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Probabilistic load flow considering distributed renewable
generation
Pedro Miguel Lousa Martins Reis Rodrigues
Dissertation submitted for obtaining the degree of
Master in Electrical and Computer Engineering
Jury
President: Doutor Gil Domingos Marques
Supervisor: Doutor Rui Manuel Gameiro de Castro
Member: Doutor José Manuel Dias Ferreira de Jesus
September 2008
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To my parents
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Acknowledgements
Acknowledgements
This work had been carried out at Technical University of Catalonia (UPC) during my stay in
Barcelona, as an Erasmus student.
I would like to specially thank my supervisor Professor Rui Castro from Technical Superior
Institute (IST) of Lisbon Technical University (UTL) and my daily supervisor Professor Roberto
Villafáfila from Technical University of Catalonia (UPC) for their support and for being always there to
help me with all my questions and problems.
I would also like to thank to all the professors, in particular to Professors Ferreira de Jesus and
Maria Eduarda that along the course have contributed for widening my knowledge.
To Gonçalo Correia, my Erasmus colleague, I thank for the friendship and the discussions
around non-technical subjects. Many thanks go to those who support me with their friendship during
my stay in Barcelona. I would also like to thank all my colleagues in Portugal for a friendly and warm
environment in the course. Thanks a lot to all of you.
I wish to deeply thank my parents António and Ascensão for their support and
encouragement.
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Abstract
Abstract
Nowadays, the implementation of environmentally-friendly and uncontrollable primary energy sources
in the electrical power system production is increasing. The incorporation of high levels of small-scale,
non-dispatchable (stochastic), distributed generation (renewables, wasted heat, etc) leads to the
transition from the traditional ‘vertical’ power system structure to a ‘horizontally-operated’ power
system, where the distribution networks contain both stochastic generation and load.
These new conditions persuade the development of new modelling and design methodologies
for the investigation of this new operational power system structure, in particular the operational
uncertainty introduced by the abatement in generation dispatchability. A large number of random
variables and complex dependencies between the system inputs are involved in the analysis of this
new power system structure. In this thesis, for the power system multivariate uncertainty analysis
problem is used a Monte-Carlo Simulation (MCS) approach, based on the modelling of the one-
dimensional marginal distributions (output spectrum of each stochastic input) and the modelling of the
multidimensional stochastic dependence structure (mutual interaction between the stochastic inputs).
First, the Stochastic Bounds Methodology is applied to model clusters of positively correlated variables
(Stochastic Plants) based on the concept of perfect positive correlation (comonotonicity). The
Stochastic Bounds Methodology can also be applied to model the dependence structure between the
clusters through the definition of the extreme dependence structures that can bound all real cases.
The modelling of the exact correlations between the clusters is based on the Joint Normal Transform
Methodology.
For the application of these methodologies are used a 5-bus/7-branch test system (Hale
Network) and the IEEE 39-bus New England test system. This approach shows that the power flows
become bidirectional, the increase in the SG penetration level in the power system leads to an
increase in the variability of the power flows and at high stochastic generation penetration levels,
reverse power flows may exceed the direct ones.
List of Tables
Keywords
Stochastic Generation (SG)
Distributed Generation (DG)
Uncertainty Analysis
Probabilistic power flow analysis
Monte Carlo Simulation (MCS)
Wind Turbine Generator (WTG)
viii
List of Tables
Resumo
Nos últimos anos, o sistema eléctrico tem vindo a aumentar a utilização de fontes de energia primária
não controláveis, nomeadamente de natureza renovável, e de menor impacto negativo no ambiente.
Esta tendência ligada à incorporação de elevados níveis de geração distribuida, não despachável
(estocástica) e de pequena dimensão, têm levado à transição da estrutura ‘vertical’ tradicional do
sistema eléctrico para uma estrutura operacional ‘horizontal’ onde os sistemas de distribuição contêm
geração estocástica e carga.
Estas novas condições fomentam o desenvolvimento de novas metodologias para a
investigação desta nova estrutura operacional, em particular a incerteza operacional introduzida pelo
abatimento na dispachabilidade da geração. A análise da nova estrutura do sistema eléctrico envolve
um elevado número de variáveis estocásticas e interdependências complexas. Nesta tese, a
metodologia utilizada é baseada no método das Simulações de Monte Carlo, envolvendo a
modelação das distribuições uni-dimensionais (espectro de saída individual de cada entrada
estocástica) e modelação da estrutura de dependência estocástica multidimensional (interacção
mútua entre as entradas estocásticas). Em primeiro lugar, a Stochastic Bounds Methodology é
aplicada para modelar os clusters de variáveis correladas positivamente (Stochastic Plants) através
dos conceito de correlação perfeita e positiva (comonotonicity). A Stochastic Bounds Methodology
pode também ser aplicada para modelar a estrutura de dependência entre os clusters através da
definição estruturas de dependência extrema que limitam todos os possíveis casos reais de
dependência. A modelação das correlações exactas entre os clusters é baseada na Joint Normal
Transform Methodology.
Para a aplicação destas metodologias é utilizada uma rede de transporte de teste de 5
barramentos e 7 linhas (Rede de Hale) e uma rede de transporte de teste do IEEE de 39 barramentos
(Rede New England). Esta abordagem mostra que o trânsito de potência nas linhas torna-se
bidireccional, o aumento do nível de penetração de geração estocástica traduz-se de uma maneira
geral, no aumento da variabilidade dos trânsitos de potência e para níveis mais elevados de geração
estocástica, os trânsitos de potência inversos podem exceder os directos.
List of Tables
Palavras-chave
Geração Estocástica
Geração Distribuida
Análise da incerteza
Trânsito de potência probabilístico
Simulações de Monte Carlo
Aerogerador
ix
Table of Contents
Acknowledgements .................................................................................................................................. v
Abstract ................................................................................................................................................... vii
Keywords ................................................................................................................................................ vii
Resumo .................................................................................................................................................. viii
Palavras-chave ...................................................................................................................................... viii
List of Figures ......................................................................................................................................... xii
List of Tables .......................................................................................................................................... xiv
List of Abbreviations ............................................................................................................................... xv
1 Introduction ....................................................................................................................................... 1
1.1 ‘Vertical’ Power System ........................................................................................................... 2
1.2 Distributed Generation (DG) .................................................................................................... 2
1.3 ‘Horizontally-Operated’ Power System (HOPS) ...................................................................... 3
1.4 Objective .................................................................................................................................. 4
1.5 Outline of the Thesis ................................................................................................................ 4
2 Power System Steady-State Uncertainty ......................................................................................... 6
2.1 Deterministic System Model (DSM) ......................................................................................... 7
2.2 Stochastic System Model (SSM) ............................................................................................. 8
2.3 Probabilistic steady-state uncertainty analysis ........................................................................ 9
2.3.1 Analytical methods ............................................................................................................... 9
2.3.2 Stochastic Simulations ......................................................................................................... 9
2.4 Power System Modelling Principles ....................................................................................... 10
2.5 Conclusions ........................................................................................................................... 10
3 Load Uncertainty vs. Stochastic Generation Uncertainty ............................................................... 11
3.1 Load Uncertainty .................................................................................................................... 12
3.2 Stochastic Generation (SG) Uncertainty ................................................................................ 14
3.3 Marginal Distributions ............................................................................................................ 15
3.3.1 Load ................................................................................................................................... 15
x
3.3.2 Stochastic Generation (SG) ............................................................................................... 21
3.4 Conclusions ........................................................................................................................... 22
4 Stochastic Dependence Modelling ................................................................................................. 23
4.1 Measure of Dependence........................................................................................................ 24
4.2 Load Dependence Modelling ................................................................................................. 25
4.3 Stochastic Generation (SG) Dependence Modelling ............................................................. 29
4.3.1 Modelling of Two-Stochastic Generators (Wind Turbine Generators – WTG) .................. 33
4.3.2 Joint Normal Transform (JNT) Methodology ...................................................................... 35
4.4 Conclusions ........................................................................................................................... 36
5 Methodologies for the modelling of Horizontally-Operated Power Systems .................................. 37
5.1 Problem Formulation .............................................................................................................. 38
5.2 Solution Formulation .............................................................................................................. 39
5.3 Stochastic Bounds Methodology (SBM) ................................................................................ 40
5.3.1 Upper bound: comonotonicity ............................................................................................ 40
5.3.2 Lower Bounds: countermonotonicity – independence ....................................................... 41
5.4 Stochastic Model Reduction .................................................................................................. 42
5.5 Joint Normal Transform (JNT) Methodology .......................................................................... 44
5.6 Stochastic Generation (SG) in Bulk Power System ............................................................... 45
5.6.1 System data ....................................................................................................................... 45
5.6.2 Marginal Distributions: Load – Wind Turbine Generators .................................................. 47
5.6.3 Stochastic Bounds Methodology (SBM) ............................................................................ 48
5.6.4 Joint Normal Transform (JNT) Methodology ...................................................................... 55
6 Application: Integration of Stochastic Generation (SG) in a Bulk Power System .......................... 62
6.1 Simulation data ...................................................................................................................... 63
6.2 System loads ......................................................................................................................... 64
6.2.1 Marginals ............................................................................................................................ 64
6.2.2 Dependence structure ........................................................................................................ 65
6.3 System wind power ................................................................................................................ 65
6.3.1 Marginals ............................................................................................................................ 65
6.3.2 Dependence structure ........................................................................................................ 66
xi
6.4 Conventional Generation (CG) Units ..................................................................................... 67
6.5 System Analysis of results ..................................................................................................... 67
6.6 Conclusions ........................................................................................................................... 78
7 Conclusions and Future Work ........................................................................................................ 79
Simulation data ...................................................................................................................................... 82
xii
List of Tables
List of Figures
Figure 3.1: Daily load for a distribution system for one month .............................................................. 12
Figure 3.2: Daily system load in 2006 in Portugal ................................................................................. 12
Figure 3.3: Daily system load in September 2006 in Portugal .............................................................. 13
Figure 3.4: Seasonal daily system load in 2006 in Portugal. ................................................................. 13
Figure 3.5: Daily power output profile of a wind park for one month. .................................................... 14
Figure 3.6: Normalized load distribution as mixture of TF-distributions (2-TF segmentation - 10000-sample MCS). ................................................................................................... 17
Figure 3.7: Normalized load distribution as mixture of TF-distributions (3-TF segmentation - 10000-sample MCS). ................................................................................................... 18
Figure 3.8: Normalized load distribution as mixture of TF-distributions (4-TF segmentation - 10000-sample MCS). ................................................................................................... 19
Figure 3.9: pdf for the system load in 2006 in Portugal (measurements). ............................................. 20
Figure 3.10: pdf for the system load in 2006 in Portugal (20000-sample MCS).................................... 20
Figure 3.11: WTG wind speed/power characteristic and wind distribution. ........................................... 21
Figure 3.12: WTG output power distribution. ......................................................................................... 22
Figure 4.1: Marginal distributions and sum distribution of two independent normal loads. ................... 26
Figure 4.2: Scatter diagram of two independent normal loads. ............................................................. 26
Figure 4.3: Scatter diagrams for correlated normal loads. .................................................................... 27
Figure 4.4: Distribution of the sum of two correlated normal loads. ...................................................... 28
Figure 4.5: Distribution of the sum of four correlated normal loads. ...................................................... 28
Figure 4.6: Aggregate power distributions. ............................................................................................ 30
Figure 4.7: Scatter diagrams and time-series data for the perfectly dependence. ............................... 31
Figure 4.8: Scatter diagrams for independence and perfect dependence between normal and Weibull distributions and their respective ranks. .......................................................... 32
Figure 4.9: (a) Wind Speed Distribution and WTG Wind Speed-Power Characteristic (b) WTG Power Output Distributions for the WTG 1. .................................................................. 33
Figure 4.10: (a) Wind Speed Distribution and WTG Wind Speed-Power Characteristic (b) WTG Power Output Distributions for the WTG 2. .................................................................. 34
Figure 4.11: Scatter diagram and time-series data for the independence case.................................... 34
Figure 4.12: Scatter diagram and time-series data for the perfect dependence case. ......................... 34
Figure 4.13: Aggregate power output for the two cases........................................................................ 35
Figure 5.1: 5-bus / 7-branch Test System (Hale Network) .................................................................... 45
Figure 5.2: Flowchart of the complete computation ............................................................................... 46
Figure 5.3: Load probability density function and load cumulative distribution of DN 4 for a 20000-sample MCS. .................................................................................................... 47
Figure 5.4 : (a) Wind Speed distribution and WTG power curve (b) WTG output distribution for a WTG of DN 4 for a 20000-sample MCS. ..................................................................... 48
Figure 5.5: System upper bound stochastic modelling: first clustering scenario................................... 49
Figure 5.6: System upper bound stochastic modelling: second clustering scenario ............................. 49
Figure 5.7: System upper bound stochastic modelling: third clustering scenario ................................. 50
Figure 5.8: System upper bound stochastic modelling: fourth clustering scenario ............................... 50
Figure 5.9: Power Flow Distributions for the system lines. .................................................................... 51
Figure 5.10: Power Flow Distributions for the system lines. ................................................................. 52
Figure 5.11: Power Flow Distributions for the system lines................................................................... 53
xiii
Figure 5.12: Clustering for the 5-bus / 7-branch test System ................................................................ 56
Figure 5.13: Load/generation cluster distributions and power injection at node 3. ............................... 57
Figure 5.14: Load/generation cluster distributions and power injection at node 4. ............................... 58
Figure 5.15: Load/generation cluster distributions and power injection at node 4. ............................... 59
Figure 5.16: Power Flow Distributions for the system lines ................................................................... 60
Figure 6.1: Single-line diagram of the 39-bus New England test system .............................................. 63
Figure 6.2: 4-TF load modelling for the New England test system (10000-sample MCS). ................... 64
Figure 6.3: Scatter diagrams for the load modelling (10000-sample MCS). ......................................... 65
Figure 6.4: WSP power output in the New England test system (10000-sample MCS). ...................... 66
Figure 6.5: Wind speed and wind power scatter diagrams (10000-sample MCS). ............................... 66
Figure 6.6: Power injection at bus 8 for the 4 wind power penetration levels (10000-sample .............. 68
Figure 6.7: Power injection at bus 24 for the 4 wind power penetration levels (10000-sample MCS). ........................................................................................................................... 68
Figure 6.8: Slack bus power injection distributions and box plots (10000-sample MCS). ..................... 69
Figure 6.9: Box-plot for the power flows in the system lines in case of no wind power penetration (10000-sample MCS). .................................................................................................. 70
Figure 6.10: Box-plot for the power flows in the system lines for 1500 MW (25%) of wind power penetration (10000-sample MCS). ............................................................................... 70
Figure 6.11: Box-plot for the power flows in the system lines for 3000 MW (50%) of wind power penetration (10000-sample MCS). ............................................................................... 71
Figure 6.12: Box-plot for the power flows in the system lines for 4500 MW (75%) of wind power penetration (10000-sample MCS). ............................................................................... 71
Figure 6.13: Some specific power flow distributions (10000-sample MCS): ......................................... 73
Figure 6.14: Some specific power flow distributions (10000-sample MCS): ......................................... 74
Figure 6.15: Some specific power flow distributions (10000-sample MCS): ......................................... 75
Figure 6.16: Some specific power flow distributions (10000-sample MCS): ......................................... 76
Figure 6.17: Distributions of system losses (10000-sample MCS). ....................................................... 77
xiv
List of Tables
List of Tables Table 3.1: Time-frames settings ............................................................................................................ 16
Table 4.1: Power Output Mean Values and Standard Deviations ......................................................... 35
Table 5.1: Test system data .................................................................................................................. 45
Table 5.2: Line Power Flows: Mean values ........................................................................................... 53
Table 5.3: Line Power Flows: Standard deviations ............................................................................... 54
Table 5.4: Mean values and standard deviations of the line power flows ............................................. 60
Table 6.1: WSPs connected to the respective node .............................................................................. 63
Table 6.2: TF settings for a 4-TF load modelling of the New England test system. .............................. 64
Table 6.3: Mean value and standard deviation for the power injections at bus 8 for the 4 wind power penetration levels. ............................................................................................. 67
Table 6.4: Mean value and standard deviation for the slack bus power injection distributions. ............ 70
Table 6.5: Mean value and standard deviation for the distributions of the system losses. ................... 77
Table A.1: Bus data of the New England 39-bus test system ............................................................... 83
Table A.2: Line data of the New England 39-bus test system ............................................................... 84
xv
List of Tables
List of Abbreviations
cdf: cumulative distribution function
CG: Conventional Generation
CHP: Combined Heat and Power
DG: Distributed Generation
DSM: Deterministic System Model
HOPS: Horizontally-Operated Power Systems
HV: High Voltage
JNT: Joint Normal Transform
LV: Low Voltage
MCS: Monte Carlo Simulation
MV: Medium Voltage
pdf: probability density function
RES: Renewable energy sources
r.v.: random variable
SBM: Stochastic Bounds Methodology
SG: Stochastic Generation
SSM: Stochastic System Model
SP: Stochastic Plant
TF: Time-frame
WSP: Wind Stochastic Plant
WTG: Wind Turbine Generator
xvi
1
Chapter 1
Introduction 1 Introduction
In recent years, a worldwide wave of radical changes in power system industries brought new
challenges for the power systems planning. Above all, the goal of a power system is to provide energy
supply to its customers in an economical and reliable manner without detriment impact on the
environment. In addition, the variety of uncertain and random factors imposes a large number of
possible alternatives to the development strategy of the system. The transition from the traditional
‘vertical’ power system structure to a ‘horizontally-operated’ power system is followed by the
incorporation of high levels of small-scale, non-dispatchable (stochastic), distributed generation
(renewables, wasted heat, etc) in the system which is one of the major challenges for the future power
systems planning. This change increases the uncertainty level but provides alternative solutions to the
planning problem. These new conditions persuade the development of new modelling and design
methodologies for the investigation of this new operational power system structure, in particular the
operational uncertainty introduced by the abatement in generation dispatchability.
2
1.1 ‘Vertical’ Power System
Until now, the power have been generated in a relatively small number of large power plants,
constructed at remote sites, close to the energy resources or supply routes and relatively far from the
load centres. This way of electric power generation, called Conventional or Centralized Generation
(CG), use synchronous generators to convert mechanical energy, obtained by the conversion of
controllable energy sources such as fossil fuels, nuclear, hydro power (large hydro-electric power
plants), etc., into electrical energy [1].
The electrical power is transformed to higher voltage level in the generation substation,
transported to the (sub) transmission substations, through the High Voltage (HV) or Very High Voltage
(VHV) transmissions systems, to be transformed to Medium Voltage (MV) level and enter in the
primary distribution systems where is transformed to Low Voltage (LV) level to be distributed to the
consumers. Thus electrical energy flows from the higher to the lower voltage levels in the network.
In this recent years, the expansion of these power plants have been limited due to socio-
economically, political, environmental and geographically considerations and have motivated the
development and implementation of a new, non-Conventional Generation (non-CG) [1].
1.2 Distributed Generation (DG)
In this thesis, the term Distributed Generation (DG) refers to non-CG units connected to the
distribution networks [1] [2]. The basic characteristics of the non-CG units can be described as follows:
- The non-CG units are small to medium scale generators which make use of new developed
power generation technologies and supporting technologies like power electronic converters
and controllers, allowing their large-scale implementation in the utility system.
- The non-CG comprises renewable energy sources as wind, biomass, sunlight, wave and
geothermal energy, and new generation technologies as the fuel cell, combined heat and
power (CHP) cogeneration, etc. The non-CG can be locally dispatched or non-dispatchable:
§ Locally Dispatchable Technology - DT (Small fossil-fuel power plants, biomass
power plants, geothermal power plants, fuel cells and CHP plants): the prime
energy sources or fuels can be controlled by the unit operator in order to regulate
the power output.
§ Non-Dispatchable Technology - NDT (small hydro, wind turbines, photovoltaic,
tidal power plants, wave power plants and CHP plants): the unit power output is
defined by prime mover availability since the prime energy source is not
3
controllable. This type of generation, called Stochastic Generation (SG), implies
power generation uncertainty.
The change to a new way of power generation is mostly related to the incorporation of sustainable
energy sources as renewable energy sources. The fossil fuels for the thermo-electric power plants are
provided by the available natural reserves which are not infinite and will be depleted in the long term.
Furthermore, the operation of fossil-fuel-fired power plants brings adverse impacts to the environment,
such as the global climate change and the greenhouse effect caused by the increase of CO2
concentration in the earth`s atmosphere. The nuclear energy brings the problem of disposal of nuclear
waste and the fear of the adverse effects of a nuclear accident. The large hydro-power plants
comprise the construction of dams and basins which brings environmental consequences, affects the
flow of rivers and the flooding of large areas. Since the renewable energy exists in huge quantities in
the nature, geographically distributed, presenting a low energy density on each generation site, the
renewable power generation is formed by small-scale converters distributed through the system that
capture the energy from existing flows of energy, such as sunshine, wind, wave power, flowing water
(hydropower), biological processes (biomass) and geothermal heat flow, to convert into electricity.
This energy is replaced by a natural process at a rate that is equal or faster than the rate at which that
resource is being consumed. These small-scale power plants are connected in different voltage levels
in the system, depending on the level of their aggregate power output and most of their contribution to
the system is defined by the non-regulated prime mover activity which introduce power generation
uncertainty in the system although several types of renewable generation are not SG (large hydro,
biomass and geothermal power plants) [2].
Nowadays, the connection of generation in the distribution systems is also increasing since
building new HV lines to solve the transmission system capacity problem related to the growth in
electricity consumption become a problem due to the rejection from the public, the investment cost
and the lack of available physical space for expansion. The generation close to the loads permits the
use of waste heat for heating or cooling and can provide standby energy during critical load periods
when the energy from the utility systems is unavailable. In addition, there is a world-wide trend
towards deregulation of the electricity markets which helps the introduction of new, more effective
forms of small scale generation (fuel cells, combined heat power plants, micro-turbines, hydrogen,
etc.) that require lower capital costs and shorter construction times.
1.3 ‘Horizontally-Operated’ Power System (HOPS)
The transition to a more ‘Horizontally-Operated’ power system (HOPS) structure may take place due
to the large-scale implementation of medium to small scale non-CG units at the MV and LV networks
i.e. the installation of generation in the distributions networks which turns the passive distribution
network into an active one. In this case, some customers also generate electricity and may supply the
network if the generation surpasses their demand. In this power system structure can be recognized
two non-dispatchable system entities, the load and Non-Dispatchable, Distributed Generation -
4
NDT/DG, and two dispatchable system entities, the CG and the Dispatchable, Distributed Generation -
DT/DG [1].
In the NDT/DG units, the power output is defined by the activity of the uncontrolled prime
mover and their control aims to maximize their energy potential. The power output of the CG units is
defined by the energy market mechanisms.
Due to the fact that the non-CG units can be locally dispatched or stochastic, leading to the
introduction of uncertainty in power system, the power flow between the transmission network and the
active distribution network is no longer uni-directional but can be bidirectional. The power can flow
‘vertically’, from higher levels to the lower voltage levels, and also ‘horizontally’, from one MV or LV
network to another or from a generator to a load within the same MV or LV network. In order to match
the total demand, the dispatchable entities adapt to the non-dispatchable entities variations taking into
account the system restrictions.
The transmission network interconnects the different active distribution systems and the large
CG units, connected at the HV and VHV networks. Theoretically, the ‘Vertical-to-Horizontal’
transformation of a power system can occur when the DG within the active distribution network is
sufficient to match the total demand of the system and thus, the large centralized power plants may be
shut down [1].
1.4 Objective
The major purpose of this dissertation is to contribute to the improvement of the knowledge of
distributed renewable generation impact in the existing power system. In order to take into
consideration the stochastic behaviour of load demands and the large integration of non-dispatchable
(stochastic), distributed generation, a probabilistic load flow program will be developed for power
system planning and operation analyses.
1.5 Outline of the Thesis
The thesis is organized as follows:
- In the 2nd
chapter are presented two main probabilistic approaches for the probabilistic
uncertainty analysis related to power system studies, namely the Analytical methods and
Stochastic Simulations (Monte-Carlo Simulation - MCS). It is shown that the MCS is the most
indicated method to solve the multivariate uncertainty problem created by the incorporation of
Stochastic Generation (SG) in the system, based on the modelling of the one-dimensional
marginal distributions and the modelling of the multidimensional stochastic dependence
structure.
5
- The 3rd
chapter presents the main differences between load uncertainty and SG uncertainty.
In this chapter, the one-dimensional marginal distributions that define the output of each
stochastic input are analyzed.
- The 4th chapter describes the Joint Normal Transform (JNT) Methodology as an adequate
method to use in the multidimensional dependence modelling. In order to measure the
dependence between non-normals distributions, the one-marginal distributions are
transformed into ranks whose product moment correlation (the rank correlation���) provides an
adequate measure of dependence. The functional relationship between the ranks is modelled
by Copula functions. The transformation of the marginals into ranks and the use of a
multidimensional normal copula for the dependence modelling is the basis for the JNT
methodology.
- The 5th
chapter presents methods to deal with high-dimensionality, i.e. model reduction
techniques to simplify the stochastic model through model approximations called Stochastic
Plants (SP’s). The Stochastic Bounds Methodology (SBM) is used for the formulation of these
approximations. This methodology proposes a stochastic modelling approach to deal with the
uncertainty introduced in the ‘Horizontally-Operated’ Power Systems (HOPS) based on the
extreme stochastic dependence structures between the system inputs. The Joint Normal
Transform Methodology is presented as a solution to the incorporation of a realistic
dependence structure to the power system modelling.
- The 6th chapter presents an application of these methodologies to solve a multivariate
uncertainty analysis problem, the integration of Stochastic Generation (SG) in a bulk power
system in order to understand better the horizontal operation of the power system.
- The 7th chapter presents the final conclusions.
6
Chapter 2
Power System Steady-State
Uncertainty 2 Power System Steady-State Uncertainty The horizontal operation of power system due to the large-scale integration of Stochastic
Generation (SG) needs an appropriate methodology that permits the incorporation of generation
uncertainty in the analysis of power system.
The incorporation of SG into the system leads to a multivariate uncertainty analysis
problem involving a large number of system inputs with different types of non-standard
distributions and complex interdependencies. The system uncertainty analysis implies the
analysis of the system steady-state for the set of all possible inputs (load/generation), involving
the definition of the Deterministic System Model (DSM) and the Stochastic System Model
(SSM).
7
2.1 Deterministic System Model (DSM)
The Deterministic System Model (DSM) combines the system variables considering the system
configuration data in order to obtain the system outputs. The steady-state or load flow analysis
concerns the determination of the voltage (magnitude and phase angle) at each bus and the
power flow (real and reactive) in each line.
In the power system model, the transmission lines k that interconnect the set of N buses
are represented by their nominal π-equivalent circuits and the numerical values for the series
impedance�����and the total line-charging admittance����. These parameters are used to
determine all the elements of the N×N bus symmetric admittance matrix of a system with N
buses whose typical elements are
� � ������� � �� ��� ��� � � �� ��� ��� ���� � ���� (2.1)
where ��� and ��� are the conductance and susceptance of the element� ��. ■ �� is equal to the sum of all admittances connected to i-th node:
��� � !�"#$ �%�&'#(
)
�*+�,� (2.2)
■ �� is equal to the negative of the admittance between the nodes i and j:
��- � �-� � . %���� (2.3)
The voltage at a typical bus i of the system is
/� � 0���1� � 0� ��� 2� � � 0���� 2� (2.4)
The current injected into the network at bus i is given by:
3� � 4�5/�5 �6� . �7�/�5 � ��-/-
8
�*� (2.5)
where 6� and 7� are the active and reactive power injections at the node i that can be computed
using the following equations:
6� � 0�0�9:�� ���;2� . 2�< � =�� ���;2� . 2�<>8
�*+ (2.6)
7� � 0�0�9:�� ���;2� . 2�<.=�� ���;2� . 2�<>8
�*+ (2.7)
8
Denoting 6�?@AB@ and 7�?@AB@ as the calculated values of 6� and 7� and the net schedule power
injected in the network at bus i as
6�?C@D � 6E� . 6F� (2.8)
where 6E� and 6F� are the schedule power generation and the schedule power demand at bus i.
The calculated values of the power generation and power demand do not always match with the
schedules ones:
G6� � 6�?C@D . 6�?@AB@ H I (2.9)
G7� � 7�?C@D . 7�?@AB@ H I (2.10)
As can been seen, the solution to the load flow problem involves the solution of non-linear
algebraic equations�6� and�7� with respect to�2� and�0� and thus, the use of iterative techniques
such as the Gauss-Seidel or Newton-Raphson procedures. Since there are four potentially
unknown variables�6�,�7�,�2� and 0�, two of them have to be specified according to the type of
bus:
- Load buses: �6� and�7� are specified, 2� and 0� are the unknown variables.
- Voltage controlled bus: �6� and 0� are specified, 2� and 7� are the unknown variables.
- Slack bus: 2� and 0� are specified, 6� and�7� are the unknown variables.
Thus, the power flow problem is to solve a system of (2N - Ng – 2) non-linear equations with (2N
- Ng – 2) state variables, where Ng is the number of voltage-controlled buses in the system and
the state variables are the unscheduled bus-voltage magnitudes and angles [10] [12].
2.2 Stochastic System Model (SSM)
The Stochastic System Model (SSM) quantifies the uncertainty related to the system inputs,
describing their behaviour and interaction. The data for the uncertainty systems inputs are
collected and an appropriate mathematical methodology is used for the representation and
quantification of the input´s uncertainty in order to provide the set of stochastic system inputs.
The system configuration data, the non-stochastic system inputs and the different sets
of stochastic system inputs are fed into the DSM in order to obtain the results corresponding to
the different operating system steady-states.
9
2.3 Probabilistic steady-state uncertainty analysis
The appropriate approach for the modelling of the power system uncertainty is the probabilistic
analysis since the uncertainty of the output of a stochastic generator or load is quantified in
numerical terms by the statistical analysis of respective data. Each uncertainty system input is
represented as a random variable (r.v.) with a specific probability density function (pdf). The
main probabilistic approaches found in the related literature are the Analytical methods and the
Stochastic Simulations (Monte-Carlo Simulation – MCS).
2.3.1 Analytical methods
For the computational efficiency advantage offered by this approach compared to MCS, are
necessary a number of simplifications:
- Linearization of the system model: is performed around an operating point that
corresponds to the mean of the system inputs which permits the representation of the
system outputs as a linear combination of the system inputs. The accuracy of this
approximation depends on the spread of the system inputs around the mean value i.e.
the dispersion of the system inputs should be limited around the mean value in order to
obtain accurate results.
- Independence: the system inputs are assumed to be statistically independent.
- Normality: the system inputs are assumed to be normally distributed which permits the
use of linearly dependent random variables (r.v.).
These assumptions have been used for the modelling of the uncertainty of the system loads [1].
The impact of these assumptions is discussed in the chapter 4, especially the normality and
independence.
2.3.2 Stochastic Simulations
The Monte Carlo Simulation (MCS) method is a numerical simulation procedure which consists
on stochastic simulations using random variables. In the MCS method, pseudo-random samples
are generated in accordance with the corresponding SSM´s probability distributions and then,
the DSM’s mathematical model is solved for each sample in order to obtain the sets of samples
for the output quantities of interest which are subjected to a statistical analysis. For the sampling
of the system inputs, new analyzing methodologies can be applied to incorporate the
dependencies between the inputs in the system analysis.
For the application of this method there is no need to simplify the mathematical models
10
since the basic computational part is deterministic. The computation time is no longer a problem
since the present computational power permits to solve the DSM in a few seconds.
2.4 Power System Modelling Principles
The use of the MCS method to treat the multivariate uncertainty analysis problem involves the
sampling of the stochastic system inputs in accordance with dependence structures provided by
SSM and the calculation of the DSM for each sample. The modelling approach involves the
following tasks [4] [8]:
1) Model the one-dimensional marginal distributions.
2) Model the multidimensional stochastic dependence structure.
The one-dimensional marginal distributions of the system inputs refer to the power output
spectrum of single units. The multidimensional stochastic dependence structure refers to the
joint behaviour of these stochastic system inputs.
2.5 Conclusions
The large-scale integration of Stochastic Generation (SG) in the system implies a new
modelling approach for the uncertainty analysis of the power system.
The system analysis implies the analysis of the system steady-state for the set of all
possible inputs (load/generation) since the system variables present most of the time, variations
so small that the power system can be considered predominately in steady-state operation. The
data for the uncertain system inputs are introduced into the Stochastic System Model (SSM),
providing the different sets of stochastic system inputs that are passed on to the Deterministic
System Model (DSM) in order to obtain the respective outputs.
The analytical methods have been applied for the modelling of the uncertainty of the
system loads, based on reasonable approximations as the assumption of independence and
normality. The incorporation of SG into the system leads to a multivariate uncertainty analysis
problem involving a large number of system inputs with different types of non-standard
distributions and complex interdependencies. In this case, the most indicated method for the
system modelling is the Monte-Carlo simulation (MCS).
The use of the MCS method for the modelling of the power system uncertainty involves
the modelling of the one-dimensional marginal distributions and the modelling of the multi-
dimensional stochastic dependence structure.
11
Chapter 3
Load Uncertainty vs.
Stochastic Generation (SG)
Uncertainty 3 Load Uncertainty vs. Stochastic Generation Uncertainty
The system load uncertainty analysis is conditioned in time since this uncertainty corresponds
to a time-dependent stochasticity. The assumptions of independence and normality are used for
the load uncertainty modelling. However, these assumptions cannot be used for modelling the
non time-dependent stochasticity introduced by the SG which involves a large number of
different types of non-standard distributions with complex interdependencies. For the power
system multivariate uncertainty analysis problem is used a MCS approach, based on the
modelling of the one-dimensional marginal distributions and the modelling of the multi-
dimensional stochastic dependence structure.
12
3.1 Load Uncertainty
In Figure 3.1, an example of the daily load profile of a distribution network based on 30-minute
load measurements for the period of one month is present. As can be seen, all measurements
of the electrical load fall in a small region for each point in the day due to the dependence of the
human activities on the time of the day. In Figure 3.2, the daily system load for the year 2006 in
Portugal is presented (30-minutes averages).
Figure 3.1: Daily load for a distribution system for one month
Figure 3.2: Daily system load in 2006 in Portugal
0 2 4 6 8 10 12 14 16 18 20 22 24100
150
200
250
300
350
400
Time of day
Syste
m L
oad (
MW
)
0 2 4 6 8 10 12 14 16 18 20 22 242000
3000
4000
5000
6000
7000
8000
9000
Time of day
Syste
m L
oad (
MW
)
13
Thus, the power consumption is not so stochastic but presents a high time-dependence on the
time of day, day of week and season. The load uncertainty can be modelled superimposing a
random noise to the mean value since all measurements of the electrical load fall in a small
region around the time conditional mean value (blue line in the Figure 3.1) for each time period.
In Figure 3.3a, the daily system load is presented for one month and in Figure 3.3b the working
days are isolated in order to distinguish the different daily load patterns.
(a) All days (b) Working days
Figure 3.3: Daily system load in September 2006 in Portugal
In Figure 3.4a, the daily system load is presented for the winter season and in Figure 3.4b the
working days are isolated.
(a) All days (b) Working days
Figure 3.4: Seasonal daily system load in 2006 in Portugal.
As can be seen in the graphics, the dependence on the time of day is cyclic and the load in
each time of the day falls in a small region around the mean value. The measured data used for
the simulated daily system load profiles above (Figure 3.1, Figure 3.2, Figure 3.3 and Figure
3.4) was obtained from REN - Rede Eléctrica Nacional, S.A..
As a conclusion, it is stressed that the load stochasticity can be modelled by
superimposing a random noise variable to the conditional mean, considering time-periods with
similar statistical characteristics.
2 4 6 8 10 12 14 16 18 20 22 243000
3500
4000
4500
5000
5500
6000
6500
7000
Time of day
Syste
m L
oad (
MW
)
2 4 6 8 10 12 14 16 18 20 22 243000
3500
4000
4500
5000
5500
6000
6500
7000
Time of day
Syste
m L
oad (
MW
)
2 4 6 8 10 12 14 16 18 20 22 243000
4000
5000
6000
7000
8000
9000
Time of day
Syste
m L
oad (
MW
)
2 4 6 8 10 12 14 16 18 20 22 243000
4000
5000
6000
7000
8000
9000
Time of day
Syste
m L
oad (
MW
)
14
3.2 Stochastic Generation (SG) Uncertainty
The incorporation of the generation uncertainty to system analysis, in addition to the
consumption uncertainty, corresponds to the integration of high levels of non-dispatchable
(stochastic) generation which is related to a new generation concept, the Distributed Generation
(SG). This implies the transition to a ‘horizontally-operated’ power system where electrical
energy generation takes place in a large number of small to medium scale, geographically
distributed power plants connected to the distribution systems, close to the system loads.
The stochastic generation can be characterized by the following aspects:
1) Small scale: the generators are small to medium scale due to the low energy density of
the stochastic prime mover in each generation site. They can be broadly seen as
‘negative loads’ that are connected in the distributions systems next to the normal ones.
2) Non-dispatchable: the stochastic generators use a stochastic prime mover i.e. an
uncontrolled primary energy source, e.g. wind-energy, solar-energy, hydro-energy,
wave-energy, waste heat, etc. The stochastic behaviour of the prime mover differs
between the different sites that are geographically dispersed through the system.
In Figure 3.5, a basic example of this uncertainty, the daily power output profile of a wind park is
presented for a period of one month. As can be seen, the power output present a high variability
during different days, i.e. the power output may vary between zero and maximum for every day.
Figure 3.5: Daily power output profile of a wind park for one month.
03:00 06:00 09:00 12:00 15:00 18:00 21:00 00:000%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Time of day
Win
d o
utp
ut
15
The measured data used for the simulated daily power output profile of a wind park (Figure 3.5)
was obtained from REN - Rede Eléctrica Nacional, S.A..
3.3 Marginal Distributions
In the Monte Carlo Simulation (MCS) process, the sampling of a single random variable (r.v.) X
with continuous invertible cumulative distribution function (cdf), )()( xXPxFX <= ,can be
described by the following steps:
1) Generation of a uniform number U in [0,1].
2) Application of the transformation )(1 uFx X
−= where u is a realization of U to obtain the
marginal distribution of the r.v. X.
It can be proof that the samples x follow the distribution XF by the following relationships:
1) By definition, the r.v. )(XFX follows a uniform distribution on the interval [0,1] where X
is a single r.v. with invertible cdf XF :
For [ ] rrFFrFXPrXFPr XXXX ==<=<∈ −− ))(())(())((:1,0 11
2) )(,)( 1 UFXUXF XX
−== . Therefore )(1 UFX
−follows the distribution of X.
These concepts are the basis for the sampling of any r.v. in MCS studies [1].
3.3.1 Load
As mentioned in section 3.1, the load stochasticity can be modelled by superimposing a random
noise variable to the conditional mean, considering time-periods, Time-frames (TF), with similar
statistical characteristics. A methodology based on mixture of normal distributions can be used
for the time-conditional modelling of single loads in a long period [1]. Usually, the normal
distribution is used to model the stochastic behaviour of the system loads for each TF, i.e. for
each group of hours with similar statistical characteristics (specific time-period of
day/week/season). This methodology can be described as follows:
1) In order to consider the cyclic time-dependence of the load, the whole concerned period
is divided in n groups of time-periods, n-Time-frames (n-TF), each one with similar
consumption statistical characteristics.
2) In each TF, it is usual practice a normal distribution be fitted to the load data through the
analysis of the statistical properties of the respective load data group.
3) Then, an aggregation procedure is applied to obtain the load distribution for the whole
concerned period resulting in a mixture of the normal distributions of the different TFs.
16
The choice of the TFs, by the analysis of the statistical properties of the load, has influence on
the accuracy of the time-conditioned load modelling by a normal distribution. The accuracy of
this approximation can be limited if a large TF is chosen.
In order to find the resulting load distribution of the whole concerned period, the MCS is
performed using an independent uniform r.v. U to sample the mixture of normals by the
following procedure. For each generated sample u, first the TF is chosen between the different
time-frames based on their relative duration. After the time-frame is chosen, a sample is drawn
from the respective normal distribution. The settings for the 2-TF, 3-TF and 4-TF modelling are
presented in Table 3.1. For example, in the 2-TF modelling, two time-periods (Time-frames –
TFs) are considered, corresponding to 0.8% (TF1) and 0.2% (TF2) of the total time. In this case,
the high-load-TF is the TF2. In the TF1, the load data is simulated by a normal distribution with
a mean value (TF1-mean) equal to 0.5% of the TF2-mean and a standard deviation equal to
0.25% of the TF1 mean.
Table 3.1: Time-frames settings
Time Ratio
Mean Load (% high-load-TF mean)
St. Deviation (% mean value)
2-TF 3-TF 4-TF 2-TF 3-TF 4-TF 2-TF 3-TF 4-TF
TF1 0.8 0.45 0.2 0.5 0.5 0.5 0.25 0.15 0.06
TF2 0.2 0.35 0.3 1 0.75 0.65 0.02 0.1 0.1
TF3 - 0.2 0.3 - 1 0.85 - 0.02 0.1
TF4 - - 0.2 - - 1 - 0.03
In Figure 3.6, Figure 3.7 and Figure 3.8, the density and cumulative distributions for a 10000-
sample MCS of a 2-TF, 3-TF and 4-TF approximation are presented. As can be seen, the cdf
provides a better approximation of the load duration curve with the increase of the number of
TFs.
If the measured data is available, the system modelling can be performed based on real
data which is an advantage because using the exact data distribution in the simulation, the
results will be more realistic. In this case, the sampling is based on the empirical cdf e
XF and
the adjacent values can be obtained by linear interpolation. For exemplifying this, in Figure 3.10
is presented the results of sampling based on the load data used in Figure 3.2 representing the
system load in 2006 for Portugal (17520 measurements). As can be seen, the pdf based on the
measurements presented in Figure 3.9 shows a highly accordance with the pdf obtained by
simulation.
The assumptions of independence and normality are adequate to model the load time-
dependent stochasticity but not to model the non-time-dependent stochasticity introduced by the
SG.
17
Figure 3.6: Normalized load distribution as mixture of TF-distributions (2-TF segmentation -
10000-sample MCS).
0 0.2 0.4 0.6 0.8 1 1.2 1.40
100
200
300
400
500
600
Load (pu)
MC
S S
am
ple
s
0 0.2 0.4 0.6 0.8 1 1.2 1.40
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Load (pu)
MC
S S
am
ple
s
CDF
18
Figure 3.7: Normalized load distribution as mixture of TF-distributions (3-TF segmentation -
10000-sample MCS).
0 0.2 0.4 0.6 0.8 1 1.2 1.40
100
200
300
400
500
600
Load (pu)
MC
S S
am
ple
s
0 0.2 0.4 0.6 0.8 1 1.2 1.40
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Load (pu)
MC
S S
am
ple
s
CDF
19
Figure 3.8: Normalized load distribution as mixture of TF-distributions (4-TF segmentation -
10000-sample MCS).
0 0.2 0.4 0.6 0.8 1 1.2 1.40
50
100
150
200
250
300
350
400
450
Load (pu)
MC
S S
am
ple
s
0 0.2 0.4 0.6 0.8 1 1.2 1.40
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Load (pu)
MC
S S
am
ple
s
CDF
20
Figure 3.9: pdf for the system load in 2006 in Portugal (measurements).
Figure 3.10: pdf for the system load in 2006 in Portugal (20000-sample MCS).
2000 3000 4000 5000 6000 7000 8000 90000
50
100
150
200
250
300
350
400
System Load (MW)
Num
ber
of
sam
ple
s
2000 3000 4000 5000 6000 7000 8000 90000
50
100
150
200
250
300
350
400
450
500
System Load (MW)
Num
ber
of
sam
ple
s
21
3.3.2 Stochastic Generation (SG)
The power output of the stochastic generator for each input value of the prime mover can be
obtained by a deterministic relationship related to the appropriate converter model. Thus, the
output power distribution of a stochastic generator is obtained by the application of the non-
monotonic function of the energy converter to the prime mover probability distribution.
For the wind power, the stochastic prime mover is the wind activity modelled as an r.v.
following, in a specific location, the wind speed distribution which may be represented as a
Weibull distribution [6]. In this case, the energy conversion system is the Wind Turbine
Generator –WTG modelled through the wind speed-power output characteristic of the WTG. In
Figure 3.11 is presented an example of a WTG characteristic and the results of a 10000-sample
MCS for the wind speed distribution. In this example is considered a pitch-controlled WTG of
1MW nominal power modelled by a characteristic with cut-in, nominal and cut-out wind speed
values of 3.5, 14 and 25 m/s, respectively. In Figure 3.12 is presented the results of a 10000-
sample MCS for the output power distributions for the WTG presented in Figure 3.11.
As can be seen, the corresponding WTG output power distribution obtained is highly
non-normal presenting a concentration of probability in the zero (wind speeds below cut-in and
above cut-out value) and nominal output power (wind speeds between nominal and cut-out
values) due to the effect of the non-monotonic WTG characteristic.
Figure 3.11: WTG wind speed/power characteristic and wind distribution.
0 5 10 15 20 25 300
100
200
300
400
500
MC
S S
am
ple
s
Wind Speed (m/s)
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
Win
d P
ow
er
(MW
)
22
Figure 3.12: WTG output power distribution.
3.4 Conclusions
The horizontal operation of the power system implies the incorporation of generation uncertainty
in the power system analysis due to the large-scale integration of Stochastic Generation (SG).
The linear approximation of the system model, the assumptions of independence between the
system inputs and perfect correlation for normally correlated system inputs have been used for
modelling of the time-dependent stochasticity of the system loads. The incorporation of SG
introduces a large number of different types of non-standard distributions with complex
interdependencies and a non time dependent stochasticity which cannot be modelled based on
the assumptions of independence and normality. Thus, a Monte Carlo Simulation (MCS)
approach is used for the power system multivariate uncertainty analysis based on the modelling
of the one marginal distributions and the modelling of the multidimensional stochastic
dependence structure. In this chapter, the marginal distributions that define the output of each
stochastic input are analyzed.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
200
400
600
800
1000
1200
1400
1600
1800
Wind Power (MW)
MC
S S
am
ple
s
23
Chapter 4
Stochastic Dependence
Modelling 4 Stochastic Dependence Modelling
As mentioned, the stochastic modelling of the system implies the modelling of the stochastic
dependence between the multiple stochastic system inputs which determine their joint
behaviour and defines their aggregate power output. The methods that have been used for
modelling the dependence structure between the correlated normal loads cannot be employed
for the modelling of the complex interdependencies introduced by the incorporation of SG in the
system. The assumptions of independence and normality lead to fallacies for the modelling of
SG and thus, new methods have to be employed for the power system uncertainty analysis.
24
4.1 Measure of Dependence
One of the possible measures of stochastic dependence is the product moment correlation
which measures the linear dependence between the r.v. The product moment correlation of the
r.v. X, Y can be calculated from the N pairs of samples (xi, yi) as follows [1]:
�JKL � M NO� . PQRN � . SQR)�*+TM NO� . PQRU)�*+ TM N � . SQRU)�*+
(4.1)
where PQ � +)M O�)�*+ and SQ � +)M �)�*+ .
The product moment correlation provides the complete representation of stochastic
dependence for r.v. with normal distributions but it does not offer a good representation of
dependence for r.v. with non-normal distributions. In this case, it offers just a measure of linear
dependence. For example, in the case of two normal r.v. X and Y perfectly dependents (one
follows perfectly the fluctuations of the other), the product moment correlation reaches the
extreme values � � %�and � � .% corresponding to a linearly dependence, i.e., they can be
written as�S � VP � �? � W X? V Y I. In the case of two non-normal r.v. X, Y perfectly dependent,
the product moment correlation values are less than 1 corresponding to a non-linearly
dependence which means that the product moment correlation fails as a measure of
dependence.
A method for measure the dependence between non-normal r.v. consists in decoupling
the dependence structure from the marginals by ranking their samples and measuring the
product moment correlation of the respective ranks. In order to transform the marginals into
ranks, the cumulative density function is applied to the r.v. and a uniform distribution Z�on the
interval [I?%\ is obtained [1]:
]^_�_ W [I?%\` 6N]aNPR b _R � 6;P b ]ac+N_R< � ]a[]ac+N_R\ � _ d�]aNPR � Z
(4.2)
In the case of two r.v. X and Y, this measure of dependence, namely rank correlation
��, can be described as the application of the product moment correlation for the ranks Za �]aNPR and Ze � ]eNSR that maintain the information concerning the dependence structure
between X and Y [1]:
�� � �;]aNPR? ]eNSR< (4.3)
where ]aNPR and ]eNSR are the cumulative density distributions of the r.v. X and Y respectively.
This technique is the basis for the stochastic dependence modelling in uncertainty analysis, by
separating the dependence structure from the one dimensional marginal distributions since this
type of measure always exists and the marginal distributions have no influence on the value of
the measure [1].
25
The relationship between product moment correlation ρ and rank correlation rρ is given by [1]:
�NP? SR � $����Nfg ��NP? SRR
(4.3)
In the case of uniform variables, the product moment correlation and rank correlation are the
same.
4.2 Load Dependence Modelling
As mentioned in the last chapter, the load distribution in the whole concerned period is obtained
by the mixture of the normal distributions that model the load in the different TFs. In a single TF
modelling, the loads may be considered independent but in the long period modelling, similar
types of loads show positive dependence due to their common coupling to the same consumer
behaviour.
Considering n independent normal loads L1, L2… Ln with mean values µ1, µ2… µn and
standard deviations σ1, σ2…σn their sum is a normal distribution with a mean and standard
deviation given by the following equations [1]:
Mean:
1
n
L i
i
µ µ=
=∑ (4.4)
Standard Deviation: 2
1
n
L i
i
σ σ=
= ∑ (4.5)
The marginal distributions and the distribution of the sum of two independent normal loads and
the scatter diagram are presented in Figure 4.1 and Figure 4.2, respectively.
In the case of the sum of n dependent normal loads, the dependence structure between them
affects radically the variance of the sum distribution but the mean is given by the same equation
as in the case of independence. The scatter diagrams for the two normal loads and the
distributions of their sum, corresponding to different correlations, are presented in
Figure 4.3 and Figure 4.4, respectively.
The results show that varying the correlation between the loads, their sum present
normal distributions with different variances around the same mean and the correlation
extremes (ρ = 1 and ρ = -1) result in a perfect linear relationship between the loads. As can be
seen, the case of maximum positive correlation, ρ = 1, corresponds to the maximum variance
around the mean value which means that the extreme values of the r.v. happen always at the
same time. The case of minimum positive correlation, ρ = -1, corresponds to the minimum
variance around the mean value which means that the extreme values never happen at the
same time.
26
Figure 4.1: Marginal distributions and sum distribution of two independent normal loads.
Figure 4.2: Scatter diagram of two independent normal loads.
10 20 30 40 50 60 700
50
100
150
200
250
300
350
400
Loads L1,L2, L1 + L2 (MW)
Num
ber
of
sam
ple
sL1
L2
L1 + L2
12 14 16 18 20 22 24 26 28 3010
15
20
25
30
35
40
45
50
L1 normal load distribution (MW)
L2 n
orm
al lo
ad d
istr
ibution (
MW
)
27
(a) ρ = -1 (b) ρ = -0.1
(c) ρ = -0.4 (d) ρ = 0.3
(e) ρ = 0.9 (f) ρ = 1
Figure 4.3: Scatter diagrams for correlated normal loads.
12 14 16 18 20 22 24 26 2815
20
25
30
35
40
45
L1 normal load distribution (MW)
L2 n
orm
al lo
ad d
istr
ibution (
MW
)
10 12 14 16 18 20 22 24 26 2815
20
25
30
35
40
45
L1 normal load distribution (MW)
L2 n
orm
al lo
ad d
istr
ibution (
MW
)
12 14 16 18 20 22 24 26 2810
15
20
25
30
35
40
45
50
L1 normal load distribution (MW)
L2 n
orm
al lo
ad d
istr
ibution (
MW
)
12 14 16 18 20 22 24 26 2815
20
25
30
35
40
45
50
L1 normal load distribution (MW)
L2 n
orm
al lo
ad d
istr
ibution (
MW
)
10 12 14 16 18 20 22 24 26 2815
20
25
30
35
40
45
50
L1 normal load distribution (MW)
L2 n
orm
al lo
ad d
istr
ibution (
MW
)
10 12 14 16 18 20 22 24 26 2810
15
20
25
30
35
40
45
50
L1 normal load distribution (MW)
L2 n
orm
al lo
ad d
istr
ibution (
MW
)
28
Figure 4.4: Distribution of the sum of two correlated normal loads.
Figure 4.5: Distribution of the sum of four correlated normal loads.
20 30 40 50 60 70 800
50
100
150
200
250
300
350
400
L1 + L2 (MW)
Num
ber
of
sam
ple
scorr = -1
corr = -0.7
corr = -0.4
corr = 0.3
corr = 0.9
corr = 1
90 100 110 120 130 140 150 160 170 180 1900
50
100
150
200
250
300
350
L1+L2+L3+L4 (MW)
Num
ber
of
sam
ple
s
corr = 0
corr = 0.2
corr = 0.4
corr = 0.6
corr = 1
29
In order to correlate more than two loads, the multivariate (joint) normal distribution can be used
as follows [1]. The mutual correlations are defined by the product moment correlation matrix R
and a random vector Y of standard normal r.v. correlated according to that matrix is obtained by
the application of the linear transformation Y = L×Z to a vector [ ]nZZZZ ,...,, 21= of
independent standard normal r.v. where L is a lower triangular matrix such that R = L×LT
(Cholesky decomposition). Then, the n-vector of standard normal r.v. correlated according to
the matrix R is transformed in a n-vector of normals l with specifics mean and standard
deviations values. The correlation matrix R has to be positive semi-definite so that the
factorization R = L×LT can be done, i.e. for all�OhXijIk? OlmO Y I. The simulation algorithm can
be described as follows:
1) Simulate an n-dimensional vector of independent standard normal samples z.
2) Calculate the matrix product y = L × z where y is a sample of the de vector Y that
follows a standard multivariate normal distribution with correlation matrix R.
3) To transform the samples y in samples drawn from normals with specific mean and
standard deviations values, the formula li = σi × yi + µi, i = 1, 2…n is applied to each yi
where σi and µi are the mean and the standard deviation of the respective load.
In Figure 4.5, the distributions of the sum of four correlated normal loads for different
correlations between them are presented. As can be seen, the distributions obtained are
normals with different spreads around the same mean as the bivariate case.
4.3 Stochastic Generation (SG) Dependence Modelling
The most difficult problem in power system modelling is the modelling of non-normal r.v. where
the product moment correlation fails as measure of dependence. The power output of a
stochastic generator is defined by two factors:
1. Stochastic Prime Mover: the type of primary energy source used for electrical power
generation.
2. Energy Conversion System: converter technology that defines the power output of the
generator for each input value of the prime mover.
For the stochastic generators, the assumption of independence between them means the
decoupling between their prime movers and positive dependence means that they are
subjected to the same prime mover activity. The independence can be observed for stochastic
generators situated in remote areas and the positive dependence for stochastic generators
situated in a small geographic area which show similar fluctuations due to their mutual
dependence on the same prime mover. The cases of independence and positive dependence
result in different joint contributions of the stochastic generators to the system i.e. the aggregate
power output of the stochastic generators is different.
30
For Wind Turbine Generators (WTGs), the output of each one follows a non-standard
distribution but assuming independence between the wind speeds in the different sites, the
aggregate power output distribution approaches a normal distribution as the number of
independent r.v. in the sum increases, according to the Central Limit Theorem [1]. In Figure 4.6,
the aggregate power distributions for different number of WTGs are presented assuming
independence between the wind speeds in the different sites.
Figure 4.6: Aggregate power distributions.
The modelling of dependent stochastic generators corresponds to the modelling of non-normals
r.v. and thus, the measure of dependence, product moment correlation, used for the normal
loads do not offer a good representation of dependence. In Figure 4.7, the scatter diagrams and
time-series data for the cases of perfectly dependence between two normal loads, two Weibull
wind speeds and a normal load and a Weibull wind speed are shown.
As can be seen, for the normal loads the perfect dependence corresponds to linear
dependence (ρL-L = 1) between them. For the Weibull speed distributions and the load perfectly
correlated to a Weibull wind speed, the co-fluctuation leads to a non-linear dependence
between them which corresponds to a product moment correlation less than one: ρW-W = 0.9952
and ρW-L = 0.9878. Thus, the cumulative density function transformation should be applied for
the transformation of the marginal into ranks in order to obtain a suitable measure of
dependence. In Figure 4.8 are presented the scatter diagrams for the cases of independence
and perfectly dependence between two normal loads, two Weibull wind speeds and their
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
200
400
600
800
1000
1200Single WTG
MC
S S
am
ple
s
Wind Power(MW)
0 0.5 1 1.5 2 2.5 30
50
100
150
200
250Three WTG
MC
S S
am
ple
s
Wind Power(MW)
0 1 2 3 4 5 60
50
100
150
200
250
300Eigth WTG
MC
S S
am
ple
s
Wind Power(MW)
0 1 2 3 4 5 6 70
50
100
150
200
250
300Ten WTG
MC
S S
am
ple
s
Wind Power(MW)
31
respective ranks. For each case, the distribution of the ranks is the same for the normal loads
distributions and Weibull wind speed distributions, containing the information about the
dependence structure since it is not affected by the marginal distributions.
Figure 4.7: Scatter diagrams and time-series data for the perfectly dependence.
0 5 10 15 20 25 30 35 40 45 5010
15
20
25
30
35
40Load-Load
Samples
Load (
MW
)
12 14 16 18 20 22 24 26 2810
15
20
25
30
35
40
45Load-Load
L1 (MW)
L2 (
MW
)
0 5 10 15 20 25 30 35 40 45 500
2
4
6
8
10
12
14
16
18Wind-Wind
Samples
Win
d s
peed(m
/s)
0 5 10 15 20 250
5
10
15
20
25Wind-Wind
W1 (m/s)
W2 (
m/s
)
0 5 10 15 20 25 30 35 40 45 500
5
10
15
20
Win
d s
peed (
m/s
)
Samples
Wind-Load
0 5 10 15 20 25 30 35 40 45 500
5
10
15
20
25
30
Load (
MW
)
0 5 10 15 20 2512
14
16
18
20
22
24
26
28Wind-Load
W1 (m/s)
L1 (
MW
)
32
Figure 4.8: Scatter diagrams for independence and perfect dependence between normal and
Weibull distributions and their respective ranks.
As mentioned, in the case of independence, for any rank of one r.v. all ranks may occur for the
other which results in a uniform distribution of the samples in the unit square. In the case of
perfect dependence, the occurrence of one rank for one r.v. implies the occurrence of the same
rank for the other which results in a order distribution of the samples in the diagonal of the unit
square.
Since the real dependencies cases in most cases fall between these extreme cases
(independence and perfect dependence) other distributions of the ranks of the r.v. should be
used to the modelling of stochastic dependence that permit separate de dependence structure
10 12 14 16 18 20 22 24 26 2810
15
20
25
30
35
40
45
50Independence
L1 (MW)
L2 (
MW
)
10 12 14 16 18 20 22 24 26 2810
15
20
25
30
35
40
45Perfect Dependence
L1 (MW)
L2 (
MW
)
0 5 10 15 20 250
5
10
15
20
25
30Independence
W1 (m/s)
W2 (
m/s
)
0 5 10 15 20 250
5
10
15
20
25
30Perfect Dependence
W1 (m/s)
W2 (
m/s
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Independence
Random generator U1
Random
genera
tor
U2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Perfect Dependence
Random generator U1
Random
genera
tor
U2
33
from the one-dimensional marginal distributions. Thus, it is necessary a way to joint distributions
of two or more r.v. interacting together in a dependence scenario. By definition, Copulas are
dependence functions that join multivariate distributions functions to their one-dimensional
marginal distributions functions. It is a multivariate distribution function defined on the unit n-
cube [0,1]n, with uniformly distributed marginals [1]. These are the marginals of the ranks
distributions of the r.v. which contain the information concerning the dependence structure
between them.
The Normal or Gaussian copula is obtained by transforming the standard normal
marginals of the multivariate standard normal distribution into uniforms (ranks). This copula is
used in the Joint Normal Transform methodology for the multidimensional dependence
modelling.
4.3.1 Modelling of Two-Stochastic Generators (Wind Turbine
Generators – WTG)
As is generally known, the wind speed at each generator site may be represented as a Weibull
distribution:
Weibull probabilistic distribution function:nNopq? rR � rq soqtrcu vwx y.sz{t
|} with scale parameter
η and shape parameter β.
This distribution is used for the modelling of the prime mover: W1 for site 1 and W2 for site 2 with
the parameters η1 = 8.39, β1 = 2.1 and η2 = 7.70, β2 = 2.00. The WTG power-wind speed
characteristic is used for the energy conversion system model of two pitch-controlled WTGs of
1MW nominal power, with the cut-in, nominal and cut-out wind speeds equal to 3, 13 and 25
m/s respectively (Figure 4.9a and Figure 4.10a). The WTG power output distributions for the
WTGs are obtained through the non-monotonic transformation of the wind speed distributions
(Figure 4.9b and Figure 4.10b).
(a) (b)
Figure 4.9: (a) Wind Speed Distribution and WTG Wind Speed-Power Characteristic (b) WTG
Power Output Distributions for the WTG 1.
0 5 10 15 20 25 300
100
200
300
400
500
MC
S S
am
ple
s
Wind Speed (m/s)
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
Win
d P
ow
er
(MW
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
200
400
600
800
1000
1200
1400
Wind Power (MW)
MC
S S
am
ple
s
34
(a) (b)
Figure 4.10: (a) Wind Speed Distribution and WTG Wind Speed-Power Characteristic (b) WTG
Power Output Distributions for the WTG 2.
Figure 4.11: Scatter diagram and time-series data for the independence case.
Figure 4.12: Scatter diagram and time-series data for the perfect dependence case.
As mentioned, the assumption of independence between wind activities in different sites may be
non-realistic in the case of WTGs situated in relatively small area due to the similar weather
conditions. In this case, a positive dependence is observed between the wind speeds in the
different locations which should be taken into account. In Figure 4.11 and Figure 4.12 are
presented the results for two 10000-sample MCS, the scatter diagrams and time-series data for
0 5 10 15 20 25 300
100
200
300
400
500
MC
S S
am
ple
s
Wind Speed (m/s)
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
Win
d P
ow
er
(MW
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
200
400
600
800
1000
1200
1400
1600
Wind Power (MW)
MC
S S
am
ple
s
0 5 10 15 20 250
5
10
15
20
25Independence
Random Variable W1(m/s)
Random
Variable
W2(m
/s)
0 5 10 15 20 25 30 35 40 45 500
2
4
6
8
10
12
14
16
18Independence
MCS Samples
W1,W
2(m
/s)
0 5 10 15 20 250
5
10
15
20
25
30Perfect Dependence
Random Variable W1(m/s)
Random
Variable
W2(m
/s)
0 5 10 15 20 25 30 35 40 45 500
5
10
15
20
25Perfect Dependence
MCS Samples
W1,W
2(m
/s)
35
the cases of independence and perfect correlation between the winds speed r.v W1 and W2 in
two sites.
In the independence case, the results show a high spreading of points and that the
generated sequences vary randomly as was expected due to decoupling between the winds
speeds in the two sites. In the case of perfect dependence, the points are perfectly correlated,
one r.v follows perfectly the fluctuations of the other which means, a high (low) value of the wind
speed r.v. in one site is combined with a high (low) value of the wind speed r.v in the other site.
In Figure 4.13 the aggregate power output for two cases is presented. In the case of
independence, extreme values of one generator can be combined with all power outputs of the
other. In the case of perfect dependence the extreme values of one implies the extreme values
of the other.
Figure 4.13: Aggregate power output for the two cases.
Table 4.1: Power Output Mean Values and Standard Deviations
MEAN P1 (MW) P2 (MW) P1 + P2 (MW)
Independence 0,43928 0,38831 0,82758
Perfect Dependence 0,43928 0,38797 0,82724
ST.DEVIATION P1 (MW) P2 (MW) P1 + P2 (MW)
Independence 0,31982 0,31122 0,4449
Perfect Dependence 0,31982 0,31074 0,6300
As was expected by the knowledge of probability theory, different aggregate power distributions
around the same mean are obtained for the different cases since the mean value of a sum of
r.v. is independent of the dependence structure and equal the sum of their means values (Table
4.1). It also can be seen that the standard deviation for the perfect dependence is much higher.
4.3.2 Joint Normal Transform (JNT) Methodology
The power system stochastic modelling involves the definition of the mutual stochastic
dependence structures between all the r.v. involved in the multivariate uncertainty analysis
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
50
100
150
200
250
300
350Independence
MC
S S
am
ple
s
Aggregated Power P1 + P2 (MW)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
200
400
600
800
1000
1200
1400Perfect Dependence
MC
S S
am
ple
s
Aggregated Power P1 + P2 (MW)
36
problem. This procedure is very complex since it involves ( 1) 2N N−
mutual dependence
structures for a system with N r.v. inputs. The Joint Normal Transform method can be employed
for multidimensional dependence modelling [1], involving the following tasks:
1) Generate an n-vector of correlated uniform rank distributions U according to a rank
correlation matrix Rr using a multivariate normal copula:
1.1) Simulate an n – dimensional vector of independent standard normal samples z.
1.2) In order to obtain the rank correlation matrix Rr is necessary an appropriate
product moment correlation matrix ~ � $ ��� s�� ~�t�since product moment
correlation and rank correlations for the joint normal distribution are not equal.
1.3) Application of the linear transformation y = L× z such that TL L R× = (Cholesky
decomposition) to the vector z, where L is a lower triangular matrix. The n –
vector y forms a sample drawn from a vector of standard normal r.v. Y correlated
according to the correlation matrix R, i.e. follows a standardized multivariate
normal distribution, with correlation matrix R.
1.4) Application of the standard normal cumulative distribution function to transform
the standard normals to uniforms, by setting ( )i iu y= Φ whereΦ is the standard
normal cumulative distribution function. The rank correlations between the
variables keep unchanged and thus, a uniform random vector n – vector U,
consistent with a specified matrix Rr, is obtained.
2) The transformation 1( )ii L ix F u−= is applied to obtain the marginal distributions Xi.
It is a mathematical requirement that the correlation matrix R be positive definite for being able
to carry out the factorization TL L R× = , i.e. for all�OhXijIk? OlmO Y I�
4.4 Conclusions
The multivariate uncertainty analysis of power systems involves decoupling between the
modelling of the marginal distributions and the modelling of the stochastic dependence
structure. This approach allows the modelling of correlated non-normal distributions whose
dependence cannot be well represented by the product moment correlation. In order to
measure the dependence between non-normals distributions, the one-marginal distributions are
transformed into ranks whose product moment correlation (the rank correlation���) provides an
adequate measure of dependence. The functional relationship between the ranks is modelled
by Copula functions. The transformation of the marginals into ranks and the use of a
multidimensional normal copula for the dependence modelling is the basis for the Joint Normal
Transform methodology. The joint distribution of the stochastic system inputs is represented
based on the available information of their mutual stochastic behaviour, involving the calculation
of the product moment/rank correlation matrix by the time-series data.
37
Chapter 5
Methodologies for the
modelling of 'Horizontally-
Operated' Power Systems 5 Methodologies for the modelling of Horizontally-Operated Power Systems
The power system industry undergoes a radical change: the transition from the ’vertical’ to a
’horizontally-operated’ power system due to the large-scale incorporation of Non-Dispatchable
(stochastic), Distributed Generation (NDT/DG) which is one of the major challenges for the
power systems of the future. For the system operational planning and design, it is necessary a
stochastic approach since the incorporation of these units introduces generation uncertainty in
the system, in addition to the uncertainty of the consumption. In the previous chapter, the Joint
Normal Transform methodology was presented for multidimensional dependence modelling. In
the case of a large number of stochastic system inputs, it is necessary to reduce the
multidimensional dependence model by specific model reduction techniques.
38
5.1 Problem Formulation
In the ‘Horizontally-Operated’ Power System (HOPS), a large share of the power generation
takes place in a large number of small to medium scale generators situated in the lower voltage
levels. This structural change leads to the abatement in system generation dispatchability, due
to the division of the total generation in the system between the conventional dispatchable
Centralized Generation (CG) and the Non-Dispatchable (stochastic), Distributed Generation
(NDT/DG) that makes use of an uncontrollable prime energy mover, as in the case of
renewables, waste heat, etc. The small to medium scale, geographically distributed power
generators can either be small-scale customer-owned conventional generators or stochastic
generators, e.g. renewable energy sources, combined heat-power plants etc.
In this new system structure, the high penetration from stochastic distributed generation in
the lower voltage levels has changed the system structure, transforming the Distribution
Networks (DNs) into active systems that contain both loads and generators and exchange
power with the Transmission Network (TN) bidirectionally, according to the generation and
consumption equilibrium in each DN.
For the system modelling, each DN can be represented as an aggregated load in parallel
with aggregated generation [8]. The net power exchange between DN and TN will be:
∑∑==
−=ii DG
k
kiDG
L
j
jiLiDN PPP1
),(
1
),()( , DNNi ,...,1=
(5.1)
where NDN is the number of DNs and Li and DGi are the number of loads and DG units
respectively, implemented in DN i.
This transformation of the power system leads to the increase of the variability in the power
flows on the system lines. When the stochastic generation inside the DN is high and the load is
low, reverse power flows are expected; in the opposite case, the TN should provide net power
to the DN equal to load minus generation. In this case, the deterministic measure may prove
insufficient for the system analysis.
The Monte Carlo Simulation (MCS) is the most indicated method to solve the multivariate
uncertainty problem created by the incorporation of Stochastic Generation (SG) in the system.
The analysis of the system steady-state for the set of all possible inputs (load/generation)
involves the definition of the Deterministic System Model (DSM) and the Stochastic System
Model (SSM). Each uncertainty system input is represented as a random variable (r.v.) with a
specific probability density function (pdf). The DSM is used in order to capture the system
steady-state operation in a specific instant in time. The SSM involves the definition of the
stochastic structures between the uncertain system inputs, i.e., the loads and the stochastic
generation. In the MCS method, pseudo-random samples of the stochastic system inputs are
generated in accordance with the respective probability distributions and dependence structures
provided by SSM and then, the DSM is solved for each sample in order to obtain the sets of
39
samples for the output quantities of interest which are subjected to a statistical analysis. Thus,
the probability distributions of the uncertain system inputs (loads/stochastic generation) are
propagated through the DSM and the respective system state (node voltage) and output (line
power flow) distributions are obtained. These distributions contain all necessary information for
the quantification of the system operational risk, i.e. lines overloading, voltage violations, etc.
Thus, the modelling approach involves the following tasks:
a) Model the one-dimensional marginal distributions,
b) Model the dependence structure.
The one-dimensional marginal distributions of the system inputs can be easily obtained
by the statistical processing of the respective real data (wind speed, solar radiation, load data),
but the modelling of the stochastic dependencies is the most cumbersome part in the modelling
procedure, due to data unavailability or modelling complexity, since the assumption of
independence is not valid. These stochastic dependencies correspond to the mutual
dependencies between loads, between stochastic generators and between the loads and the
stochastic generators. In a relatively small geographic area, covered by a DN, the prime mover
distributions are expected to be positively correlated due to similar weather conditions and so
are the loads due to similar consumer profiles. For a typical power system the number of loads
and stochastic generators may reach some thousands and thus, is necessary to define the joint
power injections of a large number of stochastic generators and loads situated in a large
geographic area with different prime movers, different converter technologies and different
consumer profiles [4] [8].
5.2 Solution Formulation
In every power system there are groups of highly correlated random inputs consisting of
stochastic generators situated in relatively small geographic areas that use the same generation
technology due to the mutual coupling to the same stochastic prime mover. Their power outputs
are expected to be strongly coupled, i.e. to follow similar fluctuations. The others groups are the
system loads of the same type due to similar consumer behaviour. Thus, the system comprises
groups, designed by clusters, of strongly positively correlated r.v., i.e. the r.v. belonging to each
of these groups present similar stochastic behaviour and the dependence structure observed
between them is close to perfect correlation. A simplified stochastic model can be used for the
modelling of the dependence structure of the cluster and the stochastic model of the system is
reduced to the definition of the stochastic dependencies between the respective clusters [4] [8].
This dependence structure between the respective clusters cannot be approximated by some
simplified model, since the dependencies are not close to perfect correlation.
In this chapter, for the solution of this problem are used two methodologies:
40
1) Stochastic Bounds Methodology: extreme dependence structures
2) Joint Normal Transform Methodology: dependence structure modelling
The stochastic modelling of such correlated clusters is a very difficult problem, since there is an
infinite stochastic dependencies number that may be applied for the system modelling.
The Stochastic Bounds Methodology is applied in order to tackle this problem through the
definition of stochastic bounds, i.e. extreme dependence structures that can bound all real
cases. The results can be used for the adequacy assessment and risk management of the
system [8].
The Joint Normal Transform Methodology is applied in order to model the dependence
structure between the clusters, based on the mutual correlations obtained by data analysis [4].
5.3 Stochastic Bounds Methodology (SBM)
In this section, is considered a methodology for the modelling and analysis of horizontally-
operated power systems to deal with the operational uncertainty introduced by the high
penetration of stochastic renewable generation.
The principles for this modelling are based on the decoupling of the single individual
behaviour (marginal) of the input random variables (wind speed/load) from the concurrent
behaviour of them (dependence structure). For the distribution functions (marginals) of
individual input r.v. can be available reasonable information while for the dependence structure
hardly any or no information is available. The stochastic dependence modelling of such random
inputs is a very difficult problem, since there is an infinite stochastic dependencies number that
may be applied for the system modelling. This methodology introduces the definition of
stochastic bounds, i.e. extreme dependence structures that can bound all real cases.
According to the Stochastic Bounds Methodology (SBM), all possible dependence
structures for a number of positively correlated r.v. can be bounded between two extreme
cases: independence (lower bound) and comonotonicity (upper bound). In the case of two r.v.,
the lower bound corresponds to the case of perfect negative correlation between them, i.e. the
case of countermonotonicity [1].
5.3.1 Upper bound: comonotonicity
Comonotonicity is a dependence concept that refers to the case of extreme positive
dependence between the r.v. and thus, includes all cases of perfect positive non-linear
dependence providing a more general dependence concept than perfect correlation.
Comonotonic r.v. are increasing functions of the same underlying random factor, i.e. they
41
always vary in the same way, meaning that if one increases, then all the others increase too. In
this case, the same random generator is used for the system modelling. Denoting by (
C C C1 2 MY ,Y ,....Y ) the comonotonic version of the random vector ( 1 2 MY ,Y ,...Y ), they can be
expressed as:
Comonotonicity: i
C -1i YY = F (U) , i = 1,2,...M,
(5.2)
where U is an r.v. that is uniformly distributed on the unit interval and 1 2 MY Y YF ,F ,...F are the
marginal distributions of both random vectors.
The sums i iY∑ and C
i iY∑ have the same expected value, but different overall
probability distributions. Comonotonicity provides the higher risk case corresponding to the
worst-case scenario, i.e. the extreme distribution of the sum of the r.v.
This case corresponds to the Fréchet-Hoeffding copula CU which offers the upper
bound to all possible copulas:
( , ) 1XY U X Y rF (x, y)= C (F (x),F (y)) X Yρ⇒ = (5.3)
The FX and FY are the cumulative distribution functions of the r.v. X and Y. The FXY is a joint
distribution function with margins FX and FY.
5.3.2 Lower Bounds: countermonotonicity – independence
The lower bound is more difficult to obtain. In the case of two r.v., the lower bound corresponds
to the case of perfect negative correlation between them, i.e. the case of countermonotonicity.
Denoting by ( CM1Y , CM
2Y ) the countermonotonic version of the bidimensional random vector ( 1Y ,
2Y ), they can be expressed as:
Countermonotonicity: i
CM -11 YY = F (U) ,
2
CM -12 YY = F (1- U) (5.4)
Countermonotonicity offers always the lower stochastic bound to the aggregate stochasticity of
two r.v.. However, in the case of more than two r.v., the application of this concept is not
possible, since by definition more than two r.v. cannot be mutually perfectly negatively
dependent.
This case corresponds to the Fréchet-Hoeffding copula CL which offers the lower bound
to all possible copulas:
( , ) 1XY L X Y rF (x, y)= C (F (x),F (y)) X Yρ⇒ = − (5.5)
The FX and FY are the cumulative distribution functions of the r.v. X and Y. The FXY is a joint
distribution function with margins FX and FY.
In the case that it is known that a weak positive dependence exists between a number
of r.v., the lower bound shifts to the case of independence. Different random generators are
42
used for each r.v. Denoting by ( I I I1 2 MY ,Y ,....Y ) the independent version of the random vector (
1 2 MY ,Y ,...Y ), they can be expressed as:
Independence: i
I -1i Y iY = F (U ) , i = 1,2,...M,
(5.6)
The case of independence provides the lower risk case, corresponding to a best-case scenario,
i.e. the distribution of the sum of the r.v. with minimum variability.
In all different dependence scenarios, corresponding to different dependence structures
between the system inputs, different distributions of their sum around the same mean are
obtained since their marginal distributions are the same for all the different operating conditions.
The different extreme dependence structures correspond to distributions for the sum of the r.v.
with the same mean, the minimum variance in the case of independence and maximum in the
case of comonotonicity. All the other dependence structures correspond to sums whose
variances lie in between those two bounds.
A higher variance implies a distribution with a larger spread around the mean, i.e. a
higher probability of obtaining extreme values. The comonotonic scenario can provide an easy
method for the stochastic design of the system in the case that the r.v. are strongly positively
correlated since this dependence concept provides the higher risk case, i.e. the extreme
distribution of the sum of these r.v..
The use of the independence scenario for the system analysis, in cases of positively
correlated r.v., is a fallacy since all realistic dependence structures corresponds to more severe
impact to the system. Thus, the extreme dependence concepts correspond to the extreme
forms of the sum of the r.v.
5.4 Stochastic Model Reduction
In power systems stochastic modelling, groups of strongly positively correlated r.v. inputs can
be defined as clusters, consisting of loads of the same type and/or stochastic generators of the
same type situated in a relatively small geographic area.
The application of this methodology involves the identification of the positively correlated
r.v., the definition of their one-dimensional marginal distributions which remain the ones
obtained by the system analysis and the substitution of the dependence structure between the
positively correlated r.v. by the extreme structure of comonotonicity [4] [8]. Thus, comonotonic
sampling can be used for the distribution of the aggregated load and for the distribution of the
Distributed Generation (DG) production in each DN. The use of this concept for the cluster
modelling corresponds to the highest-risk case for the aggregate stochasticity of the load cluster
and Stochastic Generation (SG) cluster, offering a worst case based design. In this case, based
on the Stochastic Bounds Methodology, it can be shown that this dependence concept
corresponds to the distribution with maximum spread around the mean for the sum of these r.v.,
i.e. corresponds to the extreme case of stochastic power production/consumption for the
43
cluster. This approximation is close to reality but safe, since it corresponds to a risk measure
that represents the worst case scenario for the system impact of the cluster.
The system stochastic model can be subdivided into K SG and M load clusters of positively
correlated inputs. Each SG cluster k and load cluster m contains a total of gk SG units and lm
loads respectively. For the application of comonotonic sampling, the same uniform random
number U is used for the sampling of all r.v. belonging to the same cluster. The dependence
structure in each cluster is modelled according to the following procedure:
1) As it is the general practice in MCS, first is generate K + M uniform random samples
Uk,m in [0,1] for each cluster.
2) Application of the transformation -1
Xx = F (u) to obtain the marginal distributions for each
r.v. :
SG cluster k: kg
-1kg G kG = F (U ), kg = 1,...,g for the g-th generator of the k-th
cluster
(5.6)
Load cluster m: ml
-1ml L K+mL = F (U ), ml = 1,...,l for the l-th load of the m-th
cluster
(5.7)
k = 1…K and m = 1…M.
This comonotonic approximation of the dependence structure of a cluster is defined as a
Stochastic Plant (SP). This concept provides a solution for the system modelling when no
information concerning the dependence structure is available, or when it is too cumbersome to
work with. According to the SBM, the obtained aggregate distributions kG (U )k kg∑ and
m ml mL (U )∑ correspond to the extreme power output distribution of the cluster.
When a stochastic plant comprises both stochastic generators as well as loads, the worst-
case approach in this case corresponds to a dependence scenario with a combination of high
(low) load and low (high) stochastic generation. In this case, the generation and load must be
countermonotonic to obtain the extreme distribution for net power exchange between each DN i
and the TN:
SG component k: kg
-1kg G iG = F (U ), kg = 1,...,g
(5.8)
Load component m: ml
-1ml L iL = F (1-U ), ml = 1,...,l
(5.9)
k = 1…K and m = 1…M. A single random generator Ui is used for the modelling of the cluster.
This is a worst-case approach and offers a trade off between modelling simplicity and
accuracy. In the cases of geographically small systems due to the existence of strong positive
44
dependencies between the system inputs, the closer the clusters dependence structure, chosen
by the system designer, is to positive dependence, more realistic is this approach provided by
these dependence concepts. These concepts can provide conservative modelling results when
the cluster dependence structure is not strongly positive.
The system dependence structure is reduced to the definition of the mutual stochasticity
between the comonotonic clusters of system inputs by the model approximations (Stochastic
Plants), instead of the complete dependence structure between all the system r.v. [1]. The
different clusters can be combined based on the three extreme dependence concepts [8].
5.5 Joint Normal Transform (JNT) Methodology
In each cluster, all the r.v. can be modelled using the same uniform random number since they
are fluctuating in the same way, but for r.v. belonging to different clusters such approximation is
quite conservative and their mutual correlation should be measured. Thus, each cluster can be
regarded as one stochastic entity due to the perfectly correlation between their r.v. and the
system dependence structure is reduced to the definition of the mutual stochasticity between
the clusters.
In this section, it is applied the Joint Normal Transform Methodology to define dependence
structure between the clusters, based on the mutual correlations obtained by data analysis [4].
For this, in the case of a system subdivided into K stochastic generation and M load clusters, is
created K + M uniform sampling vectors Uk +M consistent with a specified correlation matrix R.
The steps for the application of this method can be described as follows:
1) Simulate a K + M – dimensional vector of independent standard normal samples z.
2) Application of the linear transformation = L×zy
such that TL×L = R (Cholesky
decomposition) to the vector z, where L is a lower triangular matrix. The K + M – vector
y forms a sample drawn from a vector of standard normal r.v. Y correlated according to
the correlation matrix R, i.e. that follows a standardized multivariate normal distribution,
with correlation matrix R.
3) Application of the standard normal cumulative distribution function to transform the
standard normals to uniforms, by setting ( )i iu y= Φ whereΦ is the standard normal
cumulative distribution function. The rank correlations between the variables keep
unchanged and thus, a uniform random vector K +M – vector Uk + M, consistent with a
specified matrix R, is obtained.
The correlation between the clusters is produced by the way that the values of Uk + M co-vary
across the different simulations. The K + M – dimensional vector Uk+M will be used as random
numbers for the modelling of the clusters. It is a mathematical requirement that the correlation
matrix R be positive definite for being able to carry out the factorization TL×L = R , i.e. for
45
all�OhXijIk? OlmO Y I. Since this method involves the change of marginal distributions, a small
difference between the input and the output rank correlation matrices is obtained but it is in
general small enough so that the applicability of the overall method is not affected.
5.6 Stochastic Generation (SG) in Bulk Power System
5.6.1 System data
The study case, for the application of the methodologies, involves the modelling of a bulk
power system, consisting of a TN and the underlying DNs with a high penetration of wind power
[4] [8]. The 5 bus – 7 branch test network of Figure 5.1 is used as a system model. The large-
scale implementation of stochastic generation in the underlying distribution systems at nodes 3,
4 and 5 of the test system is considered by the connection of 45, 40 and 60 Wind Turbine
Generators (WTGs) of 1MW nominal power in each distribution system respectively. The
system comprises two centralized large CG units connected to the buses 1 and 2 and four loads
at the buses 2, 3, 4 and 5 representing the aggregated power demand of the four underlying
DNs. Bus 1 is the slack bus of the system. In Table 5.1, the system data are presented. The
flowchart of the complete computation is shown in Figure 5.2.
Figure 5.1: 5-bus / 7-branch Test System (Hale Network)
Table 5.1: Test system data
N
Generators High Load Mean value Line p-q
*Zpq (pu)
*Ypq (pu)
PG (MW)
QG (MVAr) PL
(MW) QL
(MVar)
1 Slack - - 1-2 2 + j6 j6
2 40(CG) 30(CG) 20 10 1-3 8 + j24 j5
3 45(DG) - 45 15 2-3 6 + j18 j4
4 40(DG) - 40 5 2-4 6 + j18 j4
5 60(DG) - 60 10 2-5 4 + j12 j3
3-4 1 + j3 j2
4-5 8 + j24 j5
*10-2
46
START
STOCHASTIC SYSTEM MODEL
Load data
Non-CG data
For 20 000 samples
Generate MCS random sample for load data at each
load bus.
Generate MCS random sample for DG data at each
load bus.
Subtract MCS sample for DG data from MCS
sample for load data at each load bus to create
MCS sample for the aggregate data at each
distribution system.
DETERMINISTIC SYSTEM MODEL
Run load flow computation
System Configuration
data
CG power output
Record load flow computation results for each
sample.
Perform statistical analysis on the load flow results and the output distribution
Check any violation on the test system
STOP
Figure 5.2: Flowchart of the complete computation
47
The stochastic analysis of the system involves the definition of the marginal distributions of the
system inputs (loads and WTGs) and the stochastic dependencies between them. The data for
the prime mover activity (Weibull wind distributions) and aggregated load in each distribution
system are considered to be known.
5.6.2 Marginal Distributions: Load – Wind Turbine Generators
A. Loads
The methodology used for the modelling of the aggregated load in each DN is based on
mixtures of normal r.v. (two or three). For the application of this technique the data are grouped
in time-frames and the load in each time-frame is modelled as a normal r.v. with parameters
derived from the analysis of the respective data group. Thus, the Load Duration Curve of each
HV/MV transformer is approximated as a two or three step ladder function, which is used for the
derivation of the parameters of the normal distributions of the mixture (mean value – standard
deviation) and their relative ratio. In this case, two time frames are considered (high – low load
state), with the low load level being 50% of the high one for 80% of the total time. Each load
level corresponds to different time periods of consumption with the consumption in each time-
frame being simulated by a normal distribution with high load mean values presented in Table
5.1 and a standard deviation equal to 25% of the low-load state mean value for the low level
and 4% of the high-load state mean value for the high one. The generated probability density
function and cumulative distribution of a 20000 - samples MCS for the load in the DN at node 4,
normalized according to the high load mean value are presented in Figure 5.3 and is
representative for all systems loads. As can be seen, the cumulative distribution obtained is a
very good approximation of the load duration curve.
Figure 5.3: Load probability density function and load cumulative distribution of DN 4 for a
20000-sample MCS.
B. Wind Turbine Generators
The output power distribution of a stochastic generator is derived by the application of the non-
monotonic function of the energy converter to the prime mover probability distribution. In the
-5 0 5 10 15 20 25 30 35 40 450
100
200
300
400
500
600
700
800
900
Load (MW)
MC
S S
am
ple
s
-5 0 5 10 15 20 25 30 35 40 450
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
4
Load (MW)
MC
S S
am
ple
s
CDF
48
case of a WTG, the prime mover activity is modelled by the wind speed distribution, which is a
Weibull distribution while the energy converter is modelled through the wind speed-power
output characteristic. The characteristic considered in this analysis is with cut-in, nominal and
cut-out wind speed values of 3.5, 14 and 25 m/s, respectively. Due to this non-monotonic
transformation, a concentration of probability masses in the values equal to zero (wind speeds
below cut-in and above cut-out value) and nominal (wind speeds between nominal and cut-out
values) is observed. The WTG characteristic, the results of a 20000-sample MCS for the wind
speed and the output power distribution for a WTG are presented in Figure 5.4.
(a) (b)
Figure 5.4 : (a) Wind Speed distribution and WTG power curve (b) WTG output distribution for a
WTG of DN 4 for a 20000-sample MCS.
5.6.3 Stochastic Bounds Methodology (SBM)
The application of the Stochastic Bounds Methodology involves the clustering of the
positively correlated r.v. and the combination of the different clusters based on the three
extreme dependence concepts [8]. The marginal distributions referred to the individual
behaviour of each stochastic generator and load, are kept constant for the different dependence
scenarios.
A. Simulation Details
A 20000-sample MCS was used for the system simulation. The simulation was programmed in
Matlab. The total number of random variables involved in the analysis is 149: 45 WTG r.v. for
DN 3, 40 WTG r.v. for DN 4, 60 WTG r.v. for DN 5 and 4 load distributions. The simulation
duration was 826 seconds on a Intel(R) Core(TM) 2 Duo CPU 2.4 GHz machine. For the test
system are defined seven clusters: four load clusters (DNs 2-3-4-5) and three WTG clusters
(DNs 3-4-5). Five different extreme dependence scenarios are considered for the system
analysis: the lower stochastic bound (LB – all system r.v. are independent from each other) and
0 5 10 15 20 25 30 350
150
300
450
600
750
MC
S S
am
ple
s
Wind Speed (m/s)
0 5 10 15 20 25 30 350
0.2
0.4
0.6
0.8
1
Win
d P
ow
er
(MW
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
500
1000
1500
2000
2500
3000
Wind Power (MW)
MC
S S
am
ple
s
49
the four upper bounds (UB). Therefore, five different distributions are obtained for each of the
system variables.
In this first scenario (Figure 5.5), seven random generators Ui are used for the system
sampling, corresponding to the case that the different clusters are considered independent and
thus, no severe positive dependence appears between the loads and the stochastic generation
in the different DNs.
In this second scenario (Figure 5.6), four random generators are used for the system sampling,
because the stochastic generation remains independent between the different DNs, but the
loads in the system are considered to be positively dependent. In this case is used the upper
bound comonotonic bound for their combined modelling.
In this third scenario (Figure 5.7), two random generators are used for the system sampling,
DN 3 WTG cluster 1
45 units Sampling (U1)
DN 4 WTG cluster 2
40 units Sampling (U2)
DN 5 WTG cluster 3
60 units Sampling (U3)
DN 2 Load cluster 1
20 MW Sampling (U4)
DN 4 Load cluster 3
40 MW Sampling (U4)
DN 3 Load cluster 2
45 MW Sampling (U4)
DN 5 Load cluster 4
60 MW Sampling (U4)
DN 3 WTG cluster 1
45 units Sampling (U1)
DN 4 WTG cluster 2
40 units Sampling (U2)
DN 5 WTG cluster 3
60 units Sampling (U3)
DN 2 Load cluster 1
20 MW Sampling (U4)
DN 4 Load cluster 3
40 MW Sampling (U6)
DN 3 Load cluster 2
45 MW Sampling (U5)
DN 5 Load cluster 4
60 MW
Sampling (U7)
Figure 5.5: System upper bound stochastic modelling: first clustering scenario
Figure 5.6: System upper bound stochastic modelling: second clustering scenario
50
because the stochastic generation is also considered to be positively correlated, but they are
not correlated to the load in the system.
In this fourth scenario (Figure 5.8), only one random generator is used for the sampling of the
system, because is considered positive correlation between the stochastic generation and the
load in the system that may occur due to their mutual dependence on weather. In this case
which corresponds to the worst case for the system stochastic modelling, is used the
countermonotonic sampling in order to define the extreme distribution for the system output.
DN 4 Load cluster 3
40 MW Sampling
(1-U1)
DN 3 Load cluster 2
45 MW
Sampling
(1-U1) DN 5 Load cluster 4
60 MW
Sampling (1-U1)
DN 2 Load cluster 1
20 MW
Sampling (1-U1) DN 4
WTG cluster 2
40 units
Sampling (U1)
DN 3 WTG cluster 1
45 units
Sampling (U1)
DN 5 WTG cluster 3
60 units Sampling (U1)
Figure 5.7: System upper bound stochastic modelling: third clustering scenario
Figure 5.8: System upper bound stochastic modelling: fourth clustering scenario
DN 3 WTG cluster 1
45 units Sampling (U1)
DN 4 WTG cluster 2
40 units Sampling (U1)
DN 5 WTG cluster 3
60 units Sampling (U1)
DN 2 Load cluster 1
20 MW Sampling (U2)
DN 4 Load cluster 3
40 MW Sampling (U2)
DN 3 Load cluster 2
45 MW Sampling (U2)
DN 5 Load cluster 4
60 MW Sampling (U2)
51
B. System Analysis of Results
As can be seen in Figure 5.9, Figure 5.10 and Figure 5.11 different power flows distributions
around the same central point are obtained, where the lower bound represents the ‘best-case
scenario’, giving distributions with minimum dispersion for all the system lines, while the upper
bounds represents the ‘worst-case scenarios’. The mean values are the same for all the
different scenarios, while the standard deviations are minimal for the lower stochastic bound
and maximal for the upper bounds (Table 5.2 and Table 5.3). Since the same marginal
distributions of the inputs of the steady-state system model are used in the different
dependence scenarios, their sum obtained in all cases will have the same mean value.
Figure 5.9: Power Flow Distributions for the system lines.
-150 -100 -50 0 50 1000
200
400
600
800
1000
1200
1400
1600
Line 1-2 power flow (MW)
MC
S S
am
ple
s
1º scenario
2º scenario
3º scenario
4º scenario
Independence
-50 -40 -30 -20 -10 0 10 20 30 40 500
200
400
600
800
1000
1200
1400
1600
Line 1-3 power flow (MW)
MC
S S
am
ple
s
1º scenario
2º scenario
3º scenario
4º scenario
Independence
52
Figure 5.10: Power Flow Distributions for the system lines.
-20 -15 -10 -5 0 5 10 15 20 25 300
200
400
600
800
1000
1200
1400
Line 2-3 power flow (MW)
MC
S S
am
ple
s
1º scenario
2º scenario
3º scenario
4º scenario
Independence
-30 -20 -10 0 10 20 300
200
400
600
800
1000
1200
1400
Line 2-4 power flow (MW)
MC
S S
am
ple
s
1º scenario
2º scenario
3º scenario
4º scenario
Independence
-60 -40 -20 0 20 40 600
200
400
600
800
1000
1200
1400
Line 2-5 power flow (MW)
MC
S S
am
ple
s
1º scenario
2º scenario
3º scenario
4º scenario
Independence
53
Figure 5.11: Power Flow Distributions for the system lines
Table 5.2: Line Power Flows: Mean values
Line p - q
LB P(MW)
UB (1) P(MW)
UB (2) P(MW)
UB (3) P(MW)
UB (4) P(MW)
1 - 2 -3,738 -3,778 -3,702 -3,538 -2,891
1 - 3 3,767 3,770 3,742 3,792 4,016
2 - 3 6,232 6,256 6,201 6,227 6,334
2 - 4 6,595 6,614 6,576 6,587 6,712
2 - 5 11,164 10,991 11,159 11,133 11,397
3 - 4 2,146 2,109 2,202 2,063 2,133
4 - 5 0,641 0,5400 0,649 0,609 0,6338
As can be seen, the mean value of the system outputs for the different dependence structures
remains almost the same (the mean values are not effectively equal because they result from a
sampling process) which is an indication of the linear behaviour of the system model and thus,
-40 -30 -20 -10 0 10 20 30 400
200
400
600
800
1000
1200
1400
Line 3-4 power flow (MW)
MC
S S
am
ple
s
1º scenario
2º scenario
3º scenario
4º scenario
Independence
-25 -20 -15 -10 -5 0 5 10 15 200
200
400
600
800
1000
1200
1400
1600
Line 4-5 power flow (MW)
MC
S S
am
ple
s
1º scenario
2º scenario
3º scenario
4º scenario
Independence
54
they can be approximated as a linear combination of the inputs due to the central limit theorem.
For the same reason, the shape of the power flow distributions in the lines 1-2, 1-3, 2-3, 2-4 and
2-5 are approximate normals although the inputs are highly non-normals.
Table 5.3: Line Power Flows: Standard deviations
Line p - q
LB P(MW)
UB (1) P(MW)
UB (2) P(MW)
UB (3) P(MW)
UB (4) P(MW)
1 - 2 25,315 31,968 33,940 43,086 50,996
1 - 3 9,826 12,650 13,387 17,557 20,765
2 - 3 4,744 6,855 7,040 8,9214 10,491
2 - 4 5,484 7,553 7,857 10,333 12,162
2 - 5 11,819 18,197 18,550 21,243 25,026
3 - 4 5,800 11,236 11,048 8,4509 10,020
4 - 5 2,807 6,998 6,660 2,884 3,420
The standard deviations however present an increasing trend while passing from the lower
bound to the upper bounds. The lower bound (’best-case scenario’) gives distributions with
minimum dispersion for all the system lines. In the lines 3-4 and 4-5, the maximum dispersion
(extreme distributions) is obtained in scenarios (1) and (2) although the worst-case scenario is
expected in the countermonotonic case (upper bound 4). For the lines connected to the system
CG, 1-2, 1-3, 2-3, 2-4 and 2-5, the dispersion of the distributions increases while passing from
the lower bound to the upper bound. Thus, different upper bounds can provide the extreme
distributions for different lines in the system. The system analysis also shows that the lines 2-3,
2-4 e 2-5 are the ones that are mostly stressed and the presence of highly bidirectional power
flows which indicated that the system design should be performed based on the higher values
and should be able to support both power flow directions. Thus, the Stochastic Bounds
Methodology propose an analysis focuses on the worst case of aggregate stochastic stress for
the system and therefore is quite conservative since all real cases correspond to lower system
stress, i.e. more moderate stochastic dependence structures. The assumption of independence
which is the ’best-case scenario’ that can be obtained in the system due to the application of the
stochastic generation, leads to underestimation of the probability of extreme outcomes in the
system since any realistic dependence assumption will result into more extreme distributions,
due to the positive dependence between the inputs. The above methodology permit to measure
the risk of exceeding the system safety margins for the worst case scenarios and take
respective actions to reduce it.
C. Conclusion
This methodology proposes a stochastic modelling approach to deal with the uncertainty
introduce in the HOPS based on the extreme stochastic dependence structures between the
system inputs. For this, this technique proposes the definition of clusters of positively dependent
random variables, the sampling of the random variables belonging to each cluster based on
extreme dependence concepts and the combination of the clusters based on the stochastic
dependence structures prevailing in the system. Based on these concepts, this method provides
the worst case scenarios for the system outputs, represented by their distributions with
55
maximum variation. For the definition of the maximum aggregate effect of the stochastic
generation to the system, the extreme dependence concepts of comonotonicity and
countermonotonicity are used while the independence concept provides the minimum effect to
the system. This methodology permit a better understanding of the impact of stochastic
generation providing important information for the adequacy assessment and risk management
of the system which can be very realistic in the cases of geographically small systems due to
the existence of strong positive dependencies between the system inputs. In large power
systems, this approach can lead to conservative results since the differences between the lower
stochastic bound and maximal stochastic bound can be very large and new techniques should
be used to specify the dependence structure in detail.
5.6.4 Joint Normal Transform (JNT) Methodology
The application of the Joint Normal Transform Methodology involves the clustering of the
positively correlated r.v. and the modelling of the exact correlations between the clusters [4].
This method is a solution to the incorporation of a realistic dependence structure to the power
system modelling.
A. Simulation Details
A 20000-sample MCS was used for the system simulation. The simulation was programmed in
Matlab. The total number of random variables involved in the analysis is 149: 45 WTG r.v. for
DN 3, 40 WTG r.v. for DN 4, 60 WTG r.v. for DN 5 and 4 load r.v. for the aggregated load at the
HV/MV transformers. The simulation duration was 42.8691 seconds on a Intel(R) Core(TM) 2
Duo CPU 2.4 GHz machine. For the test system are defined seven clusters: four load clusters
(DNs 2-3-4-5) and three WTG clusters (DNs 3-4-5).
For the modelling of the dependence structure between the clusters as is presented in
Figure 5.12, the following positive definite correlation matrix corresponding to a typical
dependence structure is used:
=
18.08.07.07.07.07.0
8.018.07.07.07.07.0
8.08.017.07.07.07.0
7.07.07.016.06.06.0
7.07.07.06.016.06.0
7.07.07.06.06.016.0
7.07.07.06.06.06.01
inR
The lower-right matrix corresponds to the mutual stochastic generation correlation. The upper-
left matrix corresponds to the mutual load correlation. The upper-right and lower-left matrixes
correspond to the stochastic generation/load cross correlation.
56
For the comonotonic modelling of each cluster, a uniform 7-dimensional sampling vector U
consistent with the specified correlation Rin, obtained by the application of the Joint Normal
Transform Methodology, is used.
B. System Analysis of Results
The rank correlation matrix measured at the resulting MCS samples is the following:
=
178.077.066.066.065.065.0
78.0178.065.065.065.066.0
77.078.0166.066.064.065.0
66.065.066.0154.053.054.0
66.065.066.054.0154.053.0
65.065.064.053.054.0154.0
65.066.065.054.053.054.01
simR
As can be seen, a small deviation between the input matrix Rin and the one obtained by the
simulations Rsim, due to the transformation of the marginals discussed above. This change is
however small enough so as not to affect the results. The obtained distributions of each
load/generation cluster connected at the same system nodes are presented in Figure 5.13,
DN 3 WTG cluster 1
45 units
Sampling (U1)
DN 4 WTG cluster 2
40 units
Sampling (U2)
DN 5 WTG cluster 3
60 units
Sampling (U3)
DN 2 Load cluster 1
20 MW Sampling (U4)
DN 4
Load cluster 3 40 MW
Sampling (U6)
DN 3 Load cluster 2
45 MW Sampling (U5)
DN 5
Load cluster 4 60 MW
Sampling (U7)
0.8 0.8
0.6 0.6
0.6 0.6
0.8
0.6
0.7
Figure 5.12: Clustering for the 5-bus / 7-branch test System
57
Figure 5.14 and Figure 5.15. The stochastic generation cluster aggregated power production,
obtained as the comonotonic sum of all the stochastic generators belonging to the cluster,
corresponds to the extreme distribution for the cluster power generation, (maximum dispersion
of the power output).
Figure 5.13: Load/generation cluster distributions and power injection at node 3.
0 10 20 30 40 50 600
50
100
150
200
250
300
350
400
Load(MW)
MC
S S
am
ple
s
Node 3
0 5 10 15 20 25 30 35 40 450
200
400
600
800
1000
1200
Wind Power(MW)
MC
S S
am
ple
s
Node 3
-30 -20 -10 0 10 20 30 40 500
50
100
150
200
250
300
350
400
Nodal Power Injection (MW)
MC
S S
am
ple
s
Node 3
58
Figure 5.14: Load/generation cluster distributions and power injection at node 4.
0 5 10 15 20 25 30 35 40 45 500
50
100
150
200
250
300
350
400
Load(MW)
MC
S S
am
ple
s
Node 4
0 5 10 15 20 25 30 35 400
200
400
600
800
1000
1200
1400
Wind Power(MW)
MC
S S
am
ple
s
Node 4
-30 -20 -10 0 10 20 30 40 500
50
100
150
200
250
300
350
400
450
Nodal Power Injection (MW)
MC
S S
am
ple
s
Node 4
59
Figure 5.15: Load/generation cluster distributions and power injection at node 4.
0 10 20 30 40 50 60 70 800
50
100
150
200
250
300
350
400
450
Load(MW)
MC
S S
am
ple
s
Node 5
0 10 20 30 40 50 600
200
400
600
800
1000
1200
1400
Wind Power(MW)
MC
S S
am
ple
s
Node 5
-40 -30 -20 -10 0 10 20 30 40 50 600
50
100
150
200
250
300
350
400
Nodal Power Injection (MW)
MC
S S
am
ple
s
Node 5
60
As the cluster load/generation distributions are similar and the cross-correlations are the same
for each node, the obtained nodal power injections distributions derived by the subtraction of
nodal load and generation, are similar.
The distributions of the line power flows in the system are presented in Figure 5.16 and
their mean values and standard deviations are listed in Table 5.4.
Figure 5.16: Power Flow Distributions for the system lines
The mean values corresponded to the central points of these distributions gives a false
impression about the system loading.
Table 5.4: Mean values and standard deviations of the line power flows
Line p - q
Mean P (MW)
Std P (MW)
1 - 2 -3.831 14.337
1 - 3 3.842 6.130
2 - 3 6.325 3.888
2 - 4 6.601 4.234
2 - 5 11.026 10.288
3 - 4 1.658 6.495
4 - 5 0.579 4.14
C. Conclusion
This methodology proposes an approach for a more realistic dependence modelling of systems
involving a large number of stochastic inputs (loads/stochastic generators) as the ‘Horizontally-
Operated’ Power Systems. First, is necessary the reduction of the system model by the
-80 -60 -40 -20 0 20 40 600
100
200
300
400
500
600
700
800
Line power flow (MW)
MC
S S
am
ple
s
Line 1-2
Line 1-3
Line 2-3
Line 2-4
Line 2-5
Line 3-4
Line 4-5
61
modelling of the random variables, belonging to each cluster of strongly positively correlated
inputs (loads/stochastic generators), based on the worst-case scenario, i.e. the dependence
concept of perfect positive correlation (comonotonicity). Then, the exact dependence structure
between the clusters is modelled based on the correlation matrix and the Joint Normal
Methodology.
This approach shows that distribution systems become active, exchanging power with
the transmission system bidirectionally, with the presence of stochastic generation in the lower
levels of the system.
62
Chapter 6
Application: Integration of
Stochastic Generation (SG) in
a Bulk Power System
6 Application: Integration of Stochastic Generation (SG) in a Bulk Power System
63
6.1 Simulation data
The system model used is the IEEE 39-bus New England test system which comprises 39
buses, 10 CG units and 46 transmission lines (100 kV). The system data are presented
Appendix. The single-line diagram of the 39-bus New England test system [1] is presented in
Figure 6.1.
Figure 6.1: Single-line diagram of the 39-bus New England test system
Table 6.1: WSPs connected to the respective node
WSP W1 W2 W3 W4 W5 W6 W7 W8
Node 4 7 8 12 15 16 20 23
WSP W9 W10 W11 W12 W13 W14 W15
node 24 26 27 28 29 31 39
The wind penetration level in the system is defined as:
��:B���B �� 6��6��AF?l��AB � %II� PDG is the nominal wind power capacity in the system and PLoad,Total is the total amount of
nominal active load (6097 MW). The Wind Stochastic Plants (WSPs) are geographically
distributed throughout the system and connected to 15 system nodes (Table 6.1) for 4
64
penetration levels: no penetration; 1500 MW, penetration level of 25% (100 MW for each WSP);
3000 MW, penetration level of 50% (200 MW for each WSP); 4500 MW, penetration level of
75% (300 MW for each WSP).
6.2 System loads
6.2.1 Marginals
For the modelling of the aggregated load in each Distribution network (DN), the data are
grouped in Time-frames (TFs) and the load in each time-frame is modelled as a normal r.v. with
parameters derived from the analysis of the respective data group. In this case, four time
frames are considered, corresponding to different time periods of consumption with the
consumption in each time-frame being simulated by a normal distribution and the resulting
distribution is obtained by an aggregation procedure as a mixture of these normals. The settings
for the 4-TF modelling are presented in Table 6.2. The load data presented in Table A.1
(Appendix) correspond to the mean value of the high-load-TF of the respective load. The load
reactive power samples are obtained from the active power samples based on a constant load
power factor. The generated probability density function and cumulative distribution of a 10000 -
samples MCS for the load in the DN at node 8, normalized according to the high load mean
value, are presented in Figure 6.2 and is representative for all systems loads. As can be seen,
the cumulative distribution obtained is a very good approximation of the load duration curve.
Table 6.2: TF settings for a 4-TF load modelling of the New England test system.
Time Ratio Mean Load
(% high-load-TF mean)
St. Deviation (% mean load)
TF1 0.2 0.5 0.06
TF2 0.3 0.65 0.1
TF3 0.3 0.85 0.1
TF4 0.2 1 0.03
Figure 6.2: 4-TF load modelling for the New England test system (10000-sample MCS).
200 250 300 350 400 450 500 550 6000
50
100
150
200
250
Load - bus 8 (MW)
Num
ber
of
sam
ple
s
200 250 300 350 400 450 500 550 6000
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Load - bus 8(MW)
MC
S S
am
ple
s
CDF
65
6.2.2 Dependence structure
For the modelling of the dependence structure between the time-conditioned loads, first the
marginal distribution is generated for all TFs, the dependence structure is defined for the
aggregate distribution and then, the Joint Normal Transform Methodology is used for the
modelling of the correlated r.v. The scatter diagram for the modelling of a rank correlation rρ =
0.7 between the loads in buses 8 and 24 is presented for a 10000-sample MCS in Figure 6.3.
The rank correlation obtained at the output samples is 0.6747.
Figure 6.3: Scatter diagrams for the load modelling (10000-sample MCS).
The system loads are considered to be independent from the wind power injections in the
system.
6.3 System wind power
6.3.1 Marginals
The clusters of Wind Turbine Generators (WTGs) connected to the respective underlying
distribution networks are presented in Table 6.1. For the dependence structure of each cluster
of WTGs is used the comonotonic approximation, defined as Wind Stochastic Plant (WSP).
Each WSP comprises 10 sites and so, the analysis comprises a total of 150 wind speed r.v. The
Weibull parameters β and η for the wind speed distributions, for the 150 generation sites are
generated as random numbers drawn in the interval: ∈β [1.9, 2.6] and ∈η [8, 10]. In Figure
6.4, the power output of the WSPs connected at the buses 8 and 24 are presented for the wind
penetration level of 1500 MW (25%).
200 250 300 350 400 450 500 550 600100
150
200
250
300
350
Load - bus 8 (MW)
Load -
bus 2
4 (
MW
)
66
Figure 6.4: WSP power output in the New England test system (10000-sample MCS).
6.3.2 Dependence structure
The Joint Normal Transform (JNT) Methodology is used for the modelling of the correlated
Weibull wind speed distributions based on the mutual correlations, equal to 0.7, between the
different stochastic plants. In Figure 6.5a, the scatter diagram for the wind speed r.v. at WTG 6
of the WSP at bus 8 and WTG 6 of the WSP at bus 24 is presented for a 10000-sample MCS.
In Figure 6.5b, the scatter diagram for the wind power output of the WSPs connected at bus 8
and 24 for the wind penetration level of 1500 MW (25%) is presented.
(a) Wind speed (b) Wind power
Figure 6.5: Wind speed and wind power scatter diagrams (10000-sample MCS).
The rank correlation measured at the output wind speed samples equals rρ = 0.7035 for all
sites belonging to different WSPs, equals 1 for sites belonging to the same WSP and the rank
correlation for the output wind power samples yields values in the interval ∈rρ [0.6711,0.6897].
The change in correlation is due to the impact of the non-increasing, non-linear wind
speed/power characteristic of the WTG and the sum of the different wind power distributions. As
the wind speed/power characteristic is monotonic for most of the wind speed values, the wind
power correlation values are close to the wind speed ones.
0 10 20 30 40 50 60 70 80 90 1000
100
200
300
400
500
600
700
800
Wind Power - bus 8 (MW)
Num
ber
of
sam
ple
s
0 10 20 30 40 50 60 70 80 90 1000
100
200
300
400
500
600
700
800
Wind Power - bus 24 (MW)
Num
ber
of
sam
ple
s
0 5 10 15 20 25 300
5
10
15
20
25
30
Wind speed - WTG 3, bus 8(m/s)
Win
d s
peed -
WT
G 3
, bus 2
4(m
/s)
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
100
Wind Power - WSP, bus 8(MW)
Win
d P
ow
er
- W
SP
, bus 2
4(M
W)
67
The reactive power generation of the wind generators is zero since the wind parks are
considered to be not responsible for the voltage support of the system.
6.4 Conventional Generation (CG) Units
The CG units are considered to be thermal units of the same type whose dispatch is modelled
depending of the sampling of the system loads and wind power. For this, the simulation
algorithm can be described as follows:
1) Calculate the system net load T Tl g
N l g
l=1 g=1
L (i) = L (i) - G (i)∑ ∑ , where gT are the SG units G1,
G2,…Gg and lt are the loads l1, l2,…lT.
2) Calculate the aggregate CG capacity without slack generatorTc
CapT c
c=1
C = C∑ , where cT is
the number of CG units.
3) Calculate the percentage N
T
L (i)w(i) =
C.
3.1) If w(i)< 0.1
each CG unit power output is 10% of the unit capacity
Capc cC (i) = 0.1×C (i) .
3.2) If 0.1< w(i)< 1 each CG unit power output is Capc cC (i) = w(i)×C (i) .
3.3) If w(i)> 1each CG unit power output is the unit capacity Capc cC (i) = C (i) .
Each CG unit minimum power output is 10% of the unit capacity, due to restrictions with
shutting down thermal units.
6.5 System Analysis of results
As can be seen, the connection of wind power at each bus leads to an increase in the variability
of the power injections, i.e. decrease of the mean value of the distribution and a subsequent
increase in the standard deviation. The power injection distributions at bus 8 and 24 of the test
system for the 4 wind power penetration levels are presented in Figure 6.6 and Figure 6.7,
respectively. At bus 8, the connection of a wind park of 300 MW of nominal power (penetration
level 4) leads to bidirectional power injections into the system. At bus 24, the connection of a
wind park of 200MW or 300 MW of nominal power (penetration level 3 or 4) leads to
bidirectional power injections into the system.
Table 6.3: Mean value and standard deviation for the power injections at bus 8 for the 4 wind power penetration levels.
Wind Penetration: 0 MW 1500 MW 3000 MW 4500 MW
Mean [MW] 392.76 346.79 296.40 250.33
St.D.[MW] 97.55 102.54 116.83 136.50
68
As can be seen in Table 6.3, the increase in wind penetration leads to a decrease of the mean
value of the distribution and a subsequent increase in the standard deviation.
Figure 6.6: Power injection at bus 8 for the 4 wind power penetration levels (10000-sample
MCS)
Figure 6.7: Power injection at bus 24 for the 4 wind power penetration levels (10000-sample
MCS).
In Figure 6.8, the power distributions and box-plots for the power injection by the slack
bus are presented for the different wind power penetration levels. In Table 6.4, the respective
mean values and standard deviations for the slack bus power injection distributions are
presented. The increase in wind power penetration leads to a radical increase in the variability
of the power flows from/to the slack bus. In particular, for the penetration levels of 3000 MW and
-100 0 100 200 300 400 500 6000
50
100
150
200
250
Net Load - bus 8 (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-200 -100 0 100 200 300 4000
50
100
150
200
250
300
Net Load - bus 24 (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
69
4500MW of wind power, the slack bus has to absorb an excess power up to 1000 MW and
2000MW, respectively. In reality, this will correspond to power exports to neighbouring systems.
In Figure 6.9, Figure 6.10, Figure 6.11 and Figure 6.12, the box-plots for the power flow
distributions in the system lines are presented.
Figure 6.8: Slack bus power injection distributions and box plots (10000-sample MCS).
-2500 -2000 -1500 -1000 -500 0 500 10000
1000
2000
3000
4000
5000
6000
7000
8000
Power output (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
1 2 3 4
-2000
-1500
-1000
-500
0
500
Pow
er
(MW
)
Penetration level
70
Table 6.4: Mean value and standard deviation for the slack bus power injection distributions.
Wind Penetration: 0 MW 1500 MW 3000 MW 4500 MW
Mean [MW] 46.78 25.02 17.65 -65.02
St.D.[MW] 76.44 24.53 40.08 279.29
Figure 6.9: Box-plot for the power flows in the system lines in case of no wind power penetration
(10000-sample MCS).
Figure 6.10: Box-plot for the power flows in the system lines for 1500 MW (25%) of wind power
penetration (10000-sample MCS).
1-2 1-39 2-3 2-25 3-4 3-18 4-5 4-14 5-6 5-8 6-7 6-11 7-8 8-9 9-39 10-11 10-13 13-14 14-15 15-16 16-17 16-19 16-21 16-24 17-18 17-27 21-22 22-23 23-24 25-26 26-27 26-28 26-29 28-29 2-30 31-6 10-32 12-11 12-13 19-20 19-33 20-34 22-35 23-36 25-37 29-38
-800
-600
-400
-200
0
200
400
600
800
Pow
er
Flo
w (
MW
)
System lines
1-2 1-39 2-3 2-25 3-4 3-18 4-5 4-14 5-6 5-8 6-7 6-11 7-8 8-9 9-39 10-11 10-13 13-14 14-15 15-16 16-17 16-19 16-21 16-24 17-18 17-27 21-22 22-23 23-24 25-26 26-27 26-28 26-29 28-29 2-30 31-6 10-32 12-11 12-13 19-20 19-33 20-34 22-35 23-36 25-37 29-38
-800
-600
-400
-200
0
200
400
600
800
1000
Pow
er
Flo
w (
MW
)
System lines
71
Figure 6.11: Box-plot for the power flows in the system lines for 3000 MW (50%) of wind power
penetration (10000-sample MCS).
Figure 6.12: Box-plot for the power flows in the system lines for 4500 MW (75%) of wind power
penetration (10000-sample MCS).
The probabilistic system analysis shows that the increase in wind power in the system leads to
an increase in the variability of the system power flows. The incorporation of wind power leads
to an increase in the standard deviations of the power flow distributions. The presence of
stochastic generation in the lower levels of the system results to highly bidirectional power
flows. These reveal the transition from a vertical to a horizontally-operated power system where
the distribution systems become active, exchanging power with the transmission system
bidirectionally. In many cases the incorporation of wind power leads to higher reverse power
1-2 1-39 2-3 2-25 3-4 3-18 4-5 4-14 5-6 5-8 6-7 6-11 7-8 8-9 9-39 10-11 10-13 13-14 14-15 15-16 16-17 16-19 16-21 16-24 17-18 17-27 21-22 22-23 23-24 25-26 26-27 26-28 26-29 28-29 2-30 31-6 10-32 12-11 12-13 19-20 19-33 20-34 22-35 23-36 25-37 29-38
-800
-600
-400
-200
0
200
400
600
800
1000
Pow
er
Flo
w (
MW
)
System lines
1-2 1-39 2-3 2-25 3-4 3-18 4-5 4-14 5-6 5-8 6-7 6-11 7-8 8-9 9-39 10-11 10-13 13-14 14-15 15-16 16-17 16-19 16-21 16-24 17-18 17-27 21-22 22-23 23-24 25-26 26-27 26-28 26-29 28-29 2-30 31-6 10-32 12-11 12-13 19-20 19-33 20-34 22-35 23-36 25-37 29-38
-1000
-500
0
500
1000
1500
2000
Pow
er
Flo
w (
MW
)
System lines
72
flows than the direct ones. The increase in wind power integration can also leads to an increase
in the power flows in the lines.
In Figure 6.13a, Figure 6.13b and Figure 6.13c are presented the transmission lines (3-
4, 4-5, 4-14) connected to the bus 4 where is incorporated wind power generation, i.e. a cluster
of WTGs (SG units). As can be seen, the power flow distributions extend to both the positive
and negative axis (bidirectional power flows) and the increase in the SG penetration level leads
to an increase in the standard deviations of the distributions corresponding to a high variability
of the system power flows. In the line 3-4 the increase in the SG penetration level leads to
higher reverse power flows than the direct ones and the probabilities of occurrence of the
reverse power flow increase.
In Figure 6.13d, Figure 6.13e and Figure 6.13f, are presented the transmission lines (6-
7, 7-8, 8-9) connected to the bus 7 and 8 where is incorporated wind power generation, i.e. a
cluster of WTGs (SG units) in each one. As can be seen, in the lines 6-7, 7-8 the power flows
with more probability of occurrence are the direct ones and the increase in the SG penetration
level leads to an increase in the standard deviations of the distributions corresponding to a high
variability of the system power flows. In the line 6-7 the increase in the SG penetration level
leads to a decrease in the power flows values with more probability of occurrence and for a
penetration level of 75%, reverse power flows can occurred since the tail of the respective
power flow distribution extend to the negative axis. In the line 7-8 the increase in the SG
penetration level leads to an increase in the power flows in the lines. In the line 8-9, with the
increase in the SG penetration level, the probabilities of occurrence of the reverse power flows
decrease but leads to higher reverse power flows than the direct ones.
In Figure 6.14 are presented the transmission lines (14-15, 15-16, 16-17, 16-19, 16-21,
16-24) connected to the buses 15 and 16 where is incorporated wind power generation, i.e. a
cluster of WTGs (SG units) in each one. As can be seen, the power flow distributions extend to
both the positive and negative axis (bidirectional power flows) and the increase in the SG
penetration level leads to an increase in the standard deviations of the distributions
corresponding to a high variability of the system power flows. In the line 16-21, the power flows
become bi-directional only for the penetration levels of 50% and 75%. In the line 16-17, the
increase in the SG penetration level leads to higher reverse power flows than the direct ones.
In Figure 6.15 are presented the transmission lines (25-26, 26-27, 26-28, 26-29, 17-27,
28-29) connected to the buses 26, 27, 28 and 29 where is incorporated wind power generation,
i.e. a cluster of WTGs (SG units) in each one. In the line 25-26, 17-27 the increase in the SG
penetration level leads to higher reverse power flows than the direct ones and to an increase in
the power flow values with more probability of occurrence. In the lines 26-29, 28-29 the power
flows become bi-directional only for the penetration levels of 50% and 75%. In the lines 25-26,
26-28, 17-27 the increase in the SG penetration level leads to an increase in the power flow
values with more probability of occurrence.
73
(a) (b)
(c) (d)
(e) (f)
Figure 6.13: Some specific power flow distributions (10000-sample MCS):
(a) Line 3-4 (b) Line 4-5 (c) Line 4-14 (d) Line 6-7 (e) Line 7-8 (f) Line 8-9
-800 -700 -600 -500 -400 -300 -200 -100 0 100 2000
100
200
300
400
500
600
Line power flow (MW)
MC
S S
am
ple
s0 MW
1500 MW
3000 MW
4500 MW
-600 -500 -400 -300 -200 -100 0 1000
50
100
150
200
250
300
350
400
450
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-400 -350 -300 -250 -200 -150 -100 -50 0 500
50
100
150
200
250
300
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-100 0 100 200 300 400 500 6000
50
100
150
200
250
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
0 50 100 150 200 250 300 350 400 4500
50
100
150
200
250
300
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-500 -400 -300 -200 -100 0 100 200 300 400 5000
100
200
300
400
500
600
700
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
74
(a) (b)
(c) (d)
(e) (f)
Figure 6.14: Some specific power flow distributions (10000-sample MCS):
(a) Line 14-15 (b) Line 15-16 (c) Line 16-17 (d) Line 16-19 (e) Line 16-21 (f) Line 16-24
-300 -200 -100 0 100 200 300 4000
50
100
150
200
250
300
350
400
450
Line power flow (MW)
MC
S S
am
ple
s0 MW
1500 MW
3000 MW
4500 MW
-500 -400 -300 -200 -100 0 100 200 3000
50
100
150
200
250
300
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-300 -200 -100 0 100 200 300 400 500 600 7000
50
100
150
200
250
300
350
400
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-800 -600 -400 -200 0 200 400 6000
50
100
150
200
250
300
350
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-500 -400 -300 -200 -100 0 100 2000
50
100
150
200
250
300
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-400 -300 -200 -100 0 100 2000
50
100
150
200
250
300
350
400
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
75
(a) (b)
(c) (d)
(e) (f)
Figure 6.15: Some specific power flow distributions (10000-sample MCS):
(a) Line 25-26 (b) Line 26-27 (c) Line 26-28 (d) Line 26-29 (e) Line 17-27 (f) Line 28-29
-800 -700 -600 -500 -400 -300 -200 -100 0 100 2000
50
100
150
200
250
300
350
400
450
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-200 -100 0 100 200 300 400 5000
50
100
150
200
250
300
350
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-350 -300 -250 -200 -150 -100 -50 0 50 1000
50
100
150
200
250
300
350
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-300 -250 -200 -150 -100 -50 0 50 1000
50
100
150
200
250
300
350
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-400 -300 -200 -100 0 100 200 3000
50
100
150
200
250
300
350
400
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-500 -400 -300 -200 -100 0 1000
50
100
150
200
250
300
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
76
(a) (b)
(c) (d)
Figure 6.16: Some specific power flow distributions (10000-sample MCS):
(a) Line 12-11 (b) Line 12-13 (c) Line 22-23 (d) Line 23-24
In Figure 6.16a and Figure 6.16b, are presented the transmission lines (12-11, 12-13)
connected to the bus 12 where is incorporated wind power generation, i.e. a cluster of WTGs
(SG units). As can be seen, the increase in the SG penetration level leads to higher reverse
power flows than the direct ones and to an increase in the standard deviations of the
distributions corresponding to a high variability of the system power flows.
In Figure 6.16c and Figure 6.16d, are presented the transmission lines (22-23, 23-24)
connected to the bus 23 and 24 where is incorporated wind power generation, i.e. a cluster of
WTGs (SG units) in each one. As can be seen, the increase in the SG penetration level leads to
an increase in the standard deviations of the distributions corresponding to a high variability of
the system power flows and to an increase in the power flow values with more probability of
occurrence. In the line 23-24, the power flows become bi-directional only for the penetration
level of 75%.
-20 0 20 40 60 80 100 120 140 1600
100
200
300
400
500
600
Transformer power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-20 0 20 40 60 80 100 120 140 1600
50
100
150
200
250
300
350
Transformer power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-200 -150 -100 -50 0 50 1000
50
100
150
200
250
300
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
-50 0 50 100 150 200 250 300 350 400 4500
50
100
150
200
250
300
Line power flow (MW)
MC
S S
am
ple
s
0 MW
1500 MW
3000 MW
4500 MW
77
Figure 6.17: Distributions of system losses (10000-sample MCS).
Thus, the integration of stochastic generation in the power system leads to an increase in the
variability of the power flows. The mean values correspond to the central points of these
distributions give a false impression about the system loading and the problems arise at the tails
of the distributions.
In Figure 6.17, the distributions of the system losses in the 4 wind power penetration
scenarios are presented. In Table 6.5, the respective means and standard deviations are
specified.
Table 6.5: Mean value and standard deviation for the distributions of the system losses.
Wind Penetration: 0 MW 1500 MW 3000 MW 4500 MW
Mean [MW] 28.664 23.646 20.367 21.058
St.D.[MW] 10.973 10.601 9.8647 10.868
As can be seen, in a vertical power system, the system losses follow the time-dependent
system loads presenting a concentration of probability at certain values (Figure 6.17).
We can see from Table 6.5 and Figure 6.17 that up to a penetration level of 50%, wind
power integration leads to a decrease in the system losses and start to rise again for higher
penetrations levels.
0 20 40 60 80 100 1200
50
100
150
200
250
300
350
400
450
500
System losses (MW)
MC
S S
am
ple
s0 MW
1500 MW
3000 MW
4500 MW
78
6.6 Conclusions
In this chapter, the vertical to horizontal transformation of a bulk power system due to the
Stochastic Generation (SG) integration is analyzed through a unified approach for the
incorporation of the time-dependent stochasticity of the system loads and the non-time-
dependent stochasticity of SG.
The power flows become bidirectional, the increase in the SG penetration level in the
power system leads to an increase in the variability of the power flows and at high stochastic
generation penetration levels, reverse power flows may exceed the direct ones.
79
Chapter 7
Conclusions and Future work 7 Conclusions and Future Work
The power system industry undergoes a radical change: the transition from the ’vertical’ to a
’horizontally-operated’ power system due to the large-scale incorporation of non-dispatchable
(stochastic), Distributed Generation (DG) in the system which is one of the major challenges for
the power systems of the future power systems planning. In this new power system structure
can be recognize two non-dispatchable system entities, the load and non-dispatchable,
distributed generation - NDT/DG, and two dispatchable system entities, the Conventional or
Centralized Generation (CG) and the dispatchable, distributed generation - DT/DG. These new
conditions persuade the development of new modelling and design methodologies for the
investigation of this new operational power system structure, in particular the operational
uncertainty introduced by the abatement in generation dispatchability.
In a Horizontally-Operated Power System (HOPS), the high penetration from stochastic
distributed generation in the lower voltage levels has changed the system structure,
transforming the Distribution Networks (DNs) into active systems that contain both loads and
generators and exchange power with the Transmission Network (TN) bidirectionally, according
to the generation and consumption equilibrium in each DN. This transformation of the power
80
system leads to the increase of the variability in the power flows on the system lines. In this
case, the deterministic measure may prove insufficient for the system analysis.
The large-scale integration of stochastic generation in the system implies a new
modelling approach for the uncertainty analysis of the power system. In the 2nd chapter is
shown that the Monte Carlo Simulation (MCS) is the most indicated method to solve the
multivariate uncertainty problem created by the incorporation of Stochastic Generation (SG) in
the system. The linear approximation of the system model, the assumptions of independence
between the system inputs and perfect correlation for normally correlated system inputs have
been used for modelling of the time-dependent stochasticity of the system loads. The
incorporation of SG introduces a large number of different types of non-standard distributions
with complex interdependencies and a non-time dependent stochasticity which cannot be
modelled based on the assumptions of independence and normality. The 3rd chapter presents
the main differences between load uncertainty and SG uncertainty. A Monte Carlo Simulation
(MCS) approach is used for the power system multivariate uncertainty analysis based on the
modelling of the one marginal distributions and the modelling of the multidimensional stochastic
dependence structure. According to the MCS theory, the application of stochastic simulations
for the system uncertainty analysis involves the definition of the deterministic and the stochastic
system model. The deterministic system model is used in order to capture the system steady-
state operation in a specific instant in time. The stochastic system model involves the definition
of the stochastic dependence structures between the uncertain system inputs, i.e. the loads and
the stochastic generation. Each sample vector of the inputs r.v. is propagated through the
steady-state system model. Thus, the probability distributions of the uncertain system inputs
(loads-stochastic generation) are propagated through the steady-state system model and the
respective system state (node voltage) and output (line power flow) distributions are obtained.
The multivariate uncertainty analysis of power systems involves decoupling between the
modelling of the marginal distributions and the modelling of the stochastic dependence
structure. This approach requires the modelling of correlated non-normal distributions whose
dependence cannot be well represented by the product moment correlation. In order to
measure the dependence between non-normals distributions, the one-marginal distributions are
transformed into ranks whose product moment correlation (the rank correlation���) provides an
adequate measure of dependence. The functional relationship between the ranks is modelled
by Copula functions. The transformation of the marginals into ranks and the use of a
multidimensional normal copula for the dependence modelling is the basis for the Joint Normal
Transform methodology application in the multidimensional dependence modelling described in
the 4th chapter.
For a typical power system the number of loads and stochastic generators may reach some
thousands and thus, is necessary to define the joint power injections of a large number of
stochastic generators and loads situated in a large geographic area with different prime movers,
different converter technologies and different consumer profiles. In every power system there
81
are groups of highly correlated random inputs consisting of stochastic generators situated in
relatively small geographic areas that use the same generation technology due to the mutual
coupling to the same stochastic prime mover. Their power outputs are expected to be strongly
coupled, i.e. to follow similar fluctuations. The others groups are the system loads of the same
type due to similar consumer behaviour. Thus, the system comprises groups, designed by
clusters, of strongly positively correlated r.v., i.e. the r.v. belonging to each of these groups
present similar stochastic behaviour and the dependence structure observed between them is
close to perfect correlation. A simplified stochastic model can be used for the modelling of the
dependence structure of the cluster and the stochastic model of the system is reduced to the
definition of the stochastic dependencies between the respective clusters. The 5th chapter
presents methods to deal with high-dimensionality, i.e. model reduction techniques to simplify
the stochastic model through model approximations called Stochastic Plants (SP’s). The
Stochastic Bounds Methodology (SBM) is used for the formulation of these approximations. The
Stochastic Bounds Methodology can also be applied to model the dependence structure
between the clusters through the definition of the extreme dependence structures that can
bound all real cases. The Joint Normal Transform Methodology is applied in order to model the
real dependence structure between the clusters, based on the mutual correlations obtained by
data analysis.
The 6th chapter presents an application of these methodologies to solve a multivariate
uncertainty analysis problem, the integration of stochastic generation in a bulk power system in
order to understand better the horizontal operation of the power system. The power flows
become bidirectional, the increase in the SG penetration level in the power system leads to an
increase in the variability of the power flows and at high stochastic generation penetration
levels, reverse power flows may exceed the direct ones.
Future Work
The impact of the large-scale implementation of distributed stochastic generation in a
distribution system should be investigated through a probabilistic analysis, in particular the
transition from the traditional passive to an active network structure. This transformation
influences the reactive power support of the network and different voltage control strategies
should be developed in order to improve the voltage quality of the system.
82
Appendix
Simulation data
Simulation data
83
Bulk power system data for the study case in Chapter 6
Table A.1: Bus data of the New England 39-bus test system
Bus (nr.) Volts (pu) Load (MW) Load (MVAr) Gen (MW)
1 - 0.0 0.0 -
2 - 0.0 0.0 -
3 - 322.0 2.4 -
4 - 500.0 184.0 -
5 - 0.0 0.0 -
6 - 0.0 0.0 -
7 - 233.8 84.0 -
8 - 522.0 176.0 -
9 - 0.0 0.0 -
10 - 0.0 0.0 -
11 - 0.0 0.0 -
12 - 7.5 88.0 -
13 - 0.0 0.0 -
14 - 0.0 0.0 -
15 - 320.0 153.0 -
16 - 329.0 32.3 -
17 - 0.0 0.0 -
18 - 158.0 30.0 -
19 - 0.0 0.0 -
20 - 628.0 103.0 -
21 - 274.0 115.0 -
22 - 0.0 0.0 -
23 - 247.5 84.6 -
24 - 308.6 -92.2 -
25 - 224.0 47.2 -
26 - 139.0 17.0 -
27 - 281.0 75.5 -
28 - 206.0 27.6 -
29 - 283.5 26.9 -
30 1.04750 0.0 0.0 -
31 1.04000 9.2 4.6 572
32 0.98310 0.0 0.0 650
33 0.99720 0.0 0.0 632
34 1.01230 0.0 0.0 508
35 1.04930 0.0 0.0 650
36 1.06350 0.0 0.0 560
37 1.02780 0.0 0.0 540
38 1.02650 0.0 0.0 830
39 1.03000 1104.0 250.0 1000
84
Table A.2: Line data of the New England 39-bus test system
Line Number
Line Resistance (p.u.)
Reactance (p.u.)
Susceptance (p.u.)
Transformer Magnitude
Tap angle p q
1 1 2 0.0035 0.0411 0.6987 0 0
2 1 39 0.0010 0.0250 0.7500 0 0
3 2 3 0.0013 0.0151 0.2572 0 0
4 2 25 0.0070 0.0086 0.1460 0 0
5 3 4 0.0013 0.0213 0.2214 0 0
6 3 18 0.0011 0.0133 0.2138 0 0
7 4 5 0.0008 0.0128 0.1342 0 0
8 4 14 0.0008 0.0129 0.1382 0 0
9 5 6 0.0002 0.0026 0.0434 0 0
10 5 8 0.0008 0.0112 0.1476 0 0
11 6 7 0.0006 0.0092 0.1130 0 0
12 6 11 0.0007 0.0082 0.1389 0 0
13 7 8 0.0004 0.0046 0.0780 0 0
14 8 9 0.0023 0.0363 0.3804 0 0
15 9 39 0.0010 0.0250 1.2000 0 0
16 10 11 0.0004 0.0043 0.0729 0 0
17 10 13 0.0004 0.0043 0.0729 0 0
18 13 14 0.0009 0.0101 0.1723 0 0
19 14 15 0.0018 0.0217 0.3660 0 0
20 15 16 0.0009 0.0094 0.1710 0 0
21 16 17 0.0007 0.0089 0.1342 0 0
22 16 19 0.0016 0.0195 0.3040 0 0
23 16 21 0.0008 0.0135 0.2548 0 0
24 16 24 0.0003 0.0059 0.0680 0 0
25 17 18 0.0007 0.0082 0.1319 0 0
26 17 27 0.0013 0.0173 0.3216 0 0
27 21 22 0.0008 0.0140 0.2565 0 0
28 22 23 0.0006 0.0096 0.1846 0 0
29 23 24 0.0022 0.0350 0.3610 0 0
30 25 26 0.0032 0.0323 0.5130 0 0
31 26 27 0.0014 0.0147 0.2396 0 0
32 26 28 0.0043 0.0474 0.7802 0 0
33 26 29 0.0057 0.0625 1.0290 0 0
34 28 29 0.0014 0.0151 0.2490 0 0
35 2 30 0.0008 0.0112 0.1476 1.025 0
36 31 6 0.0013 0.0151 0.2572 1.07 0
37 10 32 0.0070 0.0086 0.1460 1.07 0
38 12 11 0.0035 0.0411 0.6987 1.006 0
39 12 13 0.0010 0.0250 0.7500 1.006 0
40 19 20 0.0007 0.0082 0.1389 1.06 0
41 19 33 0.0013 0.0213 0.2214 1.07 0
42 20 34 0.0011 0.0133 0.2138 1.009 0
43 22 35 0.0008 0.0128 0.1342 1.025 0
44 23 36 0.0008 0.0129 0.1382 1 0
45 25 37 0.0002 0.0026 0.0434 1.025 0
46 29 38 0.0006 0.0092 0.1130 1.025 0
85
Bibliography
[1] G.Papaefthymiou, “Integration of Stochastic Generation in Power System”, PHD thesis, Delft
University of Technology, Delft, the Netherlands, 2006.
[2]M. Reza, “Stability analysis of transmission systems with high penetration of distributed
generation”, PhD thesis, Delft University of Technology, Delft, the Netherlands, 2006.
[3]G. Papaefthymiou, Jody Verboomen, P. H. Schavemaker and L. van der Sluis, “Impact of
Stochastic Generation in Power System Contingency Analysis,” in 9th International Conference
on Probabilistic Methods Applied to Power Systems KTH, Stockholm, Sweden, June 11-15,
2006.
[4] G. Papaefthymiou, A. Tsanakas, D. Kurowicka, P. H. Schavemaker, and L. van der Sluis,
“Probabilistic power flow methodology for the modelling of horizontally-operated power
systems,” in International Conference on Future Power Systems (FPS2005), November 16-18,
2005.
[5] M.R. Haghifam and M. Omidvar, “Wind Farm Modelling in Reliability Assessment on Power
System”, in 9th International Conference on Probabilistic Methods Applied to Power Systems
KTH, Stockholm, Sweden, June 11-15, 2006.
[6] M. Reza, G. Papaefthymiou, P. H. Schavemaker, W. L. Kling, and L. van der Sluis,
“Transient stability analysis of power systems with distributed energy systems”, in CIGRE 2005
Symposium ”Power Systems with Dispersed Generation”, Athens, Greece, April 17-20 2005.
[7] G. Papaefthymiou, P. H. Schavemaker, L. van der Sluis, W. L. Kling, D. Kurowicka, and R.
M. Cooke, "Integration of stochastic generation in power systems," in 15th Power Systems
Computation Conference PSCC2005, Liege, Belgium, Invited Paper, August 22-26 2005.
[8] G. Papaefthymiou, A. Tsanakas, M. Reza, P. H. Schavemaker, and L. van der Sluis,
“Stochastic modelling and analysis of horizontally-operated power systems with a high wind
energy penetration”, in 2005 St. Petersburg PowerTech Conference, St. Petersburg, Russia,
June 27-30 2005.
[9] J.Grainger and W. D. Jr. Stevenson, Power system Analysis, Electrical Engineering,
McGraw-hill International Editions, 1994
[10] G.J. Anders, Probability Concepts in Electrical Power Systems, Wiley Interscience, 1990.
[11] José Pedro Sucena Paiva, Redes de Energia Eléctrica: uma análise sistémica, IST press,
2005.