IN
DEGREE PROJECT ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS
,
STOCKHOLM SWEDEN 2017
North European Power
Systems Reliability
VIKTOR TERRIER
KTH
Master Thesis
North European Power Systems
Reliability
Author: Viktor Terrier Supervisor: Egill T´omasson Examiner: Lennart S¨oderA thesis submitted in fulfillment of the requirements for the degree of Master of Science
in the
School of Electrical Engineering (EES)
Department of Electric Power and Energy Systems (EPE) Integration of Renewable Energy Sources group (IRES)
Abstract
North European Power Systems Reliability
The North European power system (Sweden, Finland, Norway, Denmark, Estonia, Latvia and Lithuania) is facing changes in its electricity production. The increasing share of intermittent power sources, such as wind power, makes the production less predictable. The decommissioning of large plants, for environmental or market reasons, leads to a decrease of production capacity while the demand can increase, which is detrimental to the power system reliability. Investments in interconnections and new power plants can be made to strengthen the system. Evaluating the reliability becomes essential to determine the investments that have to be made.
For this purpose, a model of the power system is built. The power system is divided into areas, where the demand, interconnections between areas, and intermittent gener-ation are represented by Cumulative Distribution Functions (CDF); while conventional generation plants follow a two-state behaviour. Imports from outside the system are set equal to their installed capacity, with considering that the neighbouring countries can always provide enough power. The model is set up by using only publicly available data.
The model is used for generating numerous possible states of the system in a Monte Carlo simulation, to estimate two reliability indices: the risk (LOLP) and the size (EPNS) of a power deficit. As a power deficit is a rare event, an excessively large number of samples is required to estimate the reliability of the system with a sufficient confidence level. Hence, a pre-simulation, called importance sampling, is run beforehand in order to improve the efficiency of the simulation.
Sammanfattning
Det nordeuropeiska elsystemets tillf¨orlitlighet
Det nordeuropeiska elsystemet (Sverige, Finland, Norge, Danmark, Estland, Lettland och Litauen) st˚ar inf¨or f¨or¨andringar i sin elproduktion. Den ¨okande andelen intermit-tenta kraftk¨allor, s˚asom vindkraft, g¨or produktionen mindre f¨oruts¨agbar. Avvecklingen av stora anl¨aggningar, av milj¨o- eller marknadssk¨al, leder till en minskning av produk-tionskapaciteten, medan efterfr˚agan kan ¨oka, vilket ¨ar till nackdel f¨or kraftsystemets tillf¨orlitlighet. Investeringar i sammankopplingar och i nya kraftverk kan g¨oras f¨or att st¨arka systemet. Utv¨ardering av tillf¨orlitligheten blir n¨odv¨andigt f¨or att best¨amma vilka investeringar som beh¨ovs.
F¨or detta ¨andam˚al byggs en modell av kraftsystemet. Kraftsystemet ¨ar uppdelat i omr˚aden, d¨ar efterfr˚agan, sammankopplingar mellan omr˚aden, och intermittent pro-duktion representeras av f¨ordelningsfunktioner; medan konventionella kraftverk antas ha ett tv˚a-tillst˚andsbeteende. Import fr˚an l¨ander utanf¨or systemet antas lika med deras installerade kapaciteter, med tanke p˚a att grannl¨anderna alltid kan ge tillr¨ackligt med str¨om. Modellen bygger p˚a allm¨ant tillg¨angliga uppgifter.
Modellen anv¨ands f¨or att generera ett stort antal m¨ojliga tillst˚and av systemet i en Monte Carlo-simulering f¨or att uppskatta tv˚a tillf¨orlitlighetsindex: risken (LOLP) och storleken (EPNS) av en effektbrist. Eftersom effektbrist ¨ar en s¨allsynt h¨andelse, kr¨avs ett mycket stort antal tester av olika tillst˚and i systemet f¨or att uppskatta tillf¨orlitligheten med en tillr¨acklig konfidensniv˚a. D¨arf¨or utnyttjas en f¨or-simulering, kallad ”Importance Sampling”, vilken k¨ors i f¨orv¨ag i syfte att f¨orb¨attra effektiviteten i simuleringen.
Acknowledgements
I would first like to thank my supervisor Egill T´omasson, Department Of Electric Power and Energy Systems, for having trusted me to carry out this project, and also for his help at all the stages of the project. He was always ready to help me whenever I ran into a trouble spot or had a question about my work. I really hope that the results of my Thesis will be valuable for your research.
Another person to whom I would also like to extend my gratitude is my examiner Lennart S¨oder, Department Of Electric Power and Energy Systems, for providing me with useful documents about data I needed, for his comments at the mid-term presentation, which proved to be very helpful for the final presentation, and also for his help with the Swedish abstract.
I would also like to acknowledge Anders Nilsberth, from Svenska Kraftn¨at, and Rickard Nilsson, from Nord Pool, for their very complete answers whenever I contacted them for some information.
I would like to especially thank Lars Herre, who helped me a lot in the process of finding a Master Thesis, even though the one I eventually carried out did not require his expertise.
Finally, I would like to express my gratitude to my family and to all those who provided me support throughout my years of study and for the Master Thesis; especially to Mathilde Ridard, Enrique Cobo Jim´enez, Alfonso de la Rocha G´omez-Arevalillo and Pablo Albiol Graullera for their daily support over this last semester.
Contents
Abstract i
Sammanfattning ii
Acknowledgements iii
Contents iv
List of Figures vii
List of Tables viii
Abbreviations x
Symbols xi
1 Literary Review / Introduction 1
1.1 Background . . . 1
1.2 Goals and objectives . . . 4
1.3 Delimitation of the study . . . 4
1.4 Roadmap . . . 5
2 The North European Power System 6 2.1 Modeling a state of the system . . . 6
2.1.1 General description of the system. . . 6
2.1.2 Generation of a state of the system. . . 8
2.2 Areas . . . 8
2.3 Demand . . . 10
2.3.1 Model of the demand . . . 10
2.3.2 Variables and parameters . . . 10
2.4 Production . . . 11
2.4.1 Intermittent power sources (wind power). . . 11
2.4.2 Conventional power sources . . . 11
2.4.3 Variables and parameters . . . 14
2.4.4 Overview of power production . . . 15
2.4.5 Overview of the demand in the North European power system . . 15
2.5 Transmission inside the power system . . . 16
Contents v
2.5.1 Model . . . 16
2.5.2 Variables and parameters . . . 17
2.6 Imports from outside the system . . . 17
3 Reliability Evaluation Model 19 3.1 Model of the problem - optimization problem . . . 19
3.1.1 Maximum power flow . . . 20
3.1.2 Optimal power flow . . . 21
3.2 Monte Carlo simulation . . . 24
3.2.1 Concept and goal . . . 25
3.2.2 Sampling the system . . . 25
3.2.3 Performance of the samples of the system . . . 27
3.2.4 Reliability indices . . . 28
3.2.5 MCS algorithm . . . 30
3.3 Importance sampling . . . 31
3.3.1 Concept and goal . . . 31
3.3.2 Evaluation of the performances and optimization problem . . . 32
3.3.3 Likelihood ratio. . . 34
3.3.4 Update of the probabilities . . . 38
3.3.5 Adaptation of the performance indices . . . 43
3.3.6 Pre-simulation algorithm . . . 44
3.4 Dependence between areas . . . 45
3.4.1 Using a copula . . . 45
3.4.2 Without copula . . . 47
3.5 Algorithm . . . 47
4 Simulation Results 49 4.1 Reliability of the current system . . . 50
4.1.1 Hypothesis . . . 50
4.1.2 Evaluation of the LOLP and EPNS . . . 53
4.2 Future scenarios . . . 54
4.2.1 Scenario for 2020 . . . 54
4.2.2 Scenario for 2025 . . . 56
4.2.3 Scenario for 2030 . . . 57
4.3 Comments on the results. . . 58
5 Conclusion 60
6 Future work 63
A Data on the demand 65
B Data on the wind power generation 69
C Data on power generating units 73
Contents vi
E Data on the imports 93
F Publicly available data 94
F.1 Demand . . . 94
F.1.1 Used . . . 94
F.1.2 Not used . . . 94
F.2 Import and transmission . . . 95
F.2.1 Used . . . 95
F.2.2 Not used . . . 95
F.3 Wind generation . . . 95
F.3.1 Used . . . 95
F.3.2 Not used . . . 96
F.4 Capacity of other types of generation sources . . . 96
F.4.1 Used . . . 96
F.4.2 Not used . . . 96
F.5 Maximum capacity in the areas . . . 96
F.5.1 Used . . . 96
F.5.2 Not used . . . 97
F.6 Generation availability . . . 97
F.6.1 Used . . . 97
F.6.2 Not used . . . 97
F.7 Other general information (not used) . . . 97
F.8 Future scenarios . . . 98
F.8.1 Used . . . 98
F.8.2 Not used . . . 98
F.9 Summary . . . 99
List of Figures
2.1 The 15 areas of the current North European power system (2015), with
transmissions (blue) and imports (red) . . . 9
2.2 Shares of maximum generation capacities in Northern Europe (%) . . . . 15
2.3 Maximum generation capacities and demand in Northern Europe (MW) . 16 3.1 Principle of both optimal power flow optimization programs . . . 22
3.2 Principle of a Monte Carlo simulation . . . 26
3.3 Iterations of an Importance Sampling algorithm until the threshold is reached . . . 34
3.4 Evolution of the PDF of the demand in SE3 (MW) with the iterations of an importance sampling . . . 41
3.5 Evolution of the PDF of the wind power capacity in SE2 (MW) with the iterations of an importance sampling . . . 41
3.6 Comparison between a crude MCS and an MCS with pre-simulation (IS). 43 A.1 CDF of the demand in Sweden and Finland . . . 65
A.2 CDF of the demand in Norway . . . 66
A.3 CDF of the demand in Denmark . . . 66
A.4 CDF of the demand in the Baltic countries . . . 67
B.1 CDF of wind power capacity in Sweden and Finland . . . 69
B.2 CDF of wind power capacity in Norway . . . 70
B.3 CDF of wind power capacity in Denmark . . . 70
B.4 CDF of wind power capacity in the Baltic countries . . . 71
D.1 CDF of the transmission lines in Norway (1/2) . . . 89
D.2 CDF of the transmission lines in Norway (2/2) . . . 90
D.3 CDF of the transmission lines in Baltic countries . . . 90
D.4 CDF of the transmission lines in and with Denmark . . . 91
D.5 CDF of the transmission lines connected with Finland . . . 91
D.6 CDF of the transmission lines in Sweden . . . 92
D.7 CDF of the transmission lines between Sweden and Norway . . . 92
List of Tables
4.1 Reliability evaluation of the current power system (2015) . . . 53
4.2 Reliability evaluation of the power system model for 2020 . . . 55
4.3 Reliability evaluation of the power system model for 2025 . . . 57
4.4 Reliability evaluation of the power system model for 2030 . . . 58
A.1 Maximum demand in the different scenarios (MW) . . . 68
A.2 Matrix of linear coefficient parameters for the demand . . . 68
B.1 Installed wind power capacity in the different scenarios (MW). . . 72
B.2 Matrix of linear coefficient parameters for wind power generation . . . 72
C.1 Power plants in Sweden (1/6) . . . 73
C.2 Power plants in Sweden (2/6) . . . 74
C.3 Power plants in Sweden (3/6) . . . 75
C.4 Power plants in Sweden (4/6) . . . 76
C.5 Power plants in Sweden (5/6) . . . 77
C.6 Power plants in Sweden (6/6) . . . 78
C.7 Power plants in Finland (1/2) . . . 79
C.8 Power plants in Finland (2/2) . . . 80
C.9 Power plants in Norway (1/5) . . . 81
C.10 Power plants in Norway (2/5) . . . 82
C.11 Power plants in Norway (3/5) . . . 83
C.12 Power plants in Norway (4/5) . . . 84
C.13 Power plants in Norway (5/5) . . . 85
C.14 Power plants in Denmark . . . 86
C.15 Power plants in the Baltic countries . . . 87
C.16 Probabilities of the power plants being unavailable depending on the type of source . . . 87
C.17 Installed and expectedly available generating capacities (MW), without wind power . . . 88
E.1 Import capacities into the North European power system . . . 93
F.1 Publicly available sources: current system . . . 99
F.2 Publicly available sources: future evolutions of the system . . . 99
List of Algorithms
1 Crude Monte Carlo simulation algorithm . . . 30
2 Pre-simulation algorithm using importance sampling . . . 44
3 Main Monte Carlo simulation using results from the pre-simulation . . . . 47
Abbreviations
Countries: DK Denmark EE Estonia FI Finland LT Lithuania LV Latvia NO Norway SE Sweden Others:CDF Cumulative Distribution Function CE Cross-Entropy
CV Coefficient of Variation EPNS Expected Power Not Served IS Importance Sampling
LOLO Loss Of Load Occasion LOLP Loss Of Load Probability LR Likelihood Ratio
MCS Monte Carlo Simulation NTC Net Transfer Capacity PDF Probability Density Function PNS Power Not Served
PTDF Power Transfer Distribution Factor
Symbols
Indices:
a index of areas g index of generators
i index of power system samples
k index of the iteration in the pre-simulation algorithm t index of transmission lines
Indicators of resources: used for describing the variables, parameters and functions L the corresponding element is related to a load
V the corresponding element is related to wind power T the corresponding element is related to transmission Z generic indicator that replaces L, V and/or T
G the corresponding element is related to conventional power
Sets: T−
a set of transmission lines entering area a
T+
a set of transmission lines leaving area a
XZ
a set of possible levels for load/wind power generation, area a
XT
t set of possible levels for transmission capacity, line t
Variables:
da demand served in area a
pa power generated in area a
ft flow on transmission line t
Xi sample i of the power system
Symbols xii
XiL= (xLi,a)1≤a≤nA nA× 1 vector of load levels, sample i
XiG= (xGi,g)1≤g≤nG nG× 1 vector of conventional generator statuses, sample i
XiV = (xVi,a)1≤a≤nA nA× 1 vector of wind power generation statuses, sample i
XiT = (xTi,t)1≤t≤nT nT × 1 vector of transmission line statuses, sample i
κ load excess variable ˆ
φk threshold defining ”failure” states at iteration k
Parameters:
da demand in area a
pa available generation in area a ft upper transmission limit on line t nA number of areas
nG number of conventional generators
yg number of identical units in generator g
nT number of transmission lines
NaZ number of levels for load or wind power generation, area a NT
t number of levels for transmission capacity, line t
N generic number of samples
NIS number of samples for pre-simulation iteration (IS)
NM CS number of samples for main simulation (MCS)
M a large number
α pre-simulation smoothing parameter β coefficient of variation
ρ pre-simulation share of highest performing states Sρ (1-ρ)-quantile of the performances
φ optimal threshold defining ”failure” states
uG = (uGg)1≤g≤nG original probability that unit g is unavailable
vG = (vgG)1≤g≤nG optimal probability that unit g is unavailable
µZ = (µZa)1≤a≤nA reference parameter, mean in each area a, resource Z
σZ = (σZa)1≤a≤nA ref. parameter, standard deviation in each area a, resource Z
µT = (µTt)1≤t≤nT reference parameter for transmission, mean on line t
Symbols xiii
Functions:
S(.) state performance function
S’(.) ranked state performance function
H(.) LOLO
Hk(.) load shedding indicator function, iteration k
WZ(.) partial likelihood ratio for resource Z
WG(.) partial likelihood ratio for the conventional generators W(.) likelihood ratio
fZ(.) PDF of the original probabilities of each level cZ(.) PDF of the optimal probabilities of each level
fnorm(.|µ, σ) normal PDF, defined by mean µ and standard deviation σ
Estimates:
Wi,kZ estimate of WZ(.) for sample i and iteration k WG
i,k estimate of WG(.) for sample i and iteration k
Wi,k estimate of W(.) for sample i and iteration k
b cZ k estimate of cZ at iteration k b vGk = (bvg,kG )1≤g≤nG estimate of v G at iteration k b µZ k = (µb Z
a,k)1≤a≤nA estimate of µ
Z at iteration k, area a
b
σkZ = (σbZa,k)1≤a≤nA estimate of σ
Chapter 1
Literary Review / Introduction
1.1
Background
The North European power system (Sweden, Finland, Norway, Denmark, Estonia, Latvia, Lithuania) is facing changes in terms of electricity production. Due to environ-mental reasons, the share of intermittent power sources is increasing. The integration of intermittent sources such as wind power constitutes a challenge. Indeed, the production capacities of such sources are highly dependent on weather-related and quickly varying parameters. This makes their availability much less predictable than the ones of con-ventional power sources. Besides, the decommissioning of some large power plants is planned for market (nuclear) or environmental (coal) reasons. As a consequence, the power system is weakened. Meanwhile, the electricity demand is forecasted to increase, which contributes to make the system less reliable. The reliability of a power system is characterized by the occurrences of load shedding situations, that is, ”at least one consumer is involuntarily disconnected due to capacity limitations in the system”, as defined in the compendium [1]. To compensate this phenomenon, the system operator and the producers are investing in new interconnections and power plants. Thus, the reliability evaluation of the power system becomes necessary to determine the location and the type of the most essential investments.
The reliability of a power system can be characterized by two main indices: the risk and the size of a power deficit. They are also defined respectively as Loss Of Load Probability (LOLP) and Expected Power Not Served (EPNS) in the compendium [1].
Chapter 1. Literary Review / Introduction 2
These elements have to be evaluated with a satisfying confidence level, which can be characterized by the coefficient of the variation of the indices.
This Master Thesis aims at evaluating the reliability indices in the North European power system in order to identify its weakest areas, using the Monte Carlo method. This method is frequently used for power system reliability evaluation, as in papers [2], [3] and [4].
The principle of the Monte Carlo method is presented in the compendium [5]: instead of computing analytically the value of the reliability indices, this method aims at estimating them through simulations. It follows three major steps, presented in document [2]:
• generate a given number of samples for the study,
• analyze the performance of the samples,
• use the results of the second step to estimate the reliability indices.
Chapter 1. Literary Review / Introduction 3
For the second step, a power flow optimization program is run, in order to simulate the samples, that is, to know how much demand can be served. Several programs are possible and some of them are defined in this document (maximum and optimal power flow, in two versions), inspired from paper [6]. The programs differ on the objective function, that is to say the function to optimize, but are all constrained by the maximum available generation and transmission capacities and the load balance constraint, which corresponds to the fact that in any area, at any moment, the sum of the imported and generated electricity is equal to the sum of the consumed and exported electricity. Then, the performances are computed and analyzed by identifying the occurrences of load shedding situations and the amount of unserved power. Finally, the results are used for estimating the reliability indices and the confidence level of the results.
Chapter 1. Literary Review / Introduction 4
1.2
Goals and objectives
This Master Thesis aims at fulfilling the following objectives:
• Model the North European power system:
Use publicly available data to set up a model of the current power system, in order to generate samples of the system.
• Create a reliability test:
Implement a reliability test, based on a Monte Carlo simulation, with an impor-tance sampling pre-simulation beforehand (in Matlab). Include the possibility to simulate dependence between the demand or wind power generation capacity in different areas.
• Evaluate the reliability of the power system and find its weakest areas:
Use the reliability test to evaluate the risk and the size of a power deficit with a sufficient confidence level. Identify the weakest areas.
• Elaborate and simulate possible future evolutions of the system:
Gather publicly available data about the possible future evolutions of the system (decommissioning of power plants, increased capacity of interconnections or wind power, increased demand). Use the most relevant ones to modify the model of the system. Run the reliability test to identify the future weakest areas.
1.3
Delimitation of the study
The problem has to be limited in order to be able to carry out the study in the time period allocated to the Master Thesis. The limitations are the following:
• Geographical limitations:
Chapter 1. Literary Review / Introduction 5
• Transmission lines:
All transmission lines between two areas are represented through one equivalent interconnection, having a total capacity equal to the sum of all the capacities of the physical lines. Besides, the line reactances are not considered in the study.
• Losses:
No additional losses are considered in the study: the losses taken into account are included in the demand or the capacity limitations.
• Origin of the data:
All data used for the study has to be publicly available. The sources are presented in appendixF.
1.4
Roadmap
The report presents the study described in the introduction. It starts with the model of the system in chapter 2. The requirements of the model are presented in section 2.1. The elaboration of the model is explained in the other sections of chapter 2. The appendicesA,B,C,DandEpresent the results of the model. The appendixFlists the sources where public data can be found.
The implementation of the reliability test is presented in chapter 3. The theory be-hind the different concepts is presented in the first sections of the chapter. The test is summarized in the algorithm at the end of the chapter, in section 3.5.
The simulations are shown in chapter4. Section4.1presents the results of the simulation on the current power system, while section 4.2examines possible future scenarios.
Chapter 2
The North European Power
System
The modeling of the North European power system is presented in this chapter. Sec-tion 2.1 contains information about the requirements and the general method used to model the system. Section2.2 explains the subdivision of the whole system into areas. The other sections contain the detailed description of the modeling of demand, power generation, interconnections and imports.
2.1
Modeling a state of the system
2.1.1 General description of the system
The power systems studied in this document are composed of the power systems of several areas, so to say interconnected geographical zones. These areas are located in Northern Europe (defined in section 2.2). The areas are also connected to the power system of regions outside the studied system; they can import electricity from these zones.
Each area is characterized by various possible values for the power demand and the power generating capacity. Besides, each interconnection between areas can have dif-ferent possible available capacities. Finally, there are several possible values for the
Chapter 2. The North European Power System 7
maximum capacity that can be imported into the considered system, from the neigh-bouring countries.
The characteristic elements of the power system can thus be summarized by the following list:
• load values in each area (section 2.3),
• power generating capacity in each area: wind power generation capacity (section 2.4.1) and conventional power sources (section2.4.2),
• transmission capacity of each interconnection (section2.5),
• import capacity into each area (section2.6).
Considering their quick variability, most of these characteristic elements (load, wind power generation capacity, interconnection capacity and import capacity) can be as-signed a value for each hour of the year. In this study, such a value is defined as a level (load level, wind power generation capacity level, transmission level, import level). For a given characteristic element and a given area (or interconnection), all the levels of the corresponding set of values in one year can be ranked from the smallest to the highest and assigned a probability, equal to the percentage of their occurrence in the set. These probabilities can be represented through a curve, with the corresponding levels on the x-axis.
Another method is used to characterize the available generation capacity from conven-tional power sources. It is defined later in section 2.4.2. Indeed, it is not relevant to consider hourly variations of the capacity of these sources because their capacity is most of the time equal to their maximum capacity, and zero otherwise, in a few cases. This two state behaviour (zero or maximum capacity) is characterized only by one probability value for each generating unit of these power sources, corresponding to the share of the time the unit is offline. Moreover, some power plants are not considered individually through a probability value because their installed capacity is too small to have any influence on the reliability of the system. This is explained in section2.4.2.2.
Chapter 2. The North European Power System 8
the system reliability, considering the low share of solar power in the North European power system.
2.1.2 Generation of a state of the system
The aim of this study is to generate states (or samples) of the power system. A state (or sample) of the system is defined by the demand and available generating capacity (value of the wind power generating capacity, availability of the plants of conventional sources) in each area, as well as the available transmission capacity between areas and the level of available power import in each area. A state of the system is generated by picking randomly (following the probabilities) one level out of each curve of demand, transmission capacity and wind power generation capacity for each area or each inter-connection, as well as the available capacity of each conventional power plant. To sum up, a state of a system is obtained by randomly assigning a value (within the range of possible values) to every characteristic element for each area and interconnection of the system.
2.2
Areas
The North European power system is composed of the power systems of Norway (NO), Sweden (SE), Finland (FI), Denmark (DK), Estonia (EE), Latvia (LV) and Lithuania (LT). It has interactions with several neighbouring countries (the Netherlands, Germany, Poland, Belarus and Russia for the time being), presented in figure2.1. Interconnections with the United Kingdom are also considered for the future evolution of the power system.
Chapter 2. The North European Power System 9
From http://www.nordpoolspot.com/How-does-it-work/Bidding-areas/, modified.
Figure 2.1: The 15 areas of the current North European power system (2015), with transmissions (blue) and imports (red)
Thus, the study is done following the bidding areas, which consist of 15 interconnected zones (NO1 to NO5 in Norway, SE1 to SE4 in Sweden, DK1 and DK2 in Denmark, FI in Finland, EE in Estonia, LV in Latvia and LT in Lithuania) represented in figure 2.1. These zones are interconnected through various transmission lines, as explained in section 2.5.
Chapter 2. The North European Power System 10
areas in SE, NO, DK and FI is studied (the Baltic countries have little effect on the power system reliability). The studied system is not assumed to be isolated.
2.3
Demand
2.3.1 Model of the demand
In order to generate a state of the demand in an area, a probability has to be assigned to each demand level.
The demand levels are gathered from publicly available data on the actual hourly demand level over a past time period. This data can be converted into a Probability Density Function (PDF) or Cumulative Distribution Function (CDF, integral of PDF). The occurrence of each load level is counted and the result is normalized, so to say divided by the total number of hours for which data is used, in order to get the sum of all the values equal to one. These values can be considered as the probability that the demand is equal to the corresponding load level.
It can be noticed that it is necessary to use data about an entire number of time period (year, season, month) so that no particular importance is given to any season of the year, when the demand would be higher or lower. In this report, data on exactly one winter (January, February, March, November and December) is used for each simulation. Indeed, winter is the most critical period especially because the demand is usually higher during the concerned months.
The CDF of the demand in each area are given in appendix A. The sources where the data is found are presented in appendixF.
2.3.2 Variables and parameters
Chapter 2. The North European Power System 11
nAis the number of areas in the system. The served load in the area is written da, while
the demand in the area is da.
2.4
Production
Depending on the power sources, different types of data are required to generate a state of electricity production in a power system. The sources with quick variations of maximum capacity should be represented through hourly data (wind power), while an alternative representation can be chosen for the other sources.
2.4.1 Intermittent power sources (wind power)
The only intermittent power source treated in this study is wind power. Available wind power generation capacity varies on a fast pace because it depends on meteorological conditions, which can change a lot from one hour to another. Thus, due to this sensi-tivity, wind power generation is represented in this study through a CDF similarly to the demand, described in section 2.3.
To sum up, the characterization of wind power generation capacity requires in each area various possible capacity levels with assigned probabilities.
The CDFs of the wind power generation are given in appendix B. The sources where the data is found are presented in appendixF.
As mentioned in section 2.1, it can be noticed that solar power could be treated the same way as wind power, but its variable behaviour is not taken into account because it has only a very small effect on the system reliability, considering its low share.
2.4.2 Conventional power sources
Chapter 2. The North European Power System 12
single probability value for each generating unit of these power sources, corresponding to the share of the time the unit is offline.
Various power plants are modeled through this single probability number. They are treated in section 2.4.2.1. Some other plants are not considered individually through a probability value but their maximum capacities are summed. This is detailed in section 2.4.2.2.
In this study, a power generating unit is defined as an individual element that can generate power. It is characterized by its capacity and the two-state behaviour described above. A group of identical generating units is referred to as a power generating block. If no unit is identical to a given unit, this unit is a block in itself. A power plant is composed of one or several power generating blocks. It can be noticed that in reality, a power block can be composed of several non-identical units, but this possibility is not considered in this study for the sake of simplification. These different units are thus considered as different blocks.
2.4.2.1 Blocks considered individually
The power generating blocks that are considered individually are the ones that fulfill two conditions: enough data is available to characterize them, and their generating capacity is high enough to be relevant (in this study, the limit is set to 20 MW). The second criterion is related to the fact that the outage of a smaller unit has less effect on the whole power system reliability than the outage of a bigger one. Simulating an excessively high number of units individually turns out to be memory- and time-consuming.
The required data for each power generating block can be summarized through a table containing the following elements:
• area where the block is located,
• number of identical generating units in the block,
• generating capacity of a unit,
Chapter 2. The North European Power System 13
The data about the location, the number of units and the capacity of a unit are easy to find because they are objectively defined. The probability that a unit is offline is more difficult to determine, because it has estimated through studies and measurements. One possibility to estimate the probabilities is to gather data about the unavailabilities of the units over a time period, and compute the ratio between the time the unit was unavailable and the length of the considered period. However, this method can lead to unsatisfactory results because data is not necessarily available for all the units and all the events. The alternative chosen in this document is the use of availability values gathered in other studies (collected in [12]), and modified to fit with other constraints (see chapter 4).
The data about all power blocks considered individually is given in appendix C. The sources where the data is found are presented in appendixF.
2.4.2.2 Share of grouped capacities always available on average
In each area, the units that do not fulfill one of the two or both conditions mentioned above (enough data has to be available and high enough generating capacity) are gath-ered into a group of units. The capacity of the group is equal to the sum of the capacities of all these units. On average, a share of the capacity of this group is always unavail-able, while the rest of the capacity is always available. This capacity is referred to as the share of grouped capacities always available on average. Considering its relatively low share in the North European power systems, solar power generation ca-pacity is not considered the same way as wind power generation caca-pacity in this study; its maximum capacity is added to the share of grouped capacities always available on average.
In practice, the share of generation that is not considered as individual units is computed as the difference between the total generation capacity of an area (except wind power generation capacity) and the sum of the capacity of all the units considered individually. The share of generation capacity that is available has to be assigned an estimated value (see chapter 4for the explanation).
Chapter 2. The North European Power System 14
The sources where the data about the maximum capacity in each area is found are presented in appendixF.
2.4.2.3 Maximum available hydro power
As explained in document [13] (page 39), the maximum available hydro power is lower than the total available capacity because of several restrictions, such as low reservoir levels, shutdowns, limitations due to icing and some legal regulations. These restrictions should be represented in the model, by avoiding the samples with an excessively high hydro power generation availability. In practice, this constraint is dealt with in chapter 4.
2.4.3 Variables and parameters
2.4.3.1 Wind power
In this document, the variables and parameters indicated with the superscript V refer to the wind power. XV
a corresponds to the set of possible levels for wind power generation
capacity in area a. The PDF of the original values for wind power is represented by the function fV(.). For a given sample i, XV
i = (xVi,a)1≤a≤nA is the nA× 1 vector of wind
power capacity statuses (share of installed capacity).
2.4.3.2 Other generation
The variables and parameters indicated with the superscript G refer to conventional power generation. The original probability that a unit of block g is available is uG = (uG
g)1≤g≤nG, where nG is the number of blocks in the system. For a given sample i,
XiG = (xGi,g)1≤g≤nG is the nG× 1 vector of generator statuses. The power generated in
the area a (including wind power) is written pa, while the available generation capacity
Chapter 2. The North European Power System 15
2.4.4 Overview of power production
The share of wind power production, as well as the share of plants from conventional sources considered individually, and the share of the remaining generation capacity (from which only a percentage is always available) are presented in figure2.2. The maximum values (in MW) are available in appendixC.
0
20
40
60
80
100
LT
LV
EE
DK2
DK1
NO5
NO4
NO3
NO2
NO1
FI
SE4
SE3
SE2
SE1
Wind
Individual
Fix
Figure 2.2: Shares of maximum generation capacities in Northern Europe (%)
2.4.5 Overview of the demand in the North European power system
Chapter 2. The North European Power System 16
0
0.5
1
1.5
2
·10
4LT
LV
EE
DK2
DK1
NO5
NO4
NO3
NO2
NO1
FI
SE4
SE3
SE2
SE1
Max generation capacity (MW)
Maximum demand (MW)
Figure 2.3: Maximum generation capacities and demand in Northern Europe (MW)
2.5
Transmission inside the power system
2.5.1 Model
An interconnection from one area to another is modeled through a single line that represents the total transmission capacity through the boundary, even though most of the areas are interconnected in reality with various smaller transmission lines. The capacities of the smaller lines are summed to give the capacity of the single equivalent line.
Chapter 2. The North European Power System 17
Each transmission line is defined as being directed, which means that it can carry electric-ity from one area to another, but not in the opposite direction. Thus, the interconnection between two areas is defined by two lines, one in each direction.
In this study, the losses of the transmission lines and the line reactances are not consid-ered.
To sum up, the characterization of transmission capacity requires in each area various capacity levels with assigned probabilities.
The data about all transmission lines is given in appendix D. The interconnections are represented in blue in figure 2.1. The sources where the data is found are presented in appendix F.
2.5.2 Variables and parameters
In this document, the variables and parameters indicated with the superscript T refer to the transmission capacities. XT
t corresponds to the set of possible levels for transmission
capacity of line t. The PDF of the original values for transmission capacity is represented by the function fT(.). For a given sample i, XT
i = (xTi,t)1≤t≤nT is the nT × 1 vector of
transmission capacity statuses (share of installed capacity), where nT is the number of
directed transmission lines. The flow on transmission line t is written ft, while the upper
transmission limit on line t is ft.
2.6
Imports from outside the system
The imports are defined by all the transmission lines that enter one of the areas of the power system, coming from a zone that is outside the system.
Chapter 2. The North European Power System 18
However, for some transmission lines, only a very few values of transmission capacity levels are publicly available. Then, the import can be represented as an additional generator, always available, located in the area where the import is done, and treated the same way as the share of grouped capacity always available on average (section 2.4.2.2), with 100% availability.
Chapter 3
Reliability Evaluation Model
The modeling of the power system and the generation of a state were presented in chapter 2. In chapter 3, the model is used to generate a large number of states, in order to analyze their performances (section 3.1) and determine the reliability of the power system through a Monte Carlo simulation (section 3.2). Section 3.3 presents the importance sampling, that makes the reliability assessment more efficient. The dependence between demand or wind power in the different areas has to be taken into account in the model, but it requires some changes in the algorithm explained in section 3.4. The main algorithm is summarized in section3.5.
3.1
Model of the problem - optimization problem
As described in Chapter2, the power system to be modeled is composed of demand and generation availability in different areas, as well as by transmission capacity between the areas. The line reactances are not considered in the study. Two different methods (maximum power flow in3.1.1and optimal power flow in3.1.2) are explored to evaluate the power flow in the system, which is a necessary step to carry out the reliability assessment (section 3.2.4).
Given a state of the system (demand levels, generation and transmission capacities), an optimization problem has to be solved to determine whether all the demand can be served or not, which is a necessary step for the analysis of the performance of the state.
Chapter 3. Reliability Evaluation Model 20
3.1.1 Maximum power flow
The first approach to solve the optimization problem aims at computing the maximum power that can be transmitted from the generating units to the consuming units. It involves the use of a directed graph (digraph in Matlab) in order to run a maximum flow problem, such as maxflow in Matlab.
To create such a graph, 2 × (nA+ 1) nodes are required, where nAis the number of areas
in the power system. These nodes can be divided in two sets of nAnodes. The last two
nodes are designated as respectively source and sink. It can be noticed that the function used in Matlab requires two nodes for one area because it does not accept graphs with directed connections in both directions between two nodes. Only one connection in one direction can be represented if only one node is used. The nodes follow the rules below:
• each area of the power system is represented by two nodes: one from the first set and one from the second set; the source and the sink do not represent any area,
• the nAnodes from the first set are all connected to the source, the weights on the
nAedges are equal to the generation availability of each area,
• the nA nodes from the second set are all connected to the sink, the weights on the nAedges are equal to demand in each area,
• each node from the first set is connected to the node from the second set that corresponds to the same area in the power system; the weights of the edges are in-finite (represented by a very high number) and reflect that there is no transmission limitation between the generation and the consumption within the same area,
• the nodes from the first set are connected to the nodes from the second set accord-ing to the interconnections between areas in the power system, the weight of the edges are equal to the transmission capacity from the first area to the second. As the edges are oriented, the interconnection in both directions in the power system is reflected by two edges in the directed graph.
Chapter 3. Reliability Evaluation Model 21
availability constraints between the source and the first set of nodes, then through the transmission constraints between the two sets, and finally through the required demand between the second set and the sink.
The maximum flow optimization is sufficient to determine whether a state of the system leads to a failure or not. However, it does not provide more information, in particular about how much the load can be increased, which would require to remove the constraints on the demand and to link the evolution of the loads. The optimal power flow proves to be more practical to that extent. Thus, only optimal power flow is used in the study.
3.1.2 Optimal power flow
The optimization program used to determine the optimal power flow in the power system is based on paper [6], and slightly modified. It uses a linear DC power flow model to represent the system.
3.1.2.1 DC power flow
The DC power flow method is designed to deal with power flow in meshed grids, where the power flow can follow several paths. It tends to prioritize some paths, based on the line reactance, representing the capacity of a line to transmit power. Even though it is run on AC grids, it is called DC power flow because the equations are close to the ones used for DC grids. It relates the power production in the different areas with the power transmission between the nodes by computing the matrix of the Power Transfer Distribution Factors (PTDF). The method is detailed in [14]. In this study however, line reactances are not taken into account, contrarily to paper [6]. Thus DC power flow is not used in this study, which is one of the main differences with paper [6].
3.1.2.2 Principle
Chapter 3. Reliability Evaluation Model 22
reach is zero, so to say all demand is served. The second program, called optimization program with load excess, aims at maximizing the difference between the highest possible demand level (beyond the current demand level) before reaching load shedding and the current demand level. This difference is negative when all demand cannot be served. These elements are represented graphically in figure 3.1 (simple optimization program on the left, optimization program with load excess on the right).
d to minimize d 0 0 d to maximize d 0 -d
Simple optimization program Program with load excess
Figure 3.1: Principle of both optimal power flow optimization programs
Among the constraints of the optimization programs, both are subject to transmission limitations. In this document, the transmissions are directed, which means that the power can flow only from the beginning to the end of the transmission line, but not in the other direction (another transmission has to be designed to represent the other direction). Then, the value of the flow ft on the transmission line t has to be greater
than or equal to zero. The upper limit of the flow on the line depends on the status of the line, which is represented by XtT, equal to the share of the installed transmission capacity ft that is available in the studied state of the system.
Chapter 3. Reliability Evaluation Model 23
on one side, and the sum of the power consumed (da) and the flows leaving the area
(ft∈ Ta+) on the other side.
The power generated in each area needs to be greater than or equal to zero, and is constrained by the available generation pa in area a.
Finally, the load served in each area has to constrained by the demand da. However, this
constraint is only present in the simple optimization program. In the second program, the served load must not be constrained, as it has to reach the highest possible demand level beyond the current level. Instead, this constraint is replaced by the definition of a load excess variable, which is equal to the served load divided by the current demand.
3.1.2.3 Simple optimization program
First is presented the simple optimization program that only tells whether all the load is served or not (equations 3.1). As explained above, the aim is to minimize the objective function, which is the difference between the total demand (sum of the demand da in
each area) and the total served load (sum of the served load da in each area). Thus,
there is no load shedding if the value of the objective function is 0.
The simple optimization program is summarized by the equations3.1.
min ft,da,pa nA X a=1 da− nA X a=1 da= nA X a=1 (da− da) s.t. 0 ≤ ft≤ ftXtT ∀t, pa+ X t∈Ta− ft= da+ X t∈Ta+ ft ∀a, 0 ≤ da≤ da ∀a, 0 ≤ pa≤ pa ∀a. (3.1)
3.1.2.4 Optimization program with load excess
Chapter 3. Reliability Evaluation Model 24
the original demand before reaching a failure state. It creates a proportionality between the demand and the served load in each area. The equations of the corresponding optimization problem is close to equations3.1, with two changes, justified above:
• the objective function has now to be maximized, it is equal to the excess load served in the system beyond the given demand,
• the served load is not limited by the given demand anymore, but equal to the given demand multiplied by the load excess variable.
The optimization program is summarized by the equations3.2.
max ft,da,pa,κ κ nA X a=1 da− nA X a=1 da= (κ − 1) nA X a=1 da s.t. 0 ≤ ft≤ ftXtT ∀t, pa+ X t∈Ta− ft= da+ X t∈Ta+ ft ∀a, da= κda ∀a, 0 ≤ pa≤ pa ∀a. (3.2)
When κ > 1, the value of the objective function gives how much the load can be increased without load shedding. When κ < 1, load shedding is necessary.
3.2
Monte Carlo simulation
The reliability analysis of the power systems in this study uses a Monte Carlo simulation (MCS), which implies the generation of a large number of samples of the system, in order to estimate reliability indices with enough precision. Monte Carlo simulation has been used in various papers concerning reliability evaluation of power systems, such as papers [2], [3] and [4].
Chapter 3. Reliability Evaluation Model 25
3.2.1 Concept and goal
Monte Carlo methods are a class of methods designed to solve problems using random samples, but also to solve deterministic problems, by observing the behaviour of a ran-dom variable in a sufficiently large number of samples, as explained in compendium [5]. The expectation value of the random variable is estimated through a limited number of samples, instead of being computed mathematically. The choice of the number of samples is a key element in Monte Carlo simulation, because it determines the precision of the estimate of the random variable. The precision can be measured through the coefficient of variation, as explained in subsection 3.2.4.3. In section 3.3, a method to decrease quickly the coefficient of variation is explained. When the Monte Carlo sim-ulation is run without using this method, it is often referred to as crude Monte Carlo simulation.
Monte Carlo methods are an example of state-space-based algorithms, as explained in paper [2], which means that they follow three major steps:
• generate a given number of samples for the study (random samples of the system, following some probabilities),
• analyze the performance of the samples (specifically in this study, it requires to determine if load shedding occurs for each sample),
• use the results of the second step to estimate the reliability indices.
The execution of these steps is explained in the following parts of the section 3.2.
Figure 3.2 represents the principle of estimating indices through a Monte Carlo sim-ulation. The three steps described above are run on a system characterized by some probabilities. The resulting estimates are supposed to be correct with some confidence level.
3.2.2 Sampling the system
Chapter 3. Reliability Evaluation Model 26 Initial probabilities Estimates with good confi-dence level MCS - generate N1 samples - analyze performances - estimate indices
Figure 3.2: Principle of a Monte Carlo simulation
transmission capacities, another for conventional power generation. The sample i of the system is represented by the vector Xi = [XiG XiV XiL XiT], where:
• XG
i is the nG× 1 vector of conventional generator statuses; the element from line
g, written xG
i,g, is equal to the number of units from generator g available in sample
i,
• XV
i is the nA× 1 vector of wind power generation statuses in each area a; the
element from line a, written xV
i,a, is equal to the share of installed wind power
generation capacity in area a that is available in sample i,
• XL
i is the nA× 1 vector of load levels in each area a; the element from line a,
written xLi,a, is equal to the share of peak demand in area a that is required in sample i,
• XT
i is the nT×1 vector of transmission line statuses in each line t; the element from
line t, written xTi,t, is equal to the share of the installed transmission capacity in line t that is available in sample i (each physical transmission line has two elements in this vector since the lines are directed in the model).
3.2.2.1 Load, wind and transmission
Chapter 3. Reliability Evaluation Model 27
The choice of the level is made following the CDF of the probabilities. A random number between 0 and 1 is generated on Matlab for each set of levels, which means that there is one random number for each area for the load levels, one for each area for the wind power generation and one for each directed transmission line. The level is chosen by comparing the random number to the CDF of the distribution of the levels. The random numbers are supposed to be independent. However, a dependence can be introduced between the areas, which makes the model more realistic. The possibility to choose the degree of dependence can prove to be desirable. The use of the copula function is one way of introducing this dependence, and is explained in the section 3.4.1.2.
3.2.2.2 Power generation from conventional sources
For the available generation of conventional sources, there is no duration curve available, because the units of the power generating blocks are usually only fully available or fully disconnected, following a two-state behaviour. Then, only the probability of the unit being fully disconnected is required. Thus, for each power generating unit, a random number between 0 and 1 is generated and compared to the probability the unit is dis-connected. In the case of a power generating block having several identical units, one random number is generated for each unit, and the available capacities are summed.
Finally, there is some share of generation that is always available. Its value is added to the wind power generation availability and the generation capacity of the available conventional power units in order to get the total available generation capacity in each area.
3.2.3 Performance of the samples of the system
Chapter 3. Reliability Evaluation Model 28
3.2.3.1 Power Not Served (PNS)
The Power Not Served (PNS) is the power that cannot be delivered due to capacity limitations in the system. It is equal for each sample of the system to the difference between the original demand and the served load if maximum power flow (or optimal power flow with a load excess variable in case of load shedding, zero otherwise) is run, or to the value of the objective function in the simple optimal power flow (without load excess variable).
3.2.3.2 Loss of Load Occasion (LOLO)
The Loss of Load Occasion (LOLO), also known as power deficit, is a binary variable equal to one if load shedding occurs, which means that ”at least one consumer is in-voluntarily disconnected due to capacity limitations in the system” (from compendium [1]), and zero otherwise.
It can be computed from the PNS: the LOLO is equal to one if the PNS is different from zero, and zero otherwise.
3.2.4 Reliability indices
To evaluate the reliability of a power system, two indices in particular are computed from the PNS and the LOLO. They are described in compendium [1].
3.2.4.1 Expected Power Not Served (EPNS)
The Expected Power Not Served (EPNS) is the expected value of the PNS. The EPNS indicates the size of the power deficit that can be expected.
The EPNS is estimated from the results of a MCS as the sum of the PNS for each sample, divided by the number of samples N used for the MCS:
Chapter 3. Reliability Evaluation Model 29
3.2.4.2 Loss Of Load Probability (LOLP)
The Loss Of Load Probability (LOLP), also known as risk of power deficit, is equal to the probability that ”at least one consumer is involuntarily disconnected due to capacity limitations in the system” (from compendium [1]). It is the expected value of the LOLO. The LOLP is an indicator of the probability of the deficit situations.
The LOLP is estimated, from the results of a MCS, as the sum of the LOLO for each sample, divided by the number of samples N used for the MCS:
\ LOLP = 1 N N X i=1 LOLO(i). (3.4) 3.2.4.3 Coefficient of variation
The coefficient of variation (CV), also known as relative standard deviation, measures the dispersion of a frequency distribution. It is used to determine how much the estimates of the reliability indices can be trusted. It is equal to the ratio between the standard deviation σ and the mean µ of the population:
CV = σ
µ. (3.5)
It can be estimated through the estimates of the standard deviation σ and the studied quantity x (where N is the number of samples of the system and xithe quantity measured
for sample i):
Chapter 3. Reliability Evaluation Model 30
The coefficient of variation is usually associated to the reliability indices because it gives them a supplementary value through showing how likely it is that the estimated indices are close to the true values.
3.2.5 MCS algorithm
With all the elements described in section 3.2, the crude Monte Carlo simulation is summarized by the algorithm 1.
while CV too high do
1. for i from 1 to NM CS do
• Generate a random state Xi as explained in section3.2.2, with the probabilities uG and fZ given in the model built on publicly available data.
• Run the optimization program by solving equations3.1.
• Evaluate the PNS and the LOLO for state i as described in sections3.2.3.1 and3.2.3.2.
end
2. Estimate the EPNS, LOLP and CV with all the states generated since the beginning by using equations 3.3,3.4, and3.6.
end
Algorithm 1: Crude Monte Carlo simulation algorithm
Comments on the algorithm This algorithm is run while the coefficient of variation is not small enough (the maximum acceptable coefficient of variation has to be defined beforehand, typically around 2%). NM CS new samples are generated every time, and
Chapter 3. Reliability Evaluation Model 31
3.3
Importance sampling
Most of the equations of this section are adapted from paper [6].
3.3.1 Concept and goal
A crude MCS often turns out to be time consuming for getting a low value for the co-efficient of variation. In particular, the reliability evaluation of power systems requires a large number of samples because load shedding is a rare event, with a very low prob-ability of occurrence (for example, around 10−5). Several variance reduction techniques are developed to get similar results more efficiently. Some of them are described in compendium [5].
Importance sampling (IS) is the variance reduction technique used in this study. This technique is very frequent in reliability evaluation of power systems, such as in papers [2], [3], [4] and [6]. It aims at making the occurrence of the rare events more frequent by modifying the probability distributions in a specific way. In this study, occurrences of load shedding are made more frequent by increasing the probability of high demand and decreasing the probability of generation availability. This process is achieved by evaluating the performance of the samples in order to rank them, and update the prob-ability distributions according to the most favorable samples, which are the ones with load shedding or with the smaller margin before load shedding.
Importance sampling is based on the theory of cross-entropy (CE), which is convenient in the study of rare events. The main idea behind the theory is to modify the proba-bility density of some elements of the study (demand, generation availaproba-bility) in order to evaluate the required parameters (reliability indices) with an unbiased estimator. However, the optimal probability density for estimating the parameters is impossible to get, because it depends on the value of these parameters, which are unknown. Thus, the CE method aims at estimating a reference parameter vector v by minimizing the cross-entropy (or Kullback–Leibler divergence) between the optimal probability density g∗ and the original density following the reference parameter vector f(.;v):
D(g∗, f ) = Z
g∗· ln(g∗(x))dx − Z
Chapter 3. Reliability Evaluation Model 32
The theory of CE is more detailed in document [7].
The algorithm used for IS is run on several iterations as a pre-simulation, until the optimal probabilities to run the MCS are found.
3.3.2 Evaluation of the performances and optimization problem
As explained in section3.1.2, the evaluation of the performances requires the introduc-tion of a load excess variable κ. The equaintroduc-tions 3.2have to be used for the optimization program.
State performance function The state performance function S is defined as the excess load served in the system beyond the given demand adn multiplied by -1 for each sample. It is equal to the opposite of the objective function in the equations 3.2:
S(i) = (1 − κ(i))
nA
X
a=1
da(i) ∀i. (3.8)
(1-ρ)-quantile of the performances The state performances from the lowest to the highest, to create the ranked state performance function S’. A share of highest performing states ρ is chosen between 0 and 1 (typically around 0.05), in order to compute the (1-ρ)-quantile of the performances Sρ, where N is the number of samples:
Sρ= S0(b(1 − ρ)N c). (3.9)
This parameter is intended to measure how much the performances of the system states are close to a load shedding situation. Its value is saved in each iteration k through the parameter ˆφk. The performances of the system states are satisfactory enough when
Sρ (recorded by assigning the value to ˆφk) is greater than or equal to φ, a threshold
Chapter 3. Reliability Evaluation Model 33
Load shedding indicator function In order to be able to select the samples having the best performances (load shedding or close to it), the load shedding indicator function Hk is defined as follows for every sample i and for each iteration k:
Hk(i) = 1 if S(i) > ˆφk 0 if S(i) ≤ ˆφk ∀i, k. (3.10)
Chapter 3. Reliability Evaluation Model 34 0 200 400 600 800 1000 Samples -2.5 -2 -1.5 -1 -0.5 0 0.5 Difference (MW) ×104
Samples of the system (iteration 1): = current demand - maximum served load
-2.5 -2 -1.5 -1 -0.5 0 0.5 Difference (MW) ×104
Samples (iteration 1) ranked following the difference: current demand - maximum served load
0 200 400 600 800 1000 Samples -2.5 -2 -1.5 -1 -0.5 0 0.5 Difference (MW) ×104
Samples of the system (iteration 2): = current demand - maximum served load
-2.5 -2 -1.5 -1 -0.5 0 0.5 Difference (MW) ×104
Samples (iteration 2) ranked following the difference: current demand - maximum served load
0 200 400 600 800 1000 Samples -2.5 -2 -1.5 -1 -0.5 0 0.5 Difference (MW) ×104
Samples of the system (iteration 3): = current demand - maximum served load
-2.5 -2 -1.5 -1 -0.5 0 0.5 Difference (MW) ×104
Samples (iteration 3) ranked following the difference: current demand - maximum served load
Figure 3.3: Iterations of an Importance Sampling algorithm until the threshold is reached
3.3.3 Likelihood ratio
Chapter 3. Reliability Evaluation Model 35
The definition of the likelihood ratios (LR) depend on the way the different elements are sampled in section3.2.2: the elements defined by a duration curve on the one hand, the elements defined only by the probability that the resource is unavailable on the other hand. We consider in this section that all elements are independent. The dependence between areas requires to use the copula function and is explained in section3.4.1.3.
3.3.3.1 Likelihood ratio for the load, wind and transmission
We define WL, WV and WT as the likelihood ratios relative respectively to the load, wind power generation and transmission line capacity. We define Z as a generic index to indicate that the equations are valid for the load (Z=L), wind power generation (Z=V) and transmission lines (Z=T). The likelihood ratios for the loads and wind power generation are N × nA matrices, where N is the number of samples of the system and
nA is the number of areas in the power system. The element of row i and column a
contains the likelihood ratio corresponding to the probability curve of load/wind power in area a, for sample i. The likelihood ratio for the transmission lines is a N × nT matrix,
where nT is the number of lines in the power system. The element of row i and column
t contains the likelihood ratio corresponding to the probability curve of transmission in area t, for sample i
For each sample i, the likelihood ratio is computed as:
WZ(XiZ|µZ, σZ) = f Z(XZ i ) cZ(XZ i |µZ, σZ) ∀i, (3.11)
where (all these elements being defined more precisely in section3.3.4), with XiZ being (for sample i) the vector of load levels if Z=L, the vector of wind generation statuses if Z=V and the vector of transmission line statuses if Z=T:
• fZ(XZ
i ) is the original PDF of the load, wind power or transmission levels,
evalu-ated with the value of state i,
• µZ and σZ are n
A× 1 reference parameter vectors for the states of the loads and
wind power generation and an nT × 1 reference parameter vector for the states
Chapter 3. Reliability Evaluation Model 36
• cZ(XZ
i |µZ, σZ) is the updated PDF of the load or wind power in each area
com-puted with the value of state i, for a given iteration k; it is the matrix of PDF of the updated probability distributions of the transmission capacities on each line.
The real values of the reference parameters are unknown. Thus, they are estimated (section3.3.4) by the vectorsµbZk andσbZk for a given type of resource Z and iteration k. In practice, the likelihood ratio is then evaluated for each sample i and iteration k as follows: Wi,kZ = WZ(XiZ|bµZk,bσ Z k) = fZ(XZ i ) b cZk(XiZ|µbZk,bσ Z k) ∀i, ∀k. (3.12)
3.3.3.2 Likelihood ratio for conventional generation
We define as WG the likelihood ratio relative to conventional power sources. It is a N × 1 matrix. Each element of the matrix is computed as follows:
WG(XiG|vG) = QnG g=1(1 − uGg)x G i,g(uG g)(yg −xG i,g) QnG g=1(1 − vGg)x G i,g(vG g)(yg −xG i,g) (3.13)
where, with XiGbeing, for sample i, the vector of generation statuses of the conventional power plants:
• uG
g is the original probability that a unit of power block g is unavailable,
• vG
g(∈ vG) is the optimal probability that a unit of power block g is unavailable; it
is a reference parameter for IS,
• yg is the number of units of generator g,
• xG
i,g is the state of generator g in sample i (number of available units in generator
g).
All these elements are defined in section3.3.4.
The real values of the optimal probabilities are unknown. Thus, they are estimated (section3.3.4) byvbG
g,k(∈bv
G
k) for a given generator g and iteration k. The likelihood ratio
Chapter 3. Reliability Evaluation Model 37 Wi,kG = WG(XiG|bvkG) = QnG g=1(1 − uGg)x G i,g(uG g)(yg −xG i,g) QnG g=1(1 −bv G g,k) xG i,g( b vG g,k) (yg−xGi,g) (3.14)
The likelihood ratio for the share of fix generation that is always available in each area is a nA× 1 vector with all elements always equal to 1.
3.3.3.3 Likelihood ratio for the system
The likelihood ratio for the whole system W is a vector (N × 1) equal to the product of all likelihood ratios, for a given sample i of the system:
W (i) = WG(XiG|vG)· nA Y a=1 WL a(XiL|µL, σL) · WaV(XiV|µV, σV) · nT Y t=1 WtT(XiT|µT, σT) ∀i, (3.15) where: • WL
a and WaV are the ath column of matrices WL and WT respectively; they
cor-respond to the likelihood ratio coefficients due to the probability distribution of load/wind power in area a, for all the samples,
• WT
t is the tth column of the matrix WT; it corresponds to the likelihood ratio
coefficients due to the probability distribution of transmission on line t, for all the samples.
The likelihood ratio is estimated for each iteration k and sample i by:
Wi,k = Wi,kG · nA Y a=1 WL i,k,a· Wi,k,aV · nT Y t=1 Wi,k,tT ∀i, ∀k, (3.16) where: • WL
i,k,a and Wi,k,aV are the columns number a of matrices Wi,kL and Wi,kV respectively,
• WT
