Calculating the Risk of Power Shortage in the Nordic Power System

STOCKHOLM SWEDEN 2018,

Calculating the Risk of Power Shortage in the Nordic Power System

ALESSANDRO CROSARA

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

Master Thesis

Calculating the Risk of Power Shortage in the Nordic Power System

Author:

Alessandro Crosara

Supervisors:

Lennart S¨oder Egill T´omasson Johan Bruce Rachel Walsh

Examiner:

Mikael Amelin

A thesis submitted in fulfillment of the requirements for the Master’s degree in Electric Power Engineering

in the

School of Electrical Engineering and Computer Science (EECS) Department of Electric Power and Energy Systems (EPE)

Integration of Renewable Energy Sources (IRES) Group

July 2018

In the near future, the decommissioning of large power plants is planned in the Nordic electric power system, due to environmental and market reasons. This will be countered by an increase in the wind power installed capacity, as well as by significant investments in the transmission system. In such a context, characterized by several changes, the Nordic power system might face reliability challenges.

This thesis aims to calculate the risk of power shortage in the different price areas which constitute the Nordic power system, for three different scenarios: a base scenario 2015, scenario 2020, and scenario 2025. Different case studies, focusing on the Nordic power system and on some of its subsystems, are investigated. The reliability evaluation which is carried out follows a probabilistic approach, by means of Monte Carlo simulations.

Crude Monte Carlo, as well as an advanced variance reduction technique – namely Cross- Entropy based Importance Sampling (CEIS) – are applied and compared. An alternative sampling method based on stratified sampling is presented too.

The starting point of this thesis is Viktor Terrier’s 2017 Master thesis, “North European Power Systems Reliability” [1]. Model-wise, among the other improvements, load and wind power are sampled in a different way to account for the correlation between them.

Data-wise, more realistic assumptions are made and more accurate data are used, thanks also to the collaboration with Sweco Energuide AB, Department of Energy Markets.

From the model perspective, it is concluded that CEIS outperforms crude Monte Carlo when simulating small to medium size systems, but it cannot be successfully applied when simulating large and very reliable systems like the Nordic system as a whole. The presented alternative sampling method can however be used for such cases. From the numerical-results perspective, the drawn conclusion is that the Nordic power system is estimated to become more reliable by years 2020 and 2025. Even if partly intermittent, more generation capacity is expected to be available, and thanks to the significant in- vestments which are planned in the transmission system, it will be possible to effectively transmit more power where needed, regardless of the area where it has been generated.

The thesis is carried out at KTH Royal Institute of Technology, Department of Electric Power and Energy Systems, in collaboration with Sweco Energuide AB, Department of Energy Markets, within the frame of the North European Energy Perspectives Project (NEPP).

P˚a grund av milj¨o- och marknadsf¨orh˚allanden, planeras kommande ˚ar nedl¨aggningen av stora kraftverk i det nordiska elsystemet. F¨or att ers¨atta dessa kr¨avs ett ¨okat antal vindkraftverk, men ¨aven stora investeringar i el¨overf¨oringssystemet. Denna ¨overg˚ang kan st¨alla det nordiska elsystemet inf¨or tillf¨orlitlighetsutmaningar.

Denna avhandling har till syfte att ber¨akna risken f¨or effektbrist i de olika prisomr˚adena som utg¨or det nordiska elsystemet, f¨or tre olika scenarier: ett referensscenario som motsvarar l¨aget under 2015, scenario 2020 och scenario 2025. Olika fallstudier utf¨ors med fokus p˚a det nordiska elsystemet och n˚agra av dess delsystem. Tillf¨orlitlighetsanalysen i denna avhandling ¨ar baserad p˚a sannolikhetsmetoder och utf¨ors med hj¨alp av Monte Carlo simuleringar. B˚ade enkel Monte Carlo och en avancerad variansreduktionsteknik, den s˚a kallade Cross-Entropy-baserade samplingsmetoden (CEIS)- till¨ampas och j¨amf¨ors med varandra. Aven en alternativ samplingsmetod baserad p˚¨ a stratifierad sampling presenteras.

Utg˚angspunkten f¨or denna avhandling ¨ar Viktor Terriers examensarbete fr˚an 2017, med titeln “Nordeuropeiska elsystemets tillf¨orlitlighet” [1]. I den f¨orb¨attrade modellen pre- senterad i denna rapport ing˚ar bland annat en f¨orb¨attrad samplingsmetod f¨or last och vindkraft, som ¨aven tar h¨ansyn till korrelationen mellan dessa parametrar. Tack vare samarbetet med energimarknadsavdelningen p˚a Sweco Energuide AB, har ¨aven nog- grannheten i de data som anv¨ands, och de antaganden som dessa baseras p˚a f¨orb¨attrats.

Ur ett modellperspektiv, dras slutsatsen att CEIS levererar b¨attre resultat j¨amf¨ort med Monte Carlo n¨ar sm˚a och medelstora system simuleras, men kan inte anv¨andas f¨or att simulera stora och h¨ogt tillf¨orlitliga system, s˚asom det nordiska elsystemet. F¨or s˚adana fallstudier kan emellertid den presenterade alternativa samplingsmetoden till¨ampas. Ur det numeriska resultatperspektivet dras slutsatsen att tillf¨orlitligheten med det nordiska elsystemet f¨orv¨antas ¨oka fram till 2020 och 2025. Trots en delvis oregelbunden pro- duktion, kommer den installerade produktionskapaciteten att vara h¨ogre, och tack vare stora planerade investeringar i ¨overf¨oringssystemet, kommer den producerade elektriska effekten att kunna transporteras till omr˚aden d¨ar den beh¨ovs, oavsett var den genereras.

Detta examensarbete har utf¨orts vid avdelningen f¨or elkraftteknik p˚a Kungliga Tekniska H¨ogskolan (KTH), i samarbete med energimarknadsavdelningen p˚a Sweco Energuide AB, inom ramen f¨or North European Energy Perspectives Project (NEPP).

First and foremost, I would like to express my gratitude to my four supervisors, all of whom helped and supported me in different ways: Lennart S¨oder, Professor at KTH, for always bringing new ideas to discussion and for supporting me when defining the thesis objectives; Egill T´omasson, PhD Candidate at KTH, for all his help, especially with the model, and for always giving me concrete suggestions on how to proceed, even while in the USA; Johan Bruce and Rachel Walsh, Head of Energy Markets and Energy Markets Analyst at Sweco Energuide AB, for helping me with the scenario definition, for providing me with power system data, and for giving me the possibility to carry out the thesis from their office.

I would also like to express my appreciation to my examiner, Prof. Mikael Amelin, for providing me with insightful comments in order to improve and finalize my work.

Many thanks to Dr. Jon Olauson too, for his helpfulness in providing me with wind power and load time series built based on the MERRA data set from NASA, even though I ended up not using such data due to the time constraint of my thesis.

Three more persons to whom I would like to express my gratitude are: Anders Nilsberth, from Svenska Kraftn¨at, for the support I received and for his prompt and precise replies to my questions; David Edward Weir, from NVE, for his helpfulness in providing me with Norwegian wind power data; and Jan Fredrik Foyn, from Nord Pool, for clarifying my doubts about the Net Transfer Capacities.

Moreover, I would like to thank my fellow KTH programme mates and friends Anna- Linnea Towle and Nahal Tamadon for grammar checking my thesis and for helping me with the Swedish abstract, respectively.

Last but not least, I sincerely thank my family, who supported me in undertaking my Master programme at KTH, and my friends who supported me especially during this last semester.

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Abstract i

Sammanfattning i

Acknowledgements ii

Contents iii

List of Figures vi

List of Tables viii

Abbreviations xi

Symbols xii

1 Introduction 1

1.1 Background . . . . 1

1.2 Risk of power shortage . . . . 2

1.3 Thesis objectives . . . . 4

1.4 Thesis structure. . . . 5

2 The Nordic power system 7 2.1 Power system areas . . . . 7

2.2 General description of the power system . . . . 9

2.3 Power system states . . . 11

2.3.1 Inverse-transform method . . . 11

2.3.2 Dependence and correlation . . . 12

2.4 Model representation of the system components . . . 13

2.4.1 Load . . . 13

2.4.2 Available Wind Power Generation Capacity (AWPGC) . . . 14

2.4.3 Total residual load . . . 14

2.4.4 Available other generation capacity. . . 16

2.4.5 Interconnection NTCs . . . 19

2.4.6 Import NTCs . . . 20

iii

3 Power system reliability evaluation model 22

3.1 Power system optimization problems . . . 22

3.1.1 Maximum-power-flow optimization problem . . . 23

3.1.2 Optimal-power-flow optimization problems . . . 23

3.2 Additional output variables . . . 27

3.2.1 Power Not Supplied (PNS) . . . 27

3.2.2 Loss Of Load Occasion (LOLO). . . 28

3.3 Reliability indices. . . 28

3.3.1 Expected Power Not Supplied (EPNS) . . . 28

3.3.2 Loss Of Load Probability (LOLP) . . . 29

3.4 Crude Monte Carlo simulation . . . 29

3.4.1 Introduction to Monte Carlo methods . . . 30

3.4.2 Principles of a crude Monte Carlo simulation . . . 30

3.4.3 Stopping rules in a Monte Carlo simulation . . . 32

3.4.4 Crude Monte Carlo applied to the simulation of a power system . 33 3.5 Cross-Entropy based Importance Sampling (CEIS) . . . 38

3.5.1 Introduction to Importance Sampling . . . 38

3.5.2 Introduction to CEIS . . . 39

3.5.3 CEIS applied to the simulation of a power system . . . 39

3.6 Alternative sampling method . . . 51

4 Scenarios definition: data collection and formatting 54 4.1 Base scenario: 2015. . . 55

4.1.1 Time frame . . . 55

4.1.2 Load . . . 55

4.1.3 Available Wind Power Generation Capacity (AWPGC) . . . 56

4.1.4 Total residual load . . . 61

4.1.5 Interconnection NTCs . . . 61

4.1.6 Import NTCs . . . 63

4.1.7 Other generation . . . 64

4.2 Scenario 2020 . . . 73

4.2.1 Time frame . . . 73

4.2.2 Load . . . 73

4.2.3 Available Wind Power Generation Capacity (AWPGC) . . . 73

4.2.4 Total residual load . . . 75

4.2.5 Interconnection NTCs . . . 75

4.2.6 Import NTCs . . . 77

4.2.7 Other generation . . . 78

4.3 Scenario 2025 . . . 80

4.3.1 Time frame . . . 80

4.3.2 Load . . . 81

4.3.3 Available Wind Power Generation Capacity (AWPGC) . . . 81

4.3.4 Total residual load . . . 81

4.3.5 Interconnection NTCs . . . 82

4.3.6 Import NTCs . . . 83

4.3.7 Other generation . . . 84

5 Case studies 85

5.1 Isolated Swedish system, scenario 2015 . . . 86

5.2 Isolated Swedish system, scenario 2020 . . . 86

5.3 Isolated Swedish system, scenario 2025 . . . 87

5.4 Swedish system with imports reduced to 20%, scenario 2015 . . . 87

5.5 Swedish system with imports reduced to 20%, scenario 2020 . . . 87

5.6 Swedish system with imports reduced to 20%, scenario 2025 . . . 88

5.7 Isolated Swedish and Finnish system, scenario 2015 . . . 88

5.8 Isolated Swedish and Finnish system, scenario 2020 . . . 88

5.9 Isolated Swedish and Finnish system, scenario 2025 . . . 89

5.10 Isolated Nordic system, scenario 2015 . . . 89

5.11 Isolated Nordic system, scenario 2020 . . . 89

5.12 Isolated Nordic system, scenario 2025 . . . 90

5.13 Simulation parameters . . . 90

6 Results 91 6.1 Results of: isolated Swedish system, scenario 2015 . . . 92

6.2 Results of: isolated Swedish system, scenario 2020 . . . 92

6.3 Results of: isolated Swedish system, scenario 2025 . . . 92

6.4 Results of: Swedish system with imports reduced to 20%, scenario 2015 . 95 6.5 Results of: Swedish system with imports reduced to 20%, scenario 2020 . 95 6.6 Results of: Swedish system with imports reduced to 20%, scenario 2025 . 95 6.7 Results of: isolated Swedish and Finnish system, scenario 2015 . . . 97

6.8 Results of: isolated Swedish and Finnish system, scenario 2020 . . . 98

6.9 Results of: isolated Swedish and Finnish system, scenario 2025 . . . 98

6.10 Results of: isolated Nordic system, scenario 2015 . . . 98

6.11 Results of: isolated Nordic system, scenario 2020 . . . 101

6.12 Results of: isolated Nordic system, scenario 2025 . . . 102

7 Discussion 104 7.1 Discussion about the simulation methods . . . 104

7.2 Discussion about the numerical results . . . 106

7.3 Numerical-results comparison with reference [1] . . . 108

8 Conclusion 110

9 Future work 113

A Comparison of the assumptions in this thesis and reference [1] 115

B CDFs of the input variables for scenario 2015 123

C Other-generation data for scenario 2015 130

Bibliography 145

2.1 The Nordic power system, divided according to Nord Pool price areas (adapted from [2]). The equivalent transmission lines as of 2015 are drawn in blue. . . . . 8 2.2 Inverse-transform method . . . 12 3.1 A simulation problem format . . . 30 4.1 The Nordic power system in scenario 2020, divided according to Nord Pool

price areas (adapted from [2]). If compared to 2015, the changes in the equivalent transmission lines, as well as the new (equivalent) transmission lines, are drawn in red. . . . 79 4.2 The Nordic power system in scenario 2025, divided according to Nord Pool

price areas (adapted from [2]). If compared to 2020, the changes in the equivalent transmission lines, as well as the new (equivalent) transmission lines, are drawn in red. . . . 84 B.1 Scenario 2015: CDF of the load in SE1, SE2, SE3, SE4, DK1, and DK2. . 124 B.2 Scenario 2015: CDF of the load in FI, NO1, NO2, NO3, NO4, and NO5. . 124 B.3 Scenario 2015: CDF of the (already scaled) AWPGC in SE1, SE2, SE3,

SE4, DK1, and DK2. . . . 125 B.4 Scenario 2015: CDF of the (already scaled) AWPGC in FI, NO1, NO2,

NO3, NO4, and NO5. . . . 125 B.5 Scenario 2015: CDF of the equivalent transmission lines (PART 1/7).

An equivalent transmission line is classified as interconnection or import depending on the case study. . . . 126 B.6 Scenario 2015: CDF of the equivalent transmission lines (PART 2/7).

An equivalent transmission line is classified as interconnection or import depending on the case study. . . . 126 B.7 Scenario 2015: CDF of the equivalent transmission lines (PART 3/7).

An equivalent transmission line is classified as interconnection or import depending on the case study. . . . 127 B.8 Scenario 2015: CDF of the equivalent transmission lines (PART 4/7).

An equivalent transmission line is classified as interconnection or import depending on the case study. . . . 127 B.9 Scenario 2015: CDF of the equivalent transmission lines (PART 5/7).

An equivalent transmission line is classified as interconnection or import depending on the case study. . . . 128 B.10 Scenario 2015: CDF of the equivalent transmission lines (PART 6/7).

An equivalent transmission line is classified as interconnection or import depending on the case study. . . . 128

vi

B.11 Scenario 2015: CDF of the equivalent transmission lines (PART 7/7).

An equivalent transmission line is classified as interconnection or import depending on the case study. . . . 129

4.1 Wind power installed capacity per area. . . . 60

4.2 Interconnections in the Nordic power system, as of year 2015. . . . 61

4.3 Forced Outage Rates (FORs) per type of power source. . . . 66

4.4 Forced Outage Rates (FORs) per type of power source, as set in [1]. . . . 66

4.5 Installed other generation capacity per Swedish area. . . . 67

4.6 Expected available other generation capacity per Swedish area. . . . 68

4.7 Always Available Grouped Capacity per Swedish area. . . . 70

4.8 Installed other generation capacity in Finland. . . . 70

4.9 Always Available Grouped Capacity in Finland.. . . 71

4.10 Installed other generation capacity per Norwegian and Danish area. . . . 72

4.11 Always Available Grouped Capacity per Norwegian and Danish area.. . . 72

4.12 Per area wind power installed capacity variation in 2020, if compared to 2015-12-31. . . . 74

4.13 Installed capacity of interconnections and imports in 2020, if compared to 2016. . . . 78

4.14 Per area wind power installed capacity variation in 2025, if compared to 2015-12-31. . . . 82

4.15 Installed capacity of interconnections and imports in 2025, if compared to 2020. . . . 82

6.1 Results for the isolated Swedish system, scenario 2015 (crude Monte Carlo). 93 6.2 Results for the isolated Swedish system, scenario 2015 (CEIS). . . . 93

6.3 Results for the isolated Swedish system, scenario 2020 (crude Monte Carlo). 93 6.4 Results for the isolated Swedish system, scenario 2020 (CEIS). . . . 94

6.5 Results for the isolated Swedish system, scenario 2025 (crude Monte Carlo). 94 6.6 Results for the isolated Swedish system, scenario 2025 (CEIS). . . . 94

6.7 Results for the Swedish system with imports reduced to 20%, scenario 2015 (crude Monte Carlo). . . . 95

6.8 Results for the Swedish system with imports reduced to 20%, scenario 2015 (CEIS). . . . 96

6.9 Results for the Swedish system with imports reduced to 20%, scenario 2020 (crude Monte Carlo). . . . 96

6.10 Results for the Swedish system with imports reduced to 20%, scenario 2020 (CEIS). . . . 96

6.11 Results for the Swedish system with imports reduced to 20%, scenario 2025 (crude Monte Carlo). . . . 97

6.12 Results for the Swedish system with imports reduced to 20%, scenario 2025 (CEIS). . . . 97

viii

6.13 Results for the isolated Swedish and Finnish system, scenario 2015 (crude Monte Carlo). . . . 98 6.14 Results for the isolated Swedish and Finnish system, scenario 2015 (CEIS). 99 6.15 Results for the isolated Swedish and Finnish system, scenario 2020 (crude

Monte Carlo). . . . 99 6.16 Results for the isolated Swedish and Finnish system, scenario 2020 (CEIS).100 6.17 Results for the isolated Swedish and Finnish system, scenario 2025 (crude

Monte Carlo). . . . 100 6.18 Results for the isolated Swedish and Finnish system, scenario 2025 (CEIS).101 6.19 Results for the isolated Nordic system, scenario 2015 (alternative sam-

pling method). . . . 102 6.20 Results for the isolated Nordic system, scenario 2020 (alternative sam-

pling method). . . . 103 6.21 Results for the isolated Nordic system, scenario 2025 (alternative sam-

pling method). . . . 103 A.1 The most important differences between reference [1] and this thesis. . . 115 C.1 List of individually modelled power plants in the Nordic power system for

scenario 2015. . . . 130

1 Crude Monte Carlo algorithm . . . 37

2 CEIS pre-simulation algorithm . . . 48

3 CEIS main-Monte-Carlo-simulation algorithm . . . 50

4 Alternative sampling algorithm . . . 53

x

AAGC Always Available Grouped Capacity

AWPGC Available Wind Power Generation Capacity CDF Cumulative Distribution Function

CE Cross Entropy

CEIS Cross-Entropy based Importance Sampling CV Coefficient of Variation

FOR Forced Outage Rate

EPNS Expected Power Not Supplied LOLO Loss Of Load Occasion

LOLP Loss Of Load Probability NTC Net Transfer Capacity PDF Probability Density Function PNS Power Not Supplied

TRL Total Residual Load

xi

Indices:

a Index of areas.

t Index of interconnections.

τ Index of import links.

k Index of iteration in the CEIS pre-simulation.

System-component superscripts:

L Load.

AW Available Wind Power Generation Capacity

(AWPGC).

T RL Total Residual Load (TRL).

G Other generation.

AAGC Always Available Grouped Capacity (AAGC).

T Interconnection NTC.

IG Import NTC.

Sets:

Ω^L_a Set of load levels in area a.

Ω^AW_a Set of AWPGC levels in area a.

Ω^{T RL} Set of TRL levels.

Ω^T_t Set of interconnection-NTC levels for

interconnection t.

Ω^IG_τ Set of import-NTC levels for import link τ .

Υa Set of imports to area a.

xii

Ψ⁻_a Set of interconnections entering area a.

Ψ⁺_a Set of interconnections exiting area a.

˚Aa Set of areas belonging to the same country to which area a belongs.

Variables:

X_i^L= (x^L_i,a)1≤a≤nA Vector of load states for sample i.

X_i^AW = (x^AW_i,a )_1≤a≤nA Vector of AWPGC states for sample i.

x^RL_i,a Residual load in area a for sample i.

x^{T RL}_i Total residual load for sample i.

X_i^G= (x^G_i,g)1≤g≤nG Vector of power-generation-block states for sample i.

X_i^T = (x^T_i,t)1≤t≤nT Vector of interconnection-NTC states for sample i.

X_i^IG= (x^IG_i,τ)_{1≤τ ≤nIG} Vector of import-NTC states for sample i.

p_i,a Available generation capacity in area a for sample i.

d_i,a Served load in area a for sample i.

fi,t Power flow on interconnection t for sample i.

p_i,a Power generation in area a for sample i.

κi Load excess variable for sample i.

P N S_i,a PNS in area a for sample i.

P N Ssystem,i PNS in the system for sample i.

LOLO_i,a LOLO in area a for sample i.

LOLOsystem,i LOLO in the system for sample i.

X_i Power system state vector for sample i.

Parameters:

nA Number of areas in the power system.

n_G Number of generation blocks in the power system.

nIG Number of imports to the power system.

U^G= (u^G_g)_1≤g≤nG Vector of the original unavailability rates.

Θ^G,units= (θ^G,unitsg )1≤g≤nG Vector indicating the number of identical units contained in each generation block.

Θ^G,capa= (θg^G,capa)1≤g≤nG Vector indicating the installed capacity of a unit belonging to each generation block.

Θ^G,area= (θg^G,area)1≤g≤nG Vector indicating the area where each power generation block is located.

Θ^AAGC,capa= (θa^AAGC,capa)1≤a≤nA Vector indicating the AAGC for each area.

n Number of samples collected from the beginning

of a Monte Carlo simulation.

N_{M C} Number of samples collected in each batch of the (main) Monte Carlo simulation.

N_min Minimum number of samples to be collected in the

(main) Monte Carlo simulation.

ρ_O^† Relative tolerance for the output scalar O^†.

φ Pre-simulation threshold parameter.

N_IS Number of samples collected at each iteration of the pre-simulation.

α Pre-simulation smoothing parameter.

ρ Pre-simulation share of the highest performing

states at each iteration.

Vb_{M C}^G = (bv^G_{M C,g})_1≤g≤nG Other-generation reference parameter vector, used in the main Monte Carlo simulation.

µb^{T RL}_{M C} Mean of the optimal PDF of the TRL levels, used in the main Monte Carlo simulation.

bσ_{M C}^{T RL} Standard deviation of the optimal PDF of the TRL levels, used in the main Monte Carlo simulation.

N_sel Number of TRL/load/AWPGC time-series states

which are selected for the only considered stratum.

Ntot Total number of states in the TRL/load/AWPGC

time-series.

ωstratum Stratum weight.

C_inst,a Installed other-generation capacity in area a.

Cexp,a Expected available other-generation capacity in

area a.

Cindiv,a Expected available other-generation capacity from individually modelled power plants in area a.

θ^AAGC,capapre,a Preliminary AAGC for area a.

r_SE^AAGC Average of the ratio _C^Cexp,a

inst,a for the Swedish areas.

r_share,a^AAGC Relative expected available other-generation capacity from individually modelled power plant in area a.

Functions:

f_a^L(·) Original PDF of load levels in area a.

f_a^AW(·) Original PDF of AWPGC levels in area a.

f^{T RL}(·) Original PDF of TRL levels.

f_t^T(·) Original PDF of interconnection-NTC levels for interconnection t.

f^IG(·) Original PDF of import-NTC levels for import τ . c^{T RL}_k (·) Updated PDF of TRL levels at iteration k of the

pre-simulation.

S(·) Performance function.

H_k(·) Indicator function at iteration k of the pre- simulation.

W_k,i^{T RL}(·) Likelihood ratio for the TRL at iteration k of the pre-simulation.

W_k,i^G(·) Likelihood ratio for the other generation at iteration k of the pre-simulation.

W_k,i(·) System likelihood ratio at iteration k of the pre- simulation.

f_k^{GAU SSIAN}(·) Temporary Gaussian distribution built based on the TRL reference parameters at iteration k of the pre-simulation.

c^{T RL}_{M C}(·) Optimal PDF of TRL levels, used in the main Monte Carlo simulation.

W_i^{T RL}(·) Likelihood ratio for the TRL for sample i in the main Monte Carlo simulation.

W_i^G(·) Likelihood ratio for the other generation for sample i in the main Monte Carlo simulation.

Wi(·) System likelihood ratio for sample i in the main Monte Carlo simulation.

Estimates:

EP N S\ a Estimate of the EPNS in area a.

EP N S\ _system Estimate of the EPNS in the system.

LOLP\a Estimate of the LOLP in area a.

LOLP\_system Estimate of the LOLP in the system.

s²_X Estimate of the variance of random variable X.

a_X CV for the estimate of E[X ], where X is a

random variable.

µb^{T RL}_k Estimate of the mean of the updated TRL Gaussian distribution at iteration k of the pre- simulation.

bσ_k^{T RL} Estimate of the standard deviation of the updated TRL Gaussian distribution at iteration k of the pre-simulation.

Vb_k^G= (bv^G_k,g)1≤g≤nG Estimate of the other-generation reference parameter vector at iteration k of the pre- simulation.

Vb_k Estimate of the reference parameter vector at iteration k of the pre-simulation.

φb_k Estimate of the (1-ρ)-quantile of the performance at iteration k of the pre-simulation.

ωb_k,g^G Temporary estimate of the updated unavailability rate for generation block g at iteration k of the pre-simulation.

LOLP\stratum,a Stratum estimate of the LOLP in area a.

Introduction

1.1 Background

The Nordic electric power system, comprising the interconnected power systems of Swe- den, Finland, Norway and Denmark, is being - and will be even more in the upcoming years - subject to significant changes; different trends can be identified.

Due to environmental reasons, the decommissioning of thermal power plants is planned in the near future. In addition, low electricity prices (determined by the increased renewable generation) and market uncertainties have affected the profitability of the conventional generation, accelerating the retirement of conventional power plants [3].

As a result, the decommissioning of large thermal power plants (nuclear in particular) is being carried out or is planned especially in southern Sweden and in Finland. On the other hand, a large amount of renewable generation is expected to be built, especially in the northern parts of Sweden and in Finland [4]. A consequence of this situation could be a more volatile system, with less schedulable generation capacity, less base load generation, and fewer flexible power plants (where by flexibility we refer to the ability of a power system to maintain continuous service despite rapid and large swings in supply or demand [3]). In such a power-generation outlook, generation-adequacy challenges may arise, posing the question whether the peaks in the demand can be covered or not.

Given the aforementioned changes in the Nordic power generation, an adequate trans- mission capacity will allow for cost-effective utilization of power, will ensure access to generation capacity, will enable exchanging system services, and will be the key for a

1

well-integrated market [4]. Thus, several investment in the transmission system are planned as a result of several drivers, as discussed in [4]. Firstly, the Nordic system will likely increase its annual energy surplus, which will make it beneficial to strengthen the transmission capacity between the Nordic power system and continental Europe (and, in addition, to the UK). Secondly, linked to the increased renewable generation capac- ity and to new interconnectors to continental Europe, UK, and Baltic countries, there will be the need to strengthen the transmission capacity of the Nordic power system in the north-south direction. The decommissioning of nuclear power plants in southern Sweden and in Finland will further increment the demand for transmission capacity in the north-south direction. Thirdly, the integration of the Baltic power system (aiming to de-synchronize from the Russian power system and to synchronize to the continental European power system) may affect the power flows, which calls for further investiga- tion. Lastly, on a political level, there is the strong will to form a more interconnected European power system.

Depending on the location and size, the electrical consumption could also increase, due to new power-intensive industries in the northern Nordic system, to electrical trans- portation, and to the consumption increase in the largest cities. However, starting from the assumption made for scenarios 2020 and 2030 in [4], where it is reasoned that the actual demand growth will be slow due to energy efficiency measures, in this thesis it will be assumed that the demand will not increase in the near future (i.e. by years 2020 and 2025).

1.2 Risk of power shortage

As mentioned in the previous section, in the upcoming years several investments will be made in the Nordic power system. The reliability evaluation of the power system - in this thesis intended as a synonym of evaluating the risk of power shortage - allows for a better understanding of which could be the most problematic system areas in the future, so as to address the most urgent investments there. In other words, estimating the reliability or the risk of power shortage of a power system means analyzing the adequacy of such a system, that is its “ability [...] to supply the aggregate electric power and energy required by the customers, under steady-state conditions” [3].

Concretely, the risk of power shortage is computed in this thesis by following a prob- abilistic approach. As highlighted in [5], generally speaking, the advantage of using a probabilistic method rather than a deterministic one is that a probabilistic approach captures the uncertainty related to the different system variables: renewable genera- tion variability, conventional generation outages, transmission line outages, and demand variability. Conversely, a deterministic approach cannot capture the many different com- binations of events which could lead to adequacy problems: it simply focuses on some worst-case scenarios.

The risk of power shortage can be quantified by means of two reliability indices [1,6]:

• Loss of Load Probability (LOLP), i.e. the probability that at least one consumer is involuntarily disconnected (this situation is defined as load shedding) due to capacity limitations; and

• Expected Power Not Supplied (EPNS), i.e. the expected value of the power which cannot be delivered due to capacity limitations.

These two reliability indicators will be mathematically defined in section3.3and can be estimated both on a system basis and for each area of the system, the second option being more interesting for the purpose of estimating the risk of power shortage in each single system area. It can be noticed that while the LOLP provides us with a measure of the probability of load shedding, the EPNS provides us with a measure of the magnitude of such an event. In analogy with [7], it is worth pointing out that the LOLP and the EPNS should be seen as reliability indicators for the day-ahead market, and not as representing the expected non-delivered power to the end users. In fact, if there is not enough production capacity to meet the demand in the day-ahead market, strategic reserves and system reserves can be activated before the controlled curtailment of demand is needed¹.

Lastly, notice some important assumptions and simplifications made in this thesis. Fre- quency, voltage, reactive power and power losses are neglected. In addition, the elec- tricity price is not taken into account; it is assumed that, in a power shortage situation,

1This means that power plants used for strategic reserve and system reserve should not be considered when simulating the system.

consumers would be willing to buy electricity at a high price in order to not be discon- nected.

1.3 Thesis objectives

In this thesis two different Monte Carlo methods are applied to simulate the Nordic power system and some subsystems of it, namely crude Monte Carlo and Cross-Entropy based Importance Sampling (CEIS) (the latter being an advanced variance-reduction technique, already applied to the simulation of power systems e.g. in references [1,8,9]).

In addition, a third method based on stratified sampling² is presented; we will refer to it as alternative sampling method.

Several case studies will be carried out, investigating three different scenarios. Firstly, year 2015 is chosen as the time frame for the base scenario, with a focus on the winter months, which are the most problematic in the Nordic countries due to the highest peaks in the demand [10]. Then, two different future scenarios are investigated, having years 2020 and 2025 as time frames (focusing once again on the winter months) and aiming to estimate how the changes described in section 1.1 will impact the reliability of the Nordic power system in the near future.

The starting point of this thesis is Viktor Terrier’s 2017 Master thesis, “North European Power Systems Reliability” [1]. In that reference, a reliability evaluation of the Nordic power system was carried out, similarly to this thesis. Method-wise, the simulation methods which were applied are crude Monte Carlo and CEIS, but with a different implementation. Data-wise, only publicly available data were used.

The objectives of this thesis, which aims to continue and improve the work done in [1], can be summarized in two points.

1. Method-wise, the purpose is on the one hand to improve the simulation algorithms, and on the other hand to apply CEIS to the simulation of the Nordic power system and some of its subsystems, and compare it with crude Monte Carlo to test its performance.

2Stratified sampling is another variance reduction technique.

2. Numerical-results-wise, the aim is to obtain more realistic results if compared to [1] thanks to better assumptions and input data (which overall can be very im- pacting on the final results). Firstly, Sweco provided data for the future scenarios simulated in this thesis, i.e. scenarios 2020 and 2025, which are more complete and detailed than those used in [1]. Such data comprise: installed wind-power capac- ity, new-transmission-lines installed capacity, and reinforced-transmission-lines in- stalled capacity. Secondly, more publicly available data are found for scenario 2015 (which is still considered as base scenario and is essential to define future scenarios too) when it comes to installed wind-power capacity, wind-power-generation time series, and large power plants being commissioned or decommissioned. Thirdly, many different assumptions are made, with the purpose of mathematically repre- senting the power system in a realistic way. Some examples: more realistic and differentiated unavailability rates are used for the conventional power plants; the import NTCs are mostly modelled in a probabilistic way, instead of being assumed constantly available; some inconsistencies are found in [1] and fixed, for instance concerning power plants, interconnections, and imports which were wrongly uti- lized for scenario 2015.

A detailed summary of how this thesis differs from reference [1] in terms of assumptions, model and input data is provided in AppendixA. The results obtained in this thesis are compared with those obtained in [1] in Chapter 7.

1.4 Thesis structure

This thesis is structured as follows. In Chapter2, the Nordic power system is described and mathematically modelled as a multi-area power system; the different system vari- ables are introduced, as well as the concept of generating a power system state. In Chapter 3, the reliability evaluation model is described in detail. The system outputs and the reliability indices of interest are presented, followed by the description of two Monte Carlo methods (crude Monte Carlo and CEIS) and their application to the simu- lation of a power system. Furthermore, the alternative sampling method is presented. In Chapter4, the three different scenarios simulated in this thesis - scenarios 2015, 2020 and 2025 - are defined, aiming to clearly explain the data collection and formatting processes

too. Chapter 5 summarizes the case studies created for the Nordic power system and some subsystems of it, based on scenarios 2015, 2020 and 2025. Next, Chapter6reports the results obtained from the simulations carried out for the case studies. In Chapter 7the results are commented and discussed both model-wise and numerical-results-wise.

In addition, a comparison is made with the numerical results obtained in [1]. Chapter 8 summarizes the drawn conclusion in this thesis. Lastly, Chapter 9ends the thesis with suggestions for further work.

Further, as already mentioned, AppendixAhighlights the main differences between this thesis and [1] in terms of assumptions, model and input data. Appendix B shows the CDFs of the load, Available Wind Power Generation Capacity (AWPGC), interconnec- tion Net Transfer Capacities (NTCs), and import NTCs, for scenario 2015. In Appendix C, the power-generation-block list used for scenario 2015 is provided.

The Nordic power system

In this chapter, the Nordic power system will be described. Its area division will be clarified, followed by a general description of the main system variables. Moreover, a power system state will be defined and it will be shown how power system states can be randomly generated. Lastly, how each system component is mathematically represented in the model will be explained.

2.1 Power system areas

Even though the term Nordic usually refers to the geographical region comprising Swe- den, Denmark, Norway, Finland and Iceland, in the following by Nordic power system we will denote the set of the interconnected power systems of Sweden, Denmark, Norway and Finland only.

The choice of disregarding the Icelandic power system is motivated by the fact that, even though pre-feasibility studies have been carried out on IceLink [11] (an interconnection between Iceland and Scotland, which is part of the European Union’s list of key energy infrastructure projects [12]), the Icelandic power system is currently isolated, and thus has no effect in the power systems of the other Nordic countries.

In order to study the Nordic power system with a certain level of detail, it needs to be divided into areas and analyzed as a multi-area power system. Different possibilities would be possible, for example one could use the national borders of each country as

7

boundaries. Generally speaking, on the one hand the smaller the system areas, the more detailed the study will be. On the other hand, data availability needs to be taken into account too. The most reasonable option seems to be to consider as power system areas the bidding zones in which Nord Pool markets are divided [2], similar to the choice made in [1]. Those 12 areas are the following:

• SE1, SE2, SE3, and SE4 (Swedish areas);

• FI (all of Finland);

• NO1, NO2, NO3, NO4, and NO5 (Norwegian areas);

• DK1 and DK2 (Danish areas).

The different power system areas are interconnected through transmission lines. More- over, some of the areas are connected with neighbouring countries (meaning non-Nordic countries). Figure 2.1shows the Nordic power system, where the equivalent¹ transmis- sion lines as of year 2015 are drawn in blue.

Figure 2.1: The Nordic power system, divided according to Nord Pool price areas (adapted from [2]). The equivalent transmission lines as of 2015 are drawn in blue.

According to [1] a power system could be investigated in different ways. It could be analyzed as a whole, as well as a subsystem of it (comprising only some areas) could be chosen. Moreover, the imports from outside the chosen system could be considered or

1In the sense explained in sections2.4.5and2.4.6. An equivalent transmission line can be classified either as an interconnection or as an import, depending on the case study.

it could be instead decided to neglect them and assume that the system (or subsystem) is isolated.

In the following, for simplicity’s sake, the whole Nordic power system is described and power imports from the neighbouring countries are taken into account. However, in the case studies chapter we will also consider subsystems of the Nordic system, and systems which are isolated from the neighboring areas.

Lastly, notice that the power exports to the neighbouring countries are neglected, since it is assumed that in a power shortage situation it would not be reasonable for the power system object of the study to export power.

2.2 General description of the power system

Each of the areas of which the power system is composed has some characteristic random variables:

• a demand (or load);

• an Available Wind Power Generation Capacity (AWPGC);

• an available power generation capacity from all the other sources (that is, except for wind power), which we will denote from now on as available other generation capacity;

• a transmission capacity (or more precisely Net Transfer Capacity, NTC) for each interconnection between system areas;

• a transmission capacity (or NTC) for each import connection from an area not belonging to the system to a system area.

These variables can be modelled in different ways, depending on their nature and on the data availability. The demand, the AWPGC, and the NTC for both interconnections and imports have all the property of being highly variable in time. For example, data with hourly resolution are available for them in the “Historical Market Data” section on Nord Pool website [13], as will be described more in detail in Chapter 4. Therefore, it is possible (even for future scenarios, if making assumptions) and reasonable to model

these variables in the following way. Given one of these variables’ hourly data for a certain time frame, we denote by level each value that such a variable can assume during the specified time frame [1]. In each area and for each variable the levels can be ranked from the smallest to the largest, and a probability can be assigned to each level (each level’s probability is computed as the percentage of occurrences of that particular level during the considered time frame); in conclusion, a Probability Density Function, PDF is obtained. A Cumulative Distribution Function (CDF), defined as the integral of the PDF, can be calculated too. Thus, based on a known data set, we can compute a PDF and a CDF for all of these variables, which we can consequently treat as random variables.

However, the other generation is modelled following a different approach, similarly to [1].

According to the aforementioned definition, the other generation comprises all the power generation sources, but wind power. This means that it comprises both conventional power plants (fossil fuel power plants, nuclear power plants) and renewable energy power plants (hydro power plants, biomass power plants, solar power plants, etc.). Except for solar power, power plants belonging to these types of energy sources are available for most of the time, and when they are available their capacity usually equals their maximum capacity. Thus, it does not seem reasonable to consider their generation capacity variability on hourly basis. On the contrary, it seems more logical to model the available generation capacity of these power plants through a two-state (or on/off) behaviour; we then assign each power plant an unavailability rate, corresponding to the probability that it is offline.

Notice that not all the available other generation capacity can be modelled as just explained. As will be explained in section2.4.4, part of the power plants is not modelled individually through an unavailability rate, because their installed capacity is small (and consequently the impact on the system reliability is minor) or because information about them was not found. An Always Available Grouped Capacity (AAGC) is then calculated for each power system area.

Moreover, solar power, which could be modelled in the same way as wind power being a highly variable energy source, is taken into account in the model only indirectly and gives a contribution to the AAGCs, following an approach similar to [1]. This simplification is adopted given solar power’s small share in power generation in the Nordic power system.

2.3 Power system states

A power system state (or sample) is defined when a value is assigned or known for every component of the system. Sampling the system (or, equivalently, generating a system sample) means to randomly assign a value, within the set of the possible values and according to the corresponding probabilities, to each system input² variable [1], that is: load and AWPGC for each area, interconnection NTCs and import NTCs for each interconnection and import respectively, and available generation capacity for every power plant modelled individually. This can be done by applying the inverse-transform method (described in section2.3.1) to random numbers drawn from a continuous uniform distribution.

Next, the inverse-transform method will be described, followed by a discussion about dependence and correlation between different variables.

2.3.1 Inverse-transform method

The inverse-transform method allows us to generate a random variable from a known probability distribution [14]. Notice that the inverse transform method can only be applied to univariate distributions [15]. Keeping the same notation as in [14], which however will not be followed in the rest of this thesis, let us assume that X is a random variable, whose CDF is F. We are interested in generating a value of X. Since every CDF is a non-decreasing function, the inverse function of F, F⁻¹, can be defined as:

F⁻¹(y) = inf{x : F (x) ≥ y}, 0 ≤ y ≤ 1 (2.1)

where inf stands for the infimum. It can be shown that, if U has a standard uniform distribution (that is, U ∼ U(0,1) then the following function:

X = F⁻¹(U ) (2.2)

has exactly F as CDF.

2The reason why the following variables are considered inputs will be explained in Chapter3, where the reliability evaluation model will be described in detail and input and output variables will be clearly distinguished.

Therefore, as illustrated in Figure2.2, in order to generate a value of the random variable X, having CDF F, we can draw U ∼ U(0,1) and set X = F⁻¹(U ).

U

X ( )

F x

1

0

x

Figure 2.2: Inverse-transform method.

2.3.2 Dependence and correlation

Generally speaking, we can assume that the different random variables are independent, or that instead they (or some of them) are dependent and in that case whether they are correlated or not.

In this thesis (as explained more in detail in section 2.4), all the system variables but load and wind power generation capacity in each area will be assumed independent (and thus not correlated).

Coming to the load and AWPGC, as described in the same section, they could either be sampled independently or be assumed dependent and correlated in each area. Follow- ing the first option (which we will not consider in the case studies), we could generate random numbers from a U(0,1) and directly apply the inverse transform method to get the system variables. Following the second option instead, alternative solutions have to be pursued. A possibility is to apply the copula theory to the multi-variate joint probability distribution of the load and the AWPGC in each area, as explained in [9]:

thanks to the copula, it becomes possible to express a multi-variate joint cumulative distribution function as a function of marginal cumulative distribution functions [9,16].

This approach was followed in [1]. Another possibility instead, which is less computa- tionally expensive, is to sample the load and the AWPGC in each area based on the