Probabilistic load flow considering distributed renewable ... · Pedro Miguel Lousa Martins Reis...

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

iii

To my parents

v

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.

vii

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 3.9: 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 5.16: 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 6.13: Some specific power flow distributions (10000-sample MCS): ......................................... 73




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

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

PDF

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


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

PDF

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


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

PDF

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


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


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


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


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


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


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


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


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


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


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




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

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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

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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


DN 5 WTG cluster 3


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


DN 4 WTG cluster 2


DN 5 WTG cluster 3


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


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


DN 4 WTG cluster 2


DN 5 WTG cluster 3


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.

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200

400

600

800

1000

1200

1400

1600

Line 1-2 power flow (MW)

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4º scenario

Independence

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1600


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2º scenario

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4º scenario

Independence

52

Figure 5.10: Power Flow Distributions for the system lines.

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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,

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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.

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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

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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.


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





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


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


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


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


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


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


MC

S S

am

ple

s

0 MW

1500 MW

3000 MW

4500 MW

74

(a) (b)

(c) (d)

(e) (f)



-300 -200 -100 0 100 200 300 4000

50

100

150

200

250

300

350

400

450


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


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


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


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


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


MC

S S

am

ple

s

0 MW

1500 MW

3000 MW

4500 MW

75

(a) (b)

(c) (d)

(e) (f)



-800 -700 -600 -500 -400 -300 -200 -100 0 100 2000

50

100

150

200

250

300

350

400

450


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


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


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


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


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


MC

S S

am

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s

0 MW

1500 MW

3000 MW

4500 MW

76

(a) (b)

(c) (d)


(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


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


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.


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




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

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