Improved Planning of Wind Farms Using Dynamic Transformer Rating

DEGREE PROJECT IN ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2020

Improved planning of wind farms using dynamic transformer rating

ANDREA MOLINA GÓMEZ

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE Improved planning of wind farms using dynamic transformer rating

Förbättrad planering av vindkraftsparker med dynamisk lastbarhet hos transformatorer

Author Andrea Molina Gómez KTH Royal Institute of Technology

Program MSc Electric Power Engineering

Place and Date KTH Royal Institute of Technology, Stockholm, Sweden Hitachi ABB Power Grids, Västerås, Sweden September 2020

Examiner Patrik Hilber KTH Royal Institute of Technology

Supervisors Kateryna Morozovska KTH Royal Institute of Technology

Tor Laneryd Hitachi ABB Power Grids ii Abstract

Due to the increase in electrical demand and renewable penetration, electrical utilities need to improve and optimize the grid infrastructure. Fundamental components in this grid infrastructure are transformers, which are designed conservatively on the base of a static rated power. However, load and weather change continuously and hence, transformers are not used in the most efficient way. For this reason a new technology has been developed: Dynamic transformer rating (DTR). By applying DTR, it is possible to load transformers above the nameplate rating without affecting their life time expectancy. This project goes one step further and uses DTR for the short term and long term wind farm planning. The optimal wind farm is designed by applying DTR to the power transformer of the farm. The optimization is carried out using a Mixed-Integer Linear Programming (MILP) model. In respect of the transformer thermal analysis, the linearized top oil model of IEEE Clause 7 is selected. The model is executed for 4 different types of power transformers: 63 MVA, 100 MVA, 200 MVA and 400 MVA. As result, it is obtained that the net present value for the investment and the capacity of the wind farm increase linearly with respect to the size of the transformer. Then, a sensitivity analysis is carried out by modifying the wind speed, the electricity price, the lifetime of the transformer and the selected weather data. From this sensitivity analysis, it is possible to conclude that wind resources and electricity price are key parameters for the feasibility of the wind farm.

Keywords

Dynamic transformer rating, wind energy, loss of life, mixed-integer linear programming, wind farm planning. iii Sammanfattning

På grund av ökningen av efterfrågan av elektricitet och förnybara energin, elförsör- gingsföretag måste förbättras och elnätets infrastruktur måste optimeras. Grund- läggande komponenter i elnätet är transformatorer, som är designade konservativt efter en statisk märkeffekt. Laster och vädret ändras dock kontinuerligt, detta bety- der att transformatorer inte används på de mest effektiva sätten. Av denna anledning har en ny teknik utvecklats: Dynamisk lastbarhet hos transformatorer (DTR). Genom att applicera DTR, gör det möjligt att belasta en transformator högre än märkdata utan att påverka den förväntade livslängden. Detta projekt går ett steg längre och använder DTR för kort och lång sikts vindkraft- parkplaneringar. Den optimala vindkraftparken är designad genom att använda DLT på krafttransformatorn för vindkraftsparken. Optimeringen utförst med hjälp av Mixed-Integer Linear programming (MILP) modell. Gällande transformatorns ter- miska analys, så valdes den linjäriserade toppoljemodellen av IEEE Clause 7. Mode- llen var utförd för fyra olika krafttransformatorer: 63 MVA, 100 MVA, 200 MVA och 400 MVA. Resultatet blev att nettonuvärdet för investeringen och kapaciteten av vindkraftsparken ökade linjärt med avseende på storleken på transformatorn. En känslighetsanalys var utförd genom att ändra vindhastigheten, elpriset, livstiden av transformatorn och de valda väderdata. Från känslighetsanalysen så var det möjligt att dra slutsatsen att vindresurser och elpriset är nyckelparametrar för vindkrafts- parkens genomförbarhet.

Nyckelord

Dynamisk lastbarhet hos transformatorer, vindkraft, livstidsförlust, mixed-integer linear programming, vindkraftsparksplanering iv Acknowledgement

I would like to thank my supervisors Dr. Kateryna Morozovska and Dr. Tor Laneryd for their support, knowledge and help, even when circumstances were difficult. I would like to thank Hitachi ABB Power Grids for giving me the opportunity to join them on a project and for giving me a wonderful welcome. I enjoyed all the moments I spent in the office with my mates. I am very thankful to my examiner Dr. Patrik Hilber, who was also the director of my master, for selecting me and letting me joining KTH family. I have learnt and enjoyed a lot in this institution. I am very grateful for all the wonderful people I have met in Stockholm during these two years. Special thanks to Davide Garibaldi and Gabriel Gomes Guerreiro for making me enjoying even the long hard hours working in the lab. I would also like to thank my friends in Madrid for so many years of supporting and friendship. I would like to thank my friend Simon Lundgren for translating my abstract into Swedish and for showing me Swedish culture. I would like to thank again Davide Garibaldi for his feedback and unconditional support during quarantine time, grazie mille per questo meraviglioso viaggio. Specially, I would like to thank my parents and my brother for their support, pa- tience and for giving me always the best in life. Sin vosotros nada hubiera sido posible. Thanks also to the rest of my family, specially to my grandparents. Contents

1 Introduction 1

2 Literature review 3 2.1 Transformer review ...... 3 2.1.1 Working principle ...... 3 2.1.2 Losses in the transformer ...... 4 2.1.3 Cooling system ...... 5 2.1.4 Transformer insulation lifetime ...... 5 2.1.5 Insulation life expectancy ...... 6 2.2 Wind Power ...... 6 2.2.1 Power law profile ...... 7 2.2.2 Available wind power ...... 8 2.2.3 Power extracted by the wind turbine ...... 8 2.3 Dynamic transformer rating ...... 9 2.3.1 Studies for implementation of DTR in wind power ...... 10 2.3.2 Implemented projects ...... 11 2.4 Consequences of transformer overloading ...... 13 2.4.1 Limitations ...... 13 2.5 Transformer thermal models ...... 14 2.5.1 Thermal diagram of the transformer ...... 14 2.5.2 Comparison between thermal models ...... 15 2.6 IEC 60076-7 difference equations model ...... 16 2.6.1 Differential equations ...... 16 2.6.2 Difference equations solution ...... 18 2.6.3 Determination of time constants ...... 18 2.7 IEEE Clause 7: top oil model ...... 19 2.8 Linearized top oil model of IEEE Clause 7 ...... 20 2.9 Loss of life ...... 22 2.10 Optimization: MILP ...... 23

3 Optimization problem for one transformer 25 3.1 Wind farm system topology ...... 25 3.2 Target function ...... 26

v vi CONTENTS

3.3 Constraints ...... 27 3.3.1 Total capacity of the wind farm ...... 28 3.3.2 Power balance ...... 28 3.3.3 Power limitations ...... 28 3.3.4 Transformer constraints ...... 29 3.3.5 Thermal model constraints ...... 31 3.3.6 Transformer selection ...... 32 3.4 Optimization problem equations ...... 32 3.4.1 Optimization variables ...... 32 3.4.2 Target function ...... 33 3.4.3 Constraints ...... 33

4 Optimization problem for several transformer 35 4.1 Variable for the selection of the transformer ...... 35 4.2 Target function ...... 35 4.3 Constraints ...... 36 4.3.1 Transformer constraints ...... 36 4.3.2 Thermal model constraints ...... 37 4.3.3 Transformer selection ...... 38 4.3.4 Conclusion ...... 40

5 Input data 41 5.1 Weather data ...... 41 5.1.1 Analysis of the wind data ...... 41 5.1.2 Analysis of temperature data ...... 44 5.2 Discount rate ...... 44 5.3 Electricity price ...... 44 5.4 Electricity demand ...... 46 5.5 Wind farm ...... 46 5.5.1 Selected turbine ...... 46 5.5.2 Wind farm costs ...... 47 5.6 Transformer selection ...... 49 5.6.1 Transformer thermal parameters ...... 49 5.6.2 Transformer costs ...... 49

6 Results 51 6.1 Results using the original input data ...... 51 6.1.1 Results for 63 MVA transformer ...... 52 6.1.2 Results for 100 MVA transformer ...... 54 6.1.3 Results for 200 MVA transformer ...... 56 6.1.4 Results for 400 MVA transformer ...... 58 6.1.5 Analysis and comparison ...... 60 6.2 Results changing the wind speed ...... 61 CONTENTS vii

6.2.1 Results for 63 MVA transformer ...... 62 6.2.2 Results for 100 MVA transformer ...... 64 6.2.3 Results for 200 MVA transformer ...... 66 6.2.4 Results for 400 MVA transformer ...... 68 6.2.5 Comparison ...... 70 6.3 Results varying the electricity price ...... 71 6.4 Results varying the lifetime of the transformer ...... 76 6.4.1 Comparison ...... 79 6.5 Results varying transformer parameters ...... 80 6.5.1 Results for 63 MVA transformer ...... 81 6.5.2 Results for 200 MVA transformer ...... 83 6.5.3 Results for 400 MVA transformer ...... 86 6.6 Comparison ...... 88

7 Conclusions and future studies 90 7.1 Conclusions ...... 90 7.2 Future studies ...... 91

References 92 List of Figures

2.1 Simplified thermal diagram of a transformer. θ0 indicates the top-oil temperature and θh indicates hot-spot temperature ...... 14 2.2 Block diagram of the differential equations...... 17

3.1 System topology ...... 26 3.2 Linearization by segment of V ...... 30

5.1 Location of the weather station Kloten A ...... 42 5.2 Annual Energy Production per turbine[MWh] ...... 43 5.3 Electricity price evolution for Swedish area SE3, where the red line represents the average value...... 45 5.4 Power curve of the wind turbine ...... 47 5.5 Investment cost breakdown for a typical wind farm [1] ...... 48

6.1 Hot-spot and top-oil temperatures for 63 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 52 6.2 Current through the 63 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 52 6.3 Comparison between the wind farm production, the production from other generators and the total demand in 63 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 53 6.4 Ageing rate of 63 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 53 6.5 Hot-spot and top-oil temperatures for 100 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 54 6.6 Current through the 100 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 54 6.7 Comparison between the wind farm production, the production from other generators and the total demand in 100 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 55 6.8 Ageing rate of 100 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 55 6.9 Hot-spot and top-oil temperatures for 200 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 56

viii LIST OF FIGURES ix

6.10 Current through the 200 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 56 6.11 Comparison between the wind farm production, the production from other generators and the total demand in 200 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 57 6.12 Ageing rate of 200 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 57 6.13 Hot-spot and top-oil temperatures for 400 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 58 6.14 Current through the 400 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 58 6.15 Comparison between the wind farm production, the production from other generators and the total demand in 400 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 59 6.16 Ageing rate of 400 MVA transformer using data from 2015 and an electricity price of 100 €/MWh ...... 59 6.17 Capacity comparison between the transformers when using 2015 data and an electricity price of 100 €/MWh ...... 61 6.18 NPV comparison between the transformers when using 2015 data and an electricity price of 100 €/MWh ...... 61 6.19 Hot-spot and top-oil temperatures for 63 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 62 6.20 Current through the 63 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 62 6.21 Comparison between the wind farm production, the production from other generators and the total demand in 63 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 63 6.22 Ageing rate of 63 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 63 6.23 Hot-spot and top-oil temperatures for 100 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 64 6.24 Current through the 100 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 64 6.25 Comparison between the wind farm production, the production from other generators and the total demand in 100 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 65 6.26 Ageing rate of 100 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 65 6.27 Hot-spot and top-oil temperatures for 200 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 66 6.28 Current through the 200 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 66 x LIST OF FIGURES

6.29 Comparison between the wind farm production, the production from other generators and the total demand in 200 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 67 6.30 Ageing rate of 200 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 67 6.31 Hot-spot and top-oil temperatures for 400 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 68 6.32 Current through the 400 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 68 6.33 Comparison between the wind farm production, the production from other generators and the total demand in 400 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 69 6.34 Ageing rate of 400 MVA transformer using data from 2015 and an electricity price of 34 €/MWh ...... 69 6.35 NPV comparison between the transformers when using 2015 data and an electricity price of 34 €/MWh ...... 70 6.36 Comparison of the current through the 400 MVA transformer when varying the electricity price ...... 71 6.37 Comparison of the hot-spot temperature in 400 MVA transformer when varying the electricity price ...... 72 6.38 Comparison of the production of the wind farm using 400 MVA transformer when varying the electricity price ...... 72 6.39 Comparison of the production of the wind farm using 400 MVA transformer when varying the electricity price including also de total demand 73 6.40 Comparison of the top oil temperature in 400 MVA transformer when varying the electricity price ...... 73 6.41 Comparison of the ageing rate of 400 MVA transformer when varying the electricity price ...... 74 6.42 Capacity comparison when varying the electricity price ...... 75 6.43 NPV comparison when varying the electricity price ...... 75 6.44 Comparison of the current through the 400 MVA transformer when varying the transformer lifetime ...... 76 6.45 Comparison of the hot-spot temperature in 400 MVA transformer when varying the transformer lifetime ...... 77 6.46 Comparison of the production of the wind farm using 400 MVA transformer when varying the transformer lifetime ...... 77 6.47 Comparison of the top oil temperature in 400 MVA transformer when the transformer lifetime ...... 78 6.48 Comparison of the ageing rate of 400 MVA transformer when varying the transformer lifetime ...... 78 6.49 Capacity comparison when varying the transformer lifetime ...... 79 6.50 NPV comparison when varying the transformer lifetime ...... 80 LIST OF FIGURES xi

6.51 Comparison of the current through the 63 MVA transformer when varying the transformer parameters ...... 81 6.52 Comparison of the hot-spot temperature in 63 MVA transformer when varying the transformer parameters ...... 81 6.53 Comparison of the production of the wind farm using 63 MVA transformer when varying the transformer parameters ...... 82 6.54 Comparison of the top oil temperature in 63 MVA transformer when the transformer parameters ...... 82 6.55 Comparison of the ageing rate of 63 MVA transformer when varying the transformer parameters ...... 83 6.56 Comparison of the current through the 200 MVA transformer when varying the transformer parameters ...... 83 6.57 Comparison of the hot-spot temperature in 200 MVA transformer when varying the transformer parameters ...... 84 6.58 Comparison of the production of the wind farm using 200 MVA transformer when varying the transformer parameters ...... 84 6.59 Comparison of the top oil temperature in 200 MVA transformer when the transformer parameters ...... 85 6.60 Comparison of the ageing rate of 200 MVA transformer when varying the transformer parameters ...... 85 6.61 Comparison of the current through the 400 MVA transformer when varying the transformer parameters ...... 86 6.62 Comparison of the hot-spot temperature in 200 MVA transformer when varying the transformer parameters ...... 86 6.63 Comparison of the production of the wind farm using 400 MVA transformer when varying the transformer parameters ...... 87 6.64 Comparison of the top oil temperature in 200 MVA transformer when the transformer parameters ...... 87 6.65 Comparison of the ageing rate of 400 MVA transformer when varying the transformer parameters ...... 88 6.66 Capacity comparison when varying transformer parameters ...... 89 6.67 NPV comparison when varying transformer parameters ...... 89 List of Tables

2.1 Cooling classes ...... 5 2.2 Approximate values of surface roughness length for various types of terrain [2] ...... 8 2.3 Current and temperature limitations ...... 13 2.4 Required data comparison between IEEE Annex G model and IEC difference equation model. represents the input parameters that is necessary to provide to the model, while * represents the parameters which can be calculated from the input parameters...... 16 2.5 IEEE suggested values for m and n ...... 20

5.1 Wind data analysis ...... 42 5.2 Average wind speed per year ...... 43 5.3 Temperature data analysis ...... 44 5.4 Analysis of the electricity price area SE3 ...... 45 5.5 Analysis of the electricity demand for this project. This demand is 10 times smaller than the original SE3 demand...... 46 5.6 Characteristics of G128-5.0MW wind turbine ...... 47 5.7 Wind farm costs ...... 48 5.8 Transformers investment costs and O&M costs ...... 50

6.1 Optimal wind farm design and NPV for each transformer when using 2015 data and an electricity price of 100 €/MWh ...... 60 6.2 Optimal wind farm design and NPV for each transformer when using 2015 data and an electricity price of 34 €/MWh ...... 70 6.3 Optimal wind farm design and NPV for each transformer when using 400 MVA transformer and a varying price ...... 74 6.4 Optimal wind farm design and NPV for each transformer when using 400 MVA transformer and a varying transformer lifetime ...... 79 6.5 Comparison between the original transformer parameters (indicated with 1) and the modified transformer parameters (indicated with 2) .. 80 6.6 Comparison between the results with the original transformer parameters (indicated with 1) and the results after changing the transformer parameters (indicated with 2) ...... 88

xii Chapter 1

Introduction

Nowadays, electric utilities have to face new challenges related to the increase of electricity demand and share of renewable energy on the electricity market. Answering these challenges results in a need for upgrading the existing grid involving substan- tial investment costs and time. Therefore, utilities are searching for new methods to meet these requirements by optimizing the existing grid infrastructure. Furthermore, in September 2015, all United Nations member states adopted the 2030 Agenda for Sustainable Development which entails 17 sustainable development goals that are to be achieved by 2030 [3]. Among these goals, goal number 7 calls for clean and affordable energy and goal number 13 addresses climate action. These goals have as objective the decrease of carbon intense energy sources and the achieve of a more efficient and reliable energy production through a responsible use of ma- terials. Therefore, this is a global problem and a global responsibility and hence, it is necessary to search for solutions. Oil-immersed power transformers are key components in substations and are also the most expensive components [4]. A failure in these components impacts heavily the performance and the economics of the entire power system. Transformers operational limits are mainly established by the insulation degradation which determines the transformer ageing. This degradation primarily depends on the winding hot-spot temperature of the transformer. Currently, transformers are designed conservatively based on static transformer rating which is the nameplate rated power of the transformer. This nameplate rated power is calculated in order for transformers to withstand extreme loading and weather scenarios. However, load and weather are changing constantly. Therefore, the real transformers capacity is changing continuously. This demonstrates that transformers are not used in the most efficient way at present. As transformers are important components in substations, their optimization in terms of loading and insulation aging is crucial for the utilities. For this reason, a new technology has been developed: Dynamic transformer rating (DTR). This technology is used to further extend the capacity of transformers by using them in a more efficient way. By applying DTR, it is possible to load transformers above their name-

1 2 CHAPTER 1. INTRODUCTION plate rating without affecting their lifetime expectancy. Instead of being limited by a fixed nameplate rating, the rating of transformers is variable. In order to avoid over- temperatures in the devices, real-time monitoring and continuous recalculation of thermal parameters are needed. DTR is specially beneficial for the penetration of renewable energies in the electricity market. In particular, it is truly beneficial for transformers that connect wind power generators to the grid where the load profile is intermittent and unpredictable. Normally, power transformers for wind farms are designed for the peak load production. However, this means that they operate below its nameplate rating 90 % of the time [5]. By using DTR it is possible to optimize these transformers by reducing their size obtaining a cutback in the investment costs [6][7]. In addition, it is also possible to use DTR to expand the wind farm without adding new transformers [8]. Therefore, DTR can facilitate a move towards achieving the sustainable development goals. This thesis goes one step further and uses DTR for the short term and long term wind farm planning. The goal of the project is to plan the optimal wind farm (size and production) for a power transformer to which DTR has been applied. The optimization is carried out using a Mixed-Integer Linear Programming (MILP) model. This model is composed by equations that reflect the behaviour of the wind farm, the thermal model of the transformer and the grid performance. The criteria to select the optimal wind farm is to maximize the net present value of the wind farm, which depends on the wind farm production, the electricity price, the wind farm costs and the initial investment. In this master thesis project, Chapter 2 includes a literature review about power transformer basics, wind power principles, an introduction to dynamic transformer rating presenting some already implemented projects, the different transformer thermal models and important transformer thermal parameters and finally, an initiation to mixed-integer linear programming. In Chapter 3, the formulation of the optimization problem for one transformer is explained. In this chapter, all the necessary equations that build the optimization problem are explained and justified. Chapter 4 contains the formulation of the optimization problem with automatic selection of the transformer. This chapter explains the changes that have to be made in the previous formulation in order to select automatically the transformer by the program. In Chapter 5, the input data necessary for the performance of the program are given. Chapter 6 presents and analyses the results obtained by the optimization program. Finally, Chapter 7 provides the conclusions of the study and gives suggestions for future studies. Chapter 2

Literature review

2.1 Transformer review

IEEE defines a transformer as “a static electrical device, involving no continuously moving parts, used in electric power systems to transfer power between circuits through the use of electromagnetic induction” [9]. Inside this group, power transformers are the ones used between generators and distribution circuits in order to increase or decrease voltage. Therefore, transformers can be used for step-up operation, mainly used at generation, and for step-down operation, mainly used to feed distribution circuits. They can also be single-phase or three-phase. Power transformers are classified into three groups depending on their power rating:

• Small power transformers: 500 to 7500 kVA

• Medium power transformers: 7500 kVA to 100 MVA

• Large power transformers: 100 MVA and above

The transformer is built up by a core, normally made of stacked layers of lami- nated steel, windings, a coolant and magnetic shields, electrostatic shields or both [10]. In larger power transformers, windings are composed by copper rectangular strip conductors insulated by oil-impregnated paper and blocks of pressboard.

2.1.1 Working principle When a varying current appears in the primary winding of the transformer, a varying magnetic flux is created in the transformer core. This varying flux induces a varying electromotive force (EMF) in the secondary winding following the rules of electromagnetic induction. Equation (2.1) is the formula to calculate the rms value of the EMF force.

3 4 CHAPTER 2. LITERATURE REVIEW

2πfNaBpeak Erms = √ (2.1) 2 Where f is the supply frequency, [Hz]; N is the winding number of turns; a is the 2 2 core cross-sectional area, [m ]; and Bpeak is the peak magnetic flux density, [Wb/m ]. According to Faraday’s law, a voltage is induced in each winding. The transformer winding voltage ratio is directly proportional to the winding turns ratio, as it is possible to see in the following equation.

U N 1 = 1 (2.2) U2 N2

Where U1 and U2 are the voltage of the primary and secondary side of the transformer respectively, [V], and N1 and N2 are the number of turns of the primary and secondary winding respectively.

2.1.2 Losses in the transformer

Power transformers are among the most efficient electrical machines and large power transformers normally have an efficiency higher than 95 % [10]. However, as the transformer is not an ideal device, there are some losses. It is possible to distinguish between the losses originated in the windings, described as copper losses and those originated in the core known as iron losses. Copper losses are produced by the current when flowing through the windings. These losses are identified as heating due to the resistive part of the windings. When working at high frequencies, skin effect and proximity effect appear in the winding increasing the winding resistance and producing extra losses. With respect to core losses, there are several causes that provoke this loss. These causes are the following:

• Hysteresis losses due to magnetic field reversing

• Eddy currents losses that are a function of the square of the frequency and the square of the material thickness

• Magnetostriction losses produced by physical expanding of the core material

• Mechanical losses

• Stray losses in the transformer’s foundation

• In large transformers where a cooling system is needed, the cooling system is part of the losses 2.1. TRANSFORMER REVIEW 5

2.1.3 Cooling system Mainly all the losses in the transformer are converted in dissipated heat. The principal sources of heat are the winding and the core. There are different methods to remove this heat depending on the application, size of the transformer and amount of heat that needs to be dissipated. The cooling system is composed of two parts: the internal part and the external part. The internal part is filled with the insulating liquid, usually oil, which acts as insulator and as medium to remove the heat. This fluid absorbs the heat produced in the transformer and goes to the transformer tank through cooling ducts. The fluid in this moment has a temperature equal to the top oil temperature. In the transformer tank wall, the heat is transferred to the external environment. Radiators are usually used to increase the surface for heat transfer. The external part is directly the environment (air) or a coolant (water). The liquid can flow using natural convection or by installing some auxiliary equipment as fans or pumps which is called forced circulation. The cooling classes which specifies the cooling method for the transformer are represented by four letters. First two letters are related to the internal medium and the two other letters are related to the external medium. In both cases, first letter identifies the cooling medium and second letter the cooling mechanism. The cooling classes are presented in Table 2.1.

Table 2.1: Cooling classes

Medium Letter Code letter Description O Liquid with flash point less than or equal to 300 ◦C 1st K Liquid with flash point greater than 300 ◦C L Liquid with no measurable flash point Internal N NC cooling equipment and windings 2nd F FC cooling equipment, NC in windings D FC cooling equipment, directed flow in windings A Air 3rd W Water External N NC 4th F FC

Where NC corresponds to “Natural Convection” and FC identifies “Forced Circu- lation”.

2.1.4 Transformer insulation lifetime A power transformer is expected to operate reliably for up to 40 years [11]. There are several causes of a decrease in transformer reliability. One of the most important 6 CHAPTER 2. LITERATURE REVIEW causes is the thermal degradation of paper insulation. The paper and pressboard for electrical insulation are made of cellulose, which deteriorates slowly as the cons- tituent large molecular chains break down during service [11]. By adding various nitrogen-containing compounds the ageing characteristic of the cellulose may be improved by neutralizing the production of acids caused by the hydrolysis (thermal degradation) of the material. Different parameters can be used to characterize cellulose degradation process during aging. In IEC 60076-7 standard [12], the degree of polymerization (DP) is used in order to describe the state of the insulation paper. The usual value for the start of operation is 1000 DP. When the DP is reduced to 200 or 35% retained tensile strength, the insulating material is considered to reach its “end of life” because the quality of the paper is really low.

2.1.5 Insulation life expectancy

One way to measure the insulation life expectancy is the ageing or change in polymerization of paper insulation described by Arrhenius equation as first order polynomial equation [12]. The ageing is given by

− EA 1 1 · − = A · t · e R (θh+273) (2.3) DPend DPstart

Where DPend is the insulation DP at the moment of the sampling; DPstart is the initial value of insulation DP; A is the pre-exponential factor, [1/h]; EA is the activa- tion energy, [kJ/mol]; t is the lifetime of transformer, [h]; R is the gas constant, [/(K ◦ mol)]; and θh is the hot-spot temperature, [ C]. It is possible to rewrite the previous equation to calculate life expectancy in years as follows:

1 1 − EA DPend DPstart · t = · e R (θh+273) (2.4) exp A · 24 · 365

In order to use these equations, a valid selection of EA and A is fundamental.

2.2 Wind Power

Wind power is the use of wind for electricity production through the transformation of the kinetic energy created by air into motion by using wind turbines. This way of electricity production is one of the fastest-growing renewable energy technologies. Global installed wind power capacity jumped from 7.5 GW in 1997 to around 564 GW by 2018 and in 2016 wind power represented the 16 % of the total energy produced by renewable resources [13]. 2.2. WIND POWER 7

In the case of Sweden, the 54 % of the total energy production is from renewable sources and the wind power capacity is currently around 6500 MW (16.6 TWh) re- presenting 11 % of the total power consumption [14]. Swedish target is to get 100 % share of renewable electricity production by 2040 [15]. Once wind power and its situation have been introduced, some of its fundamental characteristics are explained below. All the equations are taken from [2].

2.2.1 Power law profile

The power law represents a simple model for the vertical speed profile. It is used to scale up the wind speeds to the hub height of the turbine. The basic form of the power law is given by Equation (2.5).

( ) U(z) z α = (2.5) U(zr) zr

Where U(z) is the wind speed, [m/s], at height z, [m]; U(zr) is the reference wind speed, [m/s], at height zr, [m]; and α is the power law exponent. There are several methods to calculate the power law exponent. The more popular empirical methods for determining α are “Correlation for the power law exponent as a function of velocity and height” proposed by Justus and “Correlation dependent on surface roughness” proposed by Counihan. Equation (2.6) gives the first proposal and Equation (2.7) gives the second proposal.

0.37 − 0.088ln(U ) α = ref (2.6) − zref 1 0.088ln( 10 )

2 α = 0.096log10z0 + 0.016(log10z0) + 0.24 (2.7)

Where Uref is the reference wind speed, [m/s]; zref is the reference altitude, [m]; and z0 is the surface roughness, [m], that varies as 0.001m < z0 < 10m. In the following table it is possible to see some approximate values of surface roughness length for different types of terrain. 8 CHAPTER 2. LITERATURE REVIEW

Table 2.2: Approximate values of surface roughness length for various types of terrain [2]

Terrain description z0 (mm) Very smooth, ice or mud 0.01 Calm open sea 0.20 Blown sea 0.50 Snow surface 3.00 Lawn grass 8.00 Rough pasture 10.00 Fallow field 30.00 Crops 50.00 Few trees 100.00 Many trees, hedges, few buildings 250.00 Forest and woodlands 500.00 Suburbs 1500.00 Centers of cities with tall buildings 3000.00

2.2.2 Available wind power By determining the mass flow of air through a rotor disc and applying the continuity equation of fluid mechanics, it is possible to calculate the kinetic energy available from wind as follows: 1 P av = ρAU 3 (2.8) w 2 Where ρ is the air density, [kg/m3]; A is the area swept by the rotor, [m2]; and U is the wind velocity, [m/s]. It is possible to calculate the density of the air as a function of the altitude. [ ] P −gz ρ(z) = s0 exp [kg/m3] (2.9) RairT RairT

Where Ps0 is the standard atmospheric pressure at sea level (101.3 kPa); Rair is the specific gas constant for air (287.05 J/kg·K); g is the gravitational constant (9.81 m/s); T is the temperature, [K]; and z is the altitude, [m].

2.2.3 Power extracted by the wind turbine It is not possible to extract all the available power from wind. The theoretical maximum power that can be extracted from the wind is set by Betz limits which is given by Equation (2.10). 1 1 P = ρAU 3C = ρAU 30.59 (2.10) Betz 2 p,Betz 2 2.3. DYNAMIC TRANSFORMER RATING 9

The average wind turbine power can be calculated as follows: ∫ Uout ¯ Pw = Pw(U)p(U)dU (2.11) 0

Where Uout is the cut-out wind speed of the turbine, [m/s]; Pw(U) is the turbine power curve, [kW]; and p(U) is the wind probability density function. Then, the annual energy production (AEP) can be calculated as follows

¯ AEP = Pw · Number of hours per year · Availability · m (2.12)

2.3 Dynamic transformer rating

Dynamic rating (DR) applied to power components is a method for assessing real- time capacity of large scale power transmission and distribution devices. In general, this method is applied to transmission lines, power transformers and underground cables. Dynamic transformer rating (DTR) is an emerging technology and can be defined as: “The maximum loading which the transformer may acceptably sustain under time-varying load and/or environmental conditions” [16]. The new transformer ampacity depends on several limiting factors as: the hot-spot temperature, bottom oil transient temperature, maximum contingency loading and ambient temperature [17]. In order to set the transformer ampacity, a thermal assessment has to be carried out. This thermal assessment is carried out by using one of the thermal models, for ins- tance one of the model specified by the principal standards: IEC 60076-7 [12] or IEEE C57.91 [18]. DTR is possible thanks to three principles: the variation in ambient temperature, oil thermal time constant and the cumulative process of insulation thermal aging [16]. In order to explain these three principles, first it is necessary to explain static transformer rating and how it is calculated. Static transformer rating is the nameplate rated power of the transformer. Accor- ding to IEEE C57.12 [19], the rated power of a 65 ◦C average winding rise transformer is the output power that the transformer can provide under rated frequency, rated secondary voltage, continuous ambient temperature of 30 ◦C and continuous winding hot-spot rise over ambient of 80 ◦C giving a continuous rated hot-spot temperature of 110 ◦C. This operation of the transformer results in, what IEEE defines as a “normal insulation life”, of 20.55 years [18]. However, a continuous rated ambient temperature of 30 ◦C is atypical. There- fore, if the ambient temperature is lower than the rated temperature, the hot-spot rise over ambient temperature can be higher without exceeding the rated hot-spot temperature and the projected normal life of the transformer. The conclusion of this effect is that the unit can run at higher load while keeping within the same normal life. 10 CHAPTER 2. LITERATURE REVIEW

This technique can be used considerably in Sweden where the ambient temperature rarely reaches 30 ◦C. Concerning the cumulative aging process, a continuous operation at rated load is also atypical because the load is changing continuously. Therefore, if the load is lower than rated load, the loss of life of the transformer is lower than normal too. As a consequence, the transformer can operate at higher load during some periods exceeding the rated hot-spot temperature, if during other periods the load is lower than normal with temperatures below 110 ◦C. In this way, a balance between both periods is achieved and the transformer loss of life stays within the limits of the normal lifetime expectancy. The oil thermal time constant also plays an important role in DTR. The insulating oil thermal time constant is on the order of hours while the winding thermal time constant is on the order of minutes. Therefore, if the peak in the load is short enough,the oil will be able to remove the heat generated in the windings and the windings will not reach the hot-spot temperature limit. In order to avoid over-temperatures in the device, real-time monitoring and continuous recalculation of thermal parameters is needed. In other words, it is necessary a real-time measure of the load, ambient temperature, top oil temperature and hot- spot temperature. For this propose, fiber optic temperature sensors inside the transformer and a SCADA system to monitor the real-time data and calculate the thermal algorithm are used [20]. Focusing on the existing literature on DTR, it is found that there is a significant higher number of scientific articles related to DLR (Dynamic Line Rating) than related to DTR [17]. Among them, the majority of the scientific papers are about models and monitoring systems. In the following subsection, the studies where DTR is applied to wind power are briefly presented. In addition, some implemented projects are also described.

2.3.1 Studies for implementation of DTR in wind power In this sub-section, the scientific studies that apply DTR to wind power are analysed. Moreover, some studies that combine wind power with optimization problems are also studied.

• [21] shows the economic benefits of implementing DTR in wind park expansion. In addition, it demonstrates that it is possible to overload the studied transformers without affecting the life expectancy while having positive economical advantages.

• [6] does an economical and risk analysis to investigate the suitability of repla- cing a 19.4 MVA with a 16 MVA dynamically rated. This study concludes that it is profitable and with an acceptable risk to change one transformer for the other. 2.3. DYNAMIC TRANSFORMER RATING 11

• In [7], a reliability and economic analysis about the effects of DTR from the component perspective is carried out. It demonstrates how by applying DTR is possible to decrease the investment cost while improving operational performance. • [22] studies the effects on the grid after implementing DTR in the transformers of a wind farm in order to increase wind power. In order to carry out this analysis, a power system analysis is executed to see how DTR affects the grid performance with respect to reliability, voltage stability and active losses. In addition, the study identifies which contingencies cause more damages when DTR is applied. • [8] shows how it is possible to expand a wind farm without installing new transformers by applying DTR to the existed transformers. • [23] is a day-ahead dispatch optimization problem combining DTR and DLR. Results show how the combination of DTR and DLR, instead of the application of these two methods separately, yields to an increase of the system capacity reducing dispatch costs. • In [24], a method to size the transformers in a planning stage using DTR is presented. • [25] shows a way to performance an optimal microgrid design with multiple energy types. This microgrid design consists on an optimization model formu- lated as a mixed-integer linear program, which determines the optimal technology portfolio, the optimal technology placement, and the associated optimal dispatch.

2.3.2 Implemented projects In this subsection, two implemented projects in real life are introduced.

UNISON project Unison Networks Limited (Unison) is the company who carried out this project. This company is an electricity distribution company that operates in New Zealand’s North Island [26]. In the financial year 2012-2013, Unison implemented DTR in a total of 50 transformers out of 58 transformers that they owned. Unison developed its DTR algorithm following IEC60076-7 [12] standard. They collected the real-time data through sensors all over the network. With these sensors, they collected the following data: transformer’s load, top oil temperature, tap position, cooling operation and surrounding ambient temperature. Using these data as inputs, the DTR algorithm calculated in real time the dynamic rating for the transformer. As monitoring system, SCADA is used. 12 CHAPTER 2. LITERATURE REVIEW

In the process, they had some challenges. One of the challenges was to customize the algorithm for each different transformer involved in the project: older transformers do not have heat-run test data, some of them do not have fans, the transformers are from different manufacturers and have different specifications, etc. Ano- ther challenge was to obtain a optimal accuracy for the calculated or measured hot spot temperature, because they are difficult to measure and their location depend on the desing of the transformer. Finally, they demonstrated that maximum capacity is reached during winter, when the calculated dynamic rating for a transformer is around 150 % of its name plate rating. During the warmer months, it is possible to achieve in average a 140 % increase above nameplate rating [27].

FALCON project

FALCON (Flexible Approaches to Low Carbon Optimised Networks) is a project carried out by the British company: Western Power Distribution (WPD). The location of the project is in Milton Keynes with the goal of increasing the capacity during peak hours and introducing renewable energy in the system. The available studies about this project extend during the years: 2014 [28], 2015 [29][30] and 2016 [31]. This project took into account five different asset types: primary ground mounted transformers (33kV/11kV), secondary ground mounted transformers (11kV/415V), underground 33kV cables, underground 11kV cables and overhead 11kV conductors. At the first stage of this project ([28]), results showed that it is possible to run the transformers up to 10 % above their nameplate during winter and below nameplate in summer to get the normal lifetime expectancy. In addition, the project studied how the different thermal models perform in real-life situations and concluded that the IEC differential model is the most appropriate model for the UK distribution power network. At the second stage of the project [29], the variability in ampacity was highlighted. The results showed variability of around 10 % of the rated current in the ampacity due to seasonal reason. This seasonal variability was due to the daily and monthly variation in temperature. The third stage of the project [30] was focused in a network vision and proposed a dynamic network rating (DNR). The project investigated the gains of implementing dynamic rating in the different assets of the network: transformers, overhead lines, and cables. It concluded that there were gains to network operators when applying dynamic rating referring to peak loading during winter. However, these gains could be lower than the static rating during hot days in summer. In addition, it also concluded that it is not necessary to monitor all the assets in the network, only the ones that have the biggest constraints. At the last stage of the project [31], the parameter values of the transformer were studied. These parameters are crucial to estimate correctly the top oil temperature 2.4. CONSEQUENCES OF TRANSFORMER OVERLOADING 13 and hence, to set the dynamic load limit. A weighted regression model was created in order to choose the best parameters which better estimate higher oil temperatures.

2.4 Consequences of transformer overloading

The consequences of loading a transformer beyond its nameplate rating are: the temperature increase in winding, cleats, leads, insulation and oil; the increase in the leakage flux density outside the core causing additional eddy-currents; change in the moisture and gas content in the insulation and in the oil; higher stresses in brush- ing, tap-changers, cable-end connections and current. All these consequences may provoke the premature transformer failure. However, as it is concluded in [8], it is possible to overload the transformer without provoking a high reduction in its lifetime while staying within the allowed limits. The sensitivity of transformers to loading beyond nameplate rating usually depends on their size. As the size increases, the tendency is that: the leakage flux density increases, the short circuit forces increase, the mass of insulation is increased and the hot-spot temperatures are more difficult to determine. Therefore, a larger transformer could be more vulnerable to overloading than a smaller one. In addition, the consequences of a transformer failure are also more severe for larger transformers.

2.4.1 Limitations

Table 2.3 shows the maximum permissible temperature and current limitations for loading beyond nameplate rating given by the standards [12][18]. It is recommended by the standards not to exceed the current limits even if the overload does not provoke an excess of the temperature limitations.

Table 2.3: Current and temperature limitations

Types of loading Parameter Normal Long-time Short-time Winding hot-spot temperature and metallic parts 120 140 160 in contact with cellulosic insulation material [◦C] Other metallic hot-spot temperature [◦C] 140 160 180 Top-oil temperature in tank [◦C] 105 115 115 Current (p.u.) 1.5 1.5 1.8

When the hot/spot temperature exceeds 140 ◦C, it is possible that gas bubbles could appear which could decrease the dielectric strength of the transformer. 14 CHAPTER 2. LITERATURE REVIEW 2.5 Transformer thermal models

When doing a thermal modeling of a transformer, the most critical variable to take into account is the hot spot temperature. However, this variable is not easy to calculate. It can be measured using direct monitoring techniques, such as ﬁber optics, temperature sensors, loading cells and laser, or it can be calculated using indirect monitoring techniques, i.e. meteorological data and transformers thermal models [32]. The direct monitoring techniques are usually expensive and cannot be applied to old already installed transformers. For those reasons, transformer thermal models are so important.

2.5.1 Thermal diagram of the transformer

Since the thermal distribution in the transformer is really complex, a simplified thermal diagram is derived by the standards. This simplified thermal diagram is shown in Figure 2.1.

Figure 2.1: Simplified thermal diagram of a transformer. θ0 indicates the top-oil temperature and θh indicates hot-spot temperature

For this simplification, the following assumptions are made: 2.5. TRANSFORMER THERMAL MODELS 15

• The oil temperature inside the tank increases linearly from bottom to the top, independently from the cooling system • The temperature rise in the conductor at any position up in the winding is assumed to increase linearly in parallel to the oil temperature rise with a constant difference (gr) with respect to the temperature rise in the conductor • The hot-spot temperature rise is higher than the temperature rise of the conductor at the top of the winding because it is necessary to take into account the increase in stray losses, the different local oil flows and the additional paper around the conductor. In order to take into account all these linearities, a hot spot factor is used (H), which is winding-specific and has to be determined for each transformer case. Therefore, the difference in temperature between the hot-spot and the top-oil tank can be calculated as follows ∆θhr = H · gr, where θhr refers to the hot-spot temperature rise above top-oil temperature in the tank at rated current.

2.5.2 Comparison between thermal models The most important and most followed thermal models are the ones given by the standards: IEEE C57.91 [18] and IEC 60076-7 [12]. IEEE C57.91 proposes two thermal models: IEEE top oil rise model or Clause 7 model and IEEE bottom oil model or Annex G model. Clause 7 is the simplest model and it does not require an iterative process. This model does not take into account changes in the ambient temperature and its model of the load is poor. As result, this method underestimates the hot-spot temperature during overloading periods which provokes an underestimation of loss of life [16]. Therefore, this method is the least accurate. Annex G model is more complex and provides more accurate specially for transient loading conditions. This model takes into account type of liquid, cooling mode, oil viscosity, resistance changes due to the temperature change, ambient temperature changes and load changes. The main difference compared to IEC model is that IEEE Annex G is suitable only for oil-immersed transformer but it can also be used for silicon and HTHC transformers. IEC 60076-7 loading guide for mineral oil-immersed power transformers considers the variation of load current and ambient temperature. It gives two possi- bilities to describe the hot-spot temperature as a function of time: the exponential equation model and the difference equation model. Both methods give to the same results, because both come from identical heat transfer differential equations. The difference between them is the application: the exponential method is more suitable, when the heat transfer parameters are calculated by tests and the difference method is more suitable for on-line monitoring. In respect of the required data needed for the model, Table 2.4 presents a comparison between IEEE Annex G model and IEC difference equation model. Therefore, IEEE Annex G requires more data than IEC difference equation model. 16 CHAPTER 2. LITERATURE REVIEW

Table 2.4: Required data comparison between IEEE Annex G model and IEC difference equation model. represents the input parameters that is necessary to provide to the model, while * represents the parameters which can be calculated from the input parameters.

Type of data IEEE IEC Top oil temperature rise at rate load Hot spot temperature rise over top oil at rated load Loss ratio at rated load Winding time constant Oil time constant Type of cooling Average winding temperature rise at rated load * Average oil temperature rise at rated load * Bottom oil temperature rise at rated load Losses (no-load, load, stray, eddy) Weight of core, coil, tank and oil * Winding and tank material Type of fluid Hot spot factor *

As different studies present, IEEE Annex G model and IEC model yield similar results in terms of hot-spot calculation [8]. Apart from IEEE and IEC, there are more thermal models in the literature. These models are: Swift model [33], Susa model [34] and a linearized top oil model of IEEE Clause 7 [35][36]. A more extensive explanation and comparison between models can be found in [8][7]. In the following sections, IEC 60076-7 difference equations model and IEEE Clause 7 and its linearized version are explained in more detail.

2.6 IEC 60076-7 difference equations model

2.6.1 Differential equations

The transfer differential equation for top-oil temperature is given by Equation (2.13). [ ] 1 + K2 · R x dθ · (∆θ ) = k · τ · 0 + [θ − θ ] (2.13) 1 + R 0r 11 0 dt 0 a

Where K is the load factor; x is the top-oil exponent; θa is the ambient tempera- ◦ ◦ ture, [ C]; θ0 is the top-oil temperature, [ C]; θ0r is the top-oil temperature at rated current, [◦C]; R is the ratio of load losses at rated current to no-load losses at rated voltage; k11 is a thermal model constant; and τ0 is the oil time constant 2.6. IEC 60076-7 DIFFERENCE EQUATIONS MODEL 17

This equation can be solved easily by rewriting it as the sum of two differential equation solutions as follows:

∆θh = ∆θh1 − ∆θh2 (2.14) These two differential equations are described as follows d∆θ k · Ky · (∆θ ) = k · τ · h1 + ∆θ (2.15) 21 hr 22 w dt h1

y τ0 d∆θh2 (k21 − 1) · K · (∆θhr) = · + ∆θh2 (2.16) k22 dt

Where τw is the winding time constant, [min]; ∆θhr is the hot-spot to top-oil gradient at rated current and y is the winding exponent. The constants k11, k21, k22 and the time constants τw and τ0 are transformer specific. In addition, the constants k21, k22 and τw can be defined only if the transformer has installed fibre optic sensors. Finally, the hot-spot temperature can be expressed as

θh = θ0 + ∆θh (2.17) It is possible to represent the differential equations using a block diagram shown in Figure 2.2. As it is possible to see, the inputs are the load factor and the ambient temperature and the output is the hot-spot temperature. In the case that the top-oil temperature can be measured, it is possible to use this measure (dashed line path) moving the switching to its right position instead of calculating the top-oil temperature (left position of the switch).

Figure 2.2: Block diagram of the differential equations. 18 CHAPTER 2. LITERATURE REVIEW

2.6.2 Difference equations solution In order to get the solution to the differential problem, the differential equations are converted to difference equations. Equation (2.13) can be rewritten as a difference equation as follows: [[ ] ] Dt 1 + K2 · R x Dθ0 = · · (∆θ0r) − [θ0 − θa] (2.18) k11 · τ0 1 + R Where the operator D indicates the difference in the associated variable that cor- th responds to each time step Dt. The n value of Dθ0 can be calculated using the following equation.

θ0(n) = θ0(n−1) + Dθ0(n) (2.19) Continuing with the transformation, Equations 2.15 and 2.16 become

Dt y D∆θh1 = · [k21 · ∆θhr · K − ∆θh1] (2.20) k22 · τw Dt · − · · y − D∆θh2 = 1 [(k21 1) ∆θhr K ∆θh2] (2.21) · τ0 k22

As in Equation (2.19), the nth values of ∆θh1 and ∆θh2 are calculated in a similar way. Therefore, the total hot-spot temperature rise at the nth time step is given by

∆θh(n) = ∆θh1(n) − ∆θh2(n) (2.22) Finally, the hot-spot temperature at the nth time step is given by

θh(n) = θ0(n) − ∆θh(n) (2.23) In order to have an accurate solution, the time step Dt should be as small as possible, not greater than one-half of the smallest time constant in the thermal model.

2.6.3 Determination of time constants The winding time constant can be determined by using the following equation.

mw · cw · g τw = , [min] (2.24) 60 · Pw

Where g is the winding-to-oil gradient at the load considered, [K]; mw is the bare mass of the winding, [kg]; cw is the specific heat of the conductor material, [Ws/(kgK)]; Pw is the winding loss at the load considered, [W]. In order to calculate the top-oil time constant, it is necessary to calculate first the thermal capacity. The thermal capacity can be calculate as 2.7. IEEE CLAUSE 7: TOP OIL MODEL 19

C = cw · mw + cFE · mFE + cT · mT + kO · cO · mO (2.25)

Where mw is the mass of coil assembly, [kg]; mF E is the mass of core, [kg]; mT is the mass of the tank and fittings (including only the parts that are in contact with heated oil), [kg]; mO is the mass of oil, [kg]; cw is the specific heat capacity of the winding material, [Ws/kgK]; cF E is the specific heat capacity of the core, [WS/kgK]; cT is the specific heat capacity of the tank and fittings, [Ws/kgK]; cO is the specific heat capacity of the oil, [Ws/kgK]; and kO is the correction factor for the oil in the ONAF, ONAN, OF and OD cooling modes. This correction factor is the ratio of average to maximum top-oil temperature rise Once the thermal capacity is calculated, the top-oil time constant at the load considered is given C · ∆θ τ = 0 0 [min] (2.26) 0 60 · P Where P is the supplied losses in W at the load considered.

2.7 IEEE Clause 7: top oil model

In this standard, the hot-spot temperature is defined as the sum of the average ambient temperature (θA), the top-oil rise over ambient temperature (∆θT 0) and hot spot ◦ temperature rise over top oil temperature (∆θH ), all of them in C.

θH = θA + ∆θT 0 + ∆θH (2.27)

∆θT 0 and ∆θH are calculated as follows. t ∆θT 0 = (∆θT 0,U − ∆θT 0,i)(1 − exp(− )) + ∆θT 0,i (2.28) τTO t ∆θH = (∆θH,U − ∆θH,i)(1 − exp(− )) + ∆θH,i (2.29) τw Where subscripts u and i represent ultimate and initial respectively. The time constants τTO and τw are oil time constant, [h], and winding time constant, [h], respectively. The initial top-oil rise over ambient temperature is given by [ ] n K2R + 1 ∆θ = ∆θ i (2.30) T 0,i T 0,R R + 1

◦ Where ∆θT 0,R is the top-oil rise over ambient temperature at rated load, [ C]; Ki is the ratio of initial load to rated load, [p.u.]; n is an empirically derived exponent known as oil exponent; and R is the ratio of load loss at rated load to no-load loss. 20 CHAPTER 2. LITERATURE REVIEW

The ultimate top-oil rise over ambient temperature is given by [ ] n K2 R + 1 ∆θ = ∆θ U (2.31) T 0,U T 0,R R + 1

Where KU is the ratio of ultimate load to rated load, [p.u.]. The oil time constant can be calculated using the following equation

C∆θT 0,R τT 0 = (2.32) PT,R

◦ Where C is the thermal capacity of the transformer, [W h/ C] and PT,R is the total losses at rated load, [W]. Concerning the hot-spot temperature, the initial and ultimate hot-spot rise over top oil is given by

2m ∆θH,i = ∆θH,RKi (2.33)

2m ∆θH,U = ∆θH,RKU (2.34)

Where ∆θH,R is the winding hot-spot rise over top-oil temperature at rated load; Ki is the ratio of initial load to rated load, [p.u.]; KU is the ratio of ultimate load to rated load, [p.u.]; and m is an empirically derived exponent which represents the impact of transformer cooling mode on the change in resistance and oil viscosity. The standard also suggests some values for the empirical exponents n and m. Those values are collected in the following table.

Table 2.5: IEEE suggested values for m and n

Type of cooling m n ONAN 0.8 0.8 ONAF 0.8 0.9 OFAF or OFWF 0.8 0.9 ODAF or ODFW 1.0 1.0

2.8 Linearized top oil model of IEEE Clause 7

Equation (2.28) is the solution of the following first-order differential equation:

dθ τ TO = −θ + ∆θ (2.35) TO dt TO T O,U 2.8. LINEARIZED TOP OIL MODEL OF IEEE CLAUSE 7 21

This equation does not take into account the variation in ambient temperature. However, in [35] it is concluded that the error in the prediction of top oil temperature is related to the change in the ambient temperature. Therefore, a better first- order differential equation, which considers both the change in loading and in ambient temperature is given by

dθ τ TO = −θ + θ + ∆θ (2.36) TO dt TO A T O,U Remembering that the ultimate top-oil rise over ambient temperature is given by [ ] n I2R + 1 ∆θ = ∆θ (2.37) T O,U T O,R R + 1

Where the notation for KU has been changed to I. Then Equation (2.37) is substituted in Equation (2.36) and the resulting equation is discretized using the backward Euler discretization rule

dθ (t) θ (t) − θ (t − 1) TO ≈ TO TO (2.38) dt ∆t Where ∆t is the time resolution. The resulting discretized equation is

[( ) ] n 2 τTO ∆t I (t)R + 1 θTO(t) = θTO(t − 1) + ∆θT O,R + θA(t) (2.39) τTO + ∆t τTO + ∆t R + 1

However, this equation is not completely lineal yet and hence, it is not suitable for the linear optimization problem due to the non-integer exponential n. Therefore, it is necessary to do a further approximation: it is assumed that the cooling mode of the transformers is ODAF, so n = 1. The final form of the equation is given by

2 θTO(t) = K1I (t) + K2θA(t) + (1 − K2)θTO(t − 1) + K3 (2.40) Where ∆t R ∆θ K1 = T O,R (2.41) (τTO + ∆t)(R + 1) ∆t K2 = (2.42) τTO + ∆t ∆t ∆θ K3 = T O,R (2.43) (τTO + ∆t)(R + 1) With respect to the hot-spot temperature, it is also necessary to linearize an equation to calculate it. This linearization can be found in [23]. Equation (2.29) is the solution of Equation. 22 CHAPTER 2. LITERATURE REVIEW

dθ τ H + θ = ∆θ + θ (2.44) w dt H H,U T 0 Where 2m ∆θH,U = ∆θH,RI (2.45) This equation is still not linear. Following the previous case, an approximation is needed: m = 1. A further approximation is introduced by considering the steady dθH state of Equation (2.44) where dt is set to 0. The accuracy of this approximation is justified in [23]. The obtained equation for the estimation of the hot-spot temperature is Equation.

2 θH = ∆θH,RI + θTO (2.46)

2.9 Loss of life

The LOL (Loss Of Life) of the transformer insulation shows how much it deteriorates over the curse of its operation. This parameter is fundamental because it is used to ensure that the transformer stays within its normal life expectancy. In order to calculate the LOL of a transformer insulation, first it is necessary to calculate the relative ageing rate of the transformer insulation. The ageing of the insulation is a time function of temperature, moisture content, oxygen content and acid content. Taking into account only the influence of the highest insulation temperature point, i.e. highest deterioration point, the relative aging rate can be defined with Equation (2.47) for non-thermally upgraded paper and with Equation (2.48) for thermally upgraded paper. These equations are given by IEC standard [12], but in IEEE [18], it is possible to find the same expression but referred to as FAA.

− θh 98 V = 2 6 (2.47)

( 15000 − 15000 ) V = e 110+273 θh+273 (2.48) If the oxygen and water effects are taken into account, the ageing rate of the paper insulation can be expressed based on Arrhenius equation as follows

− E · k = A · e R (θh+273) (2.49) Then the ageing rate can be related to a rated ageing rate at a certain temperature and at an insulation condition. Therefore, the new relative ageing rate can be calculated as follows

k A 1 ·( Er − E ) V = = · e R θh,r+273 θh+273 (2.50) kr Ar 2.10. OPTIMIZATION: MILP 23

Where the subscript r means “rated condition”. Once V is calculated, it is possible to calculate transformer LOL over a certain period of time using the following equation ∫ t2 LOL = V dt (2.51) t1

2.10 Optimization: MILP

Before describing what a Mixed-Integer Linear Programming (MILP) model is, it is necessary to give a general definition of optimization problem. An optimization model is a mathematical algorithm which has the purpose of finding the best possible solution among all the feasible solutions for a defined problem. Usually, the problem is represented mathematically by a function, called objective function, which is formed by variables that describe the issue itself. Since this function is directly rep- resentative of the problem, the optimization model is able to pick out the best solution for the problem by maximizing or minimizing the objective function. At the end of the process, the best solution is given by the set of variables related to the maximum or minimum value achieved for the function. Furthermore, the problem may define some constraints or limitation that the variable have to fulfill in order to give a correct solution. Therefore, a solution is considered feasible only if the process of optimization can maximize/minimize the function while maintaining the variable within the constraints. Since the constraints express a limitation, they are usually defined mathematically by inequalities or equations. The optimization method used in this work is defined as MILP model. MILP stands for “Mixed-Integer Linear Programming” and as the name suggests, it consists in a mathematical model used for optimization problem solving in which at least one of the variables is integer, but not all of them (Mixed-Integer). The term linear suggests instead that the objective function of the problem and all the constraints applied to the optimization have to be linear [37]. An example of MILP model is expressed by the following equations:

min{3x1 + 2x2 − 4x3} (2.52)

x1 + x2 ≤ 4 (2.53)

x1 + x3 ≥ 5 (2.54)

x1 + x2 + x3 ≤ 10 (2.55)

x1, x2 ∈ Z; x3 ∈ N

Equation (2.52) represent the objective function, while equations (2.53)-(2.55) consists in the constraints. As defined below Equation (2.55), two of the variables (x1 and x2) are integers, while the last one is not. 24 CHAPTER 2. LITERATURE REVIEW

Since objective function and all the constraints have to be linear, the MILP model has some limitations and drawbacks [38]:

• All non-linear effects cannot be taken into account and non-linear problems have to be linearized in order to be analyzed.

• The whole time horizon has to be considered at once, increasing the complexity of the model.

• In general and especially because of the second drawback which introduces many variables, MILP models have high dimensionality. This may give problems from the computational point of view. Chapter 3

Optimization problem for one transformer

In the following chapter, the formulation of the optimization problem for one transformer is explained. The equations needed to set out the problem are deduced and some of them are modified to be adapted to MILP rules. Knowing the location of the wind farm, the wind turbine used and the transformer, the goal of the program is to calculate the optimal size for the wind farm and the production in each moment. This optimal value is obtained by maximizing the Net Present Value (NPV) of the wind farm. As it was explained in the “Literature review”, an optimization problem is defined by a target function and some constraints. These parts are described below.

3.1 Wind farm system topology

The way the problem is posed is based on the system explained in Figure 3.1. In order not to add more complexity to the optimization problem, the system is considered to be in DC, the wind farm is assumed to be connected to a grid with constant voltage and only active power is considered (reactive power is not considered). The grid provides generated power from other generators [P g] and has some loads which consume power [Demand]. The power that the loads consume is supplied by the wind farm [P wind] and by the other generators in the grid.

25 26 CHAPTER 3. OPTIMIZATION PROBLEM FOR ONE TRANSFORMER

Figure 3.1: System topology

3.2 Target function

The objective of the optimization problem is to maximize the Net Present Value (NPV) of the wind farm. The NPV can be calculated as follows

∑n Cash F low NPV = i − Initial Investment [€] (3.1) (1 + r)i i=1 Where i is the cash flow period, n is the number of time periods and r is the discount or interest rate, [%]. The number of time periods is equal to the lifetime of the wind farm which is assumed to be 25 years, a typical lifetime value for a wind farm [39]. In order to calculate the initial investment, only the turbines and the transformer costs are taken into account. Therefore, the initial investment is calculated as

Initial Investment = Ctur · numtur + Ctr [€] (3.2)

Where Ctur is the cost for each wind turbine, [€], numtur is an integer optimization variable which represents the total number of installed wind turbines and Ctr represents the transformer cost, [€]. Both Ctur and Ctr are input data. The cash flow for each period is given by 3.3. CONSTRAINTS 27

Cash F lowi = Benefitsi − Costsi + Certificatei [€] (3.3)

Due to the need of high computational resources to run the program for several years, it is decided to run the program only for one year and to assume that the benefits and the costs for subsequent years would be the same. Therefore, the benefits and the costs are assumed to be constant during the cash flow period. The benefits and the costs are calculated as

∑T wind · Benefits = (Pt Elprice)[€] (3.4) t=1

Where T is the time period of the data, in this case is 1 year (8760 hours), P wind is an optimization variable which represents the production of the wind farm, [MWh] and Elprice is an input data which indicates the electricity price, [€/MWh].

∑T ∑T O&M,fix· O&M,var· wind O&M,fix O&M,var· wind Costs = Ctur numtur+ (Ctur Pt )+Ctr + (Ctr Pt )[€] t=1 t=1 (3.5) O&M,fix Where Ctur represents the fixed operational and maintenance (O & M) cost O&M,var of the turbine, [€], and Ctur indicates the variable O & M cost of the turbine O&M,fix O&M,var which depends on the production of the wind farm, [€/MWh], Ctr and Ctr represent the fixed and variable O & M costs of the transformer respectively, [€] and [€/MWh]. All these costs are input data for the optimization problem. Sweden uses a Green Certificate system that produces extra revenues and proves that electricity has been generated by a renewable (green) energy source. The price of this certificate changes every year. In this case, the price is selected as the registered one for 2019, Cprice = 0.305 [€/MW h] [40]. Therefore, the revenues from the green certificate can be calculated as follows

∑T wind · Certificate = (Pt Cprice)[€] (3.6) t=1

3.3 Constraints

In this section, the constraints needed for the optimization problem are explained. In order to facilitate the comprehension, the constraints are separated according to their topic. 28 CHAPTER 3. OPTIMIZATION PROBLEM FOR ONE TRANSFORMER

3.3.1 Total capacity of the wind farm The total capacity of the wind farm is calculated by multiplying the capacity of the wind turbines by an integer number which represents the total number of wind turbines installed in the wind farm. This is expressed in Equation (3.7).

Capwind = numtur · Captur [MW ] (3.7)

Where Capwind is an optimization variable which represents the total capacity of the wind farm, [MW] and Captur is an input value which represents the capacity of the wind turbines, [MW]. As it is possible to deduce from the equation, the total capacity of the wind farm is limited by the total number of installed wind turbines. Therefore, it is necessary to set an upper limit for the number of turbines. This limit is just defined as a really big number, as is shown in Equation (3.8).

numtur ≤ M (3.8) Where M is a really big number as, for example, 1000.

3.3.2 Power balance In every moment, there should be a perfect balance between the generated power and the consumed power in the system. The generated power is produced by the other generators in the system and by the the wind farm. The consumed power is consumed by the loads in the system. In order not to add more complexity to the optimization, load shedding is not considered. Taking everything into consideration, the active power balance in the system is the following

wind g − Pt + Pt Demandt = 0 [MW ] (3.9) g Where Pt is an optimization variable which represents the power produced by the other generators in the system, [MW] and Demandt is an input data which indicates the consumed power in the system, [MW].

3.3.3 Power limitations It is also necessary to set some power limitations in the system. The first limitation is the physical limitation on the availability of wind power generation as follows

≤ wind ≤ · av,wind 0 Pt numtur Pt (3.10) 3.3. CONSTRAINTS 29

av,wind Where Pt is the available power in the wind, [MW]. This value is an input data and is calculated using the power curve of the wind turbine provided by the manufacturer. The second limitation is that the power produced by the wind farm cannot be higher than the total capacity of the wind farm. This limitation is expressed in Equa- tion (3.11).

wind ≤ Pt Capwind [MW ] (3.11) Finally, it is necessary to set some limitations to the power produced by the other generators in the system. These limitations are expressed in Equation (3.12).

≤ g ≤ 0 Pt 2000 (3.12)

3.3.4 Transformer constraints When the transformer is loading beyond the nameplate rating, it is necessary to es- tablish the maximum permissible temperatures. In Table 2.3, it is possible to find the maximum hot-spot and top-oil temperature values set by the standards. These maximum values depend on the type of loading: long-time or short-time. As in this case the time step is 1 hour, long-time values are selected. Equations (3.13) and (3.14) represent these temperature limitations.

hst ≤ hst,max ◦ θt θ [ C] (3.13) hst Where θt is an optimization variable which represents the hot-spot temperature, ◦ hst,max ◦ [ C] and θt is the maximum hot-spot temperature, 140 C.

top ≤ top,max ◦ θt θ [ C] (3.14) top Where θt is an optimization variable which represents the top-oil temperature ◦ top,max ◦ [ C] and θt is the maximum top-oil temperature, 110 C. Another fundamental parameter for the transformer is the LOL. As it was explained in the “Literature review”, in order to calculate the LOL, first it is necessary to calculate the ageing rate of the transformer V. The ageing rate can be calculate using Equation (2.50). However, this equation is not linear and hence, it cannot be used in a MILP problem. Therefore, first it is necessary to linearize this equation. In order to linearize this equation, a linearization based on segmentation is chosen. This is done by a function written in Matlab. The inputs of this function are the original equation and a vector with the chosen segments and the output is the coefficients of the linear equation for each segment. In this case, 25 segments between 0 and 150◦C are selected. It is possible to see how this linearization by segments is accurate enough in Figure 3.2. The coefficients of the linear equations are saved in the coefficients MV and QV and Equation (3.15) is built to calculate the ageing rate. 30 CHAPTER 3. OPTIMIZATION PROBLEM FOR ONE TRANSFORMER

Figure 3.2: Linearization by segment of V 3.3. CONSTRAINTS 31

≥ · hst Vt MV θt + QV (3.15) Once the ageing rate is calculated, it is possible to calculate LOL following the standard using Equation (3.16). It is also necessary to set an upper limitation to LOL because during the lifetime of the wind farm, the transformer cannot loose more than its designed lifetime. The transformer lifetime is specified by the standard and it is 20.55 years [18]. As the program runs for 1 year, this LOL limitation can be rewritten for one year as in Equation (3.17).

∑T LOL = Vt [h] (3.16) t T ransformer′s lifetime LOL ≤ · 8760 [h] (3.17) W ind farm′s lifetime

3.3.5 Thermal model constraints The thermal model used for this optimization problem is the linearized top oil model of IEEE Clause 7 explained in the “Literature review”. By taking the final equation of that standard, Equation (2.40), and clearing I, it is possible to obtain an equation to calculate the current through the transformer which depends on the hot-spot temperature and the top-oil temperature. This equation is Equation (3.18).

2 · hst − · top − It = P1 θt P2 θt−1 P3,t [pu] (3.18) Where 1 P1 = (3.19) ∆t·R·∆θor + ∆θhr (τ0+∆t)·(R+1)

τ0 P2 = · P1 (3.20) τ0 + ∆t ( ) ∆t ∆θor P3,t = P2 · · θamb,t + (3.21) τ0 R + 1 Where (all are input data) ∆t is the time step that, in this case is 60 min; R is the loss ratio; ∆θor is the top-oil temperature rise at rated losses; τ0 is the oil time constant, [min]; ∆θhr is the hot-spot to top-oil gradient at rated current; and θamb is the ambient temperature, [◦C]. In order to estimate the hot-spot temperature, Equation (2.46) is used. This equation is adapted to the new notation in Equation (3.22).

hst · 2 top ◦ θt = ∆θhr It + θt [ C] (3.22) 32 CHAPTER 3. OPTIMIZATION PROBLEM FOR ONE TRANSFORMER

As it was shown in Table 2.3, the current also has a limitation when the transformer is working above its rating power. Therefore, it is necessary to introduce an equation that shows this current limitation. This equation is Equation (3.23).

It ≤ 1.5 [pu] (3.23)

3.3.6 Transformer selection Finally, an equation which puts together the wind farm variables, the thermal limitations and the selected transformer is needed. This equation which combines all the above considerations is Equation (3.24).

2 · ≥ wind It Tsize Pt [MW ] (3.24)

Where Tsize is the rating power of the selected transformer, [MVA]. In Equation (3.24), I2 is considered as the power that the transformer has to supply. This consideration can be done by assuming the load to be close to 1 pu. As I2 is in pu, it is necessary to transform it to MW in order to be comparable to the power produced by the wind farm. This unit transformation is done by multiplying I2 by the transformer rated power.

3.4 Optimization problem equations

In order to sum up all the equations used for the optimization problem, this section includes a summary of the optimization problem. In the MILP optimization problem, I2 is expressed as Isq in order to obtain linear equations.

3.4.1 Optimization variables The unknown variables that the program optimizes are the following:

wind • The power produced by the wind farm: Pt , domain of non-negative reals

• The total number of installed turbines: numtur, domain of non-negative integers

• The total capacity of the wind farm: Capwind, domain of non-negative reals

g • The power generated by the other generators in the system: Pt , domain of non- negative reals

hst • The hot-spot temperature: θt , domain of non-negative reals

top • The top-oil temperature: θt , domain of non-negative reals 3.4. OPTIMIZATION PROBLEM EQUATIONS 33

• Ageing rate: Vt, domain of non-negative reals • The lost of life of the transformer: LOL, domain of non-negative reals

sq • The square root of the transformer load current: It , domain of non-negative reals

3.4.2 Target function ( ) ∑n Cash F low ∑n Benefits − Costs NPV = −Initial Investment = −(C ·num +C ) (1 + r)i (1 + r)i tur tur tr i=1 i=1 (3.25)

∑T wind · Benefits = (Pt Elprice) (3.26) t=1

∑T ∑T O&M,fix · O&M,var · wind O&M,fix O&M,var · wind Costs = Ctur numtur + (Ctur Pt ) + Ctr + (Ctr Pt ) t=1 t=1 (3.27)

3.4.3 Constraints s.t.

Capwind = numtur · Captur (3.28)

numtur ≤ M (3.29)

wind g − Pt + Pt Demandt = 0 (3.30)

≤ wind ≤ · av,wind 0 Pt numtur Pt (3.31)

wind ≤ Pt Capwind (3.32)

≤ g ≤ 0 Pt 2000 (3.33)

hst ≤ hst,max θt θ (3.34)

top ≤ top,max θt θ (3.35) 34 CHAPTER 3. OPTIMIZATION PROBLEM FOR ONE TRANSFORMER

≥ · hst Vt MV θt + QV (3.36)

∑T LOL = Vt (3.37) t T ransformer′s lifetime LOL ≤ · 8760 (3.38) W ind farm′s lifetime

sq · hst − · top − It = P1 θt P2 θt−1 P3,t (3.39)

1 P1 = (3.40) ∆t·R·∆θor + ∆θhr (τ0+∆t)·(R+1)

τ0 P2 = · P1 (3.41) τ0 + ∆t ( ) ∆t ∆θor P3,t = P2 · · θamb,t + (3.42) τ0 R + 1

hst · 2 top θt = ∆θhr It + θt (3.43)

sq ≤ It 2.25 (3.44)

sq · · ≥ wind It Tsize ∆t Pt (3.45) Chapter 4

Optimization problem for several transformer

In the following chapter, the previous optimization problem formulation is modified in order to include the automatic selection of the transformer. In this case, the transformer is not set from the beginning, but the program chooses it from a list of possible transformers. Therefore, the goal of the program is to design the best wind farm, from the point of view of maximizing the NPV, by selecting the best transformer.

4.1 Variable for the selection of the transformer

A binary optimization variable is used to select the transformer. This variable is a vector with a length equal to the number of possible transformers. Once the optimization is done, this variable is 1 for the selected transformer and 0 for the rest of the transformers. Therefore, this variable can only be 1 for one of the transformer. In order to fulfill this condition, it is necessary to write a constraint about this variable. This constraint set that the sum of all vector components has to be equal to 1, so only one component can be 1. This constraint is defined using Equation (4.1).

∑J gj = 1 (4.1) j Where J is the total number of possible transformers, j is the index which identifies the transformers and g is the binary optimization variable that it is used for the selection of the transformer.

4.2 Target function

The formula to calculate the NPV is the same as in the previous case (Equation (3.1)). However, some of the components of this equation have to be modified to include

35 36 CHAPTER 4. OPTIMIZATION PROBLEM FOR SEVERAL TRANSFORMER the selection of the transformer. These components are the initial investment and the costs. The modified equations including the binary variable g are the following

∑J Initial Investment = Ctur · numtur + (Ctr,j · gj)[€] (4.2) j

∑T ∑J ∑T O&M,fix· O&M,var· wind O&M,fix· O&M,var· wind Costs = Ctur numtur+ (Ctur Pt )+ (Ctr,j gj)+ (Ctr Pt )[€] t=1 j t=1 (4.3)

4.3 Constraints

Only the constraints related to the transformer are modified. Therefore, the constraints related to the total capacity of the wind farm, the power balance and the power limitations are the same as in the previous chapter.

4.3.1 Transformer constraints The maximum limits for the hot-spot temperature and top-oil temperature are main- tained, but it is necessary to modify the constraint to specify that all the transformer have to fulfill these limits. Equations (4.4) and (4.5) are the modified equations for the temperature limits.

hst ≤ hst,max ◦ θt,j θ [ C] (4.4)

top ≤ top,max ◦ θt,j θ [ C] (4.5) It is also necessary to do a change in the notation for the ageing rate. The modified equation for the ageing rate is Equation (4.6).

≥ · hst Vt,j MV θt,j + QV (4.6) In respect of LOL, its upper limit is not modified (Equation (3.17)), but the equation that is used to calculate LOL (Equation (3.16)) is modified. In order to have as output only the value of the selected transformer, Equation (3.16) is divided into two inequalities: Equations (4.7) and (4.8). The term which has been added in order to introduce the binary variable is [(1 − gj) · M], in one of the inequalities is summed and in the other is subtracted. As it was defined previously, M is a very large number. When the transformer is the selected one, gj = 1, the term [(1 − gj) · M] is equal to zero, so both inequalities have the same equation and the same result. On the other 4.3. CONSTRAINTS 37

hand, when the transformer is not the selected one, gj = 0, in one inequality the result is a very large number and in the other is a very small number, so that LOL will not have a result and will not be selected as the correct one.

∑T LOL ≤ Vt,j + (1 − gj) · M [h] (4.7) t

∑T LOL ≥ Vt,j − (1 − gj) · M [h] (4.8) t

4.3.2 Thermal model constraints

It is necessary to calculate the current going through each transformer. This current depends on the hot-spot and top-oil temperatures and the parameters of the transformers. In order to select only the current of the selected transformer, the same method as with LOL is used: the current equation is divided into two inequalities and the term [(1 − gj) · M] is added. It works in the same way as for LOL. Therefore, the current is calculated as follows

2 ≤ · hst − · top − − · It P1,j θt,j P2,j θt−1,j P3,t,j + (1 gj) M [pu] (4.9)

2 ≥ · hst − · top − − − · It P1,j θt,j P2,j θt−1,j P3,t,j (1 gj) M [pu] (4.10) Where

1 P1,j = (4.11) ∆t·Rj ·∆θor,j + ∆θhr,j (τ0,j +∆t)·(Rj +1)

τ0,j P2,j = · P1,j (4.12) τ0,j + ∆t ( ) ∆t ∆θor,j P3,t,j = P2,j · · θamb,t + (4.13) τ0,j Rj + 1

The estimation of the hot-spot temperature and the limitation of the current is also modified as follows

hst · 2 top ◦ θt,j = ∆θhr,j It,j + θt,j [ C] (4.14)

It,j ≤ 1.5 [pu] (4.15) 38 CHAPTER 4. OPTIMIZATION PROBLEM FOR SEVERAL TRANSFORMER

4.3.3 Transformer selection The most difficult part of this adaptation is to find an equation that substitutes Equa- tion (3.24). This equation cannot be modified directly because one of the limitations of MILP is that two optimization variables cannot be multiplied, otherwise the constraint is not linear anymore. In this case, it would mean to multiply the current and the binary optimization variable for the selection of the transformer as it is shown in Equation (4.16).

2 · · ≥ wind It,j Tsize,j gj Pt (4.16) A big part of the time invested in the project has been dedicated to try to find different methods to modify this equation. The most important ones are explained below.

Classical approach The first idea is to use a classical and restrictive approach. Two new constraints are added: one that relates the power rating of the transformer to the capacity of the wind farm and the other which relates the current to the power production. The first constraint limits the power rating of the transformer to be lower or equal to the capacity of the wind farm. The second constraint limits the power production of the wind farm to be lower or equal to the power through the transformer represented by the current square multiplied by a base power. These two equations are given by

∑J (Tsize,j · gj) ≥ Capwind (4.17) j

wind ≤ 2 · Pt It,j Sbase (4.18) This method has not been successful because Equation (4.17) represents the classical approach, not the dynamic rating approach.

Current scaling Another idea is to scale the current depending on the transformer rated power and the load. The idea consists on multiplying the equations of the current (Equation (4.9) and (4.10)) by a term corresponding to the division between the load and the transformer rated power. Therefore, when the load is higher than the transformer rated power (the wind farm is producing more than the rated power of the transformer) the current will be multiplied by a factor higher than 1, so it will increase its value. On the other hand, when the load is lower than the transformer rated power, the current will be multiplied by a factor lower than 1 reducing its value. This method is represented by the following equations 4.3. CONSTRAINTS 39

(Demand − P g) 2 ≤ t t · · hst − · top − − · It,j (P1,j θt,j P2,j θt−1,j P3,t,j) + (1 gj) M (4.19) Tsize,j · gj

(Demand − P g) 2 ≥ t t · · hst − · top − − − · It,j (P1,j θt,j P2,j θt−1,j P3,t,j) (1 gj) M (4.20) Tsize,j · gj

However, these equations are not linear, so they cannot be solved using MILP. Therefore, it is necessary to linearize these equations. A way to try to solve this pro- g blem is to set a fixed value for Pt , so this variable is not anymore an optimization variable. In order to remove the other optimization variable, gj, the current equation is generated in a loop. This method has also failed because the program cannot generate the equations in g a loop. In addition, giving a fixed value to Pt meant to set the load and the production of the wind farm.

Maximum currents Calculating the maximum current for each transformer at each time is also tried. It is calculated using Equation (4.21).

2 · hst,max − · top,max − Imax,t,j = P1,j θ P2,j θ P3,t,j (4.21) The objective of this method is to differentiate the current and hence, the thermal limits among the transformers. However, it has not been successful because it is not possible to find an equation that can relate the current with the rated power of the transformer and the power production of the wind farm without transforming the equations in non-linear equations.

Method based on paper [41] Finally, a new method that linearized the standard equations in a different way and that was published recently is used. This method is explained in [41]. This paper introduces a new indicator called thermal load factor represented by Kth. This indicator is created in order to replace hot-spot thermal limit (Equation (3.13)) and it is described by Equation (4.22). In paper [41] the notation K represents the same as I in this project. 1 α K = · K + · K (4.22) th 1 + α 1 + α o K Ko = (4.23) 1 + k11 · τo · s 40 CHAPTER 4. OPTIMIZATION PROBLEM FOR SEVERAL TRANSFORMER

2R ∆θor · x · − ∆θhr · y · (K21 − 1) α = 1+R (4.24) ∆θhr · y · k21

Where Ko is a smoothed load factor defined by Equation (4.23) and α is a constant specified by each transformer defined by (4.24) and it is used to express the relative weight of variables K and Ko. s denotes that the equation is in Laplace domain. The new limit which substitutes Equation (3.13) is given by

Kth ≤ 1.5 [pu] (4.25) As the equations are in Laplace domain, it is necessary to do an inverse Laplace transformation and a linearization of the resulting equations. Finally, the transformer is selected as the minimum value of the maximum value between the transformer load current (K or I) and the new indicator (Kth) as Equa- tion (4.26) indicates.

min(max(Kt), max(Kth,t)) (4.26) This last method has not been successful either because the program is not able to calculate the maximum and the minimum of an optimization variable.

4.3.4 Conclusion After trying different methods, it is concluded that it is not possible to carry out an optimization problem in which the program selected directly the transformer with the tools used in this project. Therefore, it is necessary to substitute the MILP model for another optimization method that allows multiplications and divisions between optimization variables. The kind of optimization method that allows these mathematical operations between optimization variables is non-linear optimization methods. However, this optimization method increases the computation complexity exponen- tially. In consequence, a problem can take several hours or several days to be solved. Therefore, this method is not feasible for this project. Chapter 5

Input data

In this chapter, the input data used for the optimization problem are presented. Wea- ther data of the selected place, discount rate for the wind farm, electricity price, electricity demand, turbine specifications, transformer specifications and costs are the input data necessary to set up the problem.

5.1 Weather data

The wind farm is located between the cities of Kopparberg and Skinnskatteberg. There- fore, a weather station in that region is chosen. This station is called Kloten A and is active since August 2009 [42]. From this station, wind and temperature data are taken. The location of this station is shown in Figure 5.1. It is necessary to carry out an in-depth consolidation data analysis because some of the data are missed and some are doubled. The missed data are completed by calculating the average of adjacent data. In the case of the doubled data, this mistake occurs during the summer and winter time change.

5.1.1 Analysis of the wind data The wind data are provided hourly from 2009-08-01 to 2019-12-31. Since 2009 is not completed, this year is not taken into consideration. In Table 5.1, it is possible to see a table with the wind speed data analysis. The collected wind data is measured at 10 m, so it is necessary to scale up these wind data to the hub height using the power law as explained in Section 2.2. Due to the terrain characteristics, the value of the surface roughness length (z0) is chosen as 500 mm corresponding to a forest in Table 2.2. The new average wind speed per year is given in Table 5.2. Once the wind speed is calculated, it is possible to calculate the annual energy production per turbine. In Figure 5.2, the annual energy production per turbine is shown.

41 42 CHAPTER 5. INPUT DATA

Figure 5.1: Location of the weather station Kloten A

Table 5.1: Wind data analysis

Year Count Average [m/s] σ σ2 2010 8760 2.03 1.27 1.62 2011 8760 2.26 1.50 2.25 2012 8784 2.16 1.37 1.88 2013 8760 2.18 1.43 2.05 2014 8760 2.35 1.56 2.45 2015 8760 2.40 1.56 2.44 2016 8784 2.22 1.40 1.97 2017 8760 2.20 1.33 1.76 2018 8760 2.05 1.39 1.94 2019 8760 2.16 1.36 1.84 5.1. WEATHER DATA 43

Table 5.2: Average wind speed per year

Year Average 2010 3.45 2011 3.83 2012 3.66 2013 3.70 2014 3.98 2015 4.07 2016 3.76 2017 3.74 2018 3.47 2019 3.66

Figure 5.2: Annual Energy Production per turbine[MWh] 44 CHAPTER 5. INPUT DATA

5.1.2 Analysis of temperature data As the wind data, the temperature data are provided hourly from 2009-08-01 to 2019-12-31. For the same reason as before, year 2009 is not taken into consideration. In the following table, the temperature analysis is presented.

Table 5.3: Temperature data analysis

Year Count Average [ºC] σ σ2 2010 8760 3.15 10.38 107.75 2011 8760 5.83 8.66 75.08 2012 8784 4.48 8.22 67.57 2013 8760 5.10 8.77 76.83 2014 8760 6.33 8.38 70.22 2015 8760 5.87 6.92 47.85 2016 8784 5.44 8.37 70.03 2017 8760 5.29 7.50 56.18 2018 8760 5.99 9.89 97.75 2019 8760 5.16 8.71 75.81

5.2 Discount rate

The discount rate is an important source of information for investors in the renewable energy sector. This parameter determines the market price for the projects. However, this discount rate is extremely difficult to get. Therefore, investors have to rely on their own experience and expert advises in order to determine this data. With this purpose, Grant Thornton carried out a survey in collaboration with Clean Energy Pipeline in 2018 about renewable energy discount rate. This survey was carried out in thirteen major geographies, including the Nordic countries. This survey determined that the discount rate for onshore wind project in the Nordics is 7.5 % [43].

5.3 Electricity price

Sweden is divided into four electricity price areas: Malmö (SE4), Stockholm (SE3), Sundsvall (SE2) and Luleå (SE1) [44]. The wind farm of this project is located in area SE3. The data related to the electric market of the Nordic countries is open for every- one and can be consulted in Nordpool, the Nominated Electricity Market Operator (NEMO). In Nordpool, it is possible to find information about the electricity price, consumption, production, exchange, etc [45]. 5.3. ELECTRICITY PRICE 45

In order to find an appropriate value for the electricity price, the evolution of the electricity price for a period of 7 years (2013-2019) in the Swedish area SE3 is eva- luated. In Figure 5.3, it is possible to see the hourly evolution of the electricity price during that period. In addition, Table 5.4 shows the average, maximum and minimum values for each year.

Figure 5.3: Electricity price evolution for Swedish area SE3, where the red line represents the average value.

Table 5.4: Analysis of the electricity price area SE3

Year Average [€/MWh] Maximum [€/MWh] Minimum [€/MWh] 2013 39.45 109.55 1.38 2014 31.63 105 0.59 2015 22.00 150.06 0.32 2016 29.23 214.25 4.02 2017 31.24 130.05 1.7 2018 44.54 255.02 1.59 2019 38.36 109.45 0.12 Total 33.78 255.02 0.12

As it is possible to see in Figure 5.3, there are some really high peaks during 2016 and 2018, but in the rest of the years, the electricity price is within the range 100 €/MWh to 0 €/MWh. It would have been interesting to set a varying price for the project, but in order not to add more complexity to the optimization problem, a fixed electricity price is selected. This fixed electricity price is chosen as the average value of the study period hourly data. The resulting value is 33.78 €/MWh. 46 CHAPTER 5. INPUT DATA

Table 5.5: Analysis of the electricity demand for this project. This demand is 10 times smaller than the original SE3 demand.

Year Average [MWh] Maximum [MWh] Minimum [MWh] 2013 994 1747 506 2014 961 1591 522 2015 973 1513 547 2016 994 1768 521 2017 996 1661 551 2018 1000 1709 547 2019 983 1631 545 Total 986 1768 506

5.4 Electricity demand

In order to maintain coherence and to be as close as possible to reality, the total demand in the system is the demand registered in area SE3. This demand is obtained again thanks to Nordpool [45]. As this demand is unnecessary big for the considered system, the demand used in this project is ten times smaller than SE3 demand (SE3 demand divided by 10). In Table 5.5, it is possible to see an analysis of the demand data used for this project. It is required to carry out a consolidation data analysis for the same reasons as for weather data.

5.5 Wind farm

5.5.1 Selected turbine

The selected turbine to build the wind farm is a Gamesa wind turbine, specifically the model G128-5.0MW. This turbine is selected for being a standard wind turbine for onshore wind farms. The characteristics of this turbine are presented in Table 5.6. The power data is provided by the manufacturer. In Figure 5.4, it is possible to see the power curve of the turbine including power and power coefficient. The power coefficient (cp) is a measure of wind turbine efficiency and it is calculated as the electric power produced by the wind turbine divided by the total wind power flowing into the turbine blades. 5.5. WIND FARM 47

Table 5.6: Characteristics of G128-5.0MW wind turbine

Rated power 5 MW Cut-in wind speed 2 m/s Rated wind speed 14 m/s Cut-out wind speed 27 m/s Wind class (IEC) Ia Rotor diameter 128 m Swept area 12,868.0 m2 Nº of blades 3 Hub height 81/95/120/140 m

Figure 5.4: Power curve of the wind turbine

5.5.2 Wind farm costs The exact costs of the wind farm including the wind turbine are hard to estimate. For that reason, an approximation based on typical onshore wind farm data has been taken into account. The investment cost for a onshore wind farm in developed countries varies between 920 €/kW and 2150 €/kW [46]. In this case, the investment cost is selected as 1000 €/kW. This capital cost can be broken down into the following most important groups: 48 CHAPTER 5. INPUT DATA

Figure 5.5: Investment cost breakdown for a typical wind farm [1]

With respect to the operation and maintenance (O&M) cost of the wind farm, it is divided into two categories: fixed cost and variable cost which depends on the energy produced by the wind farm. Studies done by the Danish wind industry association on 5000 Danish wind turbines show that for newer wind turbines the fixed rate is around 1.5 % - 2 % of the investment cost [47] per year. In this case, the fixed cost is selected as 1.5 % of the capital cost. The variable cost can be estimated as 9.3 €/MWh [47]. Taking into account that the size of each wind turbine is 5 MW, the approximate costs for this wind farm project are presented in Table 5.7.

Table 5.7: Wind farm costs

Investment cost 1000 k€/installed MW Investment cost per turbine 5000 k€ Turbine cost 3200 k€ Civil works cost 650 k€ Electrical infrastructure cost 400 k€ Grid connection 300 k€ Planning & Miscellaneous 450 k€ Fixed O&M cost 75 k€ Variable O&M cost 9.3 €/MWh 5.6. TRANSFORMER SELECTION 49 5.6 Transformer selection

In order to evaluate the optimization problem, four different typical sizes of power transformer are chosen: 63 MVA, 100 MVA, 200 MVA and 400 MVA.

5.6.1 Transformer thermal parameters The same thermal parameters are selected for all the transformers. The selected values are the characteristic ones for ODAF transformer given in Annex K of IEC standard [12] and are shown below.

• Loss ratio: R = 6

• Top-oil (in tank) temperature rise: ∆θ0r = 49 K

• Hot-spot to top-oil gradient: ∆θhr = 29 K

• Hot-spot winding rise: ∆θh = ∆θhr + ∆θ0r = 78 K

• Winding time constant: τw = 7 min

• Oil time constant: τ0 = 90 min

It is necessary to clarify that ODAF cooling method is not the typical cooling method for wind applications. However, as it was explained in Chapter 2 section 2.8, in order to linearize the thermal equations, it is required to approximate the transformer cooling mode to ODAF.

5.6.2 Transformer costs The transformer price depending on the size is another parameter that is not easy to get unless you have contact with the manufacturer. However, the maximum transformer price depending on the size is regulated by the Swedish government. The Swedish energy markets inspectorate is the government agency that regulates this price. For this project, the normative values for the monitoring period 2020-2023 published by this agency are used as the transformer investment cost [48]. Apart from the investment cost, it is necessary to take into account the operation and maintenance cost. O&M cost of each transformer can be divided into fixed and variable. The fixed O&M cost is estimated to be 1 % of the transformer investment cost for new transformer and 4 % for old transformer [49]. In this case, the fixed O&M cost is selected to be during the whole period 2 % of the transformer investment cost. Concerning the variable O&M cost, a typical value is chosen for all the transformers: 7 €/MWh. The transformer investment costs and O&M costs for each transformer considered for this project are shown in Table 5.8. 50 CHAPTER 5. INPUT DATA

Table 5.8: Transformers investment costs and O&M costs

Transformer [MVA] Investment [€] Fixed O&M [€] Var. O&M [€/MWh] 63 1,782,608 35,652 100 2,253,521 45,070 7 200 3,368,532 67,371 400 4,376,786 87,536 Chapter 6

Results

In this chapter, the results obtained after solving the optimization problem given the selected transformer are analysed. In addition, a sensitivity analysis is carried out on the fundamental parameters of the optimization in order to check the importance of each parameter and their effect in the overall result of the problem. In order to execute the optimization problem, different programs have been tried: Matlab, CVX and Python. At the end, it is concluded that the program which provided a faster performance and easier execution is Python. Therefore, the optimization problem is written in Python using “Pyomo”, a Python package used for formulating, solving, and analyzing optimization models [50]. MOSEK is used as solver for the program. As it was explained in previous chapters, the program runs for each hour during one year. Therefore, the program has to solve 332883 constraints and give value to 52563 scalar variables corresponding to all the optimization variables, except for the total number of installed turbines, plus the NPV variable and 1 integer variable corresponding to the number of turbines. The program takes an average of 30-40 minutes with a normal computer to pose the problem and to solve the optimization.

6.1 Results using the original input data

First, the input data explained in Chapter 5 are implemented in the optimization program. However, it is found that the wind resources in the selected location are insuf- ficient and when running the program, the resulting NPV value is negative, which means that it is not profitable to invest in the wind farm. This shows the importance of choosing a location with high wind resources. In order for the investment to be profitable, the electricity price has to increase to 100 €/MWh. Once the electricity price is increased, the program runs in a loop for the four different transformer. Although the program provides solution every hour during one year, the results are presented for each day during one year in order to clarify the presentation. With the objective of transforming hourly results to daily results, a

51 52 CHAPTER 6. RESULTS

Matlab program is written which calculates the daily average. The obtained results for each transformer are presented below.

6.1.1 Results for 63 MVA transformer

Figure 6.1: Hot-spot and top-oil temperatures for 63 MVA transformer using data from 2015 and an electricity price of 100 €/MWh

Figure 6.2: Current through the 63 MVA transformer using data from 2015 and an electricity price of 100 €/MWh 6.1. RESULTS USING THE ORIGINAL INPUT DATA 53

Figure 6.3: Comparison between the wind farm production, the production from other generators and the total demand in 63 MVA transformer using data from 2015 and an electricity price of 100 €/MWh

Figure 6.4: Ageing rate of 63 MVA transformer using data from 2015 and an electricity price of 100 €/MWh 54 CHAPTER 6. RESULTS

6.1.2 Results for 100 MVA transformer

Figure 6.5: Hot-spot and top-oil temperatures for 100 MVA transformer using data from 2015 and an electricity price of 100 €/MWh

Figure 6.6: Current through the 100 MVA transformer using data from 2015 and an electricity price of 100 €/MWh 6.1. RESULTS USING THE ORIGINAL INPUT DATA 55

Figure 6.7: Comparison between the wind farm production, the production from other generators and the total demand in 100 MVA transformer using data from 2015 and an electricity price of 100 €/MWh

Figure 6.8: Ageing rate of 100 MVA transformer using data from 2015 and an electricity price of 100 €/MWh 56 CHAPTER 6. RESULTS

6.1.3 Results for 200 MVA transformer

Figure 6.9: Hot-spot and top-oil temperatures for 200 MVA transformer using data from 2015 and an electricity price of 100 €/MWh

Figure 6.10: Current through the 200 MVA transformer using data from 2015 and an electricity price of 100 €/MWh 6.1. RESULTS USING THE ORIGINAL INPUT DATA 57

Figure 6.11: Comparison between the wind farm production, the production from other generators and the total demand in 200 MVA transformer using data from 2015 and an electricity price of 100 €/MWh

Figure 6.12: Ageing rate of 200 MVA transformer using data from 2015 and an electricity price of 100 €/MWh 58 CHAPTER 6. RESULTS

6.1.4 Results for 400 MVA transformer

Figure 6.13: Hot-spot and top-oil temperatures for 400 MVA transformer using data from 2015 and an electricity price of 100 €/MWh

Figure 6.14: Current through the 400 MVA transformer using data from 2015 and an electricity price of 100 €/MWh 6.1. RESULTS USING THE ORIGINAL INPUT DATA 59

Figure 6.15: Comparison between the wind farm production, the production from other generators and the total demand in 400 MVA transformer using data from 2015 and an electricity price of 100 €/MWh

Figure 6.16: Ageing rate of 400 MVA transformer using data from 2015 and an electricity price of 100 €/MWh 60 CHAPTER 6. RESULTS

6.1.5 Analysis and comparison By looking at the resulting plots, it is possible to obtain some conclusions that are explained subsequently. The top-oil temperature follows the same shape as the hot-spot temperature but reduced. This is correct because the top-oil temperature is calculated based on the hot-spot temperature, as it was explained in the formulation of the optimization problem. In the same way, the current plot follows the same shape as the hot-spot temperature because this variable also depends on the hot-spot temperature. Top-oil temperature, current and hot-spot temperature never reach their maximum values. Concerning the power plots of the transformers, it is possible to see that during peaks and valleys, the production of the wind farm and the production from the other generators have an opposite behaviour. While the rest of the plots differs among their peers, the power produced by the wind farm curve follows exactly the same curve for all of the transformers. The difference is that the larger the transformer, the higher the power curve. In respect of the ageing rate, it is noticeable that there is a peak in the ageing rate when there is a peak in the hot-spot temperature. After analysing the results, it is noticed that all the results are as expected and hence, it is possible to conclude that the results are correct. Finally, a comparison of the optimal wind farm design and the final NPV for each transformer is shown in Table 6.1. In addition, two graphical comparisons of the capacity and the NPV respectively are shown in Figures 6.17 and 6.18. It is possible to see that both follows a linear evolution. Therefore, the larger the transformer, the higher NPV, but also a bigger wind farm is needed.

Table 6.1: Optimal wind farm design and NPV for each transformer when using 2015 data and an electricity price of 100 €/MWh

Transformer size [MVA] Capacity [MW] Nº turbines LOL NPV [M€] 63 145 29 7200.72 11.2 100 230 46 7200.72 18.5 200 460 92 7200.72 38.5 400 915 183 7200.72 79.5 6.2. RESULTS CHANGING THE WIND SPEED 61

Figure 6.17: Capacity comparison between the transformers when using 2015 data and an electricity price of 100 €/MWh

Figure 6.18: NPV comparison between the transformers when using 2015 data and an electricity price of 100 €/MWh

6.2 Results changing the wind speed

In this case, in order to be able to use the average electricity price of 34 €/MWh, the wind speed of 2015 is doubled. The results are presented below. 62 CHAPTER 6. RESULTS

6.2.1 Results for 63 MVA transformer

Figure 6.19: Hot-spot and top-oil temperatures for 63 MVA transformer using data from 2015 and an electricity price of 34 €/MWh

Figure 6.20: Current through the 63 MVA transformer using data from 2015 and an electricity price of 34 €/MWh 6.2. RESULTS CHANGING THE WIND SPEED 63

Figure 6.21: Comparison between the wind farm production, the production from other generators and the total demand in 63 MVA transformer using data from 2015 and an electricity price of 34 €/MWh

Figure 6.22: Ageing rate of 63 MVA transformer using data from 2015 and an electricity price of 34 €/MWh 64 CHAPTER 6. RESULTS

6.2.2 Results for 100 MVA transformer

Figure 6.23: Hot-spot and top-oil temperatures for 100 MVA transformer using data from 2015 and an electricity price of 34 €/MWh

Figure 6.24: Current through the 100 MVA transformer using data from 2015 and an electricity price of 34 €/MWh 6.2. RESULTS CHANGING THE WIND SPEED 65

Figure 6.25: Comparison between the wind farm production, the production from other generators and the total demand in 100 MVA transformer using data from 2015 and an electricity price of 34 €/MWh

Figure 6.26: Ageing rate of 100 MVA transformer using data from 2015 and an electricity price of 34 €/MWh 66 CHAPTER 6. RESULTS

6.2.3 Results for 200 MVA transformer

Figure 6.27: Hot-spot and top-oil temperatures for 200 MVA transformer using data from 2015 and an electricity price of 34 €/MWh

Figure 6.28: Current through the 200 MVA transformer using data from 2015 and an electricity price of 34 €/MWh 6.2. RESULTS CHANGING THE WIND SPEED 67

Figure 6.29: Comparison between the wind farm production, the production from other generators and the total demand in 200 MVA transformer using data from 2015 and an electricity price of 34 €/MWh

Figure 6.30: Ageing rate of 200 MVA transformer using data from 2015 and an electricity price of 34 €/MWh 68 CHAPTER 6. RESULTS

6.2.4 Results for 400 MVA transformer

Figure 6.31: Hot-spot and top-oil temperatures for 400 MVA transformer using data from 2015 and an electricity price of 34 €/MWh

Figure 6.32: Current through the 400 MVA transformer using data from 2015 and an electricity price of 34 €/MWh 6.2. RESULTS CHANGING THE WIND SPEED 69

Figure 6.33: Comparison between the wind farm production, the production from other generators and the total demand in 400 MVA transformer using data from 2015 and an electricity price of 34 €/MWh

Figure 6.34: Ageing rate of 400 MVA transformer using data from 2015 and an electricity price of 34 €/MWh 70 CHAPTER 6. RESULTS

6.2.5 Comparison

The same conclusions can be drawn about the shape and behavior of the plots as in the previous case. However, there are some differences between this case and the previous one. In this case, the transformers are working above their rated values more often than in the previous case. In addition, the plots look more homogeneous: in the previous case it is possible to notice a succession of peaks and valleys. Comparing Table 6.2 with Table 6.1, it is possible to see that the optimal wind farms for this case have less capacity and number of turbines, but a bit higher NPV. Therefore, thanks to good wind resources (high wind speed), it is possible to build a smaller wind farm and obtain higher benefits even when the electricity price is lower than the electricity price in the previous case. This highlights the importance of choosing a good location. As in the previous case, Figure 6.35 shows the linear evolution of the NPV.

Table 6.2: Optimal wind farm design and NPV for each transformer when using 2015 data and an electricity price of 34 €/MWh

Transformer size [MVA] Capacity [MW] Nº turbines LOL NPV [M€] 63 95 19 7200.72 11.2 100 155 31 7200.72 18.6 200 305 61 7200.72 38.8 400 615 123 7200.72 80.6

Figure 6.35: NPV comparison between the transformers when using 2015 data and an electricity price of 34 €/MWh 6.3. RESULTS VARYING THE ELECTRICITY PRICE 71 6.3 Results varying the electricity price

As 400 MVA transformer is the one with the highest NPV, this transformer is selected to study the effect of the variation in the electricity price. Five electricity prices are studied: 32 €/MWh, 34 €/MWh, 39 €/MWh, 43 €/MWh and 47 €/MWh. It is possible to see that between the lowest and the highest price, there is an increase of approximately 50%. Using 31 €/MWh as electricity price, the problem is unfeasible. The results are presented below.

Figure 6.36: Comparison of the current through the 400 MVA transformer when varying the electricity price 72 CHAPTER 6. RESULTS

Figure 6.37: Comparison of the hot-spot temperature in 400 MVA transformer when varying the electricity price

Figure 6.38: Comparison of the production of the wind farm using 400 MVA transformer when varying the electricity price 6.3. RESULTS VARYING THE ELECTRICITY PRICE 73

Figure 6.39: Comparison of the production of the wind farm using 400 MVA transformer when varying the electricity price including also de total demand

Figure 6.40: Comparison of the top oil temperature in 400 MVA transformer when varying the electricity price 74 CHAPTER 6. RESULTS

Figure 6.41: Comparison of the ageing rate of 400 MVA transformer when varying the electricity price

As it is possible to see in the plots, the curves follow the same shape and there are not big differences between them. However, if now the resulting capacity and NPV are compared (Table 6.3) for the lowest and the highest price, it is noticed that an increase of 50% in the electricity price produces an increase of 6.6% in the capacity of the wind farm and an increase of more than 3000% in NPV. Therefore, an increase of the electricity price produces a huge increase in the benefits without increasing the costs.

Table 6.3: Optimal wind farm design and NPV for each transformer when using 400 MVA transformer and a varying price

Electricity price [€/MWh] Capacity [MW] Nº turbines LOL NPV [M€] 32 610 122 7200.72 16.2 34 615 123 7200.72 80.6 39 620 124 7200.72 243 43 635 127 7200.72 374 47 650 130 7200.72 507

In Figures 6.42 and 6.43, it is possible to notice that the capacity is not perfectly linear while the NPV curve continues to be linear as in the previous cases. 6.3. RESULTS VARYING THE ELECTRICITY PRICE 75

Figure 6.42: Capacity comparison when varying the electricity price

Figure 6.43: NPV comparison when varying the electricity price 76 CHAPTER 6. RESULTS 6.4 Results varying the lifetime of the transformer

In the previous cases, LOL arrives to its maximum value. Therefore, it is possible to think that the lifetime of the transformer is a limiting factor for the problem and its increase will produce an increase in the NPV. This is due to the fact that a higher lifetime of the transformer will allow the transformer to work above its rated values during longer periods or more often. Therefore, this will result in a higher power production and hence, a higher NPV. For this reason, it is decided to study the effect of LOL in 400 MVA transformer by varying the lifetime of the transformer: 20 years (the one is used in the previous cases), 30 years and 40 years. The results are presented below.

Figure 6.44: Comparison of the current through the 400 MVA transformer when varying the transformer lifetime 6.4. RESULTS VARYING THE LIFETIME OF THE TRANSFORMER 77

Figure 6.45: Comparison of the hot-spot temperature in 400 MVA transformer when varying the transformer lifetime

Figure 6.46: Comparison of the production of the wind farm using 400 MVA transformer when varying the transformer lifetime 78 CHAPTER 6. RESULTS

Figure 6.47: Comparison of the top oil temperature in 400 MVA transformer when the transformer lifetime

Figure 6.48: Comparison of the ageing rate of 400 MVA transformer when varying the transformer lifetime 6.4. RESULTS VARYING THE LIFETIME OF THE TRANSFORMER 79

6.4.1 Comparison As it is possible to see in Table 6.4, an increase in transformer lifetime does not have the expected big effect in NPV. When the lifetime of the transformer is doubled from 20 years to 40 years, the NPV only increases in 7% and the capacity in 6.5%.

Table 6.4: Optimal wind farm design and NPV for each transformer when using 400 MVA transformer and a varying transformer lifetime

Transformer lifetime [yr] Capacity [MW] Nº turbines LOL NPV [M€] 20 615 123 7200.72 80.6 30 640 128 10512 83.8 40 655 131 14016 86.4

Figure 6.49: Capacity comparison when varying the transformer lifetime 80 CHAPTER 6. RESULTS

Figure 6.50: NPV comparison when varying the transformer lifetime

6.5 Results varying transformer parameters

In the previous cases, the transformer parameters are approximated to be the same for all the selected transformer using the characteristics ODAF values. However, in reality this is not true. The objective of this section is to study the effect of varying two of the transformer parameters: R and τ0. Table 6.5 shows the comparison between the original values and the new ones. As it is possible to notice, 100 MVA transformer is used as base transformer and it maintains the same parameters as before.

Table 6.5: Comparison between the original transformer parameters (indicated with 1) and the modified transformer parameters (indicated with 2)

Transformer [MVA] R 1 R 2 τ0 1 [min] τ0 2 [min] 63 6 4 90 75 100 6 6 90 90 200 6 7.6 90 175 400 6 16.78 90 235 6.5. RESULTS VARYING TRANSFORMER PARAMETERS 81

6.5.1 Results for 63 MVA transformer

Figure 6.51: Comparison of the current through the 63 MVA transformer when varying the transformer parameters

Figure 6.52: Comparison of the hot-spot temperature in 63 MVA transformer when varying the transformer parameters 82 CHAPTER 6. RESULTS

Figure 6.53: Comparison of the production of the wind farm using 63 MVA transformer when varying the transformer parameters

Figure 6.54: Comparison of the top oil temperature in 63 MVA transformer when the transformer parameters 6.5. RESULTS VARYING TRANSFORMER PARAMETERS 83

Figure 6.55: Comparison of the ageing rate of 63 MVA transformer when varying the transformer parameters

6.5.2 Results for 200 MVA transformer

Figure 6.56: Comparison of the current through the 200 MVA transformer when varying the transformer parameters 84 CHAPTER 6. RESULTS

Figure 6.57: Comparison of the hot-spot temperature in 200 MVA transformer when varying the transformer parameters

Figure 6.58: Comparison of the production of the wind farm using 200 MVA transformer when varying the transformer parameters 6.5. RESULTS VARYING TRANSFORMER PARAMETERS 85

Figure 6.59: Comparison of the top oil temperature in 200 MVA transformer when the transformer parameters

Figure 6.60: Comparison of the ageing rate of 200 MVA transformer when varying the transformer parameters 86 CHAPTER 6. RESULTS

6.5.3 Results for 400 MVA transformer

Figure 6.61: Comparison of the current through the 400 MVA transformer when varying the transformer parameters

Figure 6.62: Comparison of the hot-spot temperature in 200 MVA transformer when varying the transformer parameters 6.5. RESULTS VARYING TRANSFORMER PARAMETERS 87

Figure 6.63: Comparison of the production of the wind farm using 400 MVA transformer when varying the transformer parameters

Figure 6.64: Comparison of the top oil temperature in 200 MVA transformer when the transformer parameters 88 CHAPTER 6. RESULTS

Figure 6.65: Comparison of the ageing rate of 400 MVA transformer when varying the transformer parameters

6.6 Comparison

As it is possible to see in the resulting plots: the larger the transformer, the bigger the effect of the parameter modification. Looking at Table 6.6 and analysing transformer 400 MVA, it is possible to see that with the parameter modification the capacity and NPV increases by 9%. This is also noticeable in Figures 6.66 and 6.67. Therefore, the effect of parameter variation is not negligible.

Table 6.6: Comparison between the results with the original transformer parameters (indicated with 1) and the results after changing the transformer parameters (indicated with 2)

Trans [MVA] Cap.1 [MW] Cap.2 [MW] Tur.1 Tur.2 NPV1 [M€] NPV2 [M€] 63 95 105 19 21 11.2 12.4 200 305 335 61 67 38.8 42.5 400 615 670 123 134 80.8 87.9 6.6. COMPARISON 89

Figure 6.66: Capacity comparison when varying transformer parameters

Figure 6.67: NPV comparison when varying transformer parameters Chapter 7

Conclusions and future studies

In this chapter, the final conclusions of the project are presented. In addition, some suggestions for future studies are given.

7.1 Conclusions

In this section, the principal conclusions that have been obtained from the project are explained.

• It has not been possible to pose an optimization problem with an automatic selection of optimal transformer and wind farm design using MILP tools. Diffe- rent formulations have been studied during months and it has not been possible to obtain results without violating MILP rules. Therefore, in order to achieve this goal, non-linear programming methods should be studied. However, this type of programming requires huge computational resources and it can take hours or days to be solved.

• Results are as expected according to theory, therefore it is possible to conclude that the model is working properly.

• In all the cases, the evolution of NPV and of the capacity are linear. In other words, the larger the transformer, the bigger the optimal wind farm and the higher the benefits.

• If the location has good wind resources (high wind speed), the obtained results are more homogeneous, transformers are working above their rated values more often and it is possible to build a smaller wind farm obtaining higher benefits. Therefore, the location of the wind farm is critical.

• Electricity price has a huge impact on the NPV and hence, on the benefits. It has been obtained that by increasing the electricity price by 50% (from 32 €/MWh to 47 €/MWh), the NPV is 30 times higher.

90 7.2. FUTURE STUDIES 91

• In respect of the lifetime of the transformer, an increase of 100% only produces an increase of 7% in the capacity and in the NPV.

• The effect of the transformer parameters in the result is not negligible and hence, when transformer data are not available, it is necessary to select reasonable parameters according to each transformer size.

• By applying DTR, it is possible to install a smaller transformer reducing costs and increasing NPV.

• The method developed in this project can help to design the optimal wind farm, but it cannot select the best option, that is a decision of the designer. As it has been showed in the results, a larger transformer produces higher benefits, but also requires a bigger wind farm. Therefore, the designer should select the best transformer with its optimal wind farm design depending on more factors as the available space for the construction of the wind farm, electricity market, etc.

7.2 Future studies

The author provides some suggestions for future studies below.

• In this case, it has not been possible to write an optimization problem where the optimal transformer and wind farm design are chosen automatically by the program without violating the MILP rules. Therefore, in the future, it would be interesting to investigate other methods and tools to achieve this goal.

• The stability of the grid has not been studied in detail in order not to add complexity to the optimization problem. Consequently, this optimization problem can be modified in order to add that stability study.

• In the same way, there are more economical parameters that could have been taken into account. Another idea for future studies could be to perform a more complex economic analysis.

• The wind farm to be designed could be included in a more complex grid which contains other renewable energy farms.

• Finally, it would be interesting to include Dynamic Line Rating (DLR) in the analysis in order to see the combine effect of both technologies in the planning of wind farms. Bibliography

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