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

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

CAMILLE HAMON

Licentiate Thesis

Stockholm, Sweden 2012

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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 energisy- stem tisdagen den 11 december 2012 klockan 10.00 i E2, Lindstedsvägen 3, Kungl Tek- niska högskolan, Stockholm.

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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 green- house 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.

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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 atmo- sphere 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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List of Figures

1.1 Forecasted load and planned production: they are equal on average but not

on a continuous basis. . . 2

1.2 The different phases in power system planning and operation. Time frame considered in this thesis: Operating period (one hour or less). . . 3

1.3 Cumulative installed wind power capacities in 2010 (dark gray) and 2011 (additional capacity compared to 2010 in light gray), numbers from [46]. . . 4

1.4 Global cumulative installed wind power capacities for the period 2001-2016, numbers from [46]. Years 2012-2016 are forecasts. . . 5

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]. . . 6

1.6 Share of net electricity production coming from wind power in Sweden [%], from [40]. . . 6

1.7 Reference power system from [59]. . . 9

1.8 The problem considered in this thesis. . . 11

2.1 An example of power curve for a 1500 kW wind turbine. . . 19

2.2 The different time frames for power system operation and planning. The operating period lies in the scope of this thesis. . . 21

2.3 Influence of load forecast errors on frequency control schemes: the production is not planned optimally. . . 23

2.4 Load, wind power (WP) production and net load on Gotland, 16 March 2009, from Gotlands Energi AB (GEAB). . . 24

2.5 Generators are driven by turbines, and supply electric power to the loads. . . 24

2.6 Requirements for the automatic active reserves in Nordel: Frequency controlled normal operation reserve (solid line) and Frequency controlled dis- turbance reserve (dashed line). . . 27

2.7 The different layers of frequency control schemes, inspired by [104]. . . 29

2.8 Illustration of the σ-method with the probability distributions of load and net load (NL) variations. . . 31

2.9 Increase in reserve requirements due to wind power, from [53]. . . 32 ix

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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 SMIB system . . . 45

3.2 Newton’s method applied to the power flow problem. . . 46

3.3 Problem of projecting A onto the sphere in a given direction. . . 47

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). . . 49

3.5 Gram-Schmidt process: taking away the projection of v₂onto b₁. . . 54

3.6 The unit sphereS²and its normal n at P = (0.41,0.82,0.4). . . 56

3.7 The unit sphereS²and its tangent plane at P = (0.41,0.82,0.4) . . . 56

4.1 The two possible generic co-dimension one bifurcations. . . 62

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

4.3 Subcritical Hopf bifurcation, from [63]. . . 65

4.4 Supercritical Hopf bifurcation, from [63]. . . 66

5.1 P-V curves with SNB and SLL. . . 79

5.2 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 u0. . . 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 Fig- ure 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. P1and P2are the net loads at buses 1 and 2, respectively. . . 101

5.10 Illustration of the need for computing more stability limits. . . 102

6.1 One predictor-correction step towards the most important point. . . 114

6.2 Search after the most important point on a set of corner points . . . 116

6.3 The tangential intersection of a SNB and a SLL surface [88]. . . 117

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List of Figures xi

6.4 Computation of the distance to second-order approximations. . . 122

6.5 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). . . . 127

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). . . 129

7.3 The stability boundary and the second-order approximations of the light blue part (in gray). . . 130

7.4 Spherical coordinates defined from λ_base, adapted from [7]. . . 132

7.5 Absolute values of the distances between the approximations and the real surface. The smooth parts are colored according to Figure 7.2. . . 133

8.1 The two phases in solving the S-OPF problem. . . 141

8.2 Approach 1: monitoring and acting; ∆t is at most a few seconds. . . 143

8.3 Approach 2: repeatedly acting; ∆t is a few minutes. . . 143

8.4 An approximation of a stability boundary consisting of two smooth parts Σ^a_{i 1} and Σ^a_{i 2}. . . 147

8.5 Geometrical interpretation of V1and V2. . . 149

8.6 Case of three smooth parts intersecting. . . 151

8.7 A parameter space with two SLL surfaces and one SNB surface. . . 154

8.8 Flowchart of the methodology to solve the S-OPF problem. . . 158

A.1 Reference power system. . . 177

A.2 IEEE 9 bus system. . . 179

A.3 IEEE 39 bus system. . . 181

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List of Tables

2.1 The four main wind turbine designs. . . 18

5.1 Description of the parameters in (5.4) and (5.5). . . 73

5.2 Different types of stability limits. . . 82

5.3 Analytical expressions for the normals to different smooth parts. In all for- mulas, the normal is of unit length and points towards instability. . . 99

6.1 Analytical expressions for the derivatives dN . . . 112

7.1 Power transfer limits . . . 126

7.2 The different stability limits in the IEEE 9 bus system. . . 128

9.1 The variation of the optimal solution when varying α. . . 164

9.2 Optimal solutions when using uc= 0. . . 165

9.3 The variation of the optimal solution when varying α. . . 166

A.1 Data for the reference power system of Figure A.1. . . 178

A.2 Data for the IEEE 9 bus system . . . 180

A.3 Data for the IEEE 39 bus system . . . 182

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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 socio- economic 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 or- ganized 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

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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).

Time Operating period

(one hour or less) Load

Planned production

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 con- trol 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 corridors¹are 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].

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1.1. BACKGROUND 3

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 purposes². 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 generators³or 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.

Time Seasonal, weekly and day-ahead

planning

Market clearing Intra-day planning

Operating period

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.

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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.

0 20 40 60 80 100

Africa and Middle East Asia Europe

Latin America North America Pacific Region

Installed capacity [GW]

2010 2011

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

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1.1. BACKGROUND 5

2002 2004 2006 2008 2010 2012 2014 2016 0

100 200 300 400 500

Cumulativeinstalledcapacity[GW] Historical data Market forecasts

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

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0 2 4 6 8 10 12 14

Wind power

Hydro power

Biomass

Solar PV

Others

14

10.5

6.7

2.4

1

Share of total electricity consumption in 2020 [%]

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].

2003 2004

2005 2006

2007 2008

2009 2010

2011 0

0.5 1 1.5 2 2.5 3 3.5 4

Windpowershareof electricityproduction[%]

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.

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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 in- cludes [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 real- time 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].

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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 analy- sis less meaningful in representing the underlying system. Furthermore, and most relevant, the increase in penetration of variable generation fur- ther 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 distri- bution of scenarios rather than a specific set of points that make up the distribution.

”

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1.3. AN INTRODUCTORY EXAMPLE 9

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.

B Load

P

g3

P

_g2

P

_g1

1

2

3

4

5 6 7

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

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varies⁴, 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 re- lieve 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.

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1.4. THESIS OUTLINE 11

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

1. System too close to or beyond the

stability limits?

2. Take actions to come farther away from the stability limits, e.g.

activation of balancing bids.

Computation of stability limits.

Yes No

t = t + ∆t

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 famil- iar 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-

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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 second- order 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 as- sessed.

– 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 con- sisted 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.

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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 Chap- ter 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 Eu- ropean, 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; Lan- noye, 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.

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Part I

Background

15

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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 genera- tion 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

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18 CHAPTER 2. TECHNICAL BACKGROUND

There are four main design alternatives for wind turbines [1, p.56] whose main char- acteristics 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 Squirrel cage induction generator

No Type B Limited variable

speed

Wound rotor induction generator

No Type C Variable speed Doubly-fed induction

generator

On the rotor side only Type D Variable speed Synchronous generator 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 Sec- tion 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 Fig- ure 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

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2.1. WIND TURBINES 19

1 3 5 7 9 11 13 15 17 19 21 23 25

0 500 1,000 1,500

Cut-in wind speed

Rated wind speed

Cut-out wind speed

Wind speed [m/s]

Power[kW]

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

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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, Nor- way 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 Op- erators 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 be- fore 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.

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2.2. ELECTRICITY MARKETS 21

Balancing markets Ancillary services such as load following or balancing can be pro- cured 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 op- erating periods are financially settled.

Time Seasonal and

weekly planning

Day-ahead planning

Intra-day planning

Operating period Market clearing:

Production plans decided.

End of planning.

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

Trading for adjustments

(updated forecasts, units’

availability)

Maintaining continuous

balance between production

and consumption.

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 plans¹, 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.

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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 sched- uled 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 di- vided 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 electri- cal grid). Hence, these two quantities are always in balance. The meaning of “imbal- ance 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,

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2.3. FREQUENCY CONTROL SCHEMES 23

Production plan

0 T

Operating period (one hour or less)

Actual load

Forecasted load Market clearing:

production planned according to forecasts

End of day- ahead planning

Intra-day trading for adjustments

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. There- fore, 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. Fur- thermore, 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.

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20 40 60 80 100 120 140

0 5 10 15 20

Hour of the day

Load/Production[MW]

Load

WP production Net load

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 sup- plied to the loads through the electrical network, as depicted in Figure 2.5, where Pmis the mechanical power delivered by the turbine, converted into electric power Peby the generator, and supplied to the load.

Turbine Generator Electrical

grid Load

Pm Pe

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 com- pensated 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-

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2.3. FREQUENCY CONTROL SCHEMES 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

Mid∆ωi

dt = Pmi− Pei− PDi, (2.1)

where ∆ω_iis the speed deviation from synchronous speed, M_iis the inertia coefficient, P_miand P_eiare as defined above, and P_Di is the damping power [69]. The term on the right-hand side is called the accelerating power Pai = Pmi− Pei− PDi. 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 change². 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 dt =d∆ωi

dt =d∆ω_j

dt ⇐⇒ ∀i, j, Pmi− Pei− PDi

Mi =P_{m j}− Pe j− PD j

Mj , (2.2) 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

∆P =X

i

By (2.2)ª

=X

i

µ Mid∆ω

dt

¶

, (2.3)

so that

d∆f

dt = 2π ∆P

PiMi, (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.