameter space is drawn with a solid line, and consists of two smooth parts, one corresponding to a SNB and one corresponding to a SLL. These two smooth parts meet in a TIP. The smooth part corresponding to the SLL is inside the one corresponding to the SNB. Hence, to the right of

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the TIP, the dashed line corresponds to breaking points which are not SLL (see Remark 5.2) and the solid line corresponds to the SNB points. To the left of the TIP, the solid line corresponds to SLL and the dash dotted line to SNB points, which are not stability limits since the switching loadability limits are reached first.

5.9

SNB

5.8

λ2

5.7

5.6

d1

5.5 λ0 5.4 1.5

TIP

SLL

1.6

d2

1.7

Harmless breaking points 1.8

1.9

2 λ1

2.1

2.2

2.3

2.4

2.5

Figure 5.5: The tangential intersection of a SNB and a SLL surface [88].

The tangential intersection points can also be illustrated by P − V curves. Consider Figure 5.6a which depicts a P − V curve with a harmless breaking point. In Figure 5.5, this situation corresponds to increasing the parameters (λ1 , λ2 ) from λ0 in, for example, direction d 2 . The quantity P in Figure 5.6a is then the amount of load increase in that direction. When the parameters reach the dashed curve in Figure 5.5, this corresponds to reaching the harmless breaking point in Figure 5.6a; the parameters then continue to increase until the SNB point (solid curve in Figure 5.5) is reached, which corresponds to reaching the nose of the dashed curve in Figure 5.6a. If we now consider a load increase in direction d 1 , the solid curve in Figure 5.5 is reached first, which corresponds to reaching the SLL point in Figure 5.6b. At this point, the system loses stability. Finally, if we consider a direction of load increase leading from λ0 to the TIP in Figure 5.5, the corresponding P −V curve will be similar to the one in Figure 5.6c. The breaking point now lies exactly at the nose of the dashed curve. Hence, it is both a SNB point and a SLL.

92


V

V

Harmless breaking point SLL

P

P

(a) Harmless breaking point.

(b) Breaking point corresponding to a SLL.

V

TIP

P (c) P − V curve when reaching a TIP.

Figure 5.6: A harmless breaking point, a SLL (harmful breaking point) and a tangential intersection point.

5.3.6 Post-contingency stability boundaries After the system undergoes a contingency, the stability boundary changes, and stability limits need to be recomputed. Stability limits in the pre-contingency system are used in order to assess how robust the system is to load increases in specific load increase directions. In post-contingency systems, stability limits give an indication of the security margin to instability after contingencies occur. Figure 5.7 shows both the pre-contingency stability boundary and a post-contingency stability boundary. The latter is the most restrictive boundary in the figure, i.e. the innermost surface. The power system considered is the IEEE 9 bus system from Appendix A.2. The contingency was the removal of one of a total of three lines between buses 5 and 7. It is very clear in the figure that the system gets more stressed and, in particular, the load at bus 5 cannot be increased as much as in the pre-contingency

93

5.4. NORMAL TO THE STABILITY BOUNDARY

state. Point 1 in the figure is inside the stability boundary in both the pre- and postcontingency cases. Point 2 is inside the stability boundary in the pre-contingency case, but becomes unstable or beyond the acceptable operating limits in the post-contingency case. If the system was operated at Point 2, it would become unstable after the contingency occurred.

Point 1

Point 2

Figure 5.7: Pre- and post-contingency stability boundaries. Colors as the same as in Figure 5.4. The post-contingency stability boundary is the innermost one. When considering post-contingency stability boundaries, post-contingency corrective actions must be taken into account, because they will change the system state. Not taking into account post-contingency corrective actions would underestimate the loading margins.

5.4 Normal to the stability boundary In reality, the entire stability boundary is not known. Our knowledge of it is restricted to the few stability limits computed for different load increase directions. This is illus-

94


trated in Figure 5.8, where Figure 5.8a shows the stability boundary, which is not known, and Figure 5.8b shows some stability limits on this boundary computed by one of the methods presented in Section 5.3.2.

(a) Entire stability boundary (unknown).

(b) Determining some points on the stability boundary.

Figure 5.8: Limited knowledge of the stability boundary. When stability limits have been computed on the stability boundary, it is possible to get more information than just the stability limit itself. One such piece of information is the normal in parameter space, for different types of stability limits. The normal can be used in order to optimally choose control actions that drive the system away from the stability boundary (in order to increase the loading margin) or that bring the system back to stability if the latter has become unstable. In the following, we will give the expressions of the normals for smooth parts corresponding to SNB, SLL, Hopf bifurcation and line thermal limits.

5.4.1 Computing the normal for SNB Let ΣSNB be a smooth part corresponding to SNB. Let λ0 ∈ ΣSNB , which means that the system (5.3) has a saddle-node bifurcation point (z 0 , λ0 ) for this value of the parameter. Let w be the left row eigenvector corresponding to the zero eigenvalue of the system Jacobian F z so that wF z (z 0 , λ0 ) = 0.

(5.38)

The following theorem from [27] guarantees existence and smoothness of the Gauss map (recall from Section 3.2 that the Gauss map of a surface maps each point of the surface to the unit normal to the surface).


95

Theorem 5.1 (Existence and smoothness of the Gauss map for SNB) Assume that the following conditions hold at (z 0 , λ0 ): wF λ 6= 0,

wF zz (v, v) 6= 0,

(5.39a) (5.39b)

where v is the right eigenvector corresponding to the zero eigenvalue of F z . Then, there exists an open set V ∈ Rm such that λ0 ∈ V ∩ Σ, and a smooth map Φ : V ∩ Σ → Rn z , such that Φ(λ) = z. It follows that ΣSNB has a normal vector n SNB (λ0 ) at λ0 , and the Gauss map n SNB : ΣSNB → Sm−1 is smooth (see [27, Section 2] and references therein). ♦ The conditions given in the theorem are similar to the ones given in (4.5), when we gave the definition of the topological normal form of SNB in generic systems. We now restate the derivation from [26] of the normal to ΣSNB . Let dλ be an arbitrary infinitesimal displacement from λ0 such that λ0 + dλ ∈ ΣSNB . Since both λ0 and λ0 + dλ are on ΣSNB , the following holds: F (z 0 , λ0 ) = 0,

F (z 0 + dz, λ0 + dλ) = 0.

(5.40) (5.41)

For λ0 + dλ to be on ΣSNB , the following must hold F z (z 0 , λ0 )dz + F λ (z 0 , λ0 )dλ = 0.

(5.42)

Now, we pre-multiply this equation with the left row eigenvector w corresponding to the zero eigenvalue of F z at (z 0 , λ0 ). The first term in the equation disappears, and we get wF λ (z 0 , λ0 )dλ = 0,

(5.43)

which holds for any dλ such that (z 0 + dz, λ0 + dλ) is on ΣSNB . Therefore, wF λ (z 0 , λ0 ) is normal to ΣSNB at (z 0 , λ0 ). A unit normal vector to ΣSNB is thus given by n SN B (λ0 ) = wF λ (z 0 , λ0 ),

(5.44)

where w is normalized so that the unit normal points in the direction away from the stability domain.

5.4.2 Computing the normal for SLL Let ΣSLL be a smooth part of the stability boundary corresponding to SLL. Let λ0 ∈ Σ . At such a point, we have (see (5.29)) SLL

F (z 0 , λ0 ) = 0, with F : Rn z × Rm → Rn z +1 .

(5.45)

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Similarly to the case of SNBs (see Theorem 5.1), there exists an open set V ∈ Rm such that λ0 ∈ V ∩ Σ, and a smooth map Φ : V ∩ Σ → Rn z , such that Φ(λ) = z. Also, the surface ΣSLL has a normal vector n SLL (λ0 ) at λ0 , and the Gauss map n SLL : ΣSLL → Sm−1 is smooth (see [88, Section III.A]). The system Jacobian F z is a Rn z +1 ×Rn z matrix, and its rank is one less than its number of rows. This can be compared to the case of a SNB where F : Rn z × Rm → Rn z , and the system Jacobian is a Rn z × Rn z matrix. In a the case of SLLs, there exists therefore a row vector w ∈ Rn z +1 such that wF z (z 0 , λ0 ) = 0.

(5.46)

The same derivation as for SNB applies, so that we find that wF λ (z 0 , λ0 ) is normal to ΣSLL . The difference with SNB is that the vector w such that wF z is normal to ΣSLL exists for SLLs due to the fact that the rank of the system Jacobian F z is one less than its number of rows, whereas such an w exists for SNBs because the system Jacobian has a zero eigenvalue and w is then the corresponding left eigenvector. In particular, the dimensions of w and F z are different for SNBs and SLLs. As for SNBs, a unit normal vector can then be defined by n SLL = wF λ (z 0 , λ0 ),

(5.47)

where w is normalized so that the unit normal points in the direction away from the stability domain.

5.4.3 Computing the normal for Hopf bifurcation Let ΣHB be a smooth part of the stability boundary corresponding to Hopf bifurcations, and let λ0 ∈ ΣHB . At a Hopf bifurcation the dynamic Jacobian, A, has a pair of purely imaginary eigenvalues ± j ω0 , ω0 > 0. The following theorem from [27, Section 2] guarantees existence and smoothness of the Gauss map for smooth parts of the surface corresponding to Hopf bifurcations. Theorem 5.2 (Existence and smoothness of the Gauss map for Hopf bifurcations) Since λ0 is not a saddle-node bifurcation point, the system Jacobian F z is nonsingular, and the implicit function theorem implies that there is a smooth function Φ defined in a neighborhood of λ0 with Φ(λ0 ) = z 0 , Φ(λ) = z and F (Φ(λ), λ) = 0. There is also a smooth function µ defined in a neighborhood of λ0 with µ(λ0 ) = j ω0 and µ(λ) an eigenvalue of F z (Φ(λ), λ). Assume that at λ0 the following conditions hold l 1 (λ0 ) 6= 0, ¡ ¢ d Re µ(λ0 ) 6= 0, dλ

where l 1 is the first Lyapunov coefficient;

(5.48a) (5.48b)

Then, there ¡exists¢an open set V such that λ0 ∈ V and ΣHB ∩ V is a smooth hypersurface given by Re µ(λ) = 0. It follows that ΣHB has a normal vector n ΣHB (λ0 ) at λ0 and the Gauss map n : ΣHB → Sm−1 is smooth. ♦


97

The conditions given in the theorem are similar to the ones given in (4.9), when we gave the definition of the topological normal form of Hopf bifurcations in generic systems. Let dλ be an arbitrary infinitesimal displacement from λ0 such that λ0 + dλ ∈ ΣHB . Then, the following holds ¡ ¢ Re µ(λ0 ) = 0, ¡ ¢ Re µ(λ0 + dλ) = 0.

(5.49) (5.50)

Expanding the second equation we get ¡ ¢ d Re µ(λ0 ) dλ

¡ ¢ dλ = Re µλ (λ0 ) dλ = 0.

(5.51)

Since displacement dλ around λ0 such that λ0 + dλ ∈ ΣHB , ¡ this ¢holds for any arbitrary HB Re µλ (λ0 ) is normal to Σ at λ0 . ¡ ¢ We give now an expression of Re µλ (λ0 ) . Let v = v(λ) and w = w(λ) be the left and right eigenvectors corresponding to µ(λ), normalized such that w v = 1 and |v| = 1. The left eigenvector w can be seen as a row vector. We have that (A − µI )v = 0.

(5.52)

Differentiating this equation with respect to λ yields µ

¶ ∂A − µλ I v + (A − µI )v λ = 0. ∂λ

(5.53)

Premultiplying by w we get µλ = w

µ

¶ ∂A v, ∂λ

(5.54)

so that µ µ ¶ ¶ ¡ ¢ ∂A Re µλ (λ0 ) = Re w v . ∂λ

(5.55)

Therefore, a unit normal vector to ΣHB can be expressed as µ µ ¶ ¶ ∂A n H B (λ0 ) = α Re w v , ∂λ

(5.56)

¯ ¯ where α is a normalization constant guaranteeing that ¯n H B ¯ = 1 and that n H B points in the direction of instability.

98


5.4.4 Computing the normal for operational limits Let ΣOL be a smooth part of the stability boundary corresponding to operational limits, and let λ0 ∈ ΣOL . Since λ0 is not a saddle-node bifurcation point, the system Jacobian F z is nonsingular, and the implicit function theorem implies that there is a smooth function Φ defined in a neighborhood of λ0 with Φ(λ0 ) = z 0 , Φ(λ) = z and F (Φ(λ), λ) = 0. Let dλ be an arbitrary infinitesimal displacement from λ0 such that λ0 + dλ ∈ ΣOL . Both points belong to ΣOL so that the equations (5.32) characterizing an operational limit hold at these points:

OL

Ψ

ΨOL (z 0 , λ0 ) = 0,

(5.57)

(z 0 + dz, λ0 + dλ) = 0.

(5.58)

L i (Φ(λ0 )) = 0,

(5.60)

(5.59)

In particular, we have for some i

L i (Φ(λ0 + dλ)) = 0,

(5.61)

where i ∈ {1, . . . , n OL } is for the i -th operational limit corresponding to ΣOL . Expanding the second equation we get L iz (Φ(λ0 )) Φλ (λ0 )dλ = 0.

(5.62)

Since this holds for any arbitrary displacement dλ around λ0 such that λ0 + dλ ∈ ΣOL , L iz (Φ(λ0 )) Φλ (λ0 ) is normal to ΣOL at λ0 . A unit normal vector can thus be computed by n OL = αL iz (Φ(λ0 )) Φλ (λ0 ),

(5.63)

where α is a normalizing factor chosen so that the normal points in the direction away from the stability domain.

5.4.5 Implicit function z = Φ(λ) and first-order derivative of A In the previous sections, we saw that it was possible to express z as a function of λ, z = Φ(λ), locally around points on the stability boundary. In fact, at any point that is not a saddle-node bifurcation point (including points not belonging to the stability boundary), the system Jacobian F z is nonsingular, and the implicit function theorem implies that such a function Φ exists locally. Theorem 5.1 guarantees the existance of such a function also for SNB. Hence, in the following, it will be assumed that it is always possible to write z = Φ(λ). In addition to this, recall that F (z) was defined as ( f (x, y), F (z) = (5.64) g (x, y).

99


In generic systems where g y is nonsingular, using the implicit function theorem again with the algebraic equations g , y can be written as a function of x: y = y(x). Since £ ¤T z = x T y T is a function of λ, x itself can be expressed as a function of λ: x = ϕ(λ). So, we have the following situation: z=

·

¸ · ¸ x ϕ(λ) = = Φ(λ). y = y(x) y(ϕ(λ))

(5.65)

The derivative of this function Φ with respect to the parameters λ is needed to compute the normals since it appears in their expression. Also needed in the case of Hopf bifurcations is the first-order derivative of the dynamic Jacobian A with respect to λ, which appears in the expression for the normal in (5.56). Formulas for both Φ and the first-order derivative of A can be found in Section B.1 in Appendix B.

5.4.6 Computations of the normals: Summary The formulas derived above for the normal to each type of smooth part can be found in Table 5.3. The second column gives the expression of the normals. The third column gives which equations can be used to compute the quantities found in these expressions. For SNB and SLL, the dimensions of the quantities appearing in the expressions are given to emphasize the difference between these two types of surface. Table 5.3: Analytical expressions for the normals to different smooth parts. In all formulas, the normal is of unit length and points towards instability. Stability type

Formula

Workflow for computation

Saddle-node bifurcations

wF λ w ∈ Rn z , F λ ∈ Rn z × Rm

Directly from the formula.

Switching stability limits

wF λ w ∈ Rn z +1 , F λ ∈ Rn z +1 ×Rm

Directly from the formula.

Hopf bifurcations

Operational limits

³

´ ´ α Re w ∂A ∂λ v

αL iz Φλ

³

1. Compute ∂A ∂λ according to Section B.2.1. 2. Compute the normal. 1. Compute Φλ according to Section B.2.1. 2. Compute the normal with the formula.

100


5.5 Iterative method to get the closest point on the stability boundary In the following, an example of how the normal can be used is given. The method developed by Ian Dobson in [25] to get the point λ∗ on a smooth part which is locally closest (in terms of the norm kλ∗ − λ0 k) to a current operating point λ0 is presented. This method uses the normal to the stability boundary, analytical expressions of which were given in the previous sections, and can be found in Table 5.3. The closest point to the stability boundary can be used in addition to the loading margin in other load increase directions (given for example by load forecasts) in order to get information about how far away from the stability boundary the system is. The method relies on the following observation: at λ∗ , the vector λ∗ − λ0 is parallel to the normal vector to the stability boundary n(λ∗ ). Therefore, the method works as follows 1. Find a point λ1 on the stability boundary by solving (5.22) in a given direction of load increase. 2. Compute the normal n 1 (λ1 ) to the stability boundary at this point. 3. Find a new point λ2 on the stability boundary by solving (5.22) in the direction of n 1 (λ1 ). 4. Iterate the process until n i (λi ) = n i −1 (λi −1 ), and set λ∗ = λi .

Ian Dobson showed that if the method converges, then λ∗ is the locally closest point to λ0 . This method works if the closest point on a given smooth part does not lie at an edge with another smooth part, i.e. is not a corner point, see Definition 5.1. In the latter case, the vector λ∗ − λ0 may not be parallel to n(λ∗ ).

5.6 Summary and challenges with larger amounts of variable resources In this section, the stability boundary, consisting of smooth parts corresponding to different types of stability limit, has been presented. It can be thought of as a surface in parameter space. In the scope of this thesis, the use of the stability limits, and thus of the stability boundary, was discussed in Section 1.4: they are needed by system operators to both monitor power systems and take optimal actions if necessary to ensure an adequate level of system security. It has been seen, however, that the stability boundary is not known. Rather, points can be computed on it in different directions in parameter space. As of today, it has been possible to predict relevant directions, and compute the stability limits in these directions, because the main source of variations in parameter space were the loads, and accurate load forecast methods exist. As we saw in Chapter 2, the expected large amounts of variable resources such as wind power in future power systems will introduce new sources of variations. It becomes more relevant to consider the variations in net load, which is the part of the load not covered by these variable resources. As of today, wind power forecasts, and thus net load forecasts, are

5.6. SUMMARY AND CHALLENGES WITH LARGER AMOUNTS OF VARIABLE RESOURCES

101

not as accurate as load forecasts. Hence, it becomes all the more challenging for system operators to choose relevant net load increase directions in parameter space. Figure 5.9 depicts the situation. A two-dimensional stability boundary in a fictitious power system is shown. Figure 5.9a shows a few forecasted realizations of load increase paths in this power system. The variations in forecast are not large. Hence, the load forecast errors are small. Figure 5.9b shows what happens in the same system when large amounts of wind power have been installed. Net loads are considered instead of loads, and the figure shows that since wind power cannot be forecasted as accurately as loads, the expected variations (or forecast errors) are larger. In such a system, identifying relevant directions in which looking for stability limits is challenging.

P2

Forecasted load increase paths.

P2

Forecasted net load increase paths.

Current operating point. P1 (a) Without wind power.

P1 (b) With wind power.

Figure 5.9: Two dimensional stability boundary of a fictitious power system, and forecasted load (system without wind power) and net load (with wind power) increase paths. P 1 and P 2 are the net loads at buses 1 and 2, respectively. Therefore, as the amount of variable resources increases, there is a need for considering more operating situations (such as load increase paths) than before. Because power systems must be operated in a secure way, this entails a need for knowing larger parts of the stability boundary. This could be achieved by computing a larger number of stability limits, in many possible load increase directions. Figure 5.10a shows what is acceptable today: computing a few stability limits on a restricted part of the stability boundary, with help of accurate load forecasts. Figure 5.10b shows what the task would be with larger amounts of variable resources: computing more points on a larger part of the stability boundary. The task of computing stability limits is computationally demanding. For example, Figure 5.4 which showed the stability boundary in the IEEE 9 bus system of Appendix A.2 used 10 000 CPFs, which required several hours on a modern computer. Therefore, computing more stability limits do not seem to be the adequate method to deal with the larger variations coming from variable resources. What is proposed in

102


(a) Getting a few points on the stability boundary.

(b) Computing more stability limits to get a larger number of points on the stability boundary.

Figure 5.10: Illustration of the need for computing more stability limits.

the following is to use local approximations of the different smooth parts of the stability boundary. The computation of local approximations is not time demanding, and they can be used to approximate the actual stability limits. Also, they can be expressed analytically, whereas there is no analytical expression of the actual stability boundary. The fact that they can be expressed analytically opens new possibilities as will be seen in the following chapters.

Chapter 6

Polynomial approximations of the stability boundary

Contents 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11

Review of existing approximations of the stability boundary Second-order approximations of the stability boundary . . . Weingarten maps . . . . . . . . . . . . . . . . . . . . . . . . . . Finding the approximation point on each type of surface . . Contingencies and corrective actions . . . . . . . . . . . . . . Considering margins . . . . . . . . . . . . . . . . . . . . . . . . Note on the parameters . . . . . . . . . . . . . . . . . . . . . . The importance function . . . . . . . . . . . . . . . . . . . . . Comparison with the iterative method . . . . . . . . . . . . . Distance to the second-order approximations . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

103 104 106 111 118 119 119 120 121 121 123

Chapter 5 explained what the stability boundary is: it is a surface in parameter space beyond which the system becomes unstable or outside of the operational limits. In this chapter, we will derive useful second-order approximations of the stability boundary. It corresponds to parts of Publication III.

6.1 Review of existing approximations of the stability boundary Chapter 5 ended with a discussion on the usefulness of approximations of the stability boundary. Before proposing our approximation, we give a review of other approximations which can be found in the literature. In [29] and [48], sensitivities of the loading margin (distance to the stability boundary) to the system parameters for small-signal and voltage stability are given, respectively. The use of the sensitivities can help the system operator take optimal actions to 103

104

CHAPTER 6. POLYNOMIAL APPROXIMATIONS OF THE STABILITY BOUNDARY

either steer the system away from instability or make it stable again. In [18], a formula for a unified sensitivity of the loading margin to changes in system parameters for different types of instabilities is given, and shown to give results that are consistent with the existing sensitivities presented in, for example, [48]. In [70], the stability boundary consisting of thermal limits, voltage stability, transient stability and small-signal stability is approximated by hyperplanes from the inside, so that the approximation is conservative whenever the stability region is strictly convex. Examples are given where the approximations are used for assessing security margins. In [114], first- and second-order approximations of the small-signal stability boundary are presented. The authors of [114] use the implicit function theorem to express the relationship between the parameters on the stability boundary. In [88, 89], the normal to and the curvature tensor of the stability boundary are used to express second-order approximations of the voltage stability boundary, thus giving an intuitive, geometrical expression of the second-order approximations. The approximation of the stability boundary proposed in this chapter is a secondorder approximation. Compared to [70], the higher order of the approximation allows for a better accuracy. While only small-signal stability was dealt with in [114], we develop further the approximations from [88, 89], and take into account voltage, smallsignal and thermal stability. Furthermore, we present a new method that searches for the most important point on the stability boundary at which the approximation can be calculated. Finally, also new in our approach is that the important case of intersections of the different smooth parts of the stability boundary is treated carefully.

6.2 Second-order approximations of the stability boundary 6.2.1 Second-order approximations We recall from Section 3.2 that a second-order approximation of a surface Σ ∈ Rm at λc is given by Γλc (x c ) = λc +C x c + 12 IIλc (x c )n Σ (λc ),

(6.1)

where x c is a displacement away from λc in the tangent hyperplane, C is a basis of the tangent plane Tλc Σ of Σ at λc and IIλc is the second fundamental form of Σ at λc , ® IIλc (x c ) = − dNλc (x c ), x c = −x cT dNλc x c . (6.2)

The map dNλc : Tλc Σ → Tλc Σ is the Weingarten at λc , which is the differential of the Gauss map N : Σ → Sm−1 where Sm−1 is the unit sphere in Rm , i.e. the map that takes the point λc ∈ Σ to the normal vector n Σ (λc ) ∈ Sm−1 , of Σ at λc as defined in Section 3.2.3. The Weingarten map measures how much the normal vector changes, thus giving a measure of the curvature of the surface. In order to approximate the stability boundary Σ, we propose to use the secondorder Taylor expansion described by (6.1). In order to compute these expansions, we need

6.2. SECOND-ORDER APPROXIMATIONS OF THE STABILITY BOUNDARY

105

1. a point λc at which the approximation is calculated, denoted the approximation point in the following, 2. the normal n Σ (λc ) of Σ at this point, 3. the Weingarten map dNλc at this point. It was observed in Section 5.3.4 that for one given state of the system (pre- or post contingency state), the stability boundary is in general not smooth, but consists of different smooth parts, corresponding to different types of loadability limit, such as saddle-node bifurcations, Hopf bifurcations, switching loadability limits or operational limits. In the following, the stability boundary or entire surface always refers to the entire stability boundary. If nothing else is mentioned, a surface denotes one of the smooth parts, so that the following statements are equivalents: – The stability boundary consists of different surfaces. – The entire surface consists of different smooth parts. The reason for not keeping to strict notations such as stability boundary or smooth parts is that it becomes natural to talk about surfaces when using differential geometry.

6.2.2 Dealing with different smooth parts Since we want to get an approximation of the entire stability boundary and that the latter consists of different surfaces, we need to compute second-order approximations for each of the surfaces that form the stability boundary. Furthermore, we need to know for each of these second-order approximations when it is the valid one, i.e. in which region we can use this specific one to approximate the stability boundary. Hence, as we talk about the entire surface as opposed to the smooth parts constituting this entire surface, we will talk about the overall approximation consisting of a collection of secondorder approximations (each one corresponding to one smooth part) limited to a validity domain. The second-order approximation of the stability boundary therefore refers in the following to the overall approximation. We suppose in the following that the stability boundary consists of n a smooth parts. To summarize, for each stability boundary of interest (i.e. for each pre- or post-contingency state of the system of interest), we need the following to approximate it: 1. an approximation point for each smooth part, 2. the normal and the Weingarten map to each smooth part at the approximation points, 3. validity domains for the second-order approximations of the smooth parts. Expressions of the normal to the different types of smooth parts were given in Table 5.3 in Section 5.4. In the following, we will therefore give the expressions of Weingarten maps of different types of smooth parts using the already obtained expression for the normals. Then, we will propose a method to find a suitable approximation point, and will explain how to choose the right second-order approximations to approximate the stability boundary.

106


6.3 Weingarten maps 6.3.1 Derivative of the normal in different coordinate system Recall from (3.28) that the second fundamental form of Σ used in the second order approximation is a function II : Tλc Σ → R defined for a displacement x c in the tangent plane as II(x c ) = −x cT dN x c ,

(6.3)

where dN is the differentiation of the Gauss map of Σ in the coordinate system of the tangent plane to Σ at λc . In the following, these coordinates will be denoted the local coordinates. Hence, if the derivative of the normal N is calculated in the original coordinate system (usually Rm equipped with the canonical basis {e 1 , . . . , e m }, where e j has its j -th element equal to 1 and all the others equal to zero), we need to project it on the tangent plane by using dN = C T

dN C. dλ

(6.4)

Note that the inverse cannot be done, i.e. we cannot retrieve dN dλ from dN . The dimensions of the matrices are as follows: – dN is of dimension (m − 1) × (m − 1). – dN dλ is of dimension m × m. – C is of dimension m × (m − 1) and C T C = I m−1 , the identity matrix. The expression (6.4) will be used for the cases of Hopf bifurcations and operational limit.

6.3.2 Computing the Weingarten map for SNB smooth parts Recall the expression of the normal to a SNB surface from (5.44): N = n SNB (λ0 ) = wF λ (z 0 , λ0 ),

(6.5)

where F are the equations of the power system model, w is a left row eigenvector corresponding to the zero eigenvalue of F z (z 0 , λ0 ) and normalized so that the unit normal points in the direction away from the stability domain. Recall also that z can be written as a function of λ, z = Φ(λ), see Section 5.4.5. Note that the normal does not exist for any variation in the direction of the normal. Recall that the Weingarten map is defined as the function dN : Tλ ΣSNB → Tλ ΣSNB which gives the derivative of the normal in any direction of the tangent plane. In the following, the local coordinates, denoted λ∗ , are the coordinates on the tangent plane. Differentiation with respect to the local coordinates is given by ∂ ∂ = C. ∂λ∗ ∂λ

(6.6)

107

6.3. WEINGARTEN MAPS Differentiation of the Gauss map with respect to local coordinates gives d dN = C T Nλ∗ = C T (wF λ )T dλ∗ ! Ã T ∂F λT ∂F λT T T T T dw + Φλ∗ w + w = C Fλ dλ∗ ∂z ∂λ∗ = F λT∗

T ∂F T dw T ∂F λ∗ + Φλ∗ w T +C T λ C w T . dλ∗ ∂z ∂λ

(6.7)

The system equations are often linear in λ (this holds if, for example, λ is chosen to be the loading or generators’ active power output). Therefore, in these case, F λλ = 0, and the last term in (6.7) disappears. Now, using (B.9), we get dw F z Φλ∗ + wF λ∗ z Φλ∗ . dλ∗

(6.8)

dw F z + wF zz Φλ∗ + wF zλ∗ = 0. dλ∗

(6.9)

dN = wF zz (Φλ∗ , Φλ∗ ) + wF zλ∗ Φλ∗ + wF λ∗ z Φλ∗ .

(6.10)

dN = −

Here, the transpose sign has been dropped for simplicity. The dimensions along which the products are made can be determined from the derivation above. To get the derivative of the left eigenvector with respect to λ∗ , we differentiate wF z = 0 (w is an eigenvector associated with the zero eigenvalue of F z ) with respect to λ∗ :

Hence, (6.8) becomes

The notation wF zz (Φλ∗ , Φλ∗ ) indicates that the product wF zz is made along the first dimension of F zz , and that the two products in F zz (Φλ∗ , Φλ∗ ) are made along the second and third dimensions of F zz , respectively. It is convenient to write this in tensor form 1 : dN ji = w k F zkl z p Φl i Φ λ∗

p j

λ∗

+ w k F k j l Φl i + w k F ki λ∗ z

λ∗

λ∗ z l

Φl j . λ∗

(6.11)

It appears then clearly that dN ji is a (m − 1) × (m − 1) symmetric matrix. Often, the parameters are taken as the loads, and the system equations in F are linear in the loads, which appear only in the load flow equations, see for example Example 5.1. Other common parameters are the generators’ mechanical power, P mi in Example 5.1. The system equations are also linear in P mi . Thus, in these cases, F λ∗ z = 0 and F zλ∗ = 0, so that the expression of the Weingarten map is dN = wF zz (Φλ∗ , Φλ∗ ).

(6.12)

The quantity Φλ∗ can be computed from Φλ in (B.19) (Appendix B) by noting that Φλ∗ = ΦλC ,

(6.13)

since the local coordinates correspond to a projection from the original coordinate system onto the tangent plane. 1. The summation is done over repeated indices

108


6.3.3 Computing the Weingarten map for SLL smooth parts The expression of the normal for smooth parts corresponding to SLL is the same as for SNB, except that the quantities involved are of different dimensions, see Table 5.3. Hence, the Weingarten map for SLL can also be expressed by dN = wF zz (Φλ∗ , Φλ∗ ) + wF zλ∗ Φλ∗ + wF λ∗ z Φλ∗ ,

(6.14)

dN = wF zz (Φλ∗ , Φλ∗ ),

(6.15)

and by

in the cases where loads or generators’ mechanical power are chosen as parameters, since the system equations are then linear in the parameters so that F zλ∗ = 0 and F λ∗ z = 0. The quantity Φλ∗ , however, is computed differently from SNB. Since F z is nonsingular for non SNB smooth parts, Φλ∗ can be computed directly by using (B.10) and the relationship between Φλ∗ and Φλ in (6.13).

6.3.4 Computing the Weingarten map for Hopf smooth parts Recall the analytical expression of the normal to Hopf smooth parts from (5.56) µ ¶ ∂A HB N = n (λ0 ) = α Re w v , (6.16) ∂λ where µλ = w ∂A ∂λ v is the partial derivative of the eigenvalue µ with respect to λ, with w and v being left and right eigenvectors corresponding to the purely imaginary eigenvalue j ω, ω > 0, and normalized so that w v = 1 and |v| = 1. The derivative of the normal in the original coordinate system is then ¡ ¢ ¡ ¢ dN = α Re µλλ + Re µλ αλ . dλ

(6.17)

Before going further in the derivation, we make the following remark. Remark 6.1 The Weingarten map dN is of interest, and, as discussed in Section 6.3.1, it can be obtained from dN dλ : dN = C T

dN C, dλ

(6.18)

where C = [c 1 , . . . , c m−1 ] with {c 1 , . . . , c m−1 } a basis of the tangent plane. Hence, in the case of Hopf bifurcation, we get ¡ ¢ ¡ ¢ dN = αC T Re µλλ C +C T Re µλ αλC (6.19) ¡ ¢ 1 = αC T Re µλλ C + C T N αλC , (6.20) α

109

6.3. WEINGARTEN MAPS

¡ ¢ since the normal is N = Re µλ . Both terms on the right-hand side are (m − 1) × (m − 1) matrices. Looking more carefully at the second term, we see that ¤ £ (6.21) C T N αλC = C T N αλ1 . . . αλm C ¤ £ T T (6.22) = C N αλ1 . . . C N αλm C , where

αλ j =

∂α , ∂λ j

∀ j = 1, . . . , m.

(6.23)

Since the columns of C are the vectors in the basis of the tangent plane and N is the normal, C iT N = 0, ∀i = 1, . . . , m − 1, from which it follows that C T N αλ j C = 0 ∀ j = 1, . . . , m. Hence, when computing the Weingarten map dN , the second term on the right-hand side in (6.20) is zero. Therefore, in the following, we will only consider the first term in the expression of (6.17), corresponding to the derivative of α with respect to λ. Note that, in (6.17), the second term is not zero but we disregard it because the quantity we are interested in is dN and not in dN /dλ. In (6.17), we need an expression of the second derivative of the eigenvalue µ with respect to the parameters λ. We consider again (5.53): ¶ µ ∂A − µλ I v + (A − µI )v λ = 0, (6.24) ∂λ which was used to compute the first derivative of µ, and we differentiate this equation with respect to λ: ¶ µ ¶ µ 2 ∂2 µ ∂A ∂µ ∂v ∂ A − i j I v+ − iI i j i ∂λ ∂λ ∂λ ∂λ ∂λ ∂λ ∂λ j (6.25) µ ¶ ∂A ∂v + − D µI + (A − µI )D v = 0, j i j λ λ λ ∂λ j ∂λi where the first derivation with respect to λ is denoted by the subscript i and the second by the subscript j . Pre-multiplying by w, and recalling that w v = 1 gives ∂2 µ ∂λi ∂λ j

=w

µ

∂2 A ∂λi ∂λ j 2

¶

v +w

µ

∂A ∂λi

−

∂µ ∂λi

I

¶

∂v ∂λ j

+w

µ

∂A ∂λ j

− D λ j µI

¶

∂v ∂λi

.

(6.26)

A ∂v All quantities but ∂λ∂i ∂λ j and ∂λ j have been obtained before. Therefore, in order to compute the second-order derivative of the eigenvalues, we need the first-order derivative of the right-eigenvector v and the second-order derivatives of A with respect to λ. Formulas for the latter are given in Section B.3. The former can be obtained by considering (5.53) and rearranging the terms so that µ ¶ ∂A − µλ I v. (6.27) (A − µI )v λ = − ∂λ

110


Since µ is an eigenvalue of A, the matrix A−µI is singular, so that we cannot pre-multiply ¡ ¢−1 by A − µI to get v λ . However, it was proven by Mike S. Jankovic in [57] that the righthand side is in fact in the range of (A − µI ), so that the equation has a solution. Several methods can be used to solve the equation for v λ . Two of them are presented by John E. Condren and Thomas W. Gedra in [22]. The authors then used the obtained derivatives of eigenvalues of A to solve an optimal-power flow with small-signal stability constraints of the type Re(µ) ≤ µmax , for some µmax > 0. Here, we will use a third method, proposed in [79] and repeated in [45]. We write the derivative of the eigenvector in the following way: ∂v = r i + γi v, ∂λi

(6.28)

for some vector r i and some scalar γi , and we let M = A − µI , so that (6.27) can be rewritten ¶ µ ¡ ¢ ∂A (6.29) M r i + γi v = − − µλi I v. ∂λi Since M v = (A − µI )v = 0, this becomes with

M r i = bi ,

(6.30)

µ

(6.31)

¶ ∂A ∂µ bi = − − I v. ∂λi ∂λi

Let e the index of the element with highest magnitude in v, and let the e-th element of r i be equal to zero. The other elements of r i can then be computed by solving     [M ]11 0 [M ]13 [b i ]1  0 1 0  ri =  0  , (6.32) [M ]31 0 [M ]33 [b i ]3

where [M ]i j are the submatrices of M when omitting its e-th row and e-th column, and [b i ]1 and [b i ]2 are subvectors of b i corresponding to its first e − 1 elements and its last n x − e + 1 elements, respectively (M is an n x × n x matrix). To obtain γi , we differentiate the normalization constraint |v| = v ∗ v = 1 with respect to λi : 0 = (v ∗ v λi ) + (v ∗ v λi )∗ = v ∗ r i + γi + (v ∗ r i + γi )∗ . Since γi ∈ R, we get

¡ ¢ γi = − Re v ∗ r i .

(6.33)

(6.34)

Hence, the first derivative of the right eigenvector v λi can be obtained from (6.28), using (6.32) and (6.34). The second derivative of the eigenvalue µλλ can then be obtained using (6.26).

6.4. FINDING THE APPROXIMATION POINT ON EACH TYPE OF SURFACE

111

6.3.5 Computing the Weingarten map for OL smooth parts Recall that the normal to a OL smooth part was given by N = n OL = αL iz Φλ ,

(6.35)

where α is a normalizing factor chosen so that the normal points in the direction away from the stability domain. Hence, differentiating the normal in the original coordinate system, we get ³ ´ dN = α L izz (Φλ , Φλ ) + L iz Φλλ + αλ L iz Φλ . dλ

(6.36)

Using the same argument as in Remark 6.1 for Hopf bifurcation, the second term corresponding to αλ disappears when computing dN from (6.4). Using the tensor notation 2 , the first term on the right-hand side can be written: µ

dN j dλk

¶

1

³ ´ p = α L iz l z p (Φlλ j , Φ k ) + L iz l Φlλ j λk . λ

(6.37)

The quantities Φλ and Φλλ can be obtained from (B.12), (B.26) and (B.27). The Weingarten map for OL smooth parts is then computed by dN = C T

µ

dN dλ

¶

C.

(6.38)

1

6.3.6 Computing Weingarten maps: Summary All formulas for computing dN can be found in Table 6.1. The third column explains in detail which equations should be used to compute the expression of dN . Note that, for Hopf bifurcations and operational limits, the Weingarten map is computed from dN /dλ using (6.4).

6.4 Finding the approximation point on each type of surface The second-order approximations defined in (6.1) are computed around a given point λc on the original stability boundary. Let Σi , i = 1, . . . , n a be all smooth parts of the stability boundary. We want to get a second-order approximation of each Σi , and thus, we need to find an approximation point λic , called in the following the most important point, on each Σi . A question that arises naturally is how to choose the points λic . In order to get the most accurate approximation of the distance, the most important points λic will be chosen as being the points on the smooth parts of the stability boundary that maximize a given importance function ρ for each Σi . Examples of importance functions are given in Section 6.8. 2. The summation is done over repeated indices.

112


Table 6.1: Analytical expressions for the derivatives dN . Stability limit

Formula for dN

Saddle-node bifurcations

Switching loadability limits

Hopf tions

bifurca-

Operational limits

Workflow for computation 1. Compute Φλ according to Section B.2.1. 2. Compute Φλ∗ according to (6.13). 3. Compute dN with the formula in the second column.

wF zz (Φλ∗ , Φλ∗ )

¡

wF zz (Φλ∗ , Φλ∗ )

¢

¡ ¢ αC T Re µλλ C

¡ ¢ αC T L izz (Φλ , Φλ ) + L iz Φλλ C

1. Compute Φλ according to Section B.2.1. 2. Compute Φλ∗ according to (6.13). 3. Compute dN with the formula in the second column.

1. Compute ∂A ∂λ according to Section B.2.1. dv 2. Compute dλ according to (6.28). d2 A 3. Compute dλdλ according to Section B.3.1. 4. Compute µλλ with (6.26). 5. Compute dN with the formula in the second column. 1. Compute Φλ according to Section B.2.1. 2. Compute Φλλ with (B.26) and (B.27). 3. Compute dN with the formula in the second column.

In the following, we consider one smooth part Σi , and drop the superscript i for the corresponding most important point λic . For one given smooth part Σi , λc is the


113

solution to the following optimization problem: max ρ(λ) λ

s.t λ ∈ Σi .

(6.39a) (6.39b)

In [85], this was solved in the case of an SNB surface by an iterative method. Starting from a point on the surface, λ0 ∈ Σi , the gradient of the measure ρ at this point can be projected on the tangent hyperplane: r=

m−1 X i =1

® c i , ρ λ (λ0 ) c i ,

(6.40)

where {c 1 , . . . , c m−1 } is the basis of the tangent hyperplane to Σi at λ0 . This defines a direction of search in the tangent hyperplane. Then a step is taken to get a new point p λ1 = λ0 + δr in the tangent hyperplane, where δ can be chosen to tune the length of p the step. The new point λ1 does not belong anymore to the surface Σi , and therefore needs to be corrected. In [85], the correction step is done by running a continuation p power flow in the direction of λ1 , but starting from x = x 0 , where x 0 are the values of the state variables at λ0 . The method is then iterated until the norm of the projection of the gradient becomes sufficiently small. It can therefore be seen as a predictor-corrector method. In [51], another predictor-corrector method was given to explore a SNB surface, where the prediction direction is chosen arbitrarily in the tangent hyperplane, and the corrector step is a Newton-Raphson method using equations describing the SNB surface to project the predicted point onto the surface orthogonally to the tangent hyperplane. Here, we will use the prediction step from [85], and a correction step similar to the one in [51] to find λc . Starting from any point λ0 on Σi , we use (6.40) to go in the direction defined by the projection r of the negative gradient of ρ in the tangent hyperplane. p This gives a predicted point in this direction, λ1 . We now assume that the smooth part Σi can be characterized by Ψ(z, λ, r ) = 0,

(6.41)

where r ∈ Rt is a vector of t additional variables necessary to characterize Σi . Such characterizations were given in Section 5.3.4. In order to project the predicted point to Σi orthogonally to the tangent hyperplane, we must have Ψ(z, λ, v) = 0, p π(λ, λ1 ) = 0,

(6.42a) (6.42b)

where the second set of equations ensures that the direction of projection will be orthogonal to the tangent hyperplane, that is ¡ p p¢ πi (λ, λ1 ) = c iT λ − λ1 = 0, i = 1, . . . , m − 1, (6.43)

114

CHAPTER 6. POLYNOMIAL APPROXIMATIONS OF THE STABILITY BOUNDARY p

where {c 1 , . . . , c m−1 } is the basis of the tangent hyperplane of Σi at λ0 . Note that π(λ, λ1 ) = 0 is a set of m −1 equations. Since z ∈ Rn z , λ ∈ Rm and v ∈ Rt , Ψ(z, λ, v) = 0 must be a set of n z + t + 1 equations for the system to have a unique solution (z, λ, v). The corrector step is very similar the illustrative example 3.2 in Section 3.1.4. The solution λ1 to (6.42) can be found by using, for example, the Newton-Raphson method, and will lie on Σi . The predictor and corrector steps can be run again from p this point to get a series of predictions and corrections λ j and λ j . This is illustrated in Fig. 6.1. p

λ j +1

λj Σi

λ j +1

Figure 6.1: One predictor-correction step towards the most important point.

Note that a basis of the tangent hyperplane at the corrected points λ j must be computed in each step, which can be done using the Gram-Schmidt procedure as explained in Example 3.4. The method stops when the gain by moving towards the projection of the negative gradient onto the tangent hyperplane becomes sufficiently small. For example, we can stop when after a certain step k k

Pm−1 i =1

® c i , ρ λ (λk ) c i k kρ λ k

< ε,

(6.44)

for some small ε. We then choose λk as the most important point λc . If the system of equations (6.42) is solved by a Newton-Raphson method, the partial derivatives of Ψ and π with respect to z, λ and v will be needed. From (6.43), we get πz = 0,

πv = 0,

T

πλ = C ,

(6.45) (6.46) (6.47)

where C = [c 1 . . . c m−1 ] is the matrix with the vectors in the basis of the tangent hyperplane of Σi at λ j −1 for step j . Next, we will give the expression of the derivatives of Ψ for the four possible surfaces (SNB, SLL, Hopf and OL).


115

6.4.1 Derivatives of Ψ for SNB surfaces The expression of Ψ for smooth parts corresponding to SNB was given in (5.27). The partial derivatives of Ψ with respect to z, λ and v are 

Fz J Ψ = F zz v 0

Fλ 0 0

 0 Fz  . 2v T

(6.48)

6.4.2 Derivatives of Ψ for SLL surfaces The expression of Ψ for smooth parts corresponding to SLL was given in (5.29). The partial derivatives of Ψ with respect to z and λ are (the vector r of additional variables was empty in this case) ¡ JΨ = Fz

¢ Fλ .

(6.49)

Note that the rank of F z is one less than the number of rows [88].

6.4.3 Derivatives of Ψ for Hopf surfaces The expression of Ψ for smooth parts corresponding to Hopf bifurcations was given in (5.31). The partial derivatives of Ψ with respect to z and λ are (the vector r of additional variables was empty in this case) µ

F¡z ¢ JΨ = Re µz

¶ F¡λ ¢ . Re µλ

(6.50)

The total derivative of µ with respect of λ was given in (5.54). In the Newton-Raphson method, we need the partial derivatives of µ with respect to λ and z. Similarly as in (5.54), these partial derivatives are expressed as ∂A ∂µ =w v, ∂λ ∂λ ∂µ ∂A =w v, ∂z ∂z

(6.51) (6.52)

where w and v are left and right eigenvectors of A corresponding to the eigenvalue µ, normalized so that w v = 1 (w is a row vector). However, we will consider that A is a function of λ only, so that its partial derivative with respect to λ is equal to its total derivative with respect to λ, and its partial derivative with respect to z is zero. Our results indicate that this leads to a good convergence rate in the Newton-Raphson method. The Jacobian in (6.50) then becomes JΨ =

µ Fz 0

¶ F¡λ ¢ . Re µλ

(6.53)

116


6.4.4 Derivatives of Ψ for OL surfaces The expression of Ψ for smooth parts corresponding to Hopf bifurcations was given in (5.32). The partial derivatives of Ψ with respect to z and λ are (the vector r of additional variables was empty in this case) JΨ =

µ

Fz L iz

Fλ L iλ .

¶

(6.54)

6.4.5 Handling CPs The most important point on the entire stability boundary Σ will be situated on one smooth part Σk of Σ. When looking for the most important point on another smooth part Σi of Σ, the search will therefore often be directed towards Σk , and it will be likely to encounter another part Σ j on its way to Σk . Let W = Σi ∩ Σ j be the set of corner points corresponding to the intersection of the two smooth parts. The search for the most important point on Σi should now continue along W . Hence, the algorithm above should be modified to find a basis of the tangent hyperplane of W at the point of intersection, and project the negative gradient onto W . This is shown in Fig. 6.2. The most important point on Σi will therefore be situated at a corner point (corresponding to the intersection of Σi and Σp with Σp possibly being a different smooth part than Σ j ).

Σj

←W p

λk λk−1 Σi

λk λk+1

0

λk

Figure 6.2: Continuing the search for the most important point on Σi along the intersection W of two smooth parts Σi and Σ j of the stability boundary: Starting from λk−1 , 0 p the predicted point is λk from which the correction step gives λk which is beyond Σ j so the search continues along the set W of CPs with λk and λk+1 .


117

The tangent hyperplane of W is a manifold of codimension 2, and its basis {c 1 , . . . , c m−2 } can be found by noting that all vectors c i must be orthogonal to both the normal to Σi and the normal to Σ j . The basis can thus be obtained by starting the Gram-Schmidt process with these two normals. The special case of SNB-SLL intersections where the intersection is tangential [87] is described in the next section. For the correction stage, p we characterize a set of CPs by Ψ1 and ψ2 , and project the predicted point λ j upon the set of CPs orthogonally to the tangent hyperplane whose basis is {c 1 , . . . , c m−2 }: Ψ1 (z, λ, r ) = 0,

ψ2 (z, λ, r ) = 0, ³ ´ p c iT λ − λ j = 0,

(6.55) (6.56) i = 1, . . . , m − 2,

(6.57)

where Ψ1 is the characterization Ψ of the first surface as derived before, and ψ2 are additional equations needed to characterize the second surface which meets the first one at the CPs. Note that F is included in Ψ1 , and therefore, ψ2 contains all equations of Ψ for the second surface except the ones already included in F in Ψ1 . When the most important point of one smooth part Σi situated at a corner point with another smooth part Σ j has been found, the search is continued on Σ j to find the most important point on this second smooth part.

6.4.6 SNB-SLL intersections As was discussed in Section 5.3.5, corner points at the intersection between SNB and SLL surfaces need special attention, because the intersection is tangential. These points were called tangential intersection points. Figure 6.3 shows a two-dimensional parameter space where a SNB surface and a SLL surface intersect in a TIP.

5.8

TIP

nH

λ2

SLL 5.6

5.4

1.6

SNB

H

1.8

2 λ1

2.2

2.4

Figure 6.3: The tangential intersection of a SNB and a SLL surface [88].

118


To the left of the TIP the set of breaking points is a SLL surface, while on the right hand side they are harmless breaking points, which do not lead to instability when encountered. Let G be the surface that contains the TIPs and all points lying in the direction of the normal vector to the two surfaces from the TIPs 3 . Now by the way our second order approximations are produced there is a point λc that is a TIP where one of the surfaces has a basis for its approximation. Let H be the approximation of G obtained through the second order approximations of the SNB and SLL surfaces at λc . The surface H is called the dividing surface in the following. Starting from λc , any point lying in the direction (positive and negative) of the normal vector to the surfaces at λc is in H . Hence, the normal, n H , to H at λc is in Tλc Σ. Now, the set of TIPs is a codimension two manifold, and the breaking point is always on the inside of the SNB surface (either on the upper or the lower part of the P-V curve), see Section 5.3.5. Hence, the curvature of the SNB and the SLL surface at a point of the CP must be the same in all directions but one. This particular exceptional direction is the normal n H to H . This means that H is a codimension one hyperplane passing through λc , with normal parallel to the eigenvector corresponding to the only nonzero eigenvalue of C (dNSNB − dNSLL )C > , where C is the matrix with basis vectors for the tangent plane and dNSNB and dNSLL are the Weingarten maps for the SNB and the SLL smooth parts, respectively. Starting the Gram-Schmidt process with the normal to the two surfaces and n H gives a basis of the tangent plane of the set of TIP. Further detail about the intersections between SNB and SLL smooth parts can be found in [84].

6.5 Contingencies and corrective actions As discussed in Section 5.3.6, when contingencies occur the stability boundary changes. Hence, in contingency analysis, second-order approximations need to be computed for each smooth part of the pre-contingency stability boundary as well as for each smooth part of all post-contingency stability boundaries. Depending on the type of simulations of interest, corrective actions can be taken after contingencies. Certain corrective actions will change the stability boundary. In order to take these corrective actions in the post-contingency stability boundary, we rewrite (5.20) as ˜ y, ˜ λ, p(λ)), x˙˜ = f (x, ˜ y, ˜ λ, p(λ)), 0 = g (x,

(6.58) (6.59)

where p is a vector of parameters depending on λ. The vector p can be used to model automatic corrective actions, for example compensation switching [106]. The vectors x˜ 3. The two surfaces have the same normal at the TIP since they meet tangentially

6.6. CONSIDERING MARGINS

119

and y˜ can have different dimensions than the original vectors x and y, depending on the dynamics of the corrective actions.

6.6 Considering margins Margins to stability limits can be considered when computing the second-order approximations in the following way for the different smooth parts: SNB For SNB points, stability limits are defined as the point at which the system Jacobian J has a zero eigenvalue, or equivalently, at which the dynamic Jacobian A has a zero eigenvalue, see Remark 5.3. Let µ be this eigenvalue for A. The boundary corresponding to µ = −ε < 0 can be considered instead of the stability boundary with µ = 0. Note that if we use the eigenvalue µ J of the system ¯ ¯ Jacobian J , then the margin must be applied on the absolute value instead ¯µ J ¯ = ε > 0.

Hopf For Hopf bifurcations, as for SNB, the boundary corresponding to Re(µ) = −ε < 0 can be considered instead of the stability boundary Re(µ) = 0, where µ is the eigenvalue that becomes purely imaginary at the Hopf bifurcation point. OL For an OL characterized by L i (see (5.33)), the boundary corresponding to L i = ε > 0 can be considered instead of the stability boundary defined by L i = 0. When computing the second-order approximations, the characterizations of each smooth part given in Section 5.3.4 can be changed according to the margins defined above.

6.7 Note on the parameters Recall from Section 5.3.2 that the parameter space can be partitioned into load ζ £ ¤T and other parameters u: λ = u T ζT with u ∈ U and ζ ∈ Rl . In the following, ζ will also include other system parameters which are set exogenously. These parameters will be called the stochastic system parameters, and can for example include, in addition to the loads, wind power production. A typical choice of u can be the output of the generators whose production can be changed by the system operators, that is, the generators which have submitted bids in order to participate in tertiary control, see Section 2.3. When computing second-order approximations, the approximation point is found in the load space for a given u c (for a given initial dispatch of the generators), but the second-order approximations are computed in the parameter space (including u). The entire stability boundary consists of several smooth parts. It can happen that different smooth parts constitute the stability boundary for different u 6= u c . One possibility to tackle this problem is to compute several second-order approximations, for different values of u. Then, when using the second-order approximations, the one corresponding to the closest u is used. For a specific example of this, the reader is referred to Chapter 9.

120


6.8 The importance function The problem of searching for the most important points on smooth parts around which the second-order approximations are computed was formulated in (6.39). A given “importance function” ρ is maximized. Since second-order approximations are local approximations, the closer the parameters are to the approximation points the more accurate the approximations are. The choice of the function ρ will therefore have an impact on the accuracy of the approximations, and must be chosen according to how the approximations are used. We will now discuss this through two examples. Example 6.1 (Euclidean distance) Let ρ be the negative of the Euclidean distance to a given point in parameter space. This point can, for example, be the present operating point λ0 , and the importance function is then q (6.60) ρ(λ) = − kλ − λ0 k = − (λ − λ0 )T (λ − λ0 ). In order to solve problem (6.39), the derivative of ρ with respect to λ is needed. It is given by ρ λ (λ) = − q

(λ − λ0 ) (λ − λ0 )T (λ − λ0 )

.

(6.61)

The most important point will then be the point closest to λ0 under the Euclidean distance. It gives an indication of how far the present operating point is from the stability boundary. Example 6.2 (Normal distribution) Suppose now that the stochastic system parameters are expected (according to forecasts) to be normally distributed N (λ0 , Λ). The probability distribution function is then 4 µ ¶ 1 1 T −1 f Z (λ) = exp − (λ − λ ) Λ (λ − λ ) . (6.62) 0 0 1 m 2 (2π) 2 (det Λ) 2 In [85], for such distributions, the importance function was chosen to be q¡ ¢ ρ(λ) = − λ − λ0 )T Λ−1 (λ − λ0 ) .

(6.63)

This choice was done because the level surfaces of ρ corresponds to level surfaces of f Z . The derivative of the importance function with respect to λ is Λ−1 (λ − λ0 ) ρ λ (λ) = − q¡ ¢. λ − λ0 )T Λ−1 (λ − λ0 )

4. Here, only the stochastic system parameters are considered, i.e. λ = ζ.

(6.64)

6.9. COMPARISON WITH THE ITERATIVE METHOD

121

The most important point will then be the most likely point on the stability boundary, “likely” being understood as measured by f Z . Note that, if the stochastic system parameters are independent, the covariance matrix Λ is diagonal with the variance σ2i > 0 of each parameter ζi on the diagonal. Its inverse Λ−1 is also diagonal, The importance function then becomes v um uX 1 ¡ ¢2 ρ = −t λi − λi0 , (6.65) 2 i =1 σi which is similar to the Euclidean distance, but where all elements have been weighted by the inverse of the variance.

6.9 Comparison with the iterative method In Section 5.5, an existing iterative method to compute the distance, from a given point in parameter space, to the closest point on the stability boundary was presented. This closest point was found by iteratively computing and following the normal to the stability boundary. Compared to our method, the iterative method suffers from two main drawbacks: 1. The distance is measured by the Euclidean norm, which is not always the most adequate measure. As was seen in Example 6.2, probability distributions can be a better measure for some applications. 2. The iterative method finds the closest point on the entire stability boundary, but does not account for the different smooth parts. For most smooth parts, the most important point lies on the edge between two smooth parts as discussed in Section 6.4.5, a situation which the iterative method does not deal with.

6.10 Distance to the second-order approximations In the following chapters, we will need a distance function which computes the signed distance from any point in parameter space to the second-order approximation for one specific smooth part of pre- or post-contingency stability boundaries. We will denote by Σia , i = 1, . . . , n c the second-order approximations of the stability boundary, in parameter space, corresponding to the system after contingency i has occurred. The second-order approximation of the stability boundary of the pre-contingency system will be denoted by Σ0a . Recall from Section 6.2.2 that each Σia consists in fact of a set of second-order approximations Σiaj , j = 1, . . . , n ai since the actual stability boundary consists itself of different smooth parts corresponding to different types of loadability limits. Let J i be the set of indices which index the smooth parts of Σia so that Σia consists of the set of second-order approximations Σiaj for j ∈ J i . Recall from (6.1) that each Σiaj

122


can be analytically expressed as 1 ij Γi j (x c ) = λc +C i j x c + IIi j (x c )n i j , 2

(6.66)

where all quantities are as defined in (6.1) but now depends on the considered smooth part Σiaj . A distance d i j : U × Rl → R can now be defined as ³ ³¡ ¢ ³ ´´ ´ ij T d i j (λ) = Γi j C i j λ − λc − λ · n i j ´´ ³¡ ¢ ³ 1 ij ij T λ − λc . (6.67) = (λc − λ) · n i j + IIi j C i j 2 £ ¤T This function gives the signed distance from any point λ = u T ζT ∈ Rl ×U to Σiaj in the direction of the normal to this smooth part at the approximation point. The term ³ ´T ³ ´ ij Ci j λ − λc

(6.68)

³¡ ¢ ³ ´´ 1 ij T IIi j C i j λ − λc , 2

(6.69)

projects the point λ onto the tangent plane of Σiaj . The term

is the distance between the tangent plane and the second-order approximation. The distance d i j (λ) is negative when the considered point λ is beyond Σiaj . Figure 6.4 depicts how the distance from a point λ to a second-order approximation Σiaj is calculated. Note that the arrows are all vectors parallel to the normal n i j . ni j

³ ´ ij C iTj λc − λ

ij

Σiaj

λc

1 2 IIi j

³ ³ ´´ ij C iTj λc − λ n i j ³³ ´ ´ ij λc − λ · n i j n i j

d i j (λ)n i j

λ Figure 6.4: Computation of the distance to second-order approximations.

6.11. SUMMARY

123

6.11 Summary Figure 6.5 shows how, for pre- and post-contingency states of the system, secondorder approximations of each smooth part of the entire surface are calculated. Given one state of the system, continuation power flows are run in different directions to find points on the stability boundary (i.e. stability limits). The stability limits will often lie on different smooth parts. From each of these stability limits, the most important point on the corresponding smooth part is sought. Second-order approximations are then computed around these most important points. If a most important point is a corner point, i.e. lies on the intersection with another smooth part, the process is repeated for this other smooth part. In this way, less initial directions of search are needed 5 .

5. The number of initial directions is chosen arbitrarily. Note that it is not guaranteed that as many smooth parts as possible will be found. A trade-off must therefore be made between ensuring finding all smooth parts (by increasing the number of initial directions) and reducing the computation time (by decreasing the number of initial directions).

124


Start: pre-contingency system, list of contingencies, list of stress directions.

Computation of the second-order approximations for a specific pre- or postcontingency system.

Initial direction of stress from the operating point. CPF to find the loadability limit in this direction. Apply new contingency to the pre-contingency system.

Computation of the second-order approximation at this point.

Continue on the new surface.

Point on an intersection?

Yes

No Yes

Change direction of stress from the operating point.

Search for the most important point of this part of the surface.

Yes

More directions of stress to explore?

No

More contingencies to be considered?

No End.

Figure 6.5: Overview of the method to compute second-order approximations of all boundary surfaces of interest.

Chapter 7

Second-order approximations: case study in the IEEE 9 bus system

Contents 7.1

Setup of the case study . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.2

Objectives with the case study . . . . . . . . . . . . . . . . . . . . . . . 127

7.3

Case 1: Illustration of the method . . . . . . . . . . . . . . . . . . . . . 129

7.4

Case 2: Accuracy of the second-order approximations . . . . . . . . . 130

7.5

Computational issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

In this chapter, second-order approximations are computed in the pre-contingency IEEE 9 bus system. The accuracy of the approximations is assessed by computing the distance between the approximations and the actual stability boundary. It corresponds to parts of Publication III.

7.1 Setup of the case study The IEEE 9 bus system from Appendix A.2 is considered. The system has three generators at buses 1, 2 and 3, and three loads at buses 5, 6 and 8. Generator 1 provides primary frequency control. The generators are modeled with their one-axis model, and equipped with AVR. The equations for the models are those used in Example 5.1. The base power is 100 MW (see Example 5.3 for an explanation of the base power and the per-unit system). The power factors of the loads are assumed to be constant as defined in the base case. Hence, an active power change of ∆P i in load at bus i is associated with a re¡ ¢ active power change ∆P i Q i0 /P i0 , where P i0 and Q i0 are the active and reactive power consumptions of the load at bus i in the base case. The system parameters λ are the three active power consumption of the loads P 5 , P 6 and P 8 . 125

CHAPTER 7. SECOND-ORDER APPROXIMATIONS: CASE STUDY IN THE IEEE 9 BUS 126 SYSTEM As explained in Section 6.4, the second-order approximations are computed around most important points, which are sought by maximizing an importance function ρ. In this case study, this function is taken as the Euclidean norm between the base case loading and the parameters: ρ(λ) = kλ − λbase k ,

(7.1)

where, in per unit, £ λbase = P 50 λ = [P 5

P 60 P6

P 80

¤T T

P8] .

= [1.25 0.9 1]T ,

(7.2) (7.3)

The following stability limits are considered: 1. Saddle-node bifurcation points. 2. Switching loadability limits. 3. Hopf bifurcations. 4. Operational limits: the active power transfer limits on the lines are considered. The absolute values of the expression in (5.35) is used. The limits for each line are given in Table 7.1 both in MW and in per unit. Table 7.1: Power transfer limits Line #

From bus #

To bus #

Limit [MW]

Limit [p.u.]

1 2 3 4 5 6 7 8 9

1 4 5 4 6 7 2 8 3

4 5 7 6 9 8 7 9 9

500 250 250 250 250 250 250 250 250

5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5

In order to illustrate the process which searches for the most important point, the actual stability boundary in the parameter space for positive values of the loads was obtained by running 10 000 continuation power flows. The stability boundary is shown in Figure 7.1. Table 7.2 shows what type of stability limit the different colors correspond to. The sets G a and G b are as defined in Example 5.1; that is, set G a includes the generators under voltage control (whose AVR has not reached its limit), and set G b includes the generators under overexcitation control (whose AVR has reached its limit), denoted OXL in the table. Note that, as discussed in Section 5.1, the sets of equations f and g (see

7.2. OBJECTIVES WITH THE CASE STUDY

127

Figure 7.1: The stability boundary of the pre-contingency IEEE 9 bus system, consisting of six different smooth parts (corresponding to the six different colors).

Example 6.4 for these sets) change when one AVR reaches its limit. Hence, for the same stability limit, the cases for which this limit was reached with different sets G a and G b are dealt with separately 1 . Note also that as discussed in Remark 5.2, an AVR reaching its limit does not always lead to system instability.

7.2 Objectives with the case study This case study aims at: 1. That is, we do not consider the cases with the same type of stability limit but different sets G a and G b as subjected to the same type of stability limit, but to two different ones.

CHAPTER 7. SECOND-ORDER APPROXIMATIONS: CASE STUDY IN THE IEEE 9 BUS 128 SYSTEM Table 7.2: The different stability limits in the IEEE 9 bus system. Color

Stability limit

Comments

Set G a

Set G b

Green

SLL

{1,2}

{2,3}

Dark blue

HB

{1,2,3}

;

Light blue

HB

{1,2}

{3}

Orange

OL

{1,2,3}

;

Red

OL

{1,2}

{3}

Yellow

OL

This SLL is due to the AVR in generator 2 reaching its limit. See comment for the light blue case. Same pair of complex eigenvalues reaching the imaginary axis as in the dark blue case, but the sets G a and G b are different. The active power limit has been reached on line 2. The active power limit has been reached on line 2. Compared to the orange smooth part, the sets G a and G b are different. The active power limit has been reached on line 4.

{1,2}

{3}

Illustrating the method The second-order approximation of one stability boundary consists of different smooth parts, each corresponding to different stability limits. From the base case operating point λ0 , the method initiates searches in the parameter space in different directions, finds the stability limits for these directions, looks for the most important points from these stability limits, and computes the second-order approximations of each smooth parts at the most important points (also called approximation points). The search for the most important points will be illustrated. The IEEE 9 bus system is particularly suitable for this, because the parameter space 2 is of dimension 3, allowing us to plot the searches in R3 . Assessing the accuracy of the approximations Second-order approximations are local approximations. The errors in the approximations come from all terms of order strictly higher than 2 in the Taylor expansion of the series. These terms are indeed neglected when computing the second-order approximations. The accuracy of the approximations is expected to decrease the farther away the parameters are from the approximation points. The accuracy can be measured by computing the distance between the approximations and the actual stability boundary.

2. When only the loads are taken as the parameters.

7.3. CASE 1: ILLUSTRATION OF THE METHOD

129

7.3 Case 1: Illustration of the method 7.3.1 Illustration of the search for the most important points The searches for the most important points are illustrated in Figure 7.2. The black circles correspond to stability limits which are found by running continuation power flows from λ0 in particular directions. From these stability limits, the black lines depict the searches for the most important points on each smooth part. The most important points are represented by white circles. Two different stability limits were computed on each smooth part (the two black circles for each smooth path in Figure 7.2).

Figure 7.2: Searches to the most important points on these smooth parts (black lines) from different start points (black circles). All searches on the same smooth part converge to the same closest point (white circles). It can be observed that, as it should be, searches starting on the same smooth part converge to the same most important point. The most important point for the whole stability boundary is located on the light blue part. It corresponds to the point the closest, under the Euclidean norm, to λbase . All other most important points lie on the edge

CHAPTER 7. SECOND-ORDER APPROXIMATIONS: CASE STUDY IN THE IEEE 9 BUS 130 SYSTEM between two smooth parts. From the searches on the dark blue, orange, red and yellow parts, it can be observed that the search is able to follow the edges using the approach presented in Section 6.4.5.

7.3.2 Illustration of second-order approximations The second-order approximations of each smooth part can then be computed from the most important points. For example, Figure 7.3 shows the actual stability boundary and the second-order approximation of the light blue part in gray. The first view is from the outside, and the second from the inside. In the view from the inside, the base case operating point λbase has been plotted with a red circle. The approximation point is the white circle on this smooth part in Figure 7.2. Accuracy of the second-order approximations is studied in more detail in next section.

(a) View from the outside

(b) View from the inside with the base case loading λbase (red circle).

Figure 7.3: The stability boundary and the second-order approximations of the light blue part (in gray).

7.4 Case 2: Accuracy of the second-order approximations In this Section, the accuracy of the second-order approximations is assessed by computing the distance between the second-order approximations computed at the approximation points shown in Figure 7.2 and the actual stability boundary. Let Σ0a j , for j = 1, . . . , 6, be the second-order approximations of the six smooth parts for the precontingency IEEE 9 bus system. The overall approximation consisting of these six smooth

7.4. CASE 2: ACCURACY OF THE SECOND-ORDER APPROXIMATIONS

131

parts is denoted Σa . The actual stability boundary is denoted Σ. The post-contingency stability boundaries will not be studied here, and we therefore omit the index 0 referring to the pre-contingency case in the following. We recall the expression of distance function defined in (6.67) which gives the signed distance from any point in parameter space to one of the second-order approximation Σaj : ´´ 1 ³¡ ¢T ³ j d j (λ) = (λc − λ) · n j + II j C j λ − λc . 2

(7.4)

Note that compared with the original expression in (6.67), the index i for the pre- or post-contingency systems has been omitted as discussed above. The distance between the actual stability boundary and the overall second-order approximation Σa will be computed in the domain Λ+ defined by positive load increases from the base case loading λbase . For any λ ∈ Λ+ , λ − λbase ≥ 0. Consider now a point λ ∈ Λ+ which belongs to the actual stability boundary Σ. We have that λ = λbase + kλ − λbase k v, where v is the following unit vector £ v = cos(ϕ) sin(θ)

sin(ϕ) sin(θ)

cos(θ)

¤T

(7.5)

defining the direction from λbase in which λ lies. The angles ϕ and θ are the azimuthal and polar angles used in the spherical coordinate system (see Figure 7.4). Hence, λ can be found by running a continuation power flow from λcase in the direction defined by v. All directions v pointing from λbase to the domain Λ+ can now be discretized in the following way £ v k,l = cos(ϕk ) sin(θl )

sin(ϕk ) sin(θl )

cos(θl )

π ∈ [0, π/2] , k ∈ {0, . . . , N p } 2N p π θl = l ∈ [0, π/2] , l ∈ {0, . . . , N t } 2N t

ϕk = k

¤T

,

(7.6)

(7.7) (7.8)

where N p + 1 and N t + 1 are the numbers of samples in the domain [0, π/2] for ϕ and θ, respectively. Given this parametrization of Λ+ and a matrix D ∈ RNp +1 × RNt +1 , the procedure of calculating the distance between the actual stability boundary and its approximation will be performed as follows: 1. Let k ∈ {0, . . . , N p } and l ∈ {0, . . . , N t }. Define v k,l as above.

2. From λbase , a continuation power flow is run in direction v k,l to find λk,l ∈ Σ (Σ is the actual stability boundary). 3. Using (7.4), compute the distances d j (λk,l ) ∈ R for all smooth parts j = 1, . . . , 6. ¯ ¯ 4. Set D k,l = ¯min j =1,...,6 d j (λk,l )¯.

CHAPTER 7. SECOND-ORDER APPROXIMATIONS: CASE STUDY IN THE IEEE 9 BUS 132 SYSTEM ¡ ¢ P 8 − P 80

λ∈Σ

θ v λbase ϕ

¡ ¢ P 6 − P 60

¡ ¢ P 5 − P 50

Figure 7.4: Spherical coordinates defined from λbase , adapted from [7].

In Figure 7.5, the values D k,l in the matrix are plotted against the corresponding ϕk and θl . The colors show to which approximation of the actual surface in Figure 7.1 the distance is calculated so that, for example, green means that it is the approximation of the SLL surface, among all approximations, that is the valid one. We see that the maximum distance between the surface and its approximation is around 0.06 p.u. (with base power 100 MW), and is reached in a region far from the approximation point (corresponding to the bottom right-hand corner of Figure 7.1 where the curvature of the surface is larger than the curvature at most of the rest of the surface).

7.5 Computational issues 7.5.1 Computation time In order to find the smooth parts of the entire stability boundary, stability limits – points on the stability boundary – are found by running continuation power flows from the operating point in many initial directions. The points corresponding to the stability limits will lie on various smooth parts. The search described in Section 6.4

7.5. COMPUTATIONAL ISSUES

133

Figure 7.5: Absolute values of the distances between the approximations and the real surface. The smooth parts are colored according to Figure 7.2.

is initiated from each of these points, and ends at the most important points on the corresponding smooth parts, as illustrated in Figure 7.2. The search is a sequence of prediction and correction steps, the most time-consuming step being the correction by Newton’s method. At the most important points, second-order approximations are calculated. This procedure must be done for the pre-contingency stability boundary and each post-contingency stability boundary of interest. In this case study, the computation time per initial direction (i.e. for running one CPF, finding the most important point and computing the second-order approximation at this most important point) is about one minute. The computer used in the simulation has an Intel Core i5 CPU at 2.53 GHz with 4 GB RAM. The procedure can be parallelized for each stability boundary (pre- and post-contingency stability boundaries) and each initial direction, so that with enough computation resources, the total computation time is equal to the computation time for one initial direction.

7.5.2 Memory requirements The most requiring element in term of memory is the second derivative of the dynamic Jacobian A, a tensor whose computation requires a size n z × n z × n z × n z , necessary to compute the Weingarten map for Hopf bifurcations, see (6.1). Furthermore, at each correction step of the search on SNB or Hopf surfaces, the first derivatives of the dynamic Jacobian and of the system Jacobian, respectively, are needed as seen in (6.48) and in (6.50). These two tensors are n z × n z × n z .

CHAPTER 7. SECOND-ORDER APPROXIMATIONS: CASE STUDY IN THE IEEE 9 BUS 134 SYSTEM Thus, if not treated carefully, these three tensors can be very heavy in memory for large power systems. For example, let us consider a system with 100 generators modeled by one-axis models with a simple model of AVR so that four state variables per generator need to be considered. Then tensors of size (n x )4 = (100 · 4)4 = 4004 are needed to compute the second derivative of A. Assuming that each entry of the tensor is represented with double-precision floating-point format requiring 8 bytes, the total memory size of the tensor would be about 200 gigabytes. These tensors are however very sparse. The Jabobians themselves are sparse, and its derivatives become even more sparse. As an example, the matrix f xxx in this case study only has 9 nonzero elements. The largest matrix is the matrix g y y y with size 194 , which corresponds to 130321 entries for a memory size of about 1 megabyte, but it only has 1511 nonzero elements and thus uses only about 60 kilobytes. The Matlab Tensor Toolbox Version 2.5 [8] was used to handle sparse tensors.

Part III

Stochastic optimal power flows

135

Chapter 8

Stochastic optimal power flow

Contents 8.1 8.2 8.3 8.4 8.5

Stochastic optimal power flow for generation re-dispatch . . Usage of the S-OPF formulation within the operating period Approximation of the constraint . . . . . . . . . . . . . . . . . Solving the S-OPF problem . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

137 140 144 156 157

This chapter introduces a stochastic optimal power flow formulation which can be used for optimally activating balancing bids under uncertainty while maintaining a specified level of system security. The solution methodology to solve the S-OPF problem was developed in the scope of this work, and is recited here. It corresponds to parts of Publication IV.

8.1 Stochastic optimal power flow for generation re-dispatch In Section 2.5.4, a review of different tools for optimally re-dispatching the generation in the context of tertiary control was given. Emphasis was given on those tools which can account for the larger uncertainty that large amounts of wind power cause. Stochastic optimal power flows were identified as particularly interesting because uncertainty is included in the optimization problem defining the generation re-dispatch. In this chapter, a new S-OPF formulation is given. The S-OPF was formulated in order to address the following issues: 1. Activation of balancing bids at minimal cost, 2. Include uncertainty in the parameters (wind power production and load for example) in the optimization problem, 3. Ensure a certain level of system security considering the stability boundary described in Section 5.3, 137

138

CHAPTER 8. STOCHASTIC OPTIMAL POWER FLOW

4. Include selected contingencies and their probabilities of occurrence.

8.1.1 Formulation of the S-OPF We now give the new proposed formulation of the S-OPF: S-OPF Problem min CG (u), u∈U

s.t.

nc X

i =0

q i P [ζ ∉ D i (u)] ≤ α,

(8.1a) (8.1b)

where u ∈ U ⊂ Rk are the control variables, CG (u) : Rk → R is the cost associated with control u ∈ U , n c is the number of contingencies, q i is the probability that contingency i occurs, ζ ∈ Rl are the stochastic system parameters, D i (u) ⊂ Rl is the stable operation domain in Rl and 1−α is the desired level of system security. The case i = 0 corresponds to the pre-contingency system. Contingencies occur with a small probability so that q i ¿ 1 for i = 1, . . . , n c and q 0 ≈ 1. When applied to the optimal activation of balancing bids, the set U represents the set of submitted balancing bids and the objective function of this new formulation becomes the minimization of generation re-dispatch costs. The S-OPF can thus be used by system operators to choose which balancing bids to activate 1 . Further detail on how and at which times S-OPF can be used for real-time operations throughout the operating period is given in Section 8.2. The constraint ensures that for all contingencies, the likelihood of system instability is less than a small value α. This is ensured by constraining the probability that the operating point in Rl leaves the stable operation domain to at most α. The stable operation domain is the domain limited by the stability boundary presented in Section 5.3. The stability boundary depends on the value of the control variables u as explained in Section 5.3.2. It is important to make a difference between: – the space of stochastic system parameters Rl , termed in the following the stochastic parameter space, – the space of control variables U , – the parameter space U × Rl . £ ¤T A point λ in parameter space can be written λ = u T ζT , with u ∈ U and ζ ∈ Rl . Uncertainty in the stochastic system parameters is accounted for by considering probabilistic constraints.

8.1.2 Comparison with today’s way of operating power systems We will take the Swedish power system as an example. As explained in Section 2.5.3, three critical transmission corridors – or bottlenecks – in the power system are mon1. It is assumed here that the bids selected in the optimal solution will be activated, i.e. the cases in which the selected bids are not activated, for instance because of technical problems, are not accounted for in the S-OPF formulation. In such cases, the system operator can re-run the S-OPF at a later time excluding the bids that could not be activated from the list of balancing bids.

8.1. STOCHASTIC OPTIMAL POWER FLOW FOR GENERATION RE-DISPATCH

139

itored, and balancing bids are activated if necessary (i.e. if the power transfers come too close to the transmission limits) to redistribute the generation on one or both sides of the bottleneck and decrease the power transfer across the bottlenecks. The transmission limits are calculated based on voltage stability, considering that the load will increase in a certain way, and such that the system survive a set of selected contingencies. With the notions introduced in Section 5.3, this corresponds to increasing the loads in the system along a certain path until reaching the stability boundary for the studied post-contingency system 2 . A security margin is added for dealing with the uncertainty, which includes the cases in which the load deviates from the load increase path considered when computing the limits. The S-OPF formulation in 8.1 considers the same set of selected contingency. When solving the S-OPF problem, the entire distribution of the uncertainty is taken into account. The uncertainty is, in this case, the way in which the net load changes. With large amounts of wind power, this uncertainty will increase. Let ζ ∈ Rl be a random variable and f ζ (ζ) the probability distribution function of ζ. We now compare today’s way of operating the Swedish power system with the proposed method: Today One or a few particular outcomes of ζ (load increase in the Swedish case) are considered, and the stability limits (these are the transmission limits in the Swedish case) are calculated based on this value of ζ. In order to account for the other possible outcomes, a security margin is subtracted. If the transfer over one bottleneck comes too close to the limit, the cheapest balancing bids which can decrease this transfer are chosen. S-OPF A level 1 − α of system security is chosen. Given the stable operation domains D i , the S-OPF problem is solved using the probability distribution f ζ of ζ. The voltage stability limits are included in the boundary of D i . The difference with today’s way of operating the system is that not just one loadability limit (one point on the boundary of D i ) corresponding to one outcome of ζ is chosen. Instead, we consider the entire distribution of ζ, and choose the cheapest bids which guarantee the chosen level of system security. No security margin needs to be added since the obtained optimal setting already accounts for the uncertainty.

8.1.3 Comparison with previous S-OPF formulations In [31], two S-OPF problems were formulated. The first one seeks at solving the following problem:

“

Suppose that probability distributions for certain node injections are known for a prospective time period. For example, since loads at buses are related to temperature, node loads would be known with a certain probability distribution. Suppose that we would like to maximize power transfers be-

2. An adapted continuation power flow method is used for this purpose, see Section 5.3.3 and the detailed explanation of the Swedish transmission operator in [93] for more detail.

140


tween a subset of other buses subject to the constraint that no flowgate flow should exceed its limit with a given probability α.

”

Remember from Section 1.1.1 that flowgates are transmission corridors, that is, transmission lines or sets of transmission lines. The problem above is formulated as max c T x d £ ¤ s.t. P A k p ≤ b k ≥ 1 − ²,

(8.2a) k = 1, . . . , m,

(8.2b)

where x d are the power injections which can be controlled, c is some vector defining the power transfers to be maximized, A k is k:th row of the matrix A of power transfer distribution factors 3 , p are the power injections (including x d ) and b k are the flowgate limits. Note that the number of constraints is the number m of flowgates (number of rows in A). The second problem given in [31] seeks at solving the following problem:

“

Given a portfolio of individual generating plants that may bid into the market to meet demand, their various geographic locations correspond to different points of connection to the electrical circuit that comprises the power system. Given the limits on the power or current carrying capability of transmission lines, cables, and transformers, and given the stochastic nature of production and load, the challenge in setting limits on transmission capability so as to maintain an acceptably small probability of line overload is clear.

”

The formulation is similar to the one above, with one constraint per limit considered (line current limits, bus voltage limits and active and reactive power transfer limits), and the objective function being this time to maximize the current across a transmission corridor of interest. These two problems are thus built in the same way as the S-OPF problem in (8.1) (maximizing, or minimizing, the objective functions under probabilistic constraints), but the main difference in our approach is that there is a single constraint, i.e. that the problem is solved so that the probability that any constraint is violated stays below a predefined threshold (as opposed to the probabilities that each constraints is violated stay below a predefined threshold). Thus, the formulation in (8.1) can be used to get an optimal generation re-dispatch which ensures a given overall level of system security.

8.2 Usage of the S-OPF formulation within the operating period The S-OPF formulation can be used for optimally choosing balancing bids under uncertainty, and considering n c contingencies. It will be seen, in Section 8.3.1, that 3. The power transfer distribution factors are the first-order sensitivities of the flows across the flow gates to the power injections (load or production).

8.2. USAGE OF THE S-OPF FORMULATION WITHIN THE OPERATING PERIOD

141

second-order approximations are used to estimate the probability of the system to become unstable under the uncertainty defined in the system parameters. Hence, using the S-OPF developed here in power system operations requires two phases: 1. Phase 1: compute the second-order approximations for the pre- and post-contingency systems. 2. Phase 2: solve the S-OPF. Phase 1 is the most-time consuming and, with the method presented here, cannot be performed during real-time operation. As seen in Section 6.8, the smooth parts of the second-order approximations are computed around approximation points on the actual stability boundary, also called most important points, which are found considering forecasts for the system parameters. Phase 2, however, is performed in real-time when the system operator wants to activate balancing bids. As seen in Section 2.3.5, the balancing bids are fully activated after a time delay (for example, in the Nordic system, this time delay is at most 15 minutes). Hence, when phase 2 is performed, forecasts for the system parameters are also used, but, compared to phase 1, forecasts are done closer to the actual realizations of the stochastic system parameters. This is depicted in Figure 8.1. Before the operating period, at t = t 0 , the second-order approximations are computed using forecasts available at t = t 0 for the stochastic system parameters (forecasts F1 in the figure). During the operating period, at t = t 1 , the system operator solves the S-OPF problem, and activates the optimal balancing bids. Later during the operating period, at t = t 1 + δ, the balancing bids must be fully activated, δ time steps after the order was given at t = t 1 . Better forecasts (forecasts F2 in the figure) are available when the S-OPF is solved.

Phase 1: Get forecasts F1, and compute the second-order approximations

Phase 2: Get forecasts F2, better than F1, and solve the S-OPF for optimal activation of balancing bids.

Balancing bids fully activated.

t0

t1

Time

t1 + δ

Operating period (one hour or less) Figure 8.1: The two phases in solving the S-OPF problem. Since the second-order approximations are local approximations, the closer to the approximation points, the better the approximations. By “closer” it is here understood larger values of the importance function ρ, see Section 6.8. Hence, better forecasts will

142


improve the accuracy of the method, since the actual system parameters will then be close to the approximation points. It is therefore desirable to perform phase 1 as close as possible to the operating period. The forecasts used during phase 2 will be more accurate than those for phase 1 because they are performed closer to the time for which the system parameters are forecasted. It is thus important to note that the forecasts are different between phase 1 and phase 2. Recall the general problem defining the scope of the thesis in Section 1.4 with Figure 1.8. Using the methodology used above, two approaches for real-time operations can be considered: Approach 1 – Monitoring and acting The system operator monitors the security of the system, and takes counteractions if the power system is too close to or beyond some stability limits. This can be checked by computing the probability of the system to become unstable if no actions are taken, i.e. evaluating the value of the left-hand side of the constraint 8.1b with u = 0. If the constraint is fulfilled (value smaller than α), no actions are necessary since the power system is already running with the required level of system security 1 − α. If, on the other hand, the constraint is not fulfilled, the system operator solves the S-OPF problem in order to find the optimal balancing bids to activate in order to operate the system with the level of system security 1 − α. The general strategy in Figure 1.8 then becomes as in Figure 8.2. The time iteration ∆t is the interval between each time the system security level is updated and monitored; ∆t should be at most a few seconds. Approach 2 – Repeatedly acting Another approach would be to solve the S-OPF formulation repeatedly, e.g. every five minutes, within the operating period, and redispatch the generation for an optimal operation five minutes later. Ramp rates of the generators must then be taken into account as constraints. The optimal setting of the generators would then be the output power that these generators should ramp up or down to five minutes after the S-OPF problem has been solved. This could be used for intra operating period economic re-dispatch and help the frequency control schemes to maintain the balance between production and generation. Note that such approaches are used today, for example in Texas [36], but the economic re-dispatch problem is formulated deterministically, and it would still be beneficial to use S-OPF formulations instead. This approach is illustrated in Figure 8.3. Remark 8.1 (Frequency deviations) The approaches above assume that balancing bids are activated when the system comes too close to some stability limit – “too close” being measured by the level of system security 1−α. As seen in Section 2.3, balancing bids are also activated when the frequency deviates from its nominal value. The acceptable range of frequency deviations can be defined as an operational limit, and can thus be accounted for in the computation of the stability limits as any other operational limit, see Section 5.3.

8.2. USAGE OF THE S-OPF FORMULATION WITHIN THE OPERATING PERIOD

143

Phase 1: compute the second-order approximations.

t = 0: beginning of the operating period t = t + ∆t

No

Monitor the system security: Pn c q P [ζ ∉ D i (0)] ≤ α? i =0 i Yes Phase 2: get the forecasts F2, solve S-OPF and activate the optimal bids.

Figure 8.2: Approach 1: monitoring and acting; ∆t is at most a few seconds.

Phase 1: compute the second-order approximations.

t = 0: beginning of the operating period t = t + ∆t Phase 2: get the forecasts F2, solve S-OPF and activate the optimal bids. Figure 8.3: Approach 2: repeatedly acting; ∆t is a few minutes.

Remark 8.2 (Considering different times ahead in the future) The S-OPF formulation allows the system operator to take an optimal decision for one specific time. The actions taken for one specific time have consequences on the rest of the operating hour, and, thus, on the subsequent decisions which will be taken.

144


Consider for example the second approach where the S-OPF is run every five minutes. It can happen that bids must be activated now to ensure an optimal and secure operation for the next five minutes, and that other bids will have to be activated for the following five minute period, corresponding to the period up to ten minutes from now. Thus, during the next ten minutes, two S-OPF problems are solved: one for the coming next five minutes, and one for the last five minutes of the ten minute period. The first SOPF problem does not take into account what happens farther than five minutes ahead, and the two decisions, taken after the two S-OPF problems are solved, while optimal for their respective five-minute period are not for sure optimal for the ten-minute period. This small example reflects the fact that the decision taken by solving the S-OPF problem is only optimal for the point in time it considers, and that the expected costs for the rest of the operating period arising from this decision are not considered. Considering these expected costs would require developing other tools. This is left as future work.

8.3 Approximation of the constraint In practice, the general problem defined in 8.1 cannot unfortunately be solved as such, and it will be seen in the following that approximations of the constraint (8.1b) are necessary. We recite below the method developed within the scope of the project to solve this S-OPF problem.

8.3.1 Approximations of the stable operating domains First, the stable operating domains D i for the pre- and post-contingency systems cannot be expressed analytically. Because no analytical expressions of such domains are known, the probabilities in the constraint (8.1b) cannot be expressed analytically. To overcome this issue, we propose here to use the second-order approximations of the stability boundary presented in Chapter 6. Using the notations introduced in Section 6.10, we will denote by Σia , i = 1, . . . , n c the second-order approximations of the stability boundary, in parameter space, corresponding to the system after contingency i has occurred. Recall that each Σia in fact consists of a set of second-order approximations Σiaj , j = 1, . . . , n ai . Using the analytical expressions of the approximations of the stability boundaries, the event “the operating point ζ in stochastic parameter space is not in the stable operating region for u ∈ U ” can be approximated by the event £ ¤T “the point λ = u T ζT in parameter space is beyond the second-order approximation of the stability boundary”. The time horizon considered for these events is that of the forecast F2, see Section 8.2. In order to characterize this event, the distance function defined in Section 6.10 which

145

8.3. APPROXIMATION OF THE CONSTRAINT

gives the signed distance from any point in parameter space to the second-order approximation is used. Note that the distance functions d i j are random variables since ζ, £ ¤T and thus λ = u T ζT , are random. Consider a specific contingency i (or the pre-contingency system corresponding to i = 0). The event “Not in the stable operating domain D i ” can now be approximated by “min j ∈J i d i j (u, ζ) < 0”. The original problem in (8.1) can thus be approximated by S-OPF Problem – Approximation 1 min CG (u), u∈U

s.t.

(8.3a)

· ¸ nc X q i P min d i j (u, ζ) < 0 ≤ α.

i =0

(8.3b)

j ∈J i

In order to solve this optimization problem, we need an analytical expression of the probability in the constraint. Before giving an analytical expression of the probability of the minimum, the case of a single constraint P [d i j (u, ζ) < 0] is studied.

8.3.2 Approximating the probability of one distance being negative In the following, we consider the distance to one particular smooth part Σiaj of the approximation Σia . The probability P [d i j (u, ζ) < 0] must be computed. Recall the definition of the distance function from (6.67). Using the definition of the second fundamental form in (6.3), the distance function is a quadratic function in ζ as can be seen by re-writing (6.67) in the following manner: d i j (u, ζ) = n iTj

·

¸ 1£ ∆u − ∆u T ∆ζ 2

∆ζT

¤T

C i j dNi j C iTj

· ¸ ∆u , ∆ζ

(8.4)

£ ¤T where ∆u = u c − u, ∆ζ = ζc − ζ and λc = u cT ζTc is the approximation point at which the second-order approximations Σiaj was computed. Let · 1 (M )11 M = − C i j dNi j C iTj = (M )21 2

¸ (M )12 . (M )22

(8.5)

1 d i j (u, ζ) = a i j (u) + b i j (u)T ζ + ζT A i j ζ, 2

(8.6)

The distance function can then be written as

146


where ³¡ ¢ ´ ¡ ¢T T a i j (u) = n i j 1 ∆u + ∆u T (M )11 ∆u + n i j 2 + 2∆u T (M )12 ζc + ζTc (M )22 ζc , ¡ ¢T b i j (u) = − n i j 2 − 2(M )21 ∆u − 2(M )22 ζc , A i j = 2(M )22 ,

(8.7) (8.8) (8.9)

¡ ¢ ¡ ¢ and n i j 1 and n i j 2 are the two components of n i j corresponding to u and ζ, respectively. Remark 8.3 (Dependence on the time horizon) The probability distribution of the stochastic system parameters ζ depends on the time horizon of the forecast F2 used to solve the S-OPF problem. Hence, the distance function depends also on this time horizon. In the following, we assume that the first four cumulants (see Section 3.4) of ζ are known. They can be computed, for example, using historical £ ¤ data and the formulas (3.50). The question is thus to be able to express P d i j (u, ζ) < 0 knowing the cumulants £ ¤T of λ = u T ζT , which can be done as follows: 1. The first four cumulants of d i j are computed from the cumulants of ζ using (8.6)

2. Using the Cornish-Fisher expansion presented in Section 3.4, the function k d in (3.47) for the distance d i j can be constructed, neglecting the higher order terms (terms with cumulants of order higher than four). 3. Using k d and Ψ, the cumulative probability distribution of the normal distribution N (0, 1), £ ¤ P d i j (u, ζ) < 0 ≈ Ψ (k d (0)) .

(8.10)

Example 8.1 Assume that ζ has a Gaussian distribution with mean m and covariance matrix Λ, then d i j (u, ζ) as defined by (8.6) has mean a i j (u) + b i j (u)T m + 21 (m T A i j m + tr(A i j Λ)) and cumulants κr , for (r ≥ 2), ¡¡ ¢r ¢ 1 ¡ ¢r −2 1 κr = (r − 1)!tr A i j Λ + r !b i j (u)> Λ A i j Λ b i j (u). 2 2

(8.11)

8.3.3 Approximating the probability of the minimum of the distances being negative Consider now one contingency i ∈ {1, . . . , n c } or the pre-contingency case i = 0. The approximation Σia of the actual stability boundary corresponding to the post-contingency (or pre-contingency) system consists in general of several smooth parts. The probability that the distance to one specific smooth part is negative can be computed as described

147


above. However, in the S-OPF in (8.3), the probability to be calculated is that of the minimum of the distances to any smooth part being negative. We begin by introducing the problem of computing the probability of the minimum by studying the simple case in which the second-order approximation consists of two smooth parts. Case for two distances Consider Figure 8.4a where the approximation Σia of the stability boundary for one specific value of u consists of two smooth parts Σia1 and Σia2 . The approximation Σia consists of the parts of each smooth part drawn with a solid line. The dashed lines are parts of one smooth part which are beyond the other smooth part, and, therefore, do not correspond to a valid loadability limit (since the other smooth part is encountered first). In Figure 8.4b, the normals to these two smooth parts are n 1 and n 2 . The regions A, B and C correspond to the regions beyond Σia1 but within Σia2 , beyond Σia2 but within Σia1 and beyond both smooth parts, respectively. We consider one given value of the parameter ζ. The distances from ζ to the two smooth parts in the direction of the normals are denoted d 1 and d 2 . The problem is to calculate the probability that the minimum of these distances is negative. When d 1 (respectively d 2 ) is negative, ζ is beyond Σia1 (respectively Σia2 ). In the case depicted in the figure, the two distances are positive. Σia2 Σia1

Σia2

n1 Σia1

A d1

ζ

ζ

C

d2

B n2

(a) Validity of each smooth part: the overall approximation Σia consists of the solid lines.

(b) Distances to the two smooth parts.

Figure 8.4: An approximation of a stability boundary consisting of two smooth parts Σia1 and Σia2 . To the knowledge of the authors, no general formula exists for the probability of the minimum of several dependent random variables. Some approximations must there-

148


fore be done. The sought probability P [min (d 1 (u, ζ), d 2 (u, ζ)) < 0]

(8.12)

p1 + p2,

(8.13)

p 1 = P [d 1 (u, ζ) < 0] ,

(8.14)

could be approximated by

where

p 2 = P [d 2 (u, ζ) < 0] ,

(8.15)

which can be computed using the Cornish-Fisher expansion as seen in Section 8.3.2. In Figure 8.4b, the probability p 1 (respectively p 2 ) is the probability of ζ being in the region A ∪C (respectively B ∪C ). Since region C is included in both A ∪C and B ∪C , the approximation (8.13) of the probability of the minimum would count the probability of ζ being in region C twice. Therefore, a better approximation would be p 1 +p 2 −pC where pC is the probability of ζ being in region C i.e. that d 1 < 0 and d 2 < 0. The probability pC can be expressed as P

·· ¸ · ¸¸ d1 0 < . d2 0

(8.16)

Let k d1 (d 1 ) and k d2 (d 2 ) be the two random variables defined from (3.47) for the two distances. In Section 3.4, we saw that there exists U1 and U2 normally distributed N (0, 1) such that £ ¤ P [d 1 < 0] = P U1 < k d1 (0) , £ ¤ P [d 2 < 0] = P U2 < k d2 (0) ,

(8.17) (8.18)

so that (8.16) can be expressed as

P

·· ¸ · ¸¸ U1 k (0) < d1 . U2 k d2 (0)

(8.19)

It is now assumed that the vector [U1 U2 ]T is approximated by [V1 V2 ]T which has a multivariate Gaussian distribution with V j standard normal. In the following, let ρ = Cov(V1 ,V2 ) = Cov(k d1 (d 1 ), k d2 (d 2 )).

(8.20)

Note that, in general, the vector [U1 U2 ]T will not be a Gaussian vector. The probability in (8.19) is then approximated by P

·· ¸ · ¸¸ V1 k (0) < d1 . V2 k d2 (0)

(8.21)

149


Consider now Figure 8.5, where c 1 = k d1 (0) and c 2 = k d2 (0). The quantities c 3 and c 4 will be expressed later in (8.28) and (8.29). The correlation ρ between V1 and V2 can be interpreted as the cosine of the angle between V1 and V2 . The horizontally (respectively vertically) striped area corresponds to V1 < c 1 (respectively V2 < c 2 ), that is, to ζ being beyond Σia1 (respectively beyond Σia2 in Figure 8.4b) or, equivalently, to d 1 (u, ζ) < 0 (respectively d 2 (u, ζ) < 0).

c3

C

V2

(x0,y0)

V1

c2

c4

cos-1(ȡ)

c1

Figure 8.5: Geometrical interpretation of V1 and V2 .

The probability pC of ζ to belong to C in Figure 8.4b was approximated by (8.21), which is the probability pˆC of [V1 V2 ]T belonging to C in Figure 8.5. Since [V1 V2 ]T is assumed to be normally distributed, pˆC can be computed as 1 2π

pˆC =

Z

1

C

e − 2 (x

2 +y 2 )

dxdy.

(8.22)

The region C is an unbounded cone whose apex is the point (x 0 , y 0 ) in Figure 8.5. The cone spreads from β = cos−1 (ρ)−π/2 to π/2. Taking (x 0 , y 0 ) as origin in (8.22), the region 0 C is translated to the unbounded cone C with apex at (0, 0), and (8.22) becomes pˆC =

1 2π

Z

1

C

0

2 +(y+y

e − 2 ((x+x0 )

2 0)

) dxdy.

(8.23)

0

The region C is best expressed using polar coordinates, and we get pˆC =

1 2π

Z

π/2 Z ∞

β

0

1

2 +(r

e − 2 ((r cos θ+x0 )

sin θ+y 0 )2 )

r dr dθ.

(8.24)

150


Let h = h(θ) = x 0 cos θ + y 0 sin θ, and z = r + h. Equation (8.24) becomes pˆC =

1 − 1 (x 2 +y 2 ) e 2 0 0 2π

1 − 1 (x 2 +y 2 ) = e 2 0 0 2π 1 − 1 (x 2 +y 2 ) = e 2 0 0 2π =

1 − 1 (x 2 +y 2 ) e 2 0 0 2π

Z Z Z Z

pi /2

1

e 2h

2

β pi /2

e

1 2 2h

e

1 2 2h

β pi /2 β pi /2 ³

β

Z

∞ h

µZ

∞ h

1 2

e − 2 z (z − h)dzdθ 1 2

e − 2 z zdz −

Z

∞ h

¶ 1 2 e − 2 z hdz dθ

³ 1 2 p ´ e − 2 h − 2πhΦ(−h) dθ

´ p 1 2 1 − 2πhe − 2 h Φ(−h) dθ

(8.25)

In order to compute this integral, we need the expression of x 0 and y 0 . From Figure 8.5, we see that x0 = c1 , y 0 = c4 − c3 p

(8.26) ρ 1 − ρ2

,

(8.27)

from which x 0 can be computed, but we now need expressions for c 3 and c 4 to compute y 0 . Again from Figure 8.5, the following holds:

The expression of y 0 becomes

c 3 = c 1 − c 2 ρ, q c4 = c2 1 − ρ 2 . ¡ ¢ 1 y0 = p c2 − c1 ρ . 1 − ρ2

(8.28) (8.29)

(8.30)

The equation (8.25) above allows us to compute an approximation of the probability of ζ lying in the pairwise intersections of two regions defined as being beyond two different smooth parts Σiaj and Σiak of the second-order approximation Σia . To summarize, if only two smooth parts are considered, it is proposed to approximate the probability P [min (d 1 (u, ζ) < 0, d 2 (u, ζ) < 0]]

(8.31)

p 1 + p 2 − pˆC ,

(8.32)

by

where p 1 = P [d 1 (u, ζ) < 0], p 2 = P [d 2 (u, ζ) < 0] and pˆC is computed from (8.25). Two approximations have been made: 1. When computing the Corner-Fisher expansion, only the terms with cumulants up to the fourth order have been included.

151


2. The vector [U1 U2 ]T in (8.19) was approximated by the vector [V1 V2 ]T which has a bivariate normal distribution. The validity of these approximations will be assessed in Chapter 9. Remark 8.4 The probability of being beyond only one of the two smooth parts could also be computed as P [d 1 d 2 < 0] using the Cornish-Fisher expansion, this time applied to d 1 d 2 . However, the distribution of d 1 d 2 could be far from the normal distribution, and a good accuracy when using the Cornish-Fisher expansion would require using high order cumulants and cross cumulants. We now study how to extend the method above to the case where more than two smooth parts are intersecting. General case We begin by considering three surfaces, and studying why the method above cannot directly be applied when more than two smooth parts intersect. Consider Figure 8.6. As was the case in Figure 8.4a, overall approximation Σia consists of the innermost smooth parts. The letters A to F represents separate regions.

Σia3

D

F

A B Σia2

E

C Σia1 Figure 8.6: Case of three smooth parts intersecting.

If the same procedure as for two smooth parts was used, the probability P [min (d 1 (u, ζ) < 0, d 2 (u, ζ) < 0, d 3 (u, ζ) < 0)]

(8.33)

152


would be approximated by p 1 + p 2 + p 3 − P [{d 1 < 0} ∩ {d 2 < 0}] − P [{d 1 < 0} ∩ {d 3 < 0}] − P [{d 2 < 0} ∩ {d 3 < 0}]

≈p 1 + p 2 + p 3 − pˆF − pÊ ∪F − pˆD∪F ,

(8.34)

where, as before, £ ¤ p j = P d j (u, ζ) < 0 .

(8.35)

Since the region F is included in D ∪F and E ∪F , it would be removed three times, which is not acceptable, since it would lead to underestimate the probability of system failure. Instead, the following is proposed to approximate the sought probability, which is ¸ · (8.36) P min d i j (u, ζ) < 0 . j ∈J i

We assume that these quantities have been computed: – the probabilities p j in (8.35) for all smooth parts Σiaj , j ∈ J i , – the probabilities pˆC in (8.25) corresponding to all pairwise intersection between any two smooth parts. Then, the following procedure can be applied: 1. Start with one smooth part Σiaj , j ∈ J i , and set pˆ = p j .

2. Add another smooth part Σiak , k 6= j ∈ J i and remove the probability pˆC corresponding to the intersection with the first smooth part: pˆ = p j + p k − pˆC . 3. Continue adding one smooth part at a time, and removing the probability pˆC corresponding to the largest pairwise intersection with one of the already added smooth parts.

In this way, for each new smooth part, we only remove the intersection with one of the already considered smooth parts. Let n s = |J i | be the number of smooth parts in the approximation of the considered stability boundary. Let Υ be a Rn s × Rn s matrix such that ∀l ∈ J i , Υl l = 0 and ∀l , m ∈ J i , l 6= m, Υl m contains the probability pˆC corresponding to the intersection of the smooth parts Σial and Σiam . This matrix can be used to carry out the procedure described above, by, for each smooth part but the first one, adding p j and removing the largest element under the diagonal on row j of Υ (this element corresponds to the largest pairwise intersection of the j -th smooth part with any of the previously added smooth parts). As an example, consider Figure 8.6 one more. The matrix Υ is   0 pˆF pÊ ∪F 0 pˆD∪F  , Υ =  pˆF (8.37) pÊ ∪F pˆD∪F 0

where the first (respectively second and third) row and column correspond to Σia1 (respectively Σia2 and Σia3 ). The procedure starts with pˆ = p 1 . The second smooth part, Σia2


153

is then considered by adding p 2 and removing the largest element under the diagonal on the second row, which is pˆF , so that pˆ = p 1 + p 2 − pˆF . Finally, the third smooth part is considered by adding p 3 and removing the maximum of pÊ ∪F and pˆD∪F , so that the approximation of the sought probability becomes ¡ ¡ ¢¢ pˆ = p 1 + (p 2 − p F ) + p 3 − max pˆD∪F , pÊ ∪F . (8.38)

Note that, at this point, the procedure is not optimal, since it results in either D or E being counted twice. This is because the order in which the smooth parts are dealt with is important. The steps must therefore be carried out so that pˆ is minimized in the end, which corresponds to as few regions as possible being counted several times. Formally, this can be expressed by finding an optimal permutation among the set S n s of all permutations of the numbers 1, . . . , n s which solves the following problem µ ¶ X p j − max Υ j ,k . (8.39) min π∈S n s j ∈J i

k:π(k) pˆF + pˆD∪F , pÊ ∪F + pˆD∪F > pˆF + pÊ ∪F .

(8.40) (8.41)

Thus, the proposed permutation would consider Σia3 first, and be either π∗ = (3, 1, 2) or π∗ = (3, 2, 1). If the former is chosen, the matrix Υ becomes, after permutation,   0 pÊ ∪F pˆD∪F 0 pˆF  . Υ =  pÊ ∪F (8.42) pˆD∪F pˆF 0 Now, during the procedure, both pˆD∪F and pÊ ∪F will be removed, and we get ¡ ¢ ¡ ¢ pˆ = p 3 + p 1 − pÊ ∪F + p 2 − pˆD∪F ,

which is the right approximation for the sought probability in (8.33). 4. That is, they are large compared to that of F only.

(8.43)

154


8.3.4 Case of SNB-SLL intersections The method given in the previous section is used for computing and approximation of the probability of the minimum of the distances to the second-order approximations being negative. The method is valid if two smooth parts intersect. The special case of a smooth part corresponding to a SNB meeting a smooth part corresponding to a SLL must be investigated further, however, because the intersections between two such smooth parts is tangential as discussed in Section 5.3.5. In Section 6.4.6, the concept of a dividing surface H for SNB-SLL tangential intersections was introduced. We will now see how to the probability of the parameters being beyond one of the smooth parts can be computed in case of SLL-SNB intersections. We consider Figure 8.7 depicting two tangential intersections between one SNB surface and two SLL surfaces. In the figure, H1 and H2 are the two dividing surfaces corresponding to the two TIPS, and n 1 and n 2 the normal to these dividing surfaces.

SNB B

H1 n1

SLL1

A

H2 C n2

SLL2

Figure 8.7: A parameter space with two SLL surfaces and one SNB surface. We would like to emphasize that the surface corresponding to SLL1 (respectively SLL2 ) is the solid line to the left of H1 (respectively to the right of H2 ), whereas the dashed line continuing on the right of H1 (respectively to the left of H2 ) corresponds to a set of breaking points which are not SLL, see Sections 5.3.5 and 6.4.6 for further detail. The second-order approximations for SLL smooth parts as developed in Chapter 6 approximate the whole surface corresponding to the sets of SLL and breaking points. Thus, the probability d i j from (8.6) associated with, for example, the approximation of the first SLL surface in Figure 8.7 is the probability of the parameters to be beyond the SLL and breaking point surfaces (solid and dashed lines), and similarly for the second SLL surface. In the following, let – p SLLi be the probability of the parameters to be beyond the i -th SLL surface, including the set of breaking points (i.e. beyond the solid and dashed line corresponding to SLLi in Figure 8.7); – p SNB be the probability of the parameters to be beyond both solid and dashed parts of the SNB surface. These probabilities can be computed from (8.35). Suppose now that we use the method

155


of pairwise exclusion presented previously. Since the SNB surface is located beyond the two SLL surfaces, it will be entirely disregarded. The estimated probability would be pˆ = p SLL1 + p SLL2 − pˆSLL1 ∩SLL2 ,

(8.44)

pˆSLL1 ∩SLL2 ,

(8.45)

where

corresponds to the probability of the intersection of the two SLL surfaces computed from (8.25). Hence, the approximation of the probability of the minimum of the distances would be overestimated since the regions A, B and C (included in p SLL2 , pˆSLL1 ∩SLL2 and p SLL1 , respectively) are counted when they should not. This would lead to overestimating the probability of the system to become unstable, and thus would lead to an optimal solution of the S-OPF problem corresponding to an unnecessary secure way of operating the system. Instead of using only the second-order approximations of the SNB surface and of the two SLL surfaces, the dividing surfaces Hi will also be used in the process of pairwise exclusion. Let SLLi be one SLL surface intersecting tangentially a SNB surface as in Figure 8.7. The probabilities to be removed from p SNB + p SLLi are – pˆSLLi ∩Hi− corresponding to the region located beyond the SLL surface and on the negative side of the corresponding dividing surface Hi (in the direction opposite to that of the normal n i to Hi ). – pˆSNB∩H + corresponding to the region located beyond the SNB surface and on the i positive side of the corresponding dividing surface Hi (in the direction of the normal n i to Hi ). These two probabilities can be computed by (8.25) since Hi itself is a second-order approximation, see Section 6.4.6. In the example of Figure 8.7, using this process would lead to estimating the probability of failure (probability of the minimum of the distances being negative) by p SNB + p SLL1 − (pˆSNB∩H + + pˆSLL1 ∩H1− ) 1

+p SLL2 − (pˆSNB∩H + + pˆSLL2 ∩H2− ), 2

(8.46)

which is the sought approximation.

8.3.5 Approximation of the constraint: a summary In the previous section, approximations for the expression of the constraint in (8.1b) were presented. Recall that this constraint was given by nc X

i =0

q i P [ζ ∉ D i (u)] ≤ α.

The approximation of the constraint was built in the following way:

(8.47)

156


1. In Section 8.3.1, it was proposed to use the distances d i j , defined in Section 6.10, to the smooth parts Σiaj of the second-order approximations in order to approximate the constraint in the following way (from (8.3)) nc X

i =0

·

¸

q i P min d i j (u, ζ) < 0 ≤ α. j ∈J i

(8.48)

2. In Section 8.3.2, the distance functions were given as a second-order polynomial of the stochastic system parameters ζ (in (8.6)). Using this formula, the cumulants of the distance functions can be computed from the cumulants of the stochastic system parameters 5 . The probability of one distance function to be negative can then be approximated with help of the Cornish-Fisher expansion. The corresponding formula for doing so is (8.10). 3. In Section 8.3.3, a pairwise exclusion method was proposed for approximating the probability in (8.48) of the minimum of any of the distances to be negative. The probabilities of all pairwise intersections between two smooth parts, denoted pˆC , must be computed using the formula in (8.25). The special case of the SNB-SLL intersections was dealt with in Section 8.3.4. Using the method above allows for a deterministic formulation of the constraint (8.48) as a sum of terms (such as the obtained expressions in (8.43) and in (8.46)), each one being either the probability of the distance to a smooth part being negative or the probability of the parameters being beyond two smooth parts. The former is computed by (8.10) and the latter by (8.25). Note that in the two previous formulations presented in Section 8.1.3, the Cornish-Fisher expansion was also used to get deterministic constraints.

8.4 Solving the S-OPF problem The method developed in the previous section aimed at approximated the constraint in the S-OPF problem (8.3), repeated here: S-OPF Problem – Approximation 1 min CG (u), u∈U

s.t.

nc X

i =0

(8.49a)

· ¸ q i P min d i j (u, ζ) < 0 ≤ α. j ∈J i

(8.49b)

5. The cumulants of the stochastic system parameters such as load or wind power variations are assumed to be known. They can be computed for example from historical data.

157

8.5. SUMMARY

It was seen how to formulate an approximation of the probability of the minimum, for each i = 1, . . . , n c . Let pˆ i (u) be this expression, so that the problem is approximated by S-OPF Problem – Approximation 2 min CG (u),

(8.50a)

u∈U

s.t.

nc X

i =0

q i pˆ i (u) ≤ α.

(8.50b)

This is a nonlinear constrained optimization problem. The Lagrangian of the problem is L (u, γ) = CG (u) − γ

nc X

q i pˆ i (u),

(8.51)

i =0

where γ is the Lagrange multiplier. The cost CG is assumed to be differentiable. From the way the probabilities pˆ i are computed (using the distances to second-order approximations), it can be noted that pˆ i is smooth in u. Therefore, L u and L uu can be computed. The Karush-Kuhn-Tucker (KKT) conditions are used to find a local optimum to the problem [49].

8.5 Summary 8.5.1 Methodology for solving the S-OPF problem A flowchart illustrating the methodology for solving the S-OPF problem can be found in Figure 8.8. The inputs to the method are the production cost function C , the distance functions for all system states (pre- and post-contingency states) d i j , the value α and cumulants of the stochastic system parameters ζ used in the constraint. The distance functions are obtained from the second-order approximations developed in Chapter 6. The first four steps in the flowchart correspond to the steps described in Section 8.3.5. When the value of the constraint is known, the first- and second-order derivatives of the Lagrangian are computed and used to update u towards its optimal value. The iterations stop when the Karush-Kuhn-Tucker conditions for local optimality are fulfilled.

8.5.2 An overall summary: Second-order approximations and S-OPF problem The two phases needed to apply the S-OPF problem to power system operation were described in Section 8.2. The first phase, the most time demanding, is done before the operating period. During this phase, second-order approximations are computed for each system state (preand post-contingency states) of interest. Forecasts of the stochastic system parameters are used to search for the most important points around which the approximations are computed. This first phase is described by the flowchart in Figure 6.5.

158

CHAPTER 8. STOCHASTIC OPTIMAL POWER FLOW Inputs: C , d i j , α, cumulants of ζ. Compute the cumulants of all d i j (u, ζ) using (8.6) and the cumulants of ζ. Using Cornish-Fisher, £ ¤ estimate P d i j (u, ζ) < 0 with (8.10). Decide which intersections to subtract (Sections 8.3.3 and 8.3.4).

Update u.

Compute the value of the constraint. Compute L u and L uu .

KKT conditions fulfilled?

No

Yes u ∗ found. Figure 8.8: Flowchart of the methodology to solve the S-OPF problem.

The second phase must be carried out during real-time operation for optimally redispatching the generation (for example when activating balancing bids). The S-OPF problem is solved in this phase, using the second-order approximations computed before. Better forecasts are now available because of the shorter time horizon. Given the distance functions to the second-order approximations, the problem is solved using the methodology presented in Figure 8.8. During the two phases, approximations were used in order to render the problems

8.5. SUMMARY

159

tractable. These approximations are: 1. In the original problem formulation (8.1b), the constraint measures the probability that the stochastic system parameters do not belong to the stable operation domain. Since no parametrization of this domain is known, the second-order approximations of the stability boundary bounding this domain were used. The original probability was approximated by that of the minimum of the distances to the second-order approximations to be negative. This was explained in Section 8.3.1, and gave the approximation (8.3) of the original problem. 2. For each distance function, the probability that it is negative was approximated by using Corner-Fisher expansion as explained in Section 8.3.2. The probability of the minimum of any distance function to be negative was estimated by a method of pairwise exclusions. The approximations done for pairwise exclusion were listed in 8.3.3. This resulted in the approximation (8.50) of problem (8.3). The accuracy of the approximation (8.50) of the original problem will be studied in Section 9.

Chapter 9

Application of stochastic optimal power flow in the IEEE 39 bus system

Contents 9.1

Setup of the case study . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

9.2

Objectives with the case study . . . . . . . . . . . . . . . . . . . . . . . 162

9.3

Case 1: correct pre-distribution . . . . . . . . . . . . . . . . . . . . . . 164

9.4

Case 2: incorrect pre-distribution . . . . . . . . . . . . . . . . . . . . . 166

9.5

Computational time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

The S-OPF formulation presented in Chapter 8 is applied in the IEEE 39 bus system for optimally re-dispatching generation under uncertainty on the demand. The obtained optimal setting of the control parameters is used in Monte-Carlo simulations, from which the level of system security when running the system with this optimal re-dispatch is estimated. The latter is compared to the security level set in the S-OPF formulation. Conclusions are drawn on the validity of the approximations used when solving the S-OPF problem. It corresponds to parts of Publication IV.

9.1 Setup of the case study In the following, the IEEE 39 bus system from Appendix A.3 is considered. This system has 10 generators and 19 loads. Generator 1 provides primary frequency control. We assume that the loads vary with constant power factor, which means that if P 0 and Q 0 are the active and reactive power consumptions of one load, an increase ∆P in active power corresponds to an increase ∆Q = ∆P (Q 0 /P 0 ) in reactive power. The S-OPF presented in Chapter 8 will be applied in this case study for generation 161

CHAPTER 9. APPLICATION OF STOCHASTIC OPTIMAL POWER FLOW IN THE IEEE 162 39 BUS SYSTEM re-dispatch. We recall the S-OPF formulation from 8.3: S-OPF Problem – Approximation 1 min CG (u), u∈U

s.t.

nc X

i =0

(9.1a) ·

¸

q i P min d i j (u, ζ) < 0 ≤ α, j ∈J i

(9.1b)

where – ζ are the stochastic system parameters. – U = {u 2 , . . . , u 20 } is the set of control variables, assumed here to be the changes 0 in the mechanical power P m,i of generators 2 to 10, i.e. P m,i = P m,i + u i , for i =

0 2, . . . , 10, where P m,i is the base case production level for the IEEE 39 bus system. – CG (u) is the total generation cost associated with the generation re-dispatch. Here, we take

CG (u) =

10 ¡ X ¢ u i + 0.1u i2 .

(9.2)

i =2

– ζ are the stochastic system parameters. In this case study, the stochastic system parameters are all loads in the system. Their probability distribution will define the uncertainty to which the system is subjected. – q i , for i = 1, . . . , n c is the probability that contingency i occurs, and q 0 ≈ 1 is the probability of the system being the pre-contingency state. The contingencies considered here are all single failures of one line, represented by doubling the line impedances. Each contingency is assumed to have a probability of occurrence of 10−4 so that q i = 10−4 , for i = 1, . . . , n c , where n c is the number of lines. – α is for the level of system security 1 − α. Several values will be used for α. As explained in Section 8.2, solving the S-OPF problem is a two phase problem, with the second-order approximations of all pre- and post-contingency systems being computed during phase 1 before the operating period, and used then during the operating period in phase 2 to solve the approximated S-OPF problem (8.50). 0 ,C 0 ) distributed, In this case study, during phase 1, the load are assumed to be N (P D ¡ 0 ¢2 0 where P D is the base case loading for the IEEE 39 bus system, and C 0 = c 0 diag P D with c 0 chosen such that the total load gets a standard deviation of 2%. This forecast is used to search for the most important points as explained in Section 8.2. For the search, the importance function presented in Example 6.2 is chosen. The second-order approximations are computed for all pre- and post-contingency systems around these most important points. These are thus supposed to be known in the following.

9.2 Objectives with the case study The case study aims at answering the following questions:

9.2. OBJECTIVES WITH THE CASE STUDY

163

Validity of the approximations In Section 8.5, the approximations used to solve the original S-OPF formulation proposed in (8.1) were listed. The accuracy of the optimal solution giving by our method will be assessed in the following way. 1. First, a level of system security 1 − α will be chosen. 2. Second, the S-OPF problem will be solved by our method to obtain an opti£ ¤ ∗ T mal generation re-dispatch u ∗ = u 2∗ . . . u 10 which minimizes the re-dispatch cost while guaranteeing the chosen level of system security.

0 3. Third, the production levels in generators 2 to 10 will be set to P m,i + u i∗ for all i = 2, . . . , 10, and Monte Carlo simulations will be run using these new production levels. During the Monte Carlo simulations, the loads are distributed according to the forecasts used when solving the S-OPF problem. The contingencies occur with probability q i . For each run, we check if the system is stable or unstable, using the actual stability boundary computed beforehand by running optimal power flows. Specifically, for run number i , let P iMC be the value of the loads obtained from sampling the probability distribution, and j be the state of the system, with j = 0 if no contingency has occurred, and j ∈ {1, . . . , n c } if a contingency has occurred on one of the line. Let also Σ j be the actual stability boundary corresponding to the system in state j . If the load P iMC is on the inside of the stability boundary, the system is stable; otherwise, it is unstable. Let N be the total number of Monte Carlo simulations, N s be the number of runs in which the system was stable, and Nu the number of runs in which the system was unstable. Let then αˆ MC = Nu /N .

The value αˆ MC estimates the probability of system failure. It is compared to the value of α chosen when solving the S-OPF problem. A ratio α/αˆ MC close to one indicates that the methodology for solving the S-OPF gives an accurate solution, despite the approximations. In addition, note that the errors from using a limited number of samples in the Monte Carlo simulations influence this ratio. Imperfect forecasts As discussed above, the forecasts used to compute the secondorder approximations in phase 1, and the ones used when solving the S-OPF problem in phase 2 are different. A part of the case study is dedicated to the effect of having different forecasts in these two phases. Choice of u c for the second-order approximations Recall from Section 5.3.2 that the stability boundary is a surface in parameter space, the latter consisting of the stochastic system parameters and the control variables. When computing the second-order approximation, one specific value is chosen for the control variables, the most important point ζc is sought on the resulting stability boundary in the space of stochastic parameter, and the second-order approximations in the £ ¤T whole parameter space are computed around u cT ζc . Unless said otherwise, u c = 0 was chosen in the examples of this chapter. An important question is how the choice of u c impacts the accuracy of the method. Indeed, the optimal setting

CHAPTER 9. APPLICATION OF STOCHASTIC OPTIMAL POWER FLOW IN THE IEEE 164 39 BUS SYSTEM u ∗ of the control variables will be different from u c , and the second-order approximations computed with u c may not be accurate. In particular, new corner points (see Definition 5.1) can appear, meaning that the smooth parts constituting the stability boundary in the space of stochastic system parameters for u c may be different from the ones for u ∗ .

9.3 Case 1: correct pre-distribution This case study aims at answering the first and the third questions above (“Validity of the approximations” and “Choice of u c for the second-order approximations”).

9.3.1 Results In this case, the distribution of the load used when solving the S-OPF problem is 0 assumed to be N (P D ,C 0 ), i.e. the same as the one used for phase 1. The S-OPF problem in (9.1) is solved for different values of α, ranging from 0.01 to 10−6 . The results are given in Table 9.1, where C ∗ corresponds to the cost CG (u ∗ ) of the optimal solution. Table 9.1: The variation of the optimal solution when varying α. α u 1∗ u 2∗ u 3∗ u 4∗ u 5∗ u 6∗ u 7∗ u 8∗ u 9∗ C∗ α/αˆ MC

1e − 2

1e − 3

1e − 4

1e − 5

1e − 6

0.8289 -0.4240 0.1350 0.2037 -0.2415 -0.2484 0.0171 0.1362 0.5958 1.1450 1.0085

1.0771 -0.3507 0.2570 0.3454 -0.2292 -0.2379 0.1215 0.2763 0.7873 2.2756 1.0100

1.3183 -0.3109 0.3593 0.4690 -0.2337 -0.2429 0.1878 0.3942 0.9628 3.2454 0.9781

1.5864 -0.2888 0.4659 0.6188 -0.2573 -0.2649 0.1925 0.5037 1.1383 4.1868 1.0193

1.8371 -0.2558 0.5944 0.7533 -0.3364 -0.3514 0.1910 0.6169 1.3257 5.0521 1.0205

As discussed in Section 9.2, it is not reasonable to use the approximation of the stability boundary computed at u c = 0 when the optimal solution u ∗ differs from u c . Therefore, in the simulations for this case, approximations of the stability boundary were also computed for u c = u ∗ when α = 10−4 . These approximations were used to solve the S-OPF problem for α = 10−5 and α = 10−6 . This is indicated by the vertical bar in the table. The ratio α/αˆ MC and the distance between the optimal control setting and the base case setting°(u c = 0)°when using the approximations at u c0 = 0 are given in Table 9.2. The distance is °u ∗ − u c0 ° = ku ∗ k.

165

9.3. CASE 1: CORRECT PRE-DISTRIBUTION

Table 9.2: Optimal solutions when using u c = 0. α ∗

ku k α/αˆ MC

1e − 2

1e − 3

1e − 4

1e − 5

1e − 6

1.1918 1.0085

1.5128 1.0100

1.8479 0.9781

2.1880 0.8642

2.5283 0.8238

9.3.2 Discussion This case study aimed at answering the first and the third questions in Section 9.2: validity of the approximations and choice of u c for computing the second-order approximations. As far as the validity of the approximations is concerned, the results from Table 9.1 can be used since new second-order approximations were computed with new value of u c for the two cases (α = 10−5 and α = 10−6 ) for which the optimal settings of the parameters are the farthest away from the base case u = 0. The ratio α/αˆ MC should be as close to 1 as possible since a ratio of one means that when the system is run with the optimal generation re-dispatch, the level of system security is exactly the one defined in the S-OPF problem (up to the errors in the estimation of α by Monte-Carlo simulations). A ratio larger (respectively smaller) than one indicates that the probability that the system becomes unstable has been underestimated (respectively overestimated) when solving the S-OPF problem, and, hence, that the level of system security has been overestimated (respectively underestimated). For all values of α, the ratio is in the range [0.9781, 1.0214], which corresponds to a maximum error (α − αˆ MC )/αˆ MC of 3.2%. Therefore, the S-OPF problem is solved accurately, despite the approximations which had to be made to render it tractable. As far as the choice of u c is concerned, the results from Table 9.1 can be compared with those from Table 9.2 for α = 10−5 and α = 10−6 . The former were obtained using second-order approximations computed with u c chosen as the optimal setting u ∗ for α = 10−5 , while the latter were obtained using u c = 0. It can be observed that when using the same second-order approximations for all values of α (case u c = 0), the results are less and less accurate as the distance between the optimal setting u ∗ and the approximation point u c increases. For α = 10−6 , the corresponding error is 17.6%. One possible explanation for the decrease in accuracy is that the second-order approximations are computed for one particular value of u, u c , corresponding to a stability boundary Σ, but the stability boundaries for other values of u can have other smooth parts compared to Σ. New corner points can appear. A final comment can be made on the costs. As expected, the optimal cost increases as the required level of security increases (i.e. as α decreases).

CHAPTER 9. APPLICATION OF STOCHASTIC OPTIMAL POWER FLOW IN THE IEEE 166 39 BUS SYSTEM

9.4 Case 2: incorrect pre-distribution This case study aims at discussing the second issue of Section 9.2, the effect of having imperfect forecasts.

9.4.1 Results In this case, to reflect the fact that the forecasts are different when computing secondorder approximations and when solving the S-OPF, the distribution of the load used in 1 1 the latter is assumed to be N (P D ,C 1). The value P D is obtained from the base case loading by decreasing the active power load with 2 p.u. at node 2 and increasing the active power loads with 1 p.u. at nodes 23 and 28. ¡ 1 ¢2 The matrix C 1 is, as before, c 1 diag P D , where c 1 is chosen such that the total load gets a standard deviation of 2%. Using this probability distribution for the loads, the S-OPF problem is solved again for the same values of α as before. As seen in the previous section, using second-order approximations whose approximation point u c is far from u ∗ decreases noticeably the accuracy of the methodology. In order to assess the impact of having imperfect forecasts, second-order approximations computed around u c = u ∗ for α = 10−4 have therefore been used for solving the S-OPF problem for α = 10−5 and α = 10−6 . The results are presented in Table 9.3. Table 9.3: The variation of the optimal solution when varying α. α u 1∗ u 2∗ u 3∗ u 4∗ u 5∗ u 6∗ u 7∗ u 8∗ u 9∗ C∗ α/αˆ MC

1e − 2

1e − 3

1e − 4

1e − 5

1e − 6

0.6185 -0.4425 0.1663 0.2360 -0.1989 -0.2016 0.0328 0.2332 0.5388 1.0912 1.0214

0.8407 -0.3594 0.2879 0.3751 -0.1796 -0.1845 0.1478 0.3749 0.7195 2.2029 1.0122

1.0281 -0.2913 0.3886 0.4909 -0.1655 -0.1721 0.2427 0.4928 0.8709 3.1503 0.9885

1.2731 -0.2515 0.3708 0.2521 -0.0950 -0.1014 0.3332 0.6503 1.0770 3.8685 1.0148

1.3643 -0.1745 0.5642 0.6943 -0.1453 -0.1549 0.4073 0.7003 1.1400 4.8648 1.0012

9.4.2 Discussion The results from Table 9.3, compared to those in Table 9.1 show that the fact that imperfect forecasts when computing the second-order approximations do not have a

9.5. COMPUTATIONAL TIME

167

noticeable impact on the results, the accuracy of which is similar to the one obtained with perfect forecasts. According to the results from both study cases, the main cause of inaccuracy is the use of second-order approximations computed around points u c distant from the optimal setting u ∗ . In contrast, the uncertainty in the stochastic system parameters ζ does not have a noticeable negative influence on the accuracy of the method. For each post- and pre-contingency stability boundary, the second-order approximation is a set of second-order approximations, one for each smooth part of the stability boundary in the space of stochastic system parameters. The corner points in this space are thus accounted for, while the corner points in the extended space (space of control variables u and stochastic system parameters ζ) are not. This can explain the observations made above. In order to address the issue of disregarding corner points appearing for different values of u c , one of the two following approaches could be used. 1. For each system state – pre- and post-contingency states – second-order approximations can be computed for different values of u c . We would then obtain a collection of approximations for different approximation points u c . The approximations corresponding to the closest u c to u ∗ could then be used. 2. An iterative approach could be developed where, firstly, second-order approximations for a given u c are computed; secondly, the S-OPF to get u ∗ is solved; thirdly, new second-order approximations are computed, this time around u c = u ∗ , and the S-OPF problem is re-solved with these new approximations to get a new u ∗ . The process is iterated as long as the obtained u ∗ is not close enough to the used approximation point u c

9.5 Computational time The application of the S-OPF formulation to optimal activation of balancing bids must be carried out on real-time. The requirement is that the S-OPF problem must be solved in a few minutes. In the IEEE 39 bus system, the S-OPF problem is solved in about 10 seconds using parallel computing on 8 cores in Matlab. The most time consuming step is the computation, for all pairs of smooth parts, of the probabilities pˆc in (8.25) and its derivative with respect to u that the parameters are beyond two of the smooth parts. For very large systems, it is expected that the necessary computation time would render the method unusable for real-time operations. When analyzing how the probabilities pˆc depend on u, it can be noticed that the dependency on u appears in c 1 , c 2 and ρ (which are computed with help of the CornishFisher expansion, see (8.20) and (8.21)). Whereas the computation of the derivative of pˆc with respect to u is time demanding, the computations of derivatives of c 1 , c 2 and ρ with respect to u are not. A possible way of reducing the computational time would therefore be to build tables of pˆc and its derivatives with respect to c 1 , c 2 and ρ. The entries in the tables would correspond to different values of c 1 , c 2 and ρ depending

CHAPTER 9. APPLICATION OF STOCHASTIC OPTIMAL POWER FLOW IN THE IEEE 168 39 BUS SYSTEM on u. Both pˆc and its derivative with respect to u could then be estimated for all u by interpolation of entries in the tables.

Chapter 10

Conclusion and future work 10.1 Conclusion The thesis has dealt with the frequency control schemes in power systems with large amounts of wind power. A method has been developed for optimally choosing the generation re-dispatch with least cost while maintaining a pre-defined level of system security. Chapter 2 gave a review of the current way of operating frequency control schemes, and identified the challenges faced by today’s tools with large amounts of wind power in the system. Issues due to larger variability and diminished predictability are expected, which the tools used today will not be able to address. Hence, a need for new tools arises to account for the larger uncertainty, introduced by wind power, which the system operator faces. A state-of-the-art of the current research which addresses this need was given. Particularly fitting are stochastic optimal power flow formulations, a class of optimization problem which accounts for the uncertainty directly in the optimization problem. Whereas the current way of operating the system hedges against risk by adding security margins, the optimal decisions computed in the proposed S-OPF problems guarantee a specified level of system security while taking into account the entire probability distribution of the uncertainty, thus allowing a more flexible and optimal operation of the system. System security and stability are of high importance when operating power systems because system instability and collapse entail high costs for the society. Hence, the S-OPF problem proposed in this thesis has been formulated so that different types of system stability can be taken into account. The given formulation is general, and the problem need not be restricted to the operation of tertiary control reserves, although it is the scope chosen in this work. Several applications are discussed for future work in Section 10.2. When solving the S-OPF problem, the challenge arose of analytically characterizing the stability boundary – the set of all considered stability limits. Because no analytical expressions are known, approximations were needed. Second-order approximations of the stability boundary were therefore developed, and can be used as analytical ap169

170

CHAPTER 10. CONCLUSION AND FUTURE WORK

proximations of the actual stability boundary to solve S-OPF problems. Applications of second-order approximations are however not restricted to the sole case of S-OPF problems. Other applications are proposed in Section 10.2 for future work. The accuracy of the second-order approximations was assessed in the IEEE 9 bus system. While second-order approximations are local approximations, they approximate accurately the actual stability boundary over a large domain. Other approximations were introduced in the solution methodology for the proposed S-OPF problem as discussed in Chapter 8. The accuracy of the methodology was assessed by running Monte Carlo simulations which served as a reference. It was shown that, despite the approximations made when solving the S-OPF problem, the optimal setting of the control parameters guaranteed a level of system security close to the chosen one. This supports the use of the chosen approximations and, in particular, the use of the second-order approximations. To summarize, the following conclusions can be drawn: – Power system operation and planning will be subjected to more uncertainty as larger amounts of variable resources such as wind power are installed. – Existing tools are deterministic and do not account for the uncertainty. Security margins must then be considered, thus preventing to optimally use the existing resources. More flexible tools for decision making under uncertainty are therefore needed which directly account for the uncertainty when computing the optimal decision. – In the context of generation re-dispatch (for example for optimal activation of balancing bids), the proposed S-OPF formulation is such a tool. – Results in the IEEE 39 bus system have shown that optimal decision obtained when solving the S-OPF indeed accounts for the uncertainty, and allows the system operator to run the system at a specified level of security.

10.2 Future work The S-OPF formulation developed in this thesis has been part of the first phase of the project. Possible directions for future research are proposed below.

10.2.1 Application to systems with large amounts of wind power The developed method can be applied in generic power systems. Further applications must be done to study more specifically power systems with large amounts of wind power. Wind power production will enter the stochastic system parameter ζ in the S-OPF problem (8.3), but the question of how to model wind power production arises. Recent work suggests that unlike load variations, wind power forecast errors do not follow a Gaussian distribution [9]. A possible way of getting an approximation of the probability distribution function of wind power variations is to obtain the moments or cumulants of the variations (for example using historical data), and then use these quantities to approximate the probability distribution function as is done in probabilistic optimal power flows, see Section 2.5.4.

10.2. FUTURE WORK

171

Proper models for wind power plants must be used when studying power systems with large amounts of wind power. The stability limits presented in Section 5.3 depend on the models used. It was also discussed in Section 2.1.3 that different wind turbine designs have different impacts on voltage stability. In [39] for example, the results in a small power system indicates that doubly-fed induction generators are beneficial for small-signal stability, but adverse to voltage stability. The choice of the wind turbine designs, and their models, must therefore be made carefully in order to draw relevant conclusions when studying power systems with large amounts of wind power.

10.2.2 Comparing different settings for primary and secondary control schemes As mentioned in Section 1.2, primary and secondary frequency control schemes have been treated as given parameters. The S-OPF formulation developed here could be used to assess different settings (or designs) for primary and secondary frequency control. For a pre-defined level of system security, the generation re-dispatch costs for tertiary control could be compared for these different settings, and the optimal one could be chosen. For example, different settings for the distribution of primary control among the generators could be assessed. This is important since the location of primary control has an effect on system stability as seen in Section 1.1.3, and optimizing it would allow the system operator to run the system with lower costs for the same level of system security. The proposed S-OPF formulation could also be used to evaluate different designs for LFC, the secondary control schemes which will be introduced in the Nordic system in January 2013, see Section 2.3.4. Another issue is the possibility to pay thermal or hydro power plant owners to run below a certain levels in order to keep margins. This could also be studied with the proposed S-OPF formulation.

10.2.3 Comparison with today’s way of operating the system Particularly interesting in the scope of this project would be applications of the developed method in the Swedish power system. The way in which balancing bids are activated today, presented in Section 2.5.3 could be compared with the proposed method. The benefits of accounting for uncertainty using this method could be evaluated. For example, by directly accounting for the uncertainty when activating the balancing bids, the security margin on the critical transmission corridors could be lowered (see Section 2.5.3), thus rendering the system operation more efficient. A more efficient use of primary control as discussed in the previous section could also contribute in lowering these margins.

172

CHAPTER 10. CONCLUSION AND FUTURE WORK

10.2.4 Taking post-contingency corrective actions into account After a contingency happens, the system operator can take actions such as load shedding, if necessary, in order to stabilize the system. The proposed S-OPF formulation does not take these actions into account. With post-contingency corrective actions accounted for, the requirements on the optimal pre-contingency setting are less restrictive because the post-corrective actions can help maintain system stability, see Section 2.5.2. Therefore, the costs of generation re-dispatch would be lower. The costs of postcorrective actions are high, but these actions need to be taken only after contingencies, which have a small probability of occurrence. The total cost of generation re-dispatch and of the necessary corrective actions can thus be expected to be lower than if these actions are not considered.

10.2.5 Using wind power in frequency control schemes Another interesting application is participation of wind power in frequency control schemes. For example, wind curtailment orders could be included among the bids submitted for generation re-dispatching. In power systems with large amounts of wind power, it may be necessary to curtail wind power in order to maintain a safe level of system security. The method developed here would then give the optimal generation re-dispatch, including wind curtailment, necessary to guarantee this level of system security. When post-contingency actions are also taken into account, the requirements on the optimal generation re-dispatch are less restrictive are discussed above. Hence, in the situations in which wind power must be curtailed, lower amounts of wind power curtailment can be expected if the S-OPF formulation considers corrective actions, thus allowing for more wind power to be integrated. This further emphasizes the need of including post-contingency corrective actions.

10.2.6 Scalability of the method Further research is needed in order to study how the computations of second-order approximations and the methodology to solve the S-OPF problem in large-scale power systems. Since contingencies are taken into account, second-order approximations must be computed for all post-contingency systems, which is computationally demanding. Furthermore, in large power systems, the dimension of the parameter space becomes large, and many searches must be initiated in order to identify all smooth parts of the actual stability boundary as described in Section 6.2.2 and 7.2. Hence, the computation time needed for computing second-order approximations will rise in large systems. The two steps which take most time in these computations are the search for the stability limit in a given direction and the search for the most important point from this stability limit. The time associated with the computation of the second-order approximation around

10.2. FUTURE WORK

173

this most important point is much smaller than the time required to carry out these two steps. The most important point is sought using forecasts as explained in Section 6.8, and the closer to the most important point, the better the second-order approximations. It is thus important to compute the approximations as close to the operating period as possible to have forecasts as good as possible. The time horizon considered in these forecasts depends on the computation time of the second-order approximations. Therefore, faster methods to compute these approximations would both enhance the accuracy of the method and enable them to be used in large systems. As far as the S-OPF problem is concerned, it was discussed in Section 9.5 that the proposed methodology, although suitable on small power systems, is expected to be too time demanding to allow for real-time applications. Possible enhancements for future work were presented.

10.2.7 Optimal operation for the whole operating period The S-OPF gives one optimal solution for the specific point in time at which it is used. However, the system operator seeks at maintaining a secure operation of the power system at the least possible cost over the whole operating period. Solving this problem requires to consider not just one point in time but also the expected costs, for the rest of the operating period, associated with the actions taken, see Remark 8.2. Methods for dealing with this issue while still considering uncertainty through the stochastic system parameters must be developed. In [90] and [86], an approach to tackle this is proposed. The random parameters – the uncertainty in the system – are modeled by random processes in order to also take time into account. The authors point out that the developed approach suffers from tractability issues. Further research is required to develop numerical schemes which make this method scalable.

10.2.8 Other uses of the second-order approximations In the scope of this project, the second-order approximations were used to solve the proposed S-OPF problem. The usefulness of the approximations is however not reduced to the sole S-OPF problem. Other applications could be developed. For example, an ongoing project studies applications of these second-order approximations to load shedding. Having analytical expressions of the second-order approximations allows for the design of optimal load shedding schemes which bring the system back to the approximated stable operation domain, if necessary, at the minimum cost.

Part IV

Appendices

175

Appendix A

Power System Data

A.1 Reference Power System

The power system is depicted in Figure A.1, and is taken from [59].

Pg2 2

Pg1

1

5

6

7

4

Load B 3

Pg3 Figure A.1: Reference power system.

177

178

APPENDIX A. POWER SYSTEM DATA

Table A.1: Data for the reference power system of Figure A.1. Parameter

Description

Value

X 15 X 26 X 37 X 74 X 56 X 67 Xdi 0 Xdi r1 r2 r3 r4 E lim fi

Reactance of line 1-5 Reactance of line 2-6 Reactance of line 3-7 Reactance of line 7-4 Reactance of line 5-6 Reactance of line 6-7 d -axis reactance of generator i d -axis transient reactance of generator i Tap ratio of transformer between buses 1 and 5 Tap ratio of transformer between buses 2 and 6 Tap ratio of transformer between buses 3 and 7 Tap ratio of transformer between buses 7 and 4 Limit of exciter i

0.032 p.u. 0.032 p.u. 0.016 p.u. 0.016 p.u. 0.12 p.u. 0.005625 p.u. 0.8 p.u. 0.16 p.u. 1.05 p.u. 1.05 p.u. 1.05 p.u. 1.0 p.u. 2.5968 p.u.

Ki B Vr i

Gain of AVR i Susceptance of the capacitor at bus 4 Reference voltage of AVR i

100 p.u. 0.25 p.u 1.0 p.u.

The three generators are represented by the following steady-state equations: ! Ã 0 0 xd i xd i − xd i 0 Vi cos (δi − θi ) , (A.1) E qi = Ef i + 0 xd i xd i 0

Pg i =

E qi Vi 0

xd i

Qg i = −

sin (δi − θi ) ,

´ 0 Vi ³ Vi − E qi cos (δi − θi ) , 0 xd i

(A.2) (A.3)

where the different parameters can be found in Table A.1. Each generator is equipped with an automatic voltage regulator (AVR) with an overexcitation limiter (OXL) modeled in steady-state by E f i = K i (Vr i − Vi )

E f i = E lim fi

under AVR control

under OXL control.

(A.4) (A.5)

A.2 IEEE 9 bus system The IEEE 9 bus system is shown in Figure A.2. It consists of three generators located at buses 1, 2 and 3 supplying three loads at buses 5, 6 and 8.

179

A.2. IEEE 9 BUS SYSTEM

Figure A.2: IEEE 9 bus system.

180


Table A.2: Data for the IEEE 9 bus system Parameter

Description

Value [p.u.]

E lim f1 E lim f2 E lim f3

Limit of exciter 1

2

Limit of exciter 2

2.2

Limit of exciter 3

1.7

Vref−1 Vref−i

Reference voltage of AVR 1 Reference voltage of AVRs 2 and 3

1.1 1.05

The generators are equipped with AVR. The generators are modeled by their oneaxis model: δ˙i = ωi ,

0



0

E qi Vi 1  sin (δi − θi ) − D i ωi  , P mi − 0 Mi Xdi Ã ! 0 X d i − xd i 1 xd i 0 = 0 E f i − 0 E qi + Vi cos (δi − θi ) , 0 Td 0i Xdi Xdi

ω˙i = E˙ qi



(A.6a) (A.6b)

(A.6c) (A.6d)

and the AVRs by the following equations ¢ 1 ¡ 0 −E f i + K Ai (Vref-i − Vi ) E˙f i = Tei 0

0 = −E f i + E lim fi

under AVR control,

under OXL control.

(A.7a) (A.7b)

The values of the parameters can be found in [72, Section D.3]. The values which differ from this reference can be found in Table A.2.

181

A.3. IEEE 39 BUS SYSTEM

A.3 IEEE 39 bus system The IEEE 39 bus system is depicted in Figure A.3. It has 10 generators and 19 loads. The generators are equipped with AVR. The models are the same as the ones used in Section A.2. The values of the parameters can be found in [72, Section D.4] The values which differ from this reference can be found in Table A.3.

Figure A.3: IEEE 39 bus system.

182


Table A.3: Data for the IEEE 39 bus system Parameter

Value [p.u.]

Parameter

Value [p.u.]

E lim f1 E lim f2 E lim f3 E lim f4 E lim f5 E lim f6 E lim f7 E lim f8 E lim f9 E lim f 10

1.3677

Vref−1

2.0375

1.2052

Vref−2

1.1128

1.3520

Vref−3

2.1353

1.4029

Vref−4

2.1864

1.0297

Vref−5

2.0183

1.4489

Vref−6

2.0028

1.1107

Vref−7

2.0501

1.3105

Vref−8

2.0714

1.0636

Vref−9

2.1956

1.3168

Vref−10

1.3898

Appendix B

Derivatives of A and Φ B.1 Implicit function z = Φ(λ) We recall from Section (5.4.5) that the variables z appearing in the power system model F (z) ( f (x, y), F (z) = (B.1) g (x, y), can be expressed as a function of the parameters λ: z = Φ(λ). In addition, the algebraic variables y can be expressed as a function of x, y(x), and we thus have the following relations · ¸ · ¸ x ϕ(λ) z= = = Φ(λ). (B.2) y = y(x) y(ϕ(λ)) The first and second-order derivatives of Φ with respect to λ are needed in order to compute the normal and Weingarten maps to some smooth parts of the stability boundary, see Sections 5.4 and 6.3. In addition to these quantities, the first- and second-order derivatives of the dynamic Jacobian A are also needed for these computations. The following sections give expressions of all these necessary quantities. The beginning of each section gives a summary of which equations to use so that the reader can directly apply them without having to go through the derivations. The detailed derivations are given after these summaries.

B.2 First-order derivatives of A and Φ B.2.1 Summary The steps to compute the first-order derivatives of A and Φ with respect to λ are as follows 183

184

APPENDIX B. DERIVATIVES OF A AND Φ –

dA dλ :

1. Compute ϕλ and y x ϕλ with (B.12). 2. Compute y xx ϕλ with (B.15). 3. Compute

∂A ∂λ

with (B.8).

– Φλ for loadability limits of type SNB: Use (B.19). – Φλ for loadability limits of type other than SNB: Use (B.11).

B.2.2 Derivatives of A Recall that the dynamic Jacobian is defined by A = f x − f y g y−1 g x .

(B.3)

Differentiating g (x, y) = 0 with respect to x gives 0 = g x + g y yx ,

(B.4)

y x = −g y−1 g x .

(B.5)

A = fx + f y yx .

(B.6)

so that

Hence, we can rewrite A as

Differentiating A with respect to λ, we get ∂A = f xλ + f xx ϕλ + f x y y x ϕλ ∂λ ¡ ¢ + f yλ + f y x ϕλ + f y y y x ϕλ y x ¡ ¢ + f y y xx ϕλ .

(B.7)

The parameters λ are often chosen to be loads or generators’ mechanical power. In these general cases, the system equations are linear in the parameters λ, which means that f xλ = 0 and f yλ = 0 so that the first-order derivative becomes ∂A = f xx ϕλ + f x y y x ϕλ ∂λ ¡ ¢ + f y x ϕλ + f y y y x ϕλ y x ¡ ¢ + f y y xx ϕλ .

(B.8)

What is unknown in this expression is ϕλ and y xx . In practice, even if y x can be computed with (B.5), y x ϕλ is computed as the same time as ϕλ as will be seen in the next section.

B.2. FIRST-ORDER DERIVATIVES OF A AND Φ

185

B.2.3 First-order derivative of Φ and associated quantities Formulas for loadability limits other than SNB By differentiating the steady state equations F (z) = 0 with respect to λ, we get 0 = F z Φλ + F λ ,

(B.9)

Φλ = −F z−1 F λ .

(B.10)

¸ ϕλ . y x ϕλ

(B.11)

¸ ϕλ = −F z−1 F λ . y x ϕλ

(B.12)

so that

From (5.65), we have also that Φλ =

·

Hence Φλ =

·

To get y xx , we differentiate (B.4) with respect to x, we get ¡ ¢ 0 = g xx + g x y y x + g y x + g y y y x y x + g y y xx .

(B.13)

Therefore

¡ ¡ ¢ ¢ y xx = −g y−1 g xx + g x y y x + g y x + g y y y x y x .

In practice, y xx ϕλ is computed directly instead of y xx alone: ¡ ¡ ¢ ¢ y xx ϕλ = −g y−1 g xx ϕλ + g x y y x ϕλ + g y x + g y y y x ϕλ y x ϕλ ,

(B.14)

(B.15)

which can be done because y x ϕλ can be computed from (B.12). Formulas for SNB

The formulas given above hold for all types of loadability limits but SNB. Indeed, at a saddle-node bifurcation point, the matrix F z is singular, so that (B.10) does not hold. One additional equation is needed. Let w and v be left and right eigenvectors corresponding to the zero eigenvalue of F z at the SNB point. The vector w can be seen as a row vector, and both vectors can be chosen so that w v = 1. Since they correspond to a zero eigenvalue, the following hold wF z v = 0.

(B.16)

Differentiating this equation with respect to λ gives w (F zz Φλ + F zλ ) v +

dv dw (F z v) + (wF z ) = 0. dλ dλ

(B.17)

186

APPENDIX B. DERIVATIVES OF A AND Φ

The two last terms are equal to zero since both F z v and wF z are equal to zero, so that we get w (F zz Φλ + F zλ ) v = 0.

(B.18)

¸ ¸ · Fz Fλ = 0, Φ + λ v T wF zz v T wF zλ

(B.19)

Together with (B.9) it gives ·

from which Φλ can be obtained.

B.3 Second derivative of A and Φ B.3.1 Summary In order to compute the second-order derivative of Φ (necessary for computing the Weingarten map of smooth parts of the stability boundary corresponding to operational limits), we use (B.26) and (B.27). In order to compute the second-order derivative of A (necessary to compute the Weingarten map of smooth parts of the stability boundary corresponding to Hopf bifurcations), we proceed as follows: 1. Compute y x , ϕλ , y x ϕλ and y x xϕλ from (B.5), (B.12) and (B.15), respectively, 2. Compute ϕλλ and y xx (ϕλ , ϕλ ) + y x ϕλλ from (B.26) and (B.27), 3. Compute y xxx (ϕλ , ϕλ ) + y xx ϕλ,λ from (B.30), 4. Compute

d2 A dλdλ

from (B.21).

B.3.2 Formulas In the formulas for computing the Weingarten map for Hopf bifurcations, we need the first- and second-order derivatives of A with respect to λ, see (6.26) and (6.27). We saw in Section B.1 how to compute the first-order derivative, so it remains to compute the second-order derivative of A. To do so, we start off from the first-order derivative in (B.8) ∂A = f xx ϕλ + f x y y x ϕλ ∂λ ¡ ¢ + f y x ϕλ + f y y y x ϕλ y x + f y y xx ϕλ .

(B.20)

B.3. SECOND DERIVATIVE OF A AND Φ

187

and differentiate one more time to get d2 A = f xxx (ϕλ , ϕλ ) + f xx y (ϕλ , y x ϕλ ) + f xx ϕλλ dλdλ + f x y x (y x ϕλ , ϕλ ) + f x y y (y x ϕλ , y x ϕλ ) + f x y (y x x (ϕλ , ϕλ ) + y x ϕλλ )

+ f y xx (y x , ϕλ , ϕλ ) + f y x y (y x , ϕλ , y x ϕλ ) + f y x (y x , ϕλλ ) + f y x (y x x ϕλ , ϕλ )

(B.21)

+ f y y x (y x , y x ϕλ , ϕλ ) + f y y y (y x , y x ϕλ , y x ϕλ )

+ f y y (y x x ϕλ , y x ϕλ ) + f y y (y x , y x x (ϕλ , ϕλ ) + y x ϕλλ ) ¡ ¢ + f y x (y x x ϕλ , ϕλ ) + f y y (y x x ϕλ , y x ϕλ ) + f y y x x x (ϕλ , ϕλ ) + y x x ϕλ,λ .

Expressions for y x , ϕλ , y x ϕλ and y x xϕλ have already been given in (B.5), (B.12) and (B.15). Unknown in the equations above are the quantities in bold, that is – ϕλλ , – y xx (ϕλ , ϕλ ) + y x ϕλλ , – y xxx (ϕλ , ϕλ ) + y xx ϕλ,λ . To get ϕλλ and y xx (ϕλ , ϕλ ) + y x ϕλλ , we start off from (B.9) 0 = F z Φλ + F λ ,

(B.22)

and differentiate one more time with respect to λ: 0 = F zz (Φλ , Φλ ) + F z Φλλ ,

(B.23)

where we assumed that the system equations were linear in λ so that F zλ = 0. For Hopf bifurcations which we consider here, F z is nonsingular, so we get Φλλ = −F z−1 F zz (Φλ , Φλ ).

(B.24)

From (B.11), which we repeat here ·

¸ ϕλ , y x ϕλ

(B.25)

¸ ϕλλ . y xx (ϕλ , ϕλ ) + y x ϕλλ

(B.26)

Φλ = Φλλ can also be expressed as Φλλ =

·

Hence, we get that ·

¸ ϕλλ = −F z−1 F zz (Φλ , Φλ ), y xx (ϕλ , ϕλ ) + y x ϕλλ

(B.27)

which gives both ϕλλ and y xx (ϕλ , ϕλ ) + y x ϕλλ . In order to get the last unknown quantity y xxx (ϕλ , ϕλ )+y xx ϕλ,λ , we start off from (B.4) 0 = g x + g y yx ,

(B.28)

188

APPENDIX B. DERIVATIVES OF A AND Φ

which we differentiate twice with respect to λ. This equation is similar to A = fx − f y yx ,

(B.29)

which was used to get the first- and second-order derivatives of A with respect to λ. The right-hand sides of these two equations are equal when substituting g for f . Hence, when differentiated twice with respect to λ, the right-hand side of (B.28) will have the same expression as the right-hand side of (B.21) with g substituted for f . Then rearranging the terms, we get y xxx (ϕλ , ϕλ ) + y xx ϕλ,λ = h −g y−1 g xxx (ϕλ , ϕλ ) + g xx y (ϕλ , y x ϕλ ) + g xx ϕλλ

+g x y x (y x ϕλ , ϕλ ) + g x y y (y x ϕλ , y x ϕλ ) + g x y (y xx (ϕλ , ϕλ ) + y x ϕλλ )

+g y xx (y x , ϕλ , ϕλ ) + g y x y (y x , ϕλ , y x ϕλ ) + g y x (y x , ϕλλ ) + g y x (y xx ϕλ , ϕλ )

+g y y x (y x , y x ϕλ , ϕλ ) + g y y y (y x , y x ϕλ , y x ϕλ )

+g y y (y xx ϕλ , y x ϕλ ) + g y y (y x , y xx (ϕλ , ϕλ ) + y x ϕλλ ) i +g y x (y xx ϕλ , ϕλ ) + g y y (y xx ϕλ , y x ϕλ ) .

(B.30)

Bibliography [1]

Thomas Ackermann. Wind Power in Power Systems. Wiley, 2005.

[2]

Mark Ahlstrom, James Blatchford, Matthew Davis, Jacques Duchesne, David Edelson, Ulrich Focken, Debra Lew, Clyde Loutan, David Maggio, Melinda Marquis, Michael McMullen, Keith Parks, Ken Schuyler, Justin Sharp, and David Souder. Atmospheric Pressure. IEEE Power and Energy Magazine, 9(6):97–107, November 2011.

[3]

V. Ajjarapu and C. Christy. The continuation power flow: a tool for steady state voltage stability analysis. IEEE Transactions on Power Systems, 7(1):416–423, 1992.

[4]

O. Alsac and B. Stott. Optimal Load Flow with Steady-State Security. IEEE Transactions on Power Apparatus and Systems, PAS-93(3):745–751, May 1974.

[5]

Fernando L. Alvarado. Bifurcations in nonlinear systems-computational issues. In IEEE International Symposium on Circuits and Systems, pages 922–925. IEEE, 1990.

[6]

Fernando L. Alvarado and Ian Dobson. Computation of closest bifurcations in power systems. IEEE Transactions on Power Systems, 9(2):918–928, May 1994.

[7]

Andeggs. Spherical coordinate system. http://en.wikipedia.org/wiki/ Spherical_coordinate_system, October 2012. Accessed on 5 October 2012.

[8]

Brett W. Bader, Tamara G. Kolda, et al. Matlab tensor toolbox version 2.5. http: //www.sandia.gov/~tgkolda/TensorToolbox/, January 2012. Accessed on 24 August 2012.

[9]

H. Bludszuweit, J.A. Dominguez-Navarro, and A. Llombart. Statistical Analysis of Wind Power Forecast Error. IEEE Transactions on Power Systems, 23(3):983–991, August 2008.

[10]

Marcel Cailliau, Maro Foresti, and Cesar Villar. Winds of Change. IEEE Power and Energy Magazine, 8(5):53–62, September 2010.

[11]

C.A. Canizares and Fernando L. Alvarado. Point of collapse and continuation methods for large AC/DC systems. IEEE Transactions on Power Systems, 8(1):1– 8, 1993. 189

190

BIBLIOGRAPHY

[12]

C.A. Canizares, W. Rosehart, A. Berizzi, and C. Bovo. Comparison of voltage security constrained optimal power flow techniques. In Power Engineering Society Summer Meeting, pages 1680–1685, Vancouver, BC, Canada, 2001. IEEE.

[13]

F. Capitanescu, J.L. Martinez Ramos, P. Panciatici, D. Kirschen, A. Marano Marcolini, L. Platbrood, and L. Wehenkel. State-of-the-art, challenges, and future trends in security constrained optimal power flow. Electric Power Systems Research, 81(8):1731–1741, August 2011.

[14]

F. Capitanescu, T. Van Cutsem, and L. Wehenkel. Coupling Optimization and Dynamic Simulation for Preventive-Corrective Control of Voltage Instability. IEEE Transactions on Power Systems, 24(2):796–805, May 2009.

[15]

F. Capitanescu and L. Wehenkel. A New Iterative Approach to the Corrective Security-Constrained Optimal Power Flow Problem. IEEE Transactions on Power Systems, 23(4):1533–1541, November 2008.

[16]

Florin Capitanescu, Mevludin Glavic, Damien Ernst, and Louis Wehenkel. Applications of security-constrained optimal power flows. In In Proceedings of Modern Electric Power Systems Symposium, MEPS06, 2006.

[17]

Florin Capitanescu, Mevludin Glavic, Damien Ernst, and Louis Wehenkel. Contingency Filtering Techniques for Preventive Security-Constrained Optimal Power Flow. IEEE Transactions on Power Systems, 22(4):1690–1697, November 2007.

[18]

Florin Capitanescu and Thierry Van Cutsem. Unified Sensitivity Analysis of Unstable or Low Voltages Caused by Load Increases or Contingencies. IEEE Transactions on Power Systems, 20(1):321–329, February 2005.

[19]

J. Carpentier. Contribution à l’étude du dispatching économique. Bulletin de la Société d’Electricité, 3:431–447, 1962.

[20]

R.D. Christie and A. Bose. Load frequency control issues in power system operations after deregulation. IEEE Transactions on Power Systems, 11(3):1191–1200, 1996.

[21]

European Commission. Energy: Action Plans and Forecasts. http://ec. europa.eu/energy/renewables/action_plan_en.htm, 2009. Accessed on 31 July 2012.

[22]

J. Condren and T.W. Gedra. Expected-Security-Cost Optimal Power Flow With Small-Signal Stability Constraints. IEEE Transactions on Power Systems, 21(4):1736–1743, November 2006.

[23]

J. Condren, T.W. Gedra, and P. Damrongkulkamjorn. Optimal power flow with expected security costs. IEEE Transactions on Power Systems, 21(2):541–547, 2006.

[24]

J.F. Conroy and R. Watson. Frequency response capability of full converter wind turbine generators in comparison to conventional generation. Power Systems, IEEE Transactions on, 23(2):649–656, 2008.

BIBLIOGRAPHY

191

[25]

Ian Dobson. An iterative method to compute a closest saddle node or Hopf bifurcation instability in multidimensional parameter space. In [Proceedings] 1992 IEEE International Symposium on Circuits and Systems, volume 5, pages 2513– 2516. IEEE, 1992.

[26]

Ian Dobson. Observations on the geometry of saddle node bifurcation and voltage collapse in electrical power systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 39(3):240–243, March 1992.

[27]

Ian Dobson. Computing a closest bifurcation instability in multidimensional parameter space. Journal of Nonlinear Science, 3:307–327, 1993.

[28]

Ian Dobson. The irrelevance of load dynamics for the loading margin to voltage collapse and its sensitivities. In Bulk power system voltage phenomena III, voltage stability, security & control, Proceedings of the ECC/NSF workshop, Davos, Switzerland, 1994.

[29]

Ian Dobson, Fernando L. Alvarado, and Christopher L. DeMarco. Sensitivity of Hopf bifurcations to power system parameters. In [1992] Proceedings of the 31st IEEE Conference on Decision and Control, pages 2928–2933. IEEE, 1992.

[30]

Ian Dobson and Hsiao-Dong Chiang. Towards a theory of voltage collapse in electric power systems. Systems & Control Letters, 13(3):253–262, September 1989.

[31]

Ian Dobson, Scott Greene, Rajesh Rajaraman, Christopher L. DeMarco, Fernando L. Alvarado, Mevludin Glavic, Jianfeng Zhang, and Ray Zimmerman. Electric Power Transfer Capability: Concepts, Applications, Sensitivity, Uncertainty. Technical report 01-34, Power Systems Engineering Research Center, November 2001.

[32]

Ian Dobson and Liming Lu. Computing an optimum direction in control space to avoid stable node bifurcation and voltage collapse in electric power systems. IEEE Transactions on Automatic Control, 37(10):1616–1620, 1992.

[33]

Ian Dobson and Liming Lu. Voltage collapse precipitated by the immediate change in stability when generator reactive power limits are encountered. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 39(9):762–766, 1992.

[34]

Ian Dobson and Liming Lu. New methods for computing a closest saddle node bifurcation and worst case load power margin for voltage collapse. IEEE Transactions on Power Systems, 8(3):905–913, 1993.

[35]

Ian Dobson, Thierry Van Cutsem, Costas D. Vournas, CL DeMarco, M. Venkatasubramanian, T. Overbye, and C.A. Canizares. Voltage stability assessment: Concepts, practices and tools. IEEE Power Engineering Society, Power System Stability Subcommittee Special Publication, 2002.

[36]

Henry L. Durrwachter and Sherry K. Looney. Integration of Wind Generation Into the ERCOT Market. IEEE Transactions on Sustainable Energy, 3(4):862–867, October 2012.

192

BIBLIOGRAPHY

[37]

E-Bridge Consulting GMBH. Analysis and Review of Requirements for Automatic Reserves inthe Nordic Synchronous System. Technical Report September, September 2011.

[38]

Katherine Elkington. The Dynamic Impact of Large Wind Farms on Power System Stability. PhD thesis, Kungliga Tekniska Högskolan, 2012.

[39]

Katherine Elkington, Valerijs Knazkins, and Mehrdad Ghandhari. On the stability of power systems containing doubly fed induction generator-based generation. Electric Power Systems Research, 78(9):1477–1484, September 2008.

[40]

Energimyndigheten. Vindkraftsstatistik. Technical report, 2012.

[41]

ENTSO-E. Ucte - union for the coordination of the transmission of electricity.

https://www.entsoe.eu/the-association/history/ucte/. Accessed on 18 September 2012.

[42]

ENTSO-E. Nordel. https://www.entsoe.eu/the-association/history/ nordel/, 2009. Accessed on 16 September 2012.

[43]

European Parliament and Council of the European Union. Directive 2009/28/EC of the European Parliament and of the Council of 23 April 2009 on the promotion of the use of energy from renewable sources and amending and subsequently repealing Directives 2001/77/EC and 2003/30/EC. Official Journal of the European Union, L 140/16, 2009.

[44]

European Wind Energy Association. EU Energy Policy to 2050. Technical report, 2011.

[45]

MI Friswell. Calculation of Second-and Higher Order Eigenvector Derivatives. Journal of Guidance, Control, and Dynamics, 1995.

[46]

Global Wind Energy Council. Global Wind Report - Annual market update 2011. Technical report, 2012.

[47]

R.J. Green and D.M. Newbery. Competition in the British electricity spot market. Journal of political economy, pages 929–953, 1992.

[48]

Scott Greene, Ian Dobson, and Fernando L. Alvarado. Sensitivity of the loading margin to voltage collapse with respect to arbitrary parameters. IEEE Transactions on Power Systems, 12(1):262–272, 1997.

[49]

I Griva, SG Nash, and A Sofer. Linear and nonlinear optimization. 2009.

[50]

Timothy Heidel, John Kassakian, and Richard Schmalensee. Forward Pass: Policy Challenges and Technical Opportunities on the U.S. Electric Grid. IEEE Power and Energy Magazine, 10(3):30–37, May 2012.

[51]

I.A. Hiskens and R.J. Davy. Exploring the power flow solution space boundary. IEEE Transactions on Power Systems, 16(3):389–395, August 2001.

BIBLIOGRAPHY

193

[52]

Hannele Holttinen, Peter Meibom, Antje Orths, Bernhard Lange, Mark O’Malley, John Olav Giaever Tande, Ana Estanqueiro, Emilio Gomez, Lennart Söder, Goran Strbac, J Charles Smith, and Frans van Hulle. Impacts of large amounts of wind power on design and operation of power systems, results of IEA collaboration. Wind Energy, 14(2):179–192, 2011.

[53]

Hannele Holttinen, Peter Meibom, Antje Orths, Frans van Hulle, Bernhard Lange, Mark O’Malley, Jan Pierik, Bart Ummels, John Olav Giaever Tande, Ana Estanqueiro, Manuel Matos, Emilio Gomez, Lennart Söder, Goran Strbac, Anser Shakoor, João Ricardo, J Charles Smith, and Michael Milligan Erik Ela. IEA Wind Task 25 - Design and operation of power systems with large amounts of wind power. Technical report, 2009.

[54]

Hannele Holttinen, Michael Milligan, Brendan Kirby, Tom Acker, Viktoria Neimane, and Tom Molinski. Using Standard Deviation as a Measure of Increased Operational Reserve Requirement for Wind Power. Wind Engineering, 32(4):355– 377, June 2008.

[55]

Hannele Holttinen, Antje Orths, Peter Eriksen, Jorge Hidalgo, Ana Estanqueiro, Frank Groome, Yvonne Coughlan, Hendrik Neumann, Bernhard Lange, Frans Hulle, and Ivan Dudurych. Currents of Change. IEEE Power and Energy Magazine, 9(6):47–59, November 2011.

[56]

International Energy Agency. Technology Roadmap - Wind Energy. 2009.

[57]

Mike S. Jankovic. Exact nth derivatives of eigenvalues and eigenvectors. Journal of guidance, control, and dynamics, 17(1), 1994.

[58]

Liping Jiang, Yongning Chi, Haiyan Qin, Zheyi Pei, Qionghui Li, Mingliang Liu, Jianhua Bai, Weisheng Wang, Shuanglei Feng, Weizheng Kong, and Qiankun Wang. Wind Energy in China. IEEE Power and Energy Magazine, 9(6):36–46, November 2011.

[59]

Michael E. Karystianos, Nicholas G. Maratos, and Costas D. Vournas. Maximizing Power-System Loadability in the Presence of Multiple Binding Complementarity Constraints. IEEE Transactions on Circuits and Systems I: Regular Papers, 54(8):1775–1787, August 2007.

[60]

Svenska Kraftnät. Elområden. http://svk.se/Energimarknaden/El/ Elomraden/, November 2011. Accessed on 26 September 2012.

[61]

J.R. Kristoffersen. The Horns Rev wind farm and the operational experience with the wind farm main controller. Revue E-Société Royale Belge des électriciens, 122(2):26, 2006.

[62]

Yuri A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer Verlag, third edition, 1998.

[63]

Yuri A. Kuznetsov. Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858, October 2006.

194

BIBLIOGRAPHY

[64]

Yuri A. Kuznetsov. Saddle-node bifurcation. Scholarpedia, 1(10):1859, October 2006.

[65]

G. Lalor, A. Mullane, and Mark O’Malley. Frequency Control and Wind Turbine Technologies. IEEE Transactions on Power Systems, 20(4):1905–1913, November 2005.

[66]

Mark Lauby, Mark Ahlstrom, Daniel Brooks, Stephen Beuning, Jay Caspary, William Grant, Brendan Kirby, Michael Milligan, Mark O’Malley, Mahendra Patel, Richard Piwko, Pouyan Pourbeik, Dariush Shirmohammadi, and J. Smith. Balancing Act. IEEE Power and Energy Magazine, 9(6):75–85, November 2011.

[67]

A.M. Lyapunov. General Problem of Stability of Motion. PhD thesis, Kharkov, 1892.

[68]

Iain MacGill. Electricity market design for facilitating the integration of wind energy: Experience and prospects with the Australian National Electricity Market. Energy Policy, 38(7):3180–3191, July 2010.

[69]

J. Machowski, J. Bialek, and J. Bumby. Power system dynamics: stability and control. Wiley, 2011.

[70]

Yuri V. Makarov, Pengwei Du, Shuai Lu, Tony B. Nguyen, Xinxin Guo, J. W. Burns, Jim F. Gronquist, and M. A. Pai. PMU-Based Wide-Area Security Assessment: Concept, Method, and Implementation. IEEE Transactions on Smart Grid, pages 1–8, 2012.

[71]

JF Manwell, JG McGowan, and AL Rogers. Wind energy explained: theory, design and application. Wiley, second edition, 2009.

[72]

R.Z. Miñano. Optimal power flow with stability constraints. PhD thesis, Universidad de Castilla-La Mancha, 2010.

[73]

Federico Milano. Continuous Newton’s Method for Power Flow Analysis. IEEE Transactions on Power Systems, 24(1):50–57, February 2009.

[74]

Federico Milano. Power System Modelling and Scripting. Springer Verlag, 2010.

[75]

Michael Milligan. Wind Power Plants and System Operation in the Hourly Time Domain. In Wind Power 2003 Conference, AWEA, Austin, TX, 2003.

[76]

A. Monticelli, M. V. F. Pereira, and S. Granville. Security-Constrained Optimal Power Flow with Post-Contingency Corrective Rescheduling. IEEE Transactions on Power Systems, 2(1):175–180, 1987.

[77]

J. Morren, S.W.H. de Haan, W.L. Kling, and JA Ferreira. Wind turbines emulating inertia and supporting primary frequency control. IEEE Transactions on Power Systems, 21(1):433–434, 2006.

[78]

Näringsutskottet. Riktlinjer för energipolitiken Sammanfattning Innehållsförteckning. Technical report, 2008.

BIBLIOGRAPHY

195

[79]

R.B. Nelson. Simplified calculation of eigenvector derivatives. AIAA journal, 14:1201–1205, 1976.

[80]

Nordel. Agreement regarding operation of the interconnected Nordic power system (System Operation Agreement). Technical report, 2006.

[81]

Nordel. Desciption of Balance Regulation in the Nordic Countries. Technical report, 2008.

[82]

Ian Norheim, Elin Lindgren, Sanna Uski, Poul Sørensen, and Clemens Jauch. WILMAR Deliverable D5.1 - System Stability Analysis. Technical report, 2005.

[83]

Magnus Perninge. A Stochastic Control Approach to Include Transfer Limits in Power System Operation. PhD thesis, KTH Royal Institute of Technology, 2011.

[84]

Magnus Perninge. Approximating the loadability surface in the presence of SNBSLL corner points. Electric Power Systems Research, 2012.

[85]

Magnus Perninge and Lennart Söder. Optimal distribution of primary control participation with respect to voltage stability. Electric Power Systems Research, 2010.

[86]

Magnus Perninge and Lennart Söder. A Stochastic Control Approach to Manage Operational Risk in Power Systems. IEEE Transactions on Power Systems, 2011.

[87]

Magnus Perninge and Lennart Söder. Geometric properties of the loadability surface at SNB-SLL intersections and Tangential Intersection points. pages 1–5, Hersonissos, Greece, September 2011. IEEE.

[88]

Magnus Perninge and Lennart Söder. On the Validity of Local Approximations of the Power System Loadability Surface. IEEE Transactions on Power Systems, PP(99):1–1, March 2011.

[89]

Magnus Perninge and Lennart Söder. Risk estimation of the distance to voltage instability using a second order approximation of the saddle-node bifurcation surface. Electric Power Systems Research, 81(2):625–635, February 2011.

[90]

Magnus Perninge and Lennart Söder. Optimal activation of regulating bids to handle bottlenecks in power system operation. Electric Power Systems Research, 83(1):151–159, February 2012.

[91]

A. Pressley. Elementary differential geometry. Springer Verlag, 2010.

[92]

Regeringskansliet. The Swedish National Action Plan for the promotion of the use of renewable energy in accordance with Directive 2009 / 28 / EC and the Commission Decision. 2009.

[93]

Lars Sandberg and Klas Roudén. The real-time supervision of transmission capacity in the swedish grid. In S. C. Savulescu, editor, Real-time stability assessment in modern power system control centers, chapter 11, pages 279–306. Wiley-IEEE Press, first edition, 1992.

196

BIBLIOGRAPHY

[94]

Richard Scharff and Mikael Amelin. A Study of Electricity Market Design for Systems with High Wind Power Penetration. 2011.

[95]

A. Schellenberg, W. Rosehart, and J. Aguado. Cumulant-Based Probabilistic Optimal Power Flow (P-OPF) With Gaussian and Gamma Distributions. IEEE Transactions on Power Systems, 20(2):773–781, May 2005.

[96]

A. Schellenberg, W. Rosehart, and J. Aguado. Cumulant based stochastic optimal power flow (S-OPF) for variance optimization. In IEEE Power Engineering Society General Meeting, pages 2504–2509. IEEE, 2005.

[97]

J Charles Smith, Stephen Beuning, Henry Durrwachter, Erik Ela, David Hawkins, Brendan Kirby, Warren Lasher, Jonathan Lowell, Kevin Porter, Ken Schuyler, and Paul Sotkiewicz. The Wind at Our Backs. IEEE Power and Energy Magazine, 8(5):63–71, September 2010.

[98]

Marian Sobierajski. Optimal stochastic load flows. Electric Power Systems Research, 2(1):71–75, March 1979.

[99]

Lennart Söder. Reserve margin planning in a wind-hydro-thermal power system. IEEE Transactions on Power Systems, 8(2):564–571, 1993.

[100] Lennart Soder, Hans Abildgaard, Ana Estanqueiro, Camille Hamon, Hannele Holttinen, Eamonn Lannoye, Emilio Gomez-Lazaro, Mark O’Malley, and Uwe Zimmermann. Experience and Challenges With Short-Term Balancing in European Systems With Large Share of Wind Power. IEEE Transactions on Sustainable Energy, 3(4):853–861, October 2012. [101] B. Stott. Review of load-flow calculation methods. 62(7):916–929, 1974.

Proceedings of the IEEE,

[102] Svenska Kraftnät. Avtal om Balansansvar för el. Technical report, 2010. [103] Svenska Kraftnät. Regler för upphandling och rapportering av primärreglering FNR och FDR. Technical report, 2012. [104] UCTE. P1 - Policy 1: Load-Frequency Control and Performance [C]. Technical report, 2009. [105] Thierry Van Cutsem. A method to compute reactive power margins with respect to voltage collapse. IEEE Transactions on Power Systems, 6(1):145–156, 1991. [106] Thierry Van Cutsem and Costas D. Vournas. Voltage stability analysis in transient and mid-term time scales. IEEE Transactions on Power Systems, 11(1):146–154, 1996. [107] Thierry Van Cutsem and Costas D. Vournas. Voltage Stability of Electric Power Systems. Springer, first edition, 1998. [108] James E. Van Ness and John H. Griffin. Elimination Methods for Load-Flow Studies. Transactions of the American Institute of Electrical Engineers. Part III: Power Apparatus and Systems, 80(3):299–302, April 1961.

BIBLIOGRAPHY

197

[109] G. Viviani and G. Heydt. Stochastic Optimal Energy Dispatch. IEEE Transactions on Power Apparatus and Systems, PAS-100(7):3221–3228, July 1981. [110] Costas D. Vournas and Michael E. Karystianos. Transmission system tap changer settings to maximize loadability margin. In Proceedings of 14th Power Systems Computation Conference, pages 1–7, 2002. [111] Costas D. Vournas, Michael E. Karystianos, and Nicholas G. Maratos. Bifurcation points and loadability limits as solutions of constrained optimization problems. In 2000 Power Engineering Society Summer Meeting, volume 3, pages 1883–1888. IEEE, 2000. [112] Ivar Wangensteen. Power System Economics: The Nordic Electricity Market. Tapir Academic Press, 2007. [113] Allen J Wood and Bruce F Wollenberg. Power Generation Operation and Control. John Wiley & Sons, second edition, 1996. [114] Su Yang, Feng Liu, De Zhang, and Shengwei Mei. Polynomial approximation of the small-signal stability region boundaries and its credible region in highdimensional parameter space. European Transactions on Electrical Power, March 2012. [115] Taiyou Yong and R.H. Lasseter. Stochastic optimal power flow: formulation and solution. In 2000 Power Engineering Society Summer Meeting, volume 1, pages 237–242. IEEE, 2000. [116] H. Zhang and P. Li. Probabilistic analysis for optimal power flow under uncertainty. IET Generation, Transmission & Distribution, 4(5):553, 2010. [117] P. Zhang and S.T. Lee. Probabilistic Load Flow Computation Using the Method of Combined Cumulants and Gram-Charlier Expansion. IEEE Transactions on Power Systems, 19(1):676–682, February 2004.

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Variation in Persian Vowel Systems - DiVA portal

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Distributed Self-Triggered Control for Multi-Agent Systems - DiVA portal

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