Reactive Power Planning with Voltage Stability Constraints for Increasing Cross-Border Transmission Capacity

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DEGREE PROJECT, IN ELECTRIC POWER SYSTEMS , SECOND LEVEL STOCKHOLM, SWEDEN 2015

Reactive Power Planning with Voltage Stability Constraints for Increasing

Cross-Border Transmission Capacity

MANA FARROKHSERESHT

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL ENGINEERING

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Reactive Power Planning with Voltage Stability Constraint for Increasing the

Cross-Border Transmission Capacity

Author:

Mana Farrokhseresht

Supervisor:

Prof. Mohammad Reza Hesamzadeh Saeed Rahimi

Examiner:

Prof. Mohammad Reza Hesamzadeh

A thesis submitted in fulfilment of the requirements for the degree of Master of Science

in the

Electricity Market Research Group (EMReG) Department of Electric Power Systems

July 2015

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Abstract

In this work Reactive Power Planning (RPP) is studied. It is the method which aims at locating reactive compensators of optimal size at an optimal location in order to achieve or optimize a certain objective. In this work, the reactive compensators are placed in such a way that they keep the voltages in a grid longer stable and within an acceptable range of values while power flows through the grid. Usually, this power flow disturbs the voltages.

The RPP was applied in this work in order to allow a larger power flow from one grid area to another grid area. The first area is called the Source area and it contains generators which can produce power at a cheap price. The second area is called the Sink area and it is willing to import this cheap power so that it has to produce less power by itself. The two areas are connected to each other with a tie-line, which has a certain capacity (called Net Transfer Capacity or N T C). The capacity is restricted by stability requirements: exceeding the capacity would cause voltages to become unstable in either of the two grid areas. Installing reactive power compensators allows to increase the capacity of the line, keeping the voltages stable when the power flow over the tie-line increases. Reactive Power Planning therefore has an economic benefit, and different methods to optimize it will be investigated in this work. This work is divided into three parts.

In the first part the relationship is studied between reactive power compensation on the one hand and the increase of the N T C of the tie-line on the other hand. The grid which is used to illustrate this is the Swedish grid, connected to the grids of Denmark and Finland. It is observed that not only increasing the loads can lead to voltage instability in the grid, but that voltage problems can also arise within the Swedish grid from the exchange of power, flowing through the Swedish grid, between its neighbouring countries. It is shown that reactive power compensation is a technique which can potentially increase the N T C-value of the tie-lines between Sweden and Denmark and between Sweden and Finland. Depending on where the reactive power compensators are installed, the N T C increases with different values. In the next two parts however, we focus on the economic analysis of the reactive power compensation. In these two parts, an Optimal Power Flow (OPF) problem is designed, leading to the optimal placement and choice of the reactive power compensators. The optimal compensators increase the N T C so that the benefit of the decreased cost of power generation in the Sink area is maximized with respect to the cost of the reactive compensators. The difference between these two parts is in the algorithms that are applied for solving the OPF.

In the second part a heuristic method based on a Genetic Algorithm (GA), NSGA-II, is used to optimize this benefit. The reason why first a heuristic method is used is that the N T C of the tie-line cannot be expressed as an analytical, closed-form function of the reactive power ratings. Therefore, a heuristic optimization method is chosen to solve the OPF and the algorithm NSGA-II is used because of its good convergence properties and ease of implementation. However, the algorithm is also able to perform multi-objective optimization and this ability is used to optimize both the economic benefit and the voltage stability index of the Sink area. It is seen that there is a trade-off between voltage stability and economic benefit and it is up to the grid operators to make choices regarding this trade-off.

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In the third part of this work a mathematical method is used to maximize the economic benefit of the Sink area with respect to the cost of the reactive compensators. As much as possible mathematical expressions will be used in this part. As the N T C cannot be expressed as a closed-form function, some approximations need to be made. Two methods were proposed to approximate the N T C: approximation by piecewise linear functions and by polynomials obtained with statistical regression. This mathematical method leads to a Voltage Stability Constrained OPF (VSCOPF). It is programmed in GAMS and formulated as a Mixed-Integer Non-Linear Programming problem (MINLP).

Keywords: NTC, Optimal Power Flow, Reactive power compensation, Reactive Power Planning, Voltage stability.

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Acknowledgements

There are several people I would like to thank for helping me not only to complete this master thesis project but generally in my life in Sweden and Belgium.

I would like to express my sincere thanks to my main supervisor prof. Mohammad Reza Hesamzadeh for creating the project, his support, guidance and encouragement. I appreciate his willingness to help me any time I asked and for guiding me and keeping me on the right track. Also I would like to express my gratitude to my external supervisor, Mr. Saeed Rahimi, for his support, valuable guidance and positive energy.

I would like to express my gratitude to my master programme coordinator, Prof. Johan Driesen, for his kind help and support. In addition, special thank to EIT/KIC-InnoEnergy for funding my two years master’s programme: Energy in Smart Cities.

I thank Hossein Shahrokni, my course instructor for his help and guidance and Bert Willems, the financial coordinator of EIT-KIC who became one of my best friend too. Many thanks go to my colleagues and my friends at EMReG, KTH and at Electa, KU Leuven.

I have no words to express my gratitude to the Jacqmaer family, Frans, Hilde and Pieter.

Their emotional support during all these years was the biggest motivation and strength for me. I have spent enjoyable and unforgettable moments with this kind-hearted family and learned a lot from them. Thank you for all your help and your kindness.

Last but not least, I would like to express my ultimate thanks to my beloved family and my dear uncle, Ali Farrokhseresht, for their emotional and financial support. Without them, success was impossible. Thank you Babaee for raising me and encouraging me to dream.

Thank you Giti to show me that I should fight for my dreams. And thank you Nakisa, you have proven I am not alone.

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List of Figures

2.1 400 kV-buses of the Swedish part of the modified test grid . . . 9

2.2 Proposed algorithm for determining the PV-curves of tie-lines . . . 15

2.3 Contouring plot of the voltage in the Swedish grid before and after the load increase by 30%. . . 16

2.4 PV-curves for tie-lines between Sweden and its neighbours . . . 19

2.5 PV-curve for tie-lines with and without reactive compensation . . . 19

3.1 Multiple nondominated fronts . . . 24

3.2 Crowding-distance calculation . . . 25

3.3 Blending crossover, for various values of α, the exploration coefficient. . . 27

3.4 Creating of a new population from an old population and its offspring, in NSGA-II . . . 28

3.5 Single objective, single variable case to test the genetic algorithm implementation 29 3.6 Single-objective single-variable case: final population and their fitnesses . . . 30

3.7 Multi-objective (2), single-variable case to test the genetic algorithm implementation . . . 30

3.8 Multi-objective (2), single variable case: objective space and optimal solution 31 3.9 ZDT1: two-objective two-variable case with a convex optimal Pareto front to test the genetic algorithm implementation . . . 32

3.10 ZDT3: 2 objective, 2 variable case with a discontinuous optimal Pareto front to test the genetic algorithm implementation . . . 33

3.11 Cost function of the capacitor banks . . . 35

3.12 Flow chart explaining the procedure of calculating the benefit and net benefit of the installation of reactive compensators in Area-1 . . . 37

3.13 Flow chart explaining the procedure of calculating the N T C of the tie-line. . 38

3.14 Objective function for the 100 individuals of the last generation: convergence 39 3.15 Single-line-diagram of a 6-bus, 2-generator grid . . . 41

3.16 WECC 9-bus system, topology and data . . . 42

3.17 L-index of bus 5 of the WECC 9-bus system, versus its loading . . . 42

3.18 Optimal Pareto front for the two-objective optimization, Q-limits enforced. . 43

3.19 Optimal Pareto front for the two-objective optimization, Q-limits not enforced 44 4.1 Normalized VSIs for the 25 PQ-buses of Area-1 . . . 57

4.2 Second test grid. . . 62

4.3 Q(V )-curves for buses 2 and 3 of the second test grid . . . 63

4.4 VSIs for the second test grid . . . 63

4.5 Normalized VSIs for the second test grid . . . 64

4.6 Model of the transmission line l, used in the power flow equations. . . 67

4.7 Explanation of the power balances for each bus i . . . 68

4.8 N T C as function of the ratings of the reactive compensators, attached to bus 3 and bus 2 . . . 70

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List of Figures

4.9 N T C as function of the ratings of the reactive compensator, attached to bus 3, and a polynomial approximation . . . 70 4.10 Piecewise linear function in one variable . . . 72 4.11 Piecewise linear function in 1 variable: definition using binary variables . . . 73 4.12 Piecewise linear function in 2 variables . . . 75 E.1 Model of the transmission line l, used in the derivation of the power flow

equations . . . 107

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List of Tables

2.1 Values of line parameters for different voltage levels. . . 10

2.2 Contingency ranking . . . 16

2.3 Load increase by 10, 20 and 30 percent in the North of Sweden . . . 17

2.4 Load increase by 10, 20 and 30 percent in the Centre of Sweden . . . 18

3.1 Solution of the RPP: ratings and locations of reactive compensators that need to be added to the grid. . . 39

3.2 Solution of the RPP: optimal fuel costs, costs of the capacitor banks and powers flowing over the tie-line . . . 39

3.3 Branch data of the 6-bus, 2-generator grid, on a 100 MVA base . . . 41

3.4 Bus data of the 6-bus, 2-generator grid, on a 100 MVA base . . . 41

3.5 Comparison of L-indices calculated with the programme of this work and those listed in . . . 41

4.1 Normalized VSI for the 25 PQ-buses of Area-1 . . . 57

4.1 Normalized VSI for the 25 PQ-buses of Area-1 . . . 58

4.2 Clustering results . . . 59

4.3 Results of the F -test applied on the clustering. . . 60

4.4 Bus data . . . 60

4.4 Bus data . . . 61

4.5 Generator data . . . 61

4.6 Branch data, including transformers . . . 61

4.7 Coefficients of the quadratic generator cost functions . . . 61

4.8 Coefficients of the polynomial of Fig. 4.9 . . . 79

4.9 Coefficients of the OLS regression polynomial of the surface in Fig. 4.8 . . . . 80

4.10 Comparison between LAR- and OLS-regression applied on the surface of Fig. 4.8: determination coefficient and maximal relative error magnitude . . . 80

4.11 One-variable case, use of regression: Scheduled voltages and generator active power setpoints in the optimal case for a minimal safety margin of 0.5 . . . . 84

4.12 Results for varying the minimal safety margin . . . 84

4.13 Results for varying the minimal safety margin; capacitors have no cost . . . . 84

4.14 One-variable case: piecewise linear function: Scheduled voltages and generator active power setpoints in the optimal case for a minimal safety margin of 0.5 85 4.15 Two-variable case: Scheduled voltages and generator active power setpoints in the optimal case for a minimal safety margin of 0.5 . . . 86

A.1 Bus data. . . 93

A.1 Bus data. . . 94

A.2 Generator data . . . 94

A.2 Generator data . . . 95

A.3 Branch data, including transformers . . . 95

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List of Tables

A.4 Cost data of the Swedish generators . . . 97

D.1 Bus data. . . 104

D.1 Bus data. . . 105

D.2 Generator data . . . 105

D.3 Branch data, including transformers . . . 105

D.3 Branch data, including transformers . . . 106

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Abbreviations

CPF Conic Power Flow EA Evolutionary Algorithm EP Evolutionary Programming FACTS Flexible AC Transmission System

GA Genetic Algorithm

HVDC High Voltage Direct Current LAR Least Absolute Residuals

LP Linear Programming

MINLP Mixed-Integer Non-Linear Programming MIP Mixed Integer Programming

MOO Multi-Objective Optimization NLP Nonlinear Programming

NSGA-II Nondominated Sorting Genetic Algorithm-II NTC Net Transfer Capacity

OLS Ordinary Least Square OPF Optimal Power Flow PoC Point of Collapse

RPP Reactive Power Planning

SM Safety Margin

SVC Static VAr Compensator TSO Transmission System Operator VCPI Voltage Collapse Proximity Indicator VSA Voltage Stability Analysis

VSCOPF Voltage Stability Constrained Optimal Power Flow VSI Voltage Stability Index

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Symbols

B Susceptance [S]

C Capacitance [F]

cos φ Power factor

E Voltage [V]

φ Phase angle between voltage and current [rad]

I Current [A]

j Imaginary unit

J Jacobian of power flow equations

L Inductance [H]

P Active power [W]

P_Gi Generator active power [W]

PLi Load active power [W]

Q Reactive power [VAr]

QCi Reactive power rating of the compensation [VAr]

Q_Gi Generator reactive power [VAr]

QLi Load reactive power [VAr]

R Resistance [Ω]

S Apparent power [VA]

θ Bus voltage angle [rad]

U Voltage [V]

V Voltage [V]

X Reactance [Ω]

Y Admittance matrix

Z Impedance [Ω]

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Chapter 1

Introduction

1.1 Background

Electricity demand has been increasing in recent years. The changing consumption pattern and the increasing electricity demand yield a more vulnerable power grid [1,2]. Moreover, due to the liberalization of the electricity markets across Europe and the need to produce electricity more cleanly and sustainably, the production pattern changes. The huge prolif- eration of small renewable distributed generation units offers proof for this trend. Because of these evolutions in consumption and production, the power system is forced to operate under great amounts of stress. The influence of these effects on power system stability re- quires further investigation [2]. In order to guarantee a reliable, stable and secure grid, power systems have to be designed in such a way that a multitude of different disturbances and contingencies can be overcome in a safe and adequate manner. In order to achieve this goal, it is necessary to review and redesign the infrastructure of power systems which were generally set up over fifty years ago [2].

Voltage stability is a major concern during the planning and operating of a modern power system and has earned much attention in recent years, in view of the recent incorporation of more and more intermittent generation in the grid. Voltage stability is defined as the ability of a power system to maintain voltages with acceptable levels at all buses in the system after being subjected to a disturbance from a given initial operating condition [3].

A power system is thus unstable if a disturbance leads to a drop in the voltage that is not acceptable. Therefore, it is important to pay attention to how the voltage varies within the power system and to the methods to keep the voltage in acceptable limits in order to have a

”voltage stable” power system.

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1.1 Background Chapter 1 It is important to note that the main source of voltage collapse in power systems is the lack of reactive power [3,4] and that this cause of voltage instability occurs more in a stressed power system, i.e. a system with a high level of active power transfer [3].

As mentioned earlier, in the coming decades, several trends driven by different policy goals such as the liberalization of the electricity market and the penetration of renewable energy sources (RES) can lead to an increase in cross-border electricity flows in Europe. The transmission grid in Europe is also facing new challenges due to an expansion of the interconnection of countries. However, the increase of cross-border transmission capacity, stimulated by the use of more RES, and achievable by different novel technologies such as the installation of phase-shifting transformers or the use of HVDC-technology in order to interconnect countries with long-distance lines, suffers from several obstacles. These obstacles include a high in- vestment cost, environmental issues, public resistance due to the visual impacts, commercial problems, etc. [5]. Moreover, the lessons learned after the occurrence of the recent blackouts due to increased cross-border flows taught us the importance of making the power system more robust and stable in the presence of an increased interconnection [6].

Increasing the transfer capacity of the tie-lines leads to a decrease in the domestic generation cost if the cost of generation within a country is more expensive than the cost of importing power from neighbouring countries. An increased transfer capacity can also facilitate in meeting the renewable energy goals more easily [7].

An effective manner to increase the cross-border transmission capacity is to install reactive power compensation devices in a grid, such as capacitor banks or the different kinds of the more advanced FACTS devices. In general, compensation consists of injecting reactive power into the system, keeping the bus voltages closer to their nominal values, reducing the line currents and hence decreasing network losses. Although inductive reactors are sometimes used, compensation is often of a capacitive nature, counter-balancing the predominantly inductive nature of either the transmission system or the loads. Compensation can increase the Net Transfer Capacity (N T C) of tie-lines because it keeps the voltages of a grid stable, even when an increased power flows through the grid. Compensation thereby also avoids the obstacles that were previously mentioned [8].

The goal of an adequate power system operation is not only to fulfil the technical requirements of the power system but system operators also have to be aware of the costs of power generation in the area they are responsible for. That is why the study of the economy of power systems is becoming more and more important. Therefore, an optimization problem

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1.2 Literature review Chapter 1 can be defined such that the cost of the reactive power compensation is minimized and the economic benefits of increasing the tie-line capacity are maximized. In general, this optimal power flow problem (OPF) is a nonlinear programming problem and determines the optimal control setpoint for the system.

This work will concentrate on the Swedish national grid, which is part of the Nordic transmission network. Sweden is located between two countries, Denmark and Finland, which exchange power between each other [9, 10]. It is important to keep the voltages in acceptable limits while trying to increase the power transfer through the tie-lines with the neighbouring countries. The reason is that, due to the import of more power generated from renewable energy sources such as wind power in Denmark with zero marginal cost, and by increasing the cross-border transmission capacity, Sweden can decrease its generation cost by dispatching less of its own high-cost generators and importing more cheaper power from Denmark. The same applies for the connection with Finland, but there, the cheaper power produced in Finland comes from coal sources. Therefore, if Sweden can increase the transmission capacity of the tie-lines with its neighbours, while keeping the voltages in the grid stable, it can gain economic benefit due to importing more power from low-cost generation areas. There are different ways to formulate this optimization problem. The output of the optimization problems is the optimal place and size of the reactive power compensators.

1.2 Literature review

The concepts of voltage stability have been discussed widely in numerous books and articles [8] [11] [3]. To understand voltage stability it is important to study the following:

• The basic relationships between active power, reactive power, voltage magnitude and frequency in transmission systems [12][3].

• How power flow or load flow studies function. They are the most basic foundation to understand and analyze voltage stability [3][8][11].

• The relationship between the active power of a load and the voltage at the corresponding bus. This relationship can be depicted graphically by PV-curves which are one of the most practical tools for system operators indicating the stability of the system [8].

Some important information such as the maximum power that can be transferred over a line, can be extracted from the PV-curves [4] [8].

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1.4 Published papers Chapter 1 An effective method for increasing the maximum power that can be transferred over the lines (i.e. shifting PV-curves to the right) so that the voltage still remains in the allowed range is studied in this thesis. It consists of installing reactive compensators in the grid [13] [8].

Increasing the maximum transfer capability of a tie-line which interconnects two areas, one with a high generation cost and another with a low cost, decreases the generation cost in the high-cost area if this area imports cheap power from the low-cost area via the tie-line [14].

Optimal power flow (OPF) is central in the study of an economic operation of power systems [15]. Each OPF problem consists of an objective function and may be limited by some constraints [16]. This is also the case in Reactive Power Optimization or Reactive Power Planning (RPP) in power systems [17]. Typical objective functions and constraints of RPP problems are explained in [18]. With the RPP OPF we are able to determine the optimum places and ratings of reactive power compensators in the grid such that the generation costs are minimized, the economic benefit due to an increased N T C is maximized or the losses are minimized, etc. depending on which objective the system operators deem important to optimize [17, 18]. There are different algorithms to solve the Reactive Power Optimization and a discussion can be found in [17, 19]. Not always is a RPP solved by a mathematical approach and sometimes a heuristic method can be useful to solve the optimization problem [20,21].

1.3 Thesis objectives

This thesis attempts to answer the following research questions:

• Investigate the mechanisms causing voltage instability and study their effects on the voltage stability of the Swedish grid.

• Analyze the stability of the Swedish grid using PV-curves of the Sweden-Denmark and Sweden-Finland tie-lines. Establish the relationship between reactive power compensation and the shape of the PV-curves.

• Determine the optimal locations and sizes of reactive compensators so that the economic benefit of the increase of the N T C of a tie-line is maximized while keeping the voltages in the grid in the acceptable range. Solve this OPF with:

1. a heuristic method 2. a mathematical method

and compare the advantages and disadvantages of both methods.

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1.4 Published papers

• M. Farrokhseresht, M. R. Hesamzadeh, S. Rahimi, ”Evaluation of Reactive Power Com- pensation on Voltage Stability and NTC values in the Swedish Transmission Network”, IEEE PES International Conference on European Energy Market, Krakow, Poland, 28- 30 May 2014.

• M. Farrokhseresht, M. R. Hesamzadeh, J. Lin, S. Rahimi, ”Reactive Power Optimiza- tion to Increase Cross-Border Transmission Capacity”, 15th International Conference on Environment and Electrical Engineering , Italy, 2015.

1.5 Outline of this work

The methods developed in this work are outlined into three chapters, Chapter 2, 3 and 4.

Chapter2studies voltage stability from a technical point of view and the role of the reactive power compensators to improve voltage stability and increase the N T C. The economic study of reactive power compensation is included in Chapter 3 and Chapter 4. In each chapter a particular reactive power optimization OPF is introduced for which the output consists of the optimal locations and sizes of the compensators. In both Chapter3and4, the objective function of this optimization problem is the economic benefit of the N T C which is increased by installing reactive power compensators. However, in Chapter 3 a heuristic algorithm is employed and in Chapter 4a mathematical algorithm. All the chapters include an introduction, a description of the method and of the simulation test grid and an overview of the results and they end with a short conclusion and summary.

Finally, chapter 5, ”Conclusions and Future work”, consists of two sections. The first section discusses critically the important lessons learned in this work. It discusses the advantages, disadvantages, simplifications and limitations of each of the methods of this work. The next section provides some ideas to deal with the aforementioned disadvantages and limitations and also gives some ideas for future work in the field of Reactive Power Planning.

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Part I: Relationship between reactive compensation, voltage

stability and tie-line capacity

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Chapter 2

Relationship Between Reactive Compensation, Voltage Stability and Tie-Line Capacity

2.1 Introduction

Voltage stability is one of the main factors in the overall stability of power systems. System operators have to guard the voltage profile in their grid continuously. Two mechanisms cause instability in power systems: load increase and system disturbance. There are various methods to investigate the stability of the voltage in a system and PV-curves are typical tools applied by system operators [8]. In this chapter we seek to develop an algorithm for voltage stability analysis (VSA) in the Swedish grid by drawing PV-curves of the tie-lines.

In addition, PV-curves show important information about the maximum power that can be transferred over the tie-lines. The MW-value corresponding to the nose point of the PV- curves is the maximum transferrable power over a line. Suppose a high-cost generation area is connected with a low-cost area via a tie-line. If the maximum Net Transfer Capability (N T C) is increased by shifting the PV-curve of the tie-line to the right, the high-cost generation area can import more power from the low-cost generation area. Therefore, it has to produce less power itself and this leads to cost benefits. A possible technique to shift the PV-curve to the right is reactive power compensation, also called VAr compensation. The goal of Reactive Power Planning (RPP) is to find the best places to inject the reactive power and determine the optimal value of the reactive power to be injected.

In the next section, the stability and compensation theory is applied to the case of the Swedish grid, connected to Denmark and Finland. The influence of increasing the loads within the Swedish grid and the occurrence of a system disturbance (i.e. the power transfer between

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2.2 Test grid: Nordic-32, Denmark and Finland Chapter 2 the neighbouring countries of Sweden) on the voltage levels within Sweden is examined. PV- curves are drawn for the tie-lines with the neighbouring countries and the N T C of these tie-lines is improved by means of injecting reactive power into specific buses.

2.2 Test grid: Nordic-32, Denmark and Finland

This section presents a case study which aims to examine how power transfer may be increased by implementing additional VAr compensation in the Swedish grid. We are interested in increasing the transfer across the transmission interfaces Sweden-Denmark and Sweden- Finland.

The test grid consists of the Nordic32-A-grid, representing a scaled down version of Sweden and of the 400 kV rings of Denmark and Finland. Nordic32-A represents the Swedish grid at the scale of 1:2 from the viewpoint of the installed capacity [22]. The scale of 1:2 means that Nordic32-A has half of the installed generation capacity of the real Swedish grid and that therefore also the load of the model will be about half of the real load. However, the frequency dynamics of the model correspond well with those in reality. The Nordic32-A- grid is described in [23]. Actually, Nordic32-A does not structurally represent the Swedish grid. However, the dynamics of both grids are similar. Also, it must be taken into account that Nordic32-A was developed in 1993 and since then, the Swedish grid changed a lot. For example, Nordic32-A does not contain HVDC-lines. The Nordic32-A system consists of 32 buses of which 19 are generator buses and 21 have loads. The system can be divided into 4 main areas:

• North: mostly consists of hydro power plants and some load centers.

• Central: consists of a large amount of loads and large thermal power plants.

• Southwest: consists of some thermal power plants and some loads.

• External: connects to the area ”North”; it has a mix of generation and loads.

The Nordic32-A grid is slightly modified.

An overview of the structure of only the 400 kV buses of the Swedish part of the modified test grid is given in Fig 2.1.

The 400 kV ring of Denmark is described in [25] (files downloadable from [26]). A map of the 400 kV grid of Finland has been provided by ENTSO [27]. This map shows the 400 kV lines

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2.2 Test grid: Nordic-32, Denmark and Finland Chapter 2

Figure 2.1: 400 kV-buses of the Swedish part of the modified test grid [24]

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2.3 Voltage Stability Analysis on the test grid Chapter 2 in Finland. The total consumption in Finland is 9700 MWh per year on average [28]. The loads were distributed across the 400 kV nodes according to the population density of Finland [9]. Based on a list of the Finnish power plants and their generation, the power generated by each generator is determined [29]. Finally, by measuring the line length between two nodes on the map of Finland and multiplying it with the line parameters in /km of Table 2.1 at 400 kV¹, the total resistance R, reactance XL and line charging BC of the branches can be calculated [3]. As the Danish and Finnish 400 kV grids are connected to 400 kV buses in Sweden, interconnection transformers do not need to be included in the grid model.

Table 2.1: Values of line parameters for different voltage levels Nominal

Voltage 230 kV 345 kV 500 kV 765 kV 1100 kV

R (Ω/km)

XL= ωL (Ω/km) B_C = ωC(µS/km)

0.050 0.488 3.371

0.037 0.367 4.518

0.028 0.325 5.200

0.012 0.329 4.978

0.005 0.292 5.544

A full description of the Swedish, Danish and Finnish buses, branches and generator data is given in Appendix A. The power base is 100 MVA.

2.3 Voltage Stability Analysis with reactive power compen- sation applied on the test grid

The scope of the present section is to detect the mechanisms which may lead to voltage instability within the Swedish transmission system through a twofold approach: (1) Studying the impact of the load increase within the Swedish grid on the voltages and (2) voltage stability analysis (VSA) by considering the power exchange between neighbouring countries.

Finally, it is studied how the N T C of the tie-lines can be improved by injecting reactive power. The steps below give the sequence of the methodology:

1. VSA by increasing the load of the Swedish grid: The investigation of voltage stability is performed within the Swedish grid without considering the neighbouring countries. For this purpose, we increase the loads of the Swedish grid subsequently by 10, 20 and 30 percent in the three different areas of the model: the North, Centre and South-West. The tie-line buses of Sweden-Denmark and Sweden-Finland are of particular interest in the voltage stability analysis.

1Apply linear interpolation with the values for 345 and 500 kV in order to calculate the line parameters at 400 kV

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2.3 Voltage Stability Analysis on the test grid Chapter 2 2. VSA by examining the power exchange between neighbouring countries: In this section, the effect of power flowing through Sweden due to an exchange of power between Denmark and Finland, on the voltage stability within Sweden is analyzed.

Therefore, this study considers the grids of neighbouring countries. The first step is to evaluate the voltage stability within the Swedish grid in the presence of contingencies.

The most severe contingencies are those which are most likely to cause a voltage collapse in the grid [30]. We concentrate on the 10 most severe contingencies. For this study, the contingencies which stress the interfaces of Sweden-Finland and of Sweden-Denmark are of particular interest. In order to rank the most severe contingencies within the Swedish grid, we are evaluating how they affect the power transfer through all the branches of the Swedish grid.

Following assumptions have been made for performing the Voltage Stability Analysis in Sweden:

• Sweden is the main power system we are interested in to do VSA for, and we assume there is one tie-line between Sweden and its neighbours. In reality, there are more tie-lines with Finland and Denmark including HVDC lines, controlled by power electronic converters. Abstraction has thus been made of the way the countries are interconnected to each other. However, the method of this work is equally well applicable to more than one tie-line with a neighbouring country, and as the analysis is performed using PSS/E as the Power Flow tool, it should be mentioned that also HVDC lines can be modelled [31].

• The target is to calculate the Net Transfer Capacity (N T C). This is the maximum amount of MW on the tie-lines that Sweden can exchange with its neighbouring countries, while all the voltages in the Swedish network are stable and within the range. Also, an outage of one network element yields stable voltages that are within range. Since we are interested in calculating the PV-curve for each tie-line, we supervise the voltage of the connected bus to each tie-line, and calculate the N T C of the line.

• We assume that there is an original power flow when all these networks are connected together, and we use this as our base-case power flow. It is important to recognize that there is no voltage violation along the Swedish network in this base case. So we have a value for the flow on each tie-line which is the actual exchange with Sweden in the base case.

• Every value above or below the range of [0.9 , 1.1] is a ”violated voltage”.

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2.3 Voltage Stability Analysis on the test grid Chapter 2 We assume that the loads in Sweden are numbered from L_S1 to L_Sn. The loads in Denmark are numbered from LD1 to LDk. The loads in Finland are numbered from L_{F 1} to L_{F m}. The following steps produce the PV-curves of the tie-lines:

Step 1: We start with the current network topology for the three power systems connected together and run a power flow. Then all the voltages and the N T C of each tie-line are recorded. This is the starting point, the base case. It is clear that each power system has a total generation, a total load and a total loss, which are in balance for all the three networks together. We observe that there is no voltage violation in the Swedish network in the base case. The output of this step is the power flow in the whole network, the voltage of every bus and two MW-values for the N T C of the two tie-lines.

Step 2: We run an N − 1 contingency analysis for the Swedish network. We assume that there are N important grid components (generators, lines, transformers, loads, and shunt capacitors/reactors if any). The contingencies are ranked with respect to voltage severity and overload severity and this ranking produces the list of the 10 most severe contingencies. The output of this step is a list with the ranking of the 10 most severe contingencies.

Step 3: 25% of the N T C-value of the base case is added to the N T C value, so we get a MW value, for example 50 MW. Now we keep all the generations in the Swedish network and the external networks constant and then increase L_S1 to L_Sn by in total 50 MW. For example, if there are n = 100 loads, we increase every load by 0.5 MW. At the same time the loads in Finland (L_{F 1} to L_{F m}) and the loads in Denmark (LD1to LDk) are decreased by in total 50 MW (if there are k = 30 loads in Denmark and m = 30 in Finland, decrease each of them by (50/30) MW).

Step 4: We run a new power flow (same generation, but with loads decreased in Sweden, increased in external networks). Now we observe the power flow results and the voltage of all the nodes in the Swedish network. If there is no voltage violation along the Swedish network we go to step 5. But if there is a voltage violation (even for 1 node) we return to the step 3 and choose a smaller percentage such as 10%. Then, we change the loads of Sweden and the external networks with 10%

of the N T C-value of the base case.

Step 5: In step 4, we could run a power flow where no Swedish voltage was violated.

The total load increased by 50 MW in Sweden, and decreased by 50 MW in the external networks. Now the 10 most important contingencies identified in step

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2.3 Voltage Stability Analysis on the test grid Chapter 2 2 are examined. We apply each of them and record the voltages in the Swedish grid. If there is no voltage violation in the Swedish network for all of those 10 contingencies, this means that the system can handle an N − 1-contingency analysis by a load change of 50 MW and hence we can go to step 6. But if there is voltage violation (even for 1 contingency, even for 1 node), this means that the system cannot handle the N −1-contingency analysis where the load changed with 50 MW and we go to step 7.

Step 6: 50% of the N T C-value of the base case is added to the N T C value, so we get a MW value, for example 100 MW. Now we keep the all generations in the 3 networks constant and then increase LS1 to LSn by in total 100 MW. For example, increase each of the 100 Swedish loads by 1 MW. At the same time decrease the loads in Finland (LF 1 to LF m) and the loads in Denmark (LD1 to LDk) by in total 100 MW (if there are 30 loads, decrease each of them by (100/30) MW). Now we run a new power flow and observe the power flow results and the voltage of all the Swedish nodes. If there is no voltage violation in the Swedish network, step 5 is repeated (examine the 10 most severe contingencies) and if this still yields no voltage violation, we repeat step 6 with larger percentage such as 60%.

However, If there is voltage violation (even for 1 node) we return to step 3 and choose a smaller percentage (larger than 25% and smaller than 50%). After several iterations between the steps 3, 5 and 6, we obtain the final percentage, which gives largest possible MW value that can be transferred over the tie-lines while there will be no voltage violation when even each of the 10 most severe contingencies occur. The iteration is then stopped. The flow over the tie-line and the voltage of the Swedish bus connected to the tie-line are recorded and we go to step 8.

Step 7: In this case, the system cannot handle an N − 1-contingency analysis in a grid where the loads changed by 50 MW. Hence we have to choose a smaller percentage than 20% for example 15%. This means: change the loads by 15% of the N T C of the base case (decrease them in Denmark and Finland, increase them in Sweden) and we repeat step 3 with this new percentage.

Step 8: Draw the PV-curve of the tie-line. The output of step 6 is a percentage of the base case N T C after several iterations. This percentage corresponds with how much the loads of Sweden, Denmark or Finland need to be changed so that the largest MW value can be transferred over the tie-lines while there is no voltage violation when no or when a single element is out of service. This is the so-called N T C value. This value corresponds to an operating condition,

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2.4 Simulation and results Chapter 2 located on the stable halve of the PV-curve, lying closest to the nose point of the PV-curve. Now, in order to draw the curve, the coefficients of the set {1, 0.999, 0.99, · · · , 0.9, 0.85, 0.8, 0.7, 0.6, 0.5, 0.4} are multiplied with the value of the N T C. Thereafter, one power flow is run for each of these MW values and the corresponding voltage of the tie-line bus is obtained, yielding one point of the PV-curve.

Previous algorithm is a simplified version of a practice which is well known in VSA theory. The algorithm, used for drawing the PV-curves, is graphically illustrated in a flowchart in Fig.2.2.

3. Reactive power compensation: In this step the improvement in the N T C is investigated by placing reactive power compensators at buses in the Swedish grid. In other words, it is evaluated how much the PV-curves of the two tie-lines can be shifted to the right due to the injection of capacitive reactive power in the PQ-buses of the Swedish grid. To each PQ-bus located in the three different zones of the Swedish grid a 200 MVAr reactive compensator (implemented as a fixed shunt or capacitor bank) is attached but each time only one of them is switched on in order to find the most effective location for injecting reactive power.

2.4 Simulation and results

The work of this part is programmed in the power flow solver PSS/E with the help of the Python automation language. As was mentioned before, first the influence of increasing the loads in Sweden will be examined on the Swedish voltages. The most vulnerable part of the Swedish grid due to increasing the loads is the central area. Over 60% of the Swedish voltage buses collapse after increasing the central loads by 30%. The tie-line buses between Sweden- Denmark and Sweden-Finland are of particular interest in the voltage stability analysis. The highest effect on the voltages of these two tie-line buses comes from increasing the loads belonging to the centre part of the Swedish grid. Increasing the loads by 30% causes a 20%

decrease in the voltage of the Sweden-Denmark tie-line. Table 2.3 and Table 2.4 show the effect of the 10, 20 and 30 percent increasing of the loads in North and Centre of the Swedish grid respectively. The red cells indicate the buses with voltage below 0.9 p.u. and the yellow cells show the buses with voltage above 1.1 p.u. Fig.3.11 depicts the contouring plot of the voltage in the single line diagram of the Swedish grid, before and after the loads have been

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2.4 Simulation and results Chapter 2

Step 1:

Power flow on the combined S Fin Dk grid No voltage violation on Sgrid Record power flows on tie-lines

Base case

Step 2b:

Rank the S contingencies on voltage and overload severity Identify the 10 most severe

contingencies

Step 3:

Increase the loads in F Dk by x

% of the total active flow on the tie-lines between S and Dk

and between S and F Decrease the loads in S by x%

Step 4:

Perform power flow and observe the voltages within S

Is there at least one voltage violation

within S?

yes

no

yes no

Step 6b:

Perform power flow and observe the voltages within S Assign x to theI

value of y:x = y

no yes

STOP

Is there at least one voltage violation for at least one contingency

within S?

Is x < xmin? Decrease x

Step 5:

Perform a power flow analysis for each of the 10 most severe S contingencies see step 2b

and observe the voltages within S

Increase the loads in F Dk by y of the total active flow on the tie-lines

between S and Dk and between S and F

Decrease the loads in S by y%

Step 6a:

y>max{x,yprevious}

N-1 contingency analysis on S network

Step 2a:

+ +

+

( )

+ %

Figure 2.2: Proposed algorithm for determining the PV-curves of tie-lines

increased by 30%. It can be seen that the voltage in the Southern part of Sweden, and hence also around the tie-line with Denmark, decreases significantly when the load is increased.

Table2.2presents the contingencies, ranked from the most severe to less severe, which have the greatest effect on the voltages in the Swedish grid and on the power transfer across the Sweden-Denmark and Sweden-Finland interfaces.

Now based on the algorithm of Section 2.3, we can draw the PV-curves. Fig. 2.4(a) and Fig.2.4(b) show the PV-curves of the tie-line buses with Denmark and Finland respectively.

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2.4 Simulation and results Chapter 2 Table 2.2: Contingency ranking

Ranking Contingency Type 1 Generator at bus 4071 2 Generator at bus 4051 3 Generator at bus 4042 4 Generator unit1 at bus 4047 5 Generator unit2 at bus 4047

6 Branch 4011-4021

7 Generator at bus 1042 8 Generator at bus 4062 9 Generator at bus 4051 10 Generator at bus 4063

(a) Before increasing the loads (base case)

(b) After increasing the load by 30%

1.25

0.75 pu

pu

Figure 2.3: Contouring plot of the voltage in the Swedish grid before and after the load increase by 30%

These curves are part of the upper, stable, side of the complete PV-curve. As is shown, the maximum amount of power that can be transferred over the Sweden-Denmark and Sweden- Finland tie-lines is 1174 MW and 2295 MW respectively.

Considering the Sweden-Denmark tie-line, injecting reactive power to buses in the North of the Swedish grid does not have a lot of effect on increasing the N T C of this tie-line. However, the most significant improvement in the N T C happens when injecting reactive power at bus number 4061 in the South West. This causes the N T C of the Sweden-Finland tie-line to be increased by 135 MW. The same approach is applied for the Sweden-Finland tie-line and the most significant improvement in the N T C occurs as a result of attaching the reactive power compensator to bus number 4032. That bus is located in the North and by applying

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2.5 Conclusion Chapter 2 Table 2.3: Load increase by 10, 20 and 30 percent in the North of Sweden

Voltage (p.u.)

with different percentage of (P,Q)

Bus no. (10.10) (20.10) (30.10) (10.20) (20.20) (30.20) (10.30) (20.30) (30.30)

41 0.9855 0.9855 0.9855 0.9855 0.9855 0.9855 0.9855 0.9855 0.9855

42 0.982 0.982 0.982 0.982 0.982 0.982 0.982 0.982 0.982

43 0.9749 0.9749 0.9749 0.9749 0.9749 0.9749 0.9749 0.9749 0.9749

46 0.9706 0.9706 0.9706 0.9706 0.9706 0.9706 0.9706 0.9706 0.9706

47 1.0012 1.0012 1.0012 1.0012 1.0012 1.0012 1.0012 1.0012 1.0012

51 1.0008 1.0008 1.0008 1.0008 1.0008 1.0008 1.0008 1.0008 1.0008

61 0.9644 0.9644 0.9644 0.9644 0.9644 0.9644 0.9644 0.9644 0.9644

62 0.9818 0.9818 0.9818 0.9818 0.9818 0.9818 0.9818 0.9818 0.9818

63 0.9418 0.9418 0.9418 0.9418 0.9418 0.9418 0.9418 0.9418 0.9418

1011 1.1226 1.1228 1.123 1.1221 1.1223 1.1225 1.1215 1.1217 1.1219

1012 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13

1013 1.145 1.145 1.145 1.145 1.145 1.145 1.145 1.145 1.145

1014 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16

1021 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

1022 1.07 1.07 1.07 1.07 1.07 1.07 1.07 1.07 1.07

1041 0.9713 0.9713 0.9713 0.9713 0.9713 0.9713 0.9713 0.9713 0.9713

1042 1 1 1 1 1 1 1 1 1

1043 1 1 1 1 1 1 1 1 1

1044 0.9922 0.9922 0.9922 0.9922 0.9922 0.9922 0.9922 0.9922 0.9922

1045 0.9989 0.9989 0.9989 0.9989 0.9989 0.9989 0.9989 0.9989 0.9989

2031 1.0622 1.0622 1.0621 1.0622 1.0622 1.0621 1.0622 1.0622 1.0621

2032 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

4011 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01

4012 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01

4021 1.0378 1.0372 1.0366 1.0378 1.0372 1.0366 1.0378 1.0372 1.0366

4022 1.009 1.0087 1.0084 1.009 1.0087 1.0084 1.009 1.0087 1.0084

4031 1.0209 1.0208 1.0208 1.0209 1.0208 1.0208 1.0209 1.0208 1.0208

4032 1.0296 1.0294 1.0293 1.0296 1.0294 1.0293 1.0296 1.0294 1.0293

4041 1 1 1 1 1 1 1 1 1

4042 1 1 1 1 1 1 1 1 1

4043 0.9941 0.9941 0.9941 0.9941 0.9941 0.9941 0.9941 0.9941 0.9941

4044 0.9967 0.9967 0.9967 0.9967 0.9967 0.9967 0.9967 0.9967 0.9967

4045 1.0045 1.0045 1.0045 1.0045 1.0045 1.0045 1.0045 1.0045 1.0045

4046 0.9931 0.9931 0.9931 0.9931 0.9931 0.9931 0.9931 0.9931 0.9931

4047 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02

4051 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02

4061 0.9818 0.9818 0.9818 0.9818 0.9818 0.9818 0.9818 0.9818 0.9818

4062 1 1 1 1 1 1 1 1 1

4063 0.971 0.971 0.971 0.971 0.971 0.971 0.971 0.971 0.971

4071 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01

4072 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01

compensation to it, this configuration will lead to a rise of the maximum N T C by 35 MW.

Fig. 2.5(a) and Fig. 2.5(b) show these results schematically. It should be mentioned here that the optimal locations are sought manually, by trial and error. In the next two parts of this work, algorithms are developed to automatically find the optimal location and optimal MVAr rating.

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2.5 Conclusion Chapter 2 Table 2.4: Load increase by 10, 20 and 30 percent in the Centre of Sweden

Voltage (p.u.)

with different percentages of (P,Q)

Bus no. (10.10) (20.10) (30.10) (10.20) (20.20) (30.20) (10.30) (20.30) (30.30)

41 0,9855 0,9855 0,9148 0,9855 0.8436 0.8403 0.8436 0.8436 0.8436

42 0.982 0.9649 0.5862 0.982 0.7161 0.7099 0.7161 0.7161 0.7161

43 0.9686 0.955 0.2907 0.9664 0.5533 0.5531 0.5533 0.5533 0.5533

46 0.9666 0.9582 0.1765 0.9653 0.6984 0.6969 0.6984 0.6984 0.6984

47 1.0012 1.0012 0.3792 1.0012 0.7254 0.719 0.7254 0.7254 0.7254

51 1.0008 1.0008 0.3227 1.0008 0.3685 0.3532 0.3685 0.3685 0.3685

61 0.9643 0.9643 0.7988 0.9643 0.716 0.7104 0.716 0.716 0.716

62 0.9818 0.9818 0.7937 0.9818 0.7153 0.709 0.7153 0.7153 0.7153

63 0.9418 0.9418 0.851 0.9418 0.7632 0.7589 0.7632 0.7632 0.7632

1011 1.122 1.1222 1.1228 1.1231 1.1225 1,y1225 1.1225 1.1225 1.1225

1012 1.13 1.13 1.1325 1.13 1.137 1.1372 1.137 1.137 1.137

1013 1.145 1.145 1.145 1.145 1.145 1.145 1.145 1.145 1.145

1014 1.16 1.16 1.16 1.16 1,y16 1.16 1.16 1.16 1.16

1021 1.1 1.1 1.3295 1.1 1.1081 1.1083 1.1081 1.1081 1.1081

1022 1.0614 1.0405 1.0848 1.0682 1.097 1.0976 1.097 1.097 1.097

1041 0.9225 0.8748 0.001 0.8975 0.0001 0,y0001 0.0001 0.0001 0.0001

1042 1 1 0.4699 1 0.4469 0.434 0.4469 0.4469 0.4469

1043 0.9591 0.9168 0.108 0.9385 0.231 0.213 0.231 0.231 0.231

1044 0.9732 0.9499 0.3071 0.965 0.0497 0.0481 0.0497 0.0497 0.0497

1045 0.9782 0.9559 0.0734 0.9686 0.0179 0.0397 0,y0179 0.0179 0.0179

2031 1.0537 1.0437 1.092 1.0534 1.0499 1.0498 1.0499 1.0499 1.0499

2032 1.1 1.1 1.2147 1.1 1.1833 1.1853 1.1833 1.1833 1.1833

4011 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01

4012 1.01 1.01 1,y0287 1.01 1.0281 1.0286 1.0281 1.0281 1.0281

4021 1.013 1 0.9768 1.015 0.9861 0.9858 0.9861 0.9861 0.9861

4022 0.995 0.9733 1.0243 0.999 1.0281 1.0284 1.0281 1.0281 1.0281

4031 1.01 0.9969 1.0168 1.01 0.9782 0.9774 0.9782 0.9782 0.9782

4032 1.0149 0.9969 0.9371 1.0146 0.9085 0.9083 0.9085 0.9085 0.9085

4041 1 1 0.9161 1 0.853 0,y8495 0.853 0.853 0.853

4042 1 0.9833 0.599 1 0.7207 0.7141 0.7207 0.7207 0.7207

4043 0.988 0.9748 0.3474 0.9859 0.5661 0.5655 0.5661 0.5661 0.5661

4044 0.9824 0.9621 0.3623 0.9772 0.2638 0.2702 0.2638 0.2638 0.2638

4045 0.988 0.9688 0.1775 0.9812 0 0.0277 0 0 0

4046 0.9893 0.9811 0.278 0.988 0,y6745 0.6722 0.6745 0.6745 0.6745

4047 1.02 1.02 0.4239 1.02 0.7408 0.7343 0.7408 0.7408 0.7408

4051 1.02 1.02 0.3715 1.02 0.3839 0.369 0.3839 0.3839 0.3839

4061 0.9818 0.9817 0.821 0,y9818 0.7415 0.7362 0.7415 0,y7415 0.7415

4062 1 1 0.817 1 0.7411 0,y735 0.7411 0.7411 0.7411

4063 0.971 0.971 0,y8834 0.971 0.7991 0.7951 0.7991 0.7991 0,y7991

4071 1.01 1.01 1.01 1.01 1.01 1,y01 1.01 1.01 1.01

4072 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01

2.5 Conclusion

In this chapter a voltage stability analysis is performed on the Swedish grid and the role of reactive power compensation on voltage stability is analyzed. First the factors of the voltage instability mechanism are quantitatively investigated. Increasing the load, the first factor, is studied by increasing the loads inside Sweden without considering the role of the Denmark and Finland. The results show that increasing the loads in different areas of the Swedish grid has different effects on voltage stability. Increasing the loads located in the centre of

Reactive Power Planning with Voltage Stability Constraints for Increasing Cross-Border Transmission Capacity