stochastic investment in power system flexibility: a ...930844/fulltext01.pdf · yaser tohidi a...

INDEGREE PROJECT , SECOND CYCLE, 30 CREDITS,STOCKHOLM SWEDEN 2016

Stochastic Investment in Power System Flexibility: A Benders Decomposition Approach

FERNANDO GARCIA MARIÑO

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ELECTRICAL ENGINEERING

KTH Royal Institute of Technology

Master Thesis

STOCHASTIC INVESTMENT INPOWER SYSTEM FLEXIBILITY: A

BENDERS DECOMPOSITIONAPPROACH.

Author:

Fernando Garcia Marino

Supervisor:

Dr. Mohammad Reza Hesamzadeh

Mahir Sarfati

Examiner:

Dr. Mohammad Reza Hesamzadeh

Yaser Tohidi

A thesis submitted in fulfilment of the requirements

for the degree of Master of Science

in the

Electricity Market Research Group

Department of Electric Power and Energy Systems

March 2016

http://www.hesamzadeh.com/

[email protected]

http://www.hesamzadeh.com/

[email protected]

”Life is like riding a bicycle. To keep your balance, you must keep moving.”

Albert Einstein

ROYAL INSTITUTE OF TECHNOLOGY KTH

Abstract

KTH Royal Institute of Technology

Electric Power and Energy Systems

Master of Science

STOCHASTIC INVESTMENT IN POWER SYSTEM FLEXIBILITY: A

BENDERS DECOMPOSITION APPROACH

by Fernando Garcia Marino

The efficient use of the available assets is the goal of the liberalized electricity market.

Nowadays, the development of new technologies of renewable production results in a

significant increase in the total installed capacity of this type of generation in the power

system. However, the unpredictable nature of this resources results in a changing and

non controllable generation that forces the power system to be constantly adapting to

these new levels of production. Thermal units, that are the base of generation, are

the responsible to replace this changing generation, but their ramp rates may not be

fast enough to adapt it. Thus, other resources must be developed to overcome these

inconveniences. The degree to which these resources help system stability is called

flexibility. In this thesis, depending on operational short-term or investment long-term

decisions, different points of view about flexibility are studied. Short-term includes

sources as adaptable demand or storage availability while long-term is examined with

the investment. To study the influence of sources in short-term planning, a model of the

National Electricity Market (NEM) of Australia is developed. Flexibility in long-term

is analyzed with IEEE-6 and IEEE-30 node systems, applying Benders decomposition.

System Flexibility Index and economic benefit are calculated to measure flexibility. This

indicators show the utility of developed model in forecasting the required ramp service

in future power systems.

http://www.kth.se

Faculty Web Site URL Here (include http://)

Department or School Web Site URL Here (include http://)

Acknowledgements

First, I would like to thank Mahir Sarfati for the useful comments, remarks and engage-

ment through the learning process of this master thesis, as well for his support in all the

way.

I would like to express my gratitude to my family, because without them I could not have

lived this experience and their support helped me in those moments I needed. Specially

thanks to my parents Fernando and Rosa and my grandmother Pili, who has been with

me since my memory allows me to remember.

How can I forget all the people I have met in Stockholm. During these two years, I have

had the opportunity to establish a strong friendship with people all around the world,

from France to Germany, from Italy to US. Thank you all for being here. However,

there are some people that deserve to have an special mention, as they have been living

with me. Perello, Simone, Guli and Elenas, I would never forget your support. Mention

apart demand Enrique, as we decided to start this adventure together and we are still

both here. This is just the beginning.

By last, I would like to thanks my Spanish friends, as this master thesis is just the last

step of my career but however it started long time ago, where their help facilitated me

to have the best years of my life.

For those of you that helped me directly or indirectly, many thanks.

iii

Contents

Abstract ii

Acknowledgements iii

Contents iv

List of Figures vii

List of Tables ix

Abbreviations xi

Nomenclature xii

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Definition of Flexibility . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Need for flexibility: Impact of variable renewables . . . . . . . . . 4

1.2 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Simulation Platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Short-Run Economic Dispatch 8

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Ramp-Rate Providers 13

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Flexibility model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Flexibility from Supply Side . . . . . . . . . . . . . . . . . . . . . . 14

3.2.2 Flexibility from Demand Side . . . . . . . . . . . . . . . . . . . . . 15

3.2.3 Flexibility from Storage Availability . . . . . . . . . . . . . . . . . 17

3.2.4 Final Objective Function and Balancing Conditions with the in-tegrated flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Investment Planning 22

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

iv

Contents v

4.2 Investment Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Investment Planning Model . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Benders Decomposition 30

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Benders Decomposition for MILP problems . . . . . . . . . . . . . . . . . 31

5.2.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . 32

5.2.2 Benders Decomposition applied to a 2-node system with unit com-mitment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.3 Benders Decomposition for Investment Planning . . . . . . . . . . . . . . 39

5.3.1 Model of Benders Decomposition for Investment Planning . . . . . 40

5.3.2 Benders Decomposition applied to a 2-node system with invest-ment and unit commitment . . . . . . . . . . . . . . . . . . . . . . 43

6 Case Study 46

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.2 Study of Flexible Demand and Storage Availability . . . . . . . . . . . . . 46

6.2.1 5-node model of the Australian National Electricity Market . . . . 47

6.2.1.1 Flexible Demand . . . . . . . . . . . . . . . . . . . . . . . 50

6.2.1.2 Storage Availability . . . . . . . . . . . . . . . . . . . . . 51

6.2.1.3 Flexible Demand and Storage Availability . . . . . . . . . 53

6.2.2 Sensitivity of the SRED to the Interest Rate . . . . . . . . . . . . 60

6.3 Study of Investment Decisions . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.3.1 IEEE 6-node system with investment . . . . . . . . . . . . . . . . . 63

6.3.1.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.3.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.3.2 IEEE 30-node system with investment . . . . . . . . . . . . . . . . 74

6.3.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.3.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.3.3 IEEE 6-node system with investment and unit commitment . . . . 84

6.3.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7 Conclusions and Future Work 87

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

A Integrating Renewables in Electricity Market 90

A.1 Initial Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 90

A.2 Mathematical Model for Flexible Demands . . . . . . . . . . . . . . . . . 94

A.3 Flexibility from Storage Availability . . . . . . . . . . . . . . . . . . . . . 96

B Data for 5-node model of the Australian NEM 99

C Data for IEEE-6 Bus System 101

D Data for IEEE-30 Bus System 103

Contents vi

Bibliography 105

List of Figures

1.1 Equilibrium generation-demand. . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Global Cumulative Installed Renewable Power Generation Capacity 2000-2013 [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Daily Renewable Power Australia in Summer and Winter [8]. . . . . . . . 4

2.1 The modelling approach for short-run economic dispatch [12]. . . . . . . . 9

4.1 Generation in South Australia by fuel type between 22 June and 5 July2014. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.1 Flowchart of the Benders Decomposition. . . . . . . . . . . . . . . . . . . 34

5.2 2-Node system, 1: Existent generator, 2: Existent generator, 3: Demandresponse, 4: Demand response. . . . . . . . . . . . . . . . . . . . . . . . . 35

5.3 Algorithm for Benders Decomposition with Multiple Scenarios. . . . . . . 36

5.4 Benders Decomposition Flowchart with Feasibility Cuts [31]. . . . . . . . 40

5.5 2-Node system with investment, 1: Existent generator, 2: Existent gener-ator, 3: Demand response, 4: Demand response, 5: Candidate generator,6: Candidate generator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.1 The 5-node model of Australian National Electricity Market [39]. . . . . . 47

6.2 Dispatch Cost with Flexible Demand. . . . . . . . . . . . . . . . . . . . . 50

6.3 Benefit of Flexible Demand. . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.4 Dispatch Cost with Storage Availability. . . . . . . . . . . . . . . . . . . . 53

6.5 Benefit of Storage Availability. . . . . . . . . . . . . . . . . . . . . . . . . 54

6.6 Dispatch Cost with Storage Availability and Flexible Demand. . . . . . . 55

6.7 Benefit of Storage Availability and Flexible Demand. . . . . . . . . . . . . 56

6.8 Generation for Storage Availability and Flexible Demand for three cases,scenario 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.9 Generation for Storage Availability and Flexible Demand for three cases,scenario 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.10 Deviation from Dispatch Cost. . . . . . . . . . . . . . . . . . . . . . . . . 61

6.11 Difference from Dispatch Cost. . . . . . . . . . . . . . . . . . . . . . . . . 62

6.12 The modified IEEE 6-Node example system, 1: Coal, 2: Hydro, 3: Gas,4: Coal 5: Hydro, 6: Gas, 7: Gas, 8: Wind, 9: Wind, 10: Wind. . . . . . . 64

6.13 Electric load curve: New England 22/10/2010 [55]. . . . . . . . . . . . . . 65

6.14 Duck Curve Prediction [56]. . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.15 Duck Curve for 9 scenarios (ss) in IEEE 6-node system with investment. . 66

6.16 Benders Decomposition Convergence for IEEE 6-node system with invest-ment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

vii

List of Figures viii

6.17 Generation level in scenario 1 for IEEE 6-node system . . . . . . . . . . . 72

6.18 Generation level in scenario 9 for IEEE 6-node system . . . . . . . . . . . 73

6.19 The modified IEEE 30-Node example system, 1: Coal, 2: Gas-C, 3:Pump-storage hydro, 4: Waste 5: Peat, 6: Gas-A, 7: Distillate, 8: Biogas,9: Gas-B, 10: Hydro , 11: Wind. . . . . . . . . . . . . . . . . . . . . . . . 75

6.20 Benders Decomposition Convergence for IEEE 30-node system with in-vestment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.21 Generation level in scenario 1 for IEEE 30-node system . . . . . . . . . . 82

6.22 Generation level in scenario 9 for IEEE 30-node system . . . . . . . . . . 83

6.23 Gap for Benders Decomposition Convergence for IEEE 6-node systemwith investment and unit commitment. . . . . . . . . . . . . . . . . . . . . 85

List of Tables

5.1 Final load for three different scenarios. . . . . . . . . . . . . . . . . . . . . 35

5.2 Data for Generation Units. . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.3 Dispatch Cost with Non-Optimal Master Problem. . . . . . . . . . . . . . 38

5.4 Calculation times with Non-Optimal Master Problem. . . . . . . . . . . . 38

5.5 Dispatch Cost with Optimal Master Problem. . . . . . . . . . . . . . . . . 38

5.6 Calculation times with Optimal Master Problem. . . . . . . . . . . . . . . 38

5.7 Data for Generation Units in 2-node system with investment. . . . . . . . 44

5.8 Final load for three different scenarios in 2-node system with investment. 44

5.9 Dispatch cost solution for Feasibility Benders and Standard Benders. . . . 44

6.1 Dispatch Cost with Flexible Demand. . . . . . . . . . . . . . . . . . . . . 50

6.2 Benefit of Flexible Demand. . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.3 Dispatch Cost with Storage Availability. . . . . . . . . . . . . . . . . . . . 52

6.4 Benefit of Storage Availability. . . . . . . . . . . . . . . . . . . . . . . . . 53

6.5 Dispatch Cost with Storage Availability and Flexible Demand. . . . . . . 54

6.6 Benefit of Storage Availability and Flexible Demand. . . . . . . . . . . . . 55

6.7 Variation Interest Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.8 Load data for IEEE 6-node system with investment. . . . . . . . . . . . . 66

6.9 Coefficients for Duck curve. . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.10 Line invest decisions for IEEE 6-node system with investment. . . . . . . 68

6.11 Generation invest decisions for IEEE 6-node system with investment. . . . 68

6.12 Generation investment costs (GIC), transmission investment costs (TIC),normal- operation costs (NC) and adjustment costs (AC) for IEEE 6-nodesystem with investment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.13 Efficiency Benefit (EB), Flexibility Benefit (FB) and Total Benefit (TB)for IEEE 6-node system with investment. . . . . . . . . . . . . . . . . . . 70

6.14 SFI for all cases in IEEE 6-node system with investment. . . . . . . . . . 71

6.15 Load data for IEEE 30-node system with investment. . . . . . . . . . . . . 76

6.16 Line invest decisions for IEEE 30-node system with investment. . . . . . . 77

6.17 Generation invest decisions for IEEE 30-node system with investment. . . 78

6.18 Generation investment costs (GIC), transmission investment costs (TIC),normal- operation costs (NC) and adjustment costs (AC) for IEEE 30-node system with investment. . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.19 Efficiency Benefit (EB), Flexibility Benefit (FB) and Total Benefit (TB)for IEEE 30-node system with investment. . . . . . . . . . . . . . . . . . . 80

6.20 SFI for all cases in IEEE 30-node system with investment. . . . . . . . . . 80

B.1 NEM Conventional Generation. . . . . . . . . . . . . . . . . . . . . . . . . 99

ix

List of Tables x

B.2 NEM Flexible Demand. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

B.3 NEM Line Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

B.4 NEM Hydro-Pumped Generators. . . . . . . . . . . . . . . . . . . . . . . . 100

B.5 NEM Wind Generators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

B.6 NEM Wind Contingencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

C.1 Generators data for IEEE 6-node system. . . . . . . . . . . . . . . . . . . 101

C.2 Generators costs for IEEE 6-node system. . . . . . . . . . . . . . . . . . . 101

C.3 Lines Data for IEEE 6-node system. . . . . . . . . . . . . . . . . . . . . . 102

C.4 Generators data for IEEE 6-node system with unit commitment. . . . . . 102

C.5 Generators costs for IEEE 6-node system with unit commitment. . . . . . 102

D.1 The locations, types, data and costs of existing generators for the modifiedIEEE 30-node system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

D.2 Types, data and costs of candidates generators for the modified IEEE30-node system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

D.3 Data of existing lines for the modified IEEE 30-node system. . . . . . . . 104

D.4 Data of candidate lines for the modified IEEE 30-node system. . . . . . . 104

Abbreviations

SRED Short-Run Echonomic Dispatch

IP Integer Programming

MILP Mixed Integer Linear Programming

LP Linear Programming

NLP Non-Linear Programming

MINLP Mixed Integer Non-Linear Programming

GAMS General Algebraichic Modeling System

GLPK GNU Linear Programming Kit

NEM National Electricity Market

QLD QueensLanD

NSW New South Wales

SA South Austarlia

VIC VICtoria

TAS TASmania

DC Dispatch Cost

PTDF Power Transmission Distribution Factor

MP Master Problem

SP SubProblem

LOL Lost Of Load

SFI System Flexibility Index

xi

Nomenclature

Indices

i(u) Existing (candidate) generator,

n Power system node,

l(v) Existing (candidate) transmission line,

t Time period,

ω Scenario,

k Probable contingency,

it Iterations for Benders decomposition.

Parameters

T Number of time periods,

I(U) Number of existing (candidate) generators,

L(V ) Number of existing (candidate) lines,

K Number of possible contingencies,

Ω Number of scenarios,

πω Probability of realization of scenario ω,

Dn,ω Demand at node n in scenario ω under normal system operation,

Dn,k,ω Demand at node n in contingency k in scenario ω

RDi(u) Ramp down rate of generator i(u),

RUi(u) Ramp up rate of generator i(u),

GMi(u) Minimum stable generation of generator i(u),

Gi(u),ω Capacity of generator i(u) in scenario ω,

Gi(u),k,ω Capacity of generator i(u) in contingency k in scenario ω,

ci(u) Production cost of generator i(u),

cSPi(u) Start-up cost of generator i(u),

xii

Nomenclature xiii

cSDi(u) Shut-down cost of generator i(u),

Bl(v) Susceptance of transmission line l(v),

Fl(v) Capacity of transmission line l(v),

Fl(v),k,ω Capacity of transmission line l(v) in contingency k in scenario ω.

TICv Transmission investment cost for candidate line v,

GICu Generation investment cost for candidate generator u,

Υi,n Matrix linking generators i and nodes n ,

Φl,n Matrix linking lines l and nodes n ,

Hl,n PTDF matrix,

Ξ Very big number,

r Short term interest rate,

pk Probability of contingency k,

Mi Energy limit of hydro plant i,

Qi,0 Amount of energy stored before the operation in the reservoir

of hydro generator i,

Qmaxi Capacity of reservoir of hydro generator i,

ηgi Efficiency of pump-storage generator i,

ηpi Efficiency of pump-storage motor i,

αdown Minimum value of αit,

xcont Fixed value for candidate line decision,

ycont Fixed value for candidate generator decision.

Variables

xv Binary variable of transmission line v,

yu Binary variable of generator u,

gi(u),ω Dispatch of generator i (u) under normal operation, in scenario ω

gi(u),t,k,ω Dispatch of generator i (u) at time t for contingency k

in scenario ω,

g′i(u),k,ω Dispatch of generator i (u) after clearing of contingency k

in scenario ω,

ppi(u),ω Power pumped (consumed) by i (u) under normal operation, in scenario ω

ppi(u),t,k,ω Power pumped (consumed) by i (u) at time t for contingency k

in scenario ω,

Nomenclature xiv

pp′i(u),k,ω Power pumped (consumed) by i (u) after clearing of contingency k

in scenario ω,

pgi(u),ω Power turbinated (produced) by i (u) under normal operation, in scenario ω

ˆgpgi(u),t,k,ω Power turbinated (produced) by i (u) at time t for contingency k

in scenario ω,

pg′i(u),k,ω Power turbinated (produced) by i (u) after clearing of contingency k

in scenario ω,

Oi(u),ω Over produced energy by unit i (u), in scenario ω

Oi(u),t,k,ω Over produced energy by unit i (u) at time t for contingency k

in scenario ω,

O′i(u),k,ω Over produced energy by unit i (u) after clearing of contingency k

in scenario ω,

LLn,ω Lost of Load in node n , in scenario ω

LLn,t,k,ω Lost of Load in node n at time t for contingency k

in scenario ω,

LL′n,k,ω Lost of Load in node n after clearing of contingency k

in scenario ω,

Un,ω Underdelivery in node n , in scenario ω

Un,t,k,ω Underdelivery in node n at time t for contingency k

in scenario ω,

U ′n,k,ω Underdelivery in node n after clearing of contingency k

in scenario ω,

Dn,ω Demand in node n , in scenario ω

Dn,t,k,ω Demand in node n at time t for contingency k

in scenario ω,

D′n,k,ω Demand in node n after clearing of contingency k

in scenario ω,

fl(v),ω Power flow of line l (v) under normal operation in scenario ω,

fl(v),t,k,ω Power flow of line l (v) at time t for contingency k

in scenario ω,

f ′l(v),k,ω Power flow of line l (v) after clearing of contingency k

in scenario ω,

θn,ω Voltage angle of node n under normal operation, in scenario ω,

Nomenclature xv

θn,t,k,ω Voltage angle of node n at time t for contingency k

in scenario ω,

θ′n,k,ω Voltage angle of node n after clearing of contingency k

in scenario ω,

si(u),ω Start-up binary variable of generator i(u) under normal operation

in scenario ω,

si(u),t,k,ω Start-up binary variable of generator i(u) at time t for contingency

k in scenario ω,

s′i(u),k,ω Start-up binary variable of generator i(u) after clearing of

contingency k in scenario ω,

wi,ω Shut-down binary variable of generator i(u) under normal

operation in scenario ω,

wi,t,k,ω Shut-down binary variable of generator i(u) at time t for contingency

k in scenario ω,

w′i,k,ω Shut-down binary variable of generator i(u) after clearing of


zi(u),ω On-line/off-line binary variable of generator i(u) under normal


z′i(u),t,k,ω On-line/off-line binary variable of generator i(u) at time t


zi(u),k,ω On-line/off-line binary variable of generator i(u) after clearing

of contingency k in scenario ω,

hi(u),ω On-line/off-line binary variable of generator u under normal


h′i(u),t,k,ω On-line/off-line binary variable of generator u at time t


hi(u),k,ω On-line/off-line binary variable of generator u after clearing

of contingency k in scenario ω,

Qi,ω Stored water of hydro generator i in scenario ω,

Q′i,t,k,ω Stored water of hydro generator i

for contingency k in scenario ω,

Qi,k,ω Stored water of hydro generator i after clearing of contingency k,

in scenario ω,

Nomenclature xvi

αit Auxiliary term for Benders decomposition,

zit,SC Objective function of Security Check Subproblem in iteration it,

zit,OO Objective function of Optimal Operation Subproblem in iteration it,

λv,it Marginal value for candidate line in iteration it,

λu,it Marginal value for candidate generator in iteration it,

Chapter 1

Introduction

1.1 Background

In the wholesale electricity market, demand is satisfied by generation suppliers that

results in a given electricity price. Figure 1.1 shows that equilibrium point establish

the price and quantity of energy supplied. Consumers follow the Law of the Demand:

the higher the price, the lower the quantity demanded, while generation side are guided

by the Law of Supply: the higher the price, the higher the quantity supplied. This

equilibrium is not only dependant on the decision of consumers and suppliers but it is

affected by other factors. Unpredictability of renewable and stochastic generation may

change this equilibrium point. This non controllable generation establish a stochastic

production which determines the future of power systems. However, this effect can be

reduced when the system is more able to be adapted to it, i.e. if the system is more

flexible.

Flexibility is the key for the reliable operation in near future power systems. In recent

years, many countries and governments have established policies to drive more renewable

energy into the power systems, which results in a high growth in this type of generation

(Figure 1.2). With the introduction of huge amounts of cheap and environmental-

friendly but stochastic generation, big changes in the stochastic production side may

happen due to their unpredictable nature, precluding demand satisfaction [1]. Tradi-

tionally, mentioned changes has been provided almost entirely by controlling the supply

side, using more amount of dispatchable generation when it is needed. However, it

1

Chapter 1. Introduction 2

Figure 1.1: Equilibrium generation-demand.

will not be worth enough in the future, where the amount of stochastic generation will

be so great that other assets are needed to be developed to improve electricity market

efficiency.

Figure 1.2: Global Cumulative Installed Renewable Power Generation Capacity 2000-2013 [2].

This variability affects all different power system operational timeframes, since day ahead

market to actual state [3]. Thus, a transformation of the operational planning of the


power system is expected. The question of having sufficient resources to meet demand is

changed to having sufficient flexibility resources to balance demand forecast errors and

fluctuations. Increasing the level of penetration of stochastic generation into the power

system, the impacts to more long-term timeframes become more visible [4]. This affects

the choice of suitable flexibility options: in shorter timeframes, response times are of

more importance; in longer timeframes, the ability to offer large storage content and

long shifting periods would be of more importance.

Therefore different flexibility options are best suited to different operational timeframes

[5]. Operational level is given by short-term actions. For this short-term decisions,

the key is the demand response from industrial side, where they could adapt their

demand levels to the actual generation. Another option is regulating pump-hydro power

plants. When discussing about long term decisions, invest in new lines for expanding the

transmission network or new dispatchable units may help to increase flexibility. Finally,

the development of new technologies that improve the effect of stochastic production

and stabilize the market may lead to a more optimal operation of the power systems.

1.1.1 Definition of Flexibility

Power system flexibility represents the extent to which a power system can adapt electric-

ity generation and consumption as needed to maintain system stability in a cost-effective

manner [6]. Flexibility is the ability of a power system to maintain continuous service

in the face of rapid and large swings in supply or demand.

Flexibility suppliers include ”down regulation” and ”up regulation”. Depending on

which side is considered, there are taken different actions. Producers carries out down

regulation by reducing the production whereas a consumer carries out down regulation

by increasing the consumptions. Similarly, up regulation means that producers increase

production and consumers decrease the consumptions. Ramp rates of both sides are

thus extremely important for covering contingencies fast enough in order to limit its

effect and stabilize the power system [7].


1.1.2 Need for flexibility: Impact of variable renewables

The power system needs to be in balance to operate properly, i.e. demand and power

supplied to the electricity network have to match in every moment. Introducing variable

generation such as wind and solar power may increase the need of energy system flexi-

bility, which could be accomplished through additional measures on the supply or/and

demand side.

Figure 1.3: Daily Renewable Power Australia in Summer and Winter [8].

Figure 1.3 shows that renewable power production is not constant. Peaks of produc-

tion followed by a fast lost of wind/sun makes power systems hard to stabilize. This

contingencies stress the system leading to failures that are needed to be controlled. In-

creasing the penetration of this type of generation into the power system, the stress in it

is more critical, so it is necessary to implement some techniques before the contingency

occurs [9]. To make the system deal with this, preliminary studies are needed before the

installation of new plants and thus contributing to system reliability.

1.2 Problem Definition

Introduction of stochastic generation sources implies many challenges. When renewable

and stochastic energy is subsidized by governments or private entities, installed capacity

of this type increases dramatically. However, power system needs to be adapted at the

same time to this production or the existent decompensation may lead to problems as

load shedding or even blackouts.


The electric system is built in such a way that it has up to a certain point a capability

to cope with uncertainty and variability in both demand and supply of power. For

example, on the supply side, the kind of flexibility is accomplished through power plants

with different response time. From the electricity system point of view, flexibility relates

closely to grid frequency and voltage control, delivery uncertainty and variability and

power ramping rates.

The level of production of stochastic generation may change rapidly in a short period of

time. Thermal dispatchable units are best placed to replace this production due to their

good capabilities when controlling their levels of power supplied. Ramp rates of this units

plays an important role as this feature will determine if it is possible for the system to

adapt to this stochastic generation. Normally, dispatchable generators with high ramp

rates but expensive variable cost are dealing to offset changes in production. However,

if introduction of stochastic generation in the system is higher than the dispatchable

units can handle, blackouts or load shedding may happen with the consequences that

this entails. Thus, investing in new dispatchable generation may be an option as the

system will be more prepared to deal with this problem.

1.3 Thesis Objectives

The thesis aim is to analyze how the power grid has to be prepared to overcome fast

changes in stochastic production. This feature is measured with the power system

flexibility. Either from long-term (investment) or short-term (demand or pumped-hydro

generation) decisions, the system has to be ready for changes in the stochastic generation

side and be able to adapt to it before achieving a critical point. From this two different

points of view, flexibility sources are analyzed.

For short-term decisions, following features has been developed:

• Evaluating the impact of the introduction of new stochastic generation sources for

different levels of flexible load and storage availability.

• Measure the benefit achieved with the introduction of this flexible resources.

For long-term decisions:


• Evaluating the impact of investment decisions with the benefit.

• Evaluating the impact of investment decisions with the System Flexibility Index.

To achieve this goal, some implementations are done. For operational decisions, flexible

demand and storage availability is added to a given Short Run Economic Dispatch

(SRED) model. For long term issues, investment in lines and generators is added to

SRED and the resultant model is solved using Benders Decomposition.

1.4 Simulation Platforms

For this study, two solvers has been used to apply Benders Decomposition into SRED

model and compare its results. GAMS [10] and GLPK [11] are chosen because of their

different calculation capabilities.

The General Algebraic Modeling System (GAMS) is specifically designed for modeling

linear, nonlinear and mixed integer optimization problems. The system is especially use-

ful with large, complex problems. GAMS setup is very simple and intuitive, so the user

does not need to learn a new and difficult programming language. Timing and results

are available after the simulation without having to write complex codes. GAMS lets the

user concentrate on modeling. By eliminating the need to think about purely technical

machine-specific problems such as address calculations, storage assignments, subrou-

tine linkage, and input-output and flow control, GAMS increases the time available for

conceptualizing and running the model, and analyzing the results. GAMS structures

good modeling habits itself by requiring concise and exact specification of entities and

relationships.

The GLPK (GNU Linear Programming Kit) package is a new and revolutionary calcu-

lation method, intended for solving large-scale linear programming (LP), mixed integer

programming (MIP), and other related problems. It is a set of routines written in ANSI

C and organized in the form of a callable library. GLPK can also be used as a C library,

and here it will be where the system will be modelled. An ”.lp” file will be created from

a GAMS file and, from here, statements in C++ language will be written to implement

Benders Decomposition to the system.


1.5 Thesis Structure

Chapter 2 summarizes Short-Run Economic Dispatch, the electricity market model that

is used in the simulations. In Chapter 3, flexibility from Demand Side and from Storage

Availability is introduced into SRED. Chapter 4 is oriented to long-term decisions, where

investment in lines and generators are added to SRED. Benders decomposition is applied

in (Chapter 5) for the investment model. This old mathematical implementation for

transforming Mixed-Integer Linear Programming (MILP) to Linear Programming (LP)

is coded in GAMS and GLPK through C++ language and it is checked in a small 2-

node system. After explanation about Benders, some cases are studied. Chapter 6

shows those explained in Chapter 3 in a 5-node system which represent the National

Electricity Market (NEM) of Australia. Then, investment with Benders decomposition

is tested on 6-node and 30-node systems and to end, unit commitment is introduced in

the 6-node. Finally, Chapter 7 summarizes the conclusions achieved and some actions

to be taken for future work are suggested.

Chapter 2

Short-Run Economic Dispatch

2.1 Introduction

With the introduction of intermittent power sources, generation fluctuates substantially.

To keep the supply-demand balance in the system, some generators participate in the

frequency control, incrementing operation cost. If the balance generation-demand is

constantly controlled, generation units would be producing their optimal generation

continually, leading to a better utilization of the resources. Using this idea, Short-run

Economic Dispatch model (SRED) is developed [12], where generation-demand balance

is controlled every few minutes. In liberalized electricity markets, both controlled gen-

eration and demand have to be defined in some intervals called ”dispatch intervals”.

These periods can be set in hours (Europe) or minutes (Australia), leading to different

approach of modelling power systems.

Hourly dispatch intervals can be modelled as Appendix A shows [13]. From this model

A.1-A.23, that it will be referred as conventional model, has been taken the required in-

formation to introduce in SRED the resources that this thesis study. The main difference

between the conventional model and the model explained in this chapter is that SRED

is composed by three different states: steady-state equilibrium at the outset, transition

when a contingency occurs and the new steady-state equilibrium reached (Figure 2.1).

In the first state, system operator takes preventive actions, while in transient and final

states these actions are corrective. Let’s assume that there is a probability pk that a

contingency occurs, while a probability 1−pk that this contingency does not occur. If the

8

Chapter 2. Short-Run Economic Dispatch with Mathematical Modelling on theAdjustment Cost 9

Figure 2.1: The modelling approach for short-run economic dispatch [12].

contingency appears, the system moves from initial steady state to transition intervals

to overcome this variation, changing generation-demand balance until new steady state

is achieved. If no contingency occurs, the system remains in the initial steady state.

System operator task is not only keeping the balance generation-demand, but do it at

the lowest cost. Let’s say that DC(s) is the dispatch cost of the initial steady state s

and DC(s′) the dispatch cost in the final state reached s′. In the meanwhile, transition

states have been reached while the final state is achieved, resulting in an additional

dispatch cost for transition states DC(st) that gives an adjustment cost AC(s→ s′)[12]

of:

AC(s→ s′) =

Nt∑t=1

(DC(st)−DC(s′))

(1 + r)t−1(2.1)

Where Nt is the number transition states until final state is reached, t is the time period

after the contingency occurs and r is the interest rate per period. Thus, the objective

function of this model is to minimize the operational cost if no contingency occurs plus

the adjustment cost to cover this contingency:

PV (DC) =1 + p

1 + r

[(1− p)DC(s) + p

(AC(s→ s′) +

DC(s′)

r

)](2.2)


Where PV(DC) is the Present Value of the dispatch cost. It is possible to see that, if

no contingency occurs, i.e. p = 0, dispatch cost is the value of the initial steady state

equilibrium.

2.2 Mathematical Model

Before explaining the notation of the mathematical model, the assumptions taken for

developing SRED are exposed:

• Lines are lossless.

• No contingency occurs while system is not in steady state equilibrium.

• Lines and generators are always available.

• Perfect competition, none of the owners use market power.

• Marginal pricing, the price is set by the maximum production cost.

System operator task is to decide the dispatch level of controllable units with the objec-

tive of minimizing the present value of the expected dispatch cost subject to the physical

limit of the power system. Thus, system operator function is to solve the following op-

timization problem:

Minimize (1−Nk∑k=1

pk)DC(Gi, LLn) +Nk∑k=1

Nt∑t=1

pk(1 + r)t−1

DC(Gn,t,k, LLn,t,k)+

γNk∑k=1

pkDC(Gi,k, LLi,k) (2.3)

Subject to :

Energy balance constraints :

Ng∑i=1

(Gi −Oi)Υi,n +Nl∑l=1

flΦl,n + Un −Dn = 0 (2.4)


Ng∑i=1

(Gi,t,k −Oi,t,k)Υi,n +

Nl∑l=1

fl,t,kΦl,n + Un,t,k −Dn,t,k = 0 (2.5)

Ng∑i=1

(Gi,k −Oi,k)Υi,n +Nl∑l=1

fl,kΦl,n + Un,k −Dn,k = 0 (2.6)

Transmission flow balance constraints :

Nn∑n=1

[Hl,n

(Ng∑i=1

(Gi −Oi)Υi,n − (Dn − Un)

)]= fl (2.7)

Nn∑n=1

[Hl,n

(Ng∑i=1

(Gi,t,k −Oi,t,k)Υi,n − (Dn,t,k − Un,t,k)

)]= fl,t,k (2.8)

Nn∑n=1

[Hl,n

(Ng∑i=1

(Gi,k −Oi,k)Υi,n − (Dn,k − Un,k)

)]= fl,k (2.9)

Lost− load constraints :

LLn = Un +

Ng∑i=1

Oi ∗ Υi,n (2.10)

LLn,t,k = Un,t,k +

Ng∑i=1

Oi,t,k ∗ Υi,n (2.11)

LLn,k = Un,k +

Ng∑i=1

Oi,k ∗ Υi,n (2.12)

Transmission flow limits :

− fi ≤ fi ≤ fi (2.13)

− fi ≤ fl,t,k ≤ fi (2.14)

− fi ≤ fl,k ≤ fi (2.15)

Generation production limits :

0 ≤ Gi ≤ Gi (2.16)

0 ≤ Gi,t,k ≤ Gi (2.17)

0 ≤ Gi,k ≤ Gi (2.18)


Ramp− rate limits :

0 ≤ |Gi,t,k −Gi,t−1,k| ≤ RRi (2.19)

0 ≤ |Gi,k −Gi,T0,k| ≤ RRi (2.20)

Hydro energy production limits :

∀i ∈ Nh,Nt∑t=1

Gi,t,k ·∆t ≤Mi (2.21)

Chapter 3

Ramp-Rate Providers

3.1 Introduction

This chapter describes the need of flexibility sources in the electricity market. It is

referred to ”flexibility sources” all those actions that allow to adapt stochastic generation

to the system.

The fluctuation of stochastic generation leads to changing power production. As this

generation is not controllable, ramp rate providers plays an important role in system

stability (spilling could be an action to control it, but it is not considered as it reduces the

efficiency of the system because cheap resources are not being used). The problem with

it is that these ramp rates may not be high enough to backup all changing generation,

so actions have to be taken before contingencies occur and thus reduce their effect.

Ramp rate providers can be classify depending on where they come from. Thus, dis-

patchable units, demand or pump-hydro storage plants are analyzed in this part.

• Dispatchable units: Dispatchable units are the easiest way to control generation-

demand balance. However, the speed with which these units can be adapted to

the levels requested by the system operator is conditioned by their ramp rates.

• Demand: If demand could be also ”controllable”, i.e., variate its value when

stochastic power changes, the responsibility of dispatchable generation would be

smaller. However, flexible demand have its own ramp rates too, which means that

13

Chapter 3. Ramp-Rate Providers 14

this flexible demand may not be fast enough to cover the problems caused by the

contingencies.

• Pump-hydro storage plants Finally, hydro generation is a good resource to

increase system flexibility. Pump-hydro storage plants stores energy in the form

of gravitational potential energy of water, pumping water from a lower elevation

reservoir to a higher elevation. Low-cost off-peak electric power is used to run

the pumps. During periods of high electrical demand, the stored water is released

through turbines to produce electric power. Although the losses of the pumping

process makes the plant a net consumer of energy overall, the system increases

revenue by selling more electricity during periods of peak demand, when electricity

prices are highest.

In the following sections, demand and pump hydro power plants are introduced in SRED.

Appendix A shows the updating of flexible resources into the conventional electricity

market model.

3.2 Flexibility model

As seen, ramp rate providers are possible to classify in Flexibility from Supply Side

(dispatchable units 3.2.1), Flexibility from Demand Side (3.2.2) and Flexibility from

Storage Availability (pump-hydro storage plants 3.2.3). Expanding the transmission

network may also increase flexibility, as improving the power flow through the lines

can facilitate power transfer from places with cheap prices to places with higher prices.

However, in this part, it is only explained demand and storage availability, while the

other actions are studied deeply in the following chapter, as they are more associated

with investment decisions.

3.2.1 Flexibility from Supply Side

Supply providers can be composed by different types of generators: thermal, hydro, hy-

dro with the possibility of pumping water to the upper reservoir and renewable (stochas-

tic). For modelling our study cases, let’s consider that the stochastic generators are the

cause of contingencies due to loss of power in the different areas of the power system.


Thus, this generation is constant and it will not participate in the flexibility of the

system.

Thermal generators, as it is known, can be easily controlled with the addition or sub-

traction of combustible material, so this kind of production plays an important role in

the balance of the market. Thus, the improvement of these kind of units may increase

the power system flexibility. This feature is deeply studied in the next chapter, where

investment decisions are applied.

Hydro generators have a limitation with the maximum energy that they can deliver.

This limitation is considered in the constraint of maximum/minimum power 2.16-2.18

and in the ramp rate restriction 2.19-2.20.

Finally, there are some hydro power plants that can work pumping water to the upper

reservoir and work as power storage. These types of generators are explained below.

3.2.2 Flexibility from Demand Side

Analysis

Demand can contribute in many ways to the system flexibility. The most interesting

are:

1. By lowering the rate of increase at periods of high demand increase.

2. By lowering the rate of decrease at periods of high demand decrease.

3. By lowering the peak demand.

4. By increasing the valley demand.

5. By shifting energy from high demand to low demand periods.

When high demand of power consumption is required (peak), reducing the demand

contribute to decrease the stress of the generators that have to cover that peak [14]. It

also occurs conversely in the times of less demand (valley), when increasing the demand

makes that drop of demand lower and the flexible generators that have to reduce their

production will suffer less stress. Of course, shiftable demands also help to the operation

of the system. When transferring demand from high demand periods to low demand


periods (only those demand that it is not critical to be feeded in a certain moment of

time), operation costs are reduced as there will be smaller changes in the production.

A simple mathematical model that shows all explained above is developed in the follow-

ing lines.

Mathematical Formulation for Demand Side

Starting from the basis of the conventional model showed in the Appendix A, flexibility

from demand side is introduced into SRED. As in SRED model, unlike the conventional

model, there are three states (initial steady, transition and final steady state), those

equations found in A.24-A.31 have to be modify . Thus, it is necessary to add initial

load to the initial steady-state balancing condition (2.4), initial dispatch cost (2.3) and

the first step of the iteration in time (t0) of those equations that are dependent of

temporal increase as load ”ramp rate” (3.5),(3.6). With that, seven more constants

(the initial load, the utility and those explained in the Mathematical Model for Flexible

Demand in the conventional model A.2) and four more variables are added.

The main difference with the model presented in the appendix reside in the equations,

as only transition states compose the conventional model (Appendix A). As it has

been commented above, the three-stage model differentiates between transition periods

and the steady-state, so the equations presented for flexible demand in A.2 have to be

formulated thereby. However, as the initial load is known, only those equations belonging

to load shedding are different regarding the conventional model A.24-A.31:

dn,t,k = dn,t + cDn,t,k − cUn,t,k (3.1)

0 ≤ cDn,t,k ≤ Dmaxn,t − dn,t (3.2)

0 ≤ cUn,t,k ≤ dn,t −Dmaxn,t (3.3)

Nt∑t=1

dn,t,k ≥ Edayn (3.4)

dn,t,k − dn,t−1,k ≤ DUn (3.5)

dn,t−1,k − dn,t,k ≤ DDn (3.6)


And the load shedding (in this case, undelivered load Un) constraint is modified as:

Un ≤ Linitial (3.7)

Un,t,k ≤ dn,t,k + cDn,t,k − cUn,t,k (3.8)

Un,k ≤ dn,t=Nt,k (3.9)

The objective function and the balancing conditions are also modified, but they are

presented at the end of the chapter when all the flexible actions are introduced into

SRED.

3.2.3 Flexibility from Storage Availability

Analysis

Pump-hydro storage plants allow shifting the demand in time, producing energy at high-

price periods and consuming it at low-price periods, greatly improving in the power

system performance. There are another sources of storing power, as compressed air

units, but for this study only pump-hydro power plants are considered. At valley load,

when the demand and prices are low, they work as motor, pumping water to the upper

reservoir, and in those periods with high demand and prices, they turbine water and

generate power. In this manner, storage units play more a role of transmission system

than as a pure generation device.

For this flexibility source, the effect of the three-stage model is more noticeable, as

most of the equations are divided into initial-transition-final periods (A.32)(A.33) and

(A.36)-(A.41). In the rest of them (A.34)(A.35), only transition states are necessary to

be introduced.

The simple mathematical model that explains this feature is the following:

Mathematical Formulation for Storage Availability

P Tp = σTp × qTp (3.10)

P Tp,t,k = σTp × qTp,t,k (3.11)

P Tp,k = σTp × qTp,k (3.12)


PPp = σPp × qPp (3.13)

PPp,t,k = σPp × qPp,t,k (3.14)

PPp,k = σPp × qPp,k (3.15)

That represent the initial-transition-final production when turbine-pumping water re-

spectively. The water balance for the upper and the lower reserve is formulated as:

νUp,t,k = νUp,t−1,k + qPp,t−1,k − qTp,t−1,k (3.16)

νLp,t,k = νLp,t−1,k + qTp,t−1,k − qPp,t−1,k (3.17)

The limit of content of water for the upper and lower reserve in the hydro power plant

depends on only two stages (initial and transition), as the final state of this variable is

the same as the last value of the transition state:

V U,minp ≤ νUp ≤ V U,max

p (3.18)

V U,minp ≤ νUp,t,k ≤ V U,max

p (3.19)

V L,minp ≤ νLp ≤ V L,max

p (3.20)

V L,minp ≤ νLp,t,k ≤ V L,max

p (3.21)

Finally, the limit for water extracted from upper to lower reserve depends on the three-

stages as there are three variables for turbine or pumping water:

0 ≤ qTp ≤ Qmaxp (3.22)

0 ≤ qTp,t,k ≤ Qmaxp (3.23)

0 ≤ qTp,k ≤ Qmaxp (3.24)

0 ≤ qPp ≤ Qmaxp (3.25)

0 ≤ qPp,t,k ≤ Qmaxp (3.26)

0 ≤ qPp,k ≤ Qmaxp (3.27)


Variables and constants are easily identified comparing them with the conventional

model A.3.

3.2.4 Final Objective Function and Balancing Conditions with the in-

tegrated flexibility

As it has been mentioned, the objective function of SRED (2.3) has to be upgraded. For

that, it is only acted in DC(Gi, LLn), DC(Gn,t,k, LLn,t,k) and DC(Gn,k, LLn,k) as the

dispatch cost for the different states. Also notice that the up/down reserves available

for every generator with their respective costs and the appearance of the wind power

(set q) has been added to the dispatch costs. However, wind power does not contribute

to the optimal solution, as it is taken as known for every period of time (and cause of

the contingency).

• Initial Dispatch Cost

DC(Gi, LLn) =

Ng∑i=1

(CiGi + CruR

Ui + CrdR

Di

)+

Nn∑n=1

V LOLUn+ (3.28)

Np∑p=1

CpPTp +

Nq∑q=1

CqWq −Nn∑n=1

UnLinitial

• Transition Dispatch Cost

DC(Gi,t,k, LLn,t,k) =Nt∑t=1

Nk∑k=1

Ng∑i=1

(CiGi,t,k + CruR

Ui,t,k + CrdR

Di,t,k

)+

Nn∑n=1

V LOLUn,t,k +

Np∑p=1

CpPTp,t,k +

Nq∑q=1

CqWq,t,k− (3.29)

Nn∑n=1

Un(dn,t + 1.05 · cDn,t,k − 0.95 · cUn,t,k)


• Final Dispatch Cost

DC(Gi,k, LLn,k) =Nk∑k=1

Ng∑i=1

(CiGi,k + CruR

Ui,k + CrdR

Di,k

)+

Nn∑n=1

V LOLUn,k+

Np∑p=1

CpPTp,k +

Nq∑q=1

CqWq,k −Nn∑n=1

Un(dn,t=Nt + 1.05 · cDn,t=Nt,k − 0.95 · cUn,t=Nt,k)

(3.30)

After updating the objective function, balancing conditions (2.5) are reformulated, in-

cluding theses new terms (3.32). Flexible sources variables are added to the flow equa-

tions (2.8) too:

• Balancing Conditions

Ng∑i=1

(Gi −Oi)Υi,n +

Nl∑l=1

PlΦl,n +

Nq∑q=1

WqΨq,n +

Np∑p=1

(P Tp − PPp )χp,n + Un − Linitial = 0

(3.31)

Ng∑i=1


Nl∑l=1

Pl,t,kΦl,n +

Nq∑q=1

Wq,t,kΨq,n +

Np∑p=1

(P Tp,t,k − PPp,t,k)χp,n

+ Un,t,k − dn,t,k = 0 (3.32)

Ng∑i=1

(Gi,k −Oi,k)Υi,n +Nl∑l=1

Pl,kΦl,n +

Nq∑q=1

Wq,kΨq,n +

Np∑p=1

(P Tp,k − PPp,k)χp,n

+ Un,k − dn,t=Nt,k = 0 (3.33)

• Transmission Flow Balance

Nn∑n=1

[Hl,n

(Ng∑i=1

(Gi −Oi)Υi,n +

Nq∑q=1

WqΨq,n +

Np∑p=1

(P Tp − PPp )χp,n − (Linitial − Un)

)]= Pl

(3.34)


Nn∑n=1

[Hl,n

(Ng∑i=1


Nq∑q=1

Wq,t,kΨq,n

+

Np∑p=1

(P Tp,t,k − PPp,t,k)χp,n − (dn,t,k − Un,t,k)

)]= Pl,t,k (3.35)

Nn∑n=1

[Hl,n

(Ng∑i=1

(Gi,k −Oi,k)Υi,n +

Nq∑q=1

Wq,kΨq,n +

Np∑p=1

(P Tp,k − PPp,k)χp,n

− (dn,t=Nt,k − Un,k)

)]= Pl,k (3.36)

Two new constants are introduced, Ψq,n and χp,n, that represent the node in which

the stochastic and pump-hydro storage plants respectively are located. It is possible to

notice that in the transition and final balance, the actual load is who acts instead the

scheduled one, as it is in the optimization function.

Chapter 4

Investment Planning

4.1 Introduction

The big challenge with renewables (and stochastic) energies is to deal with their unpre-

dictable nature. Regulation and contingency reserves are dispatched to fix the alternat-

ing power output that this kind of generation produce, so a discussion of this study is

necessary to minimize the problems caused by it. As illustrative example, Figure 4.1

shows generation in South Australia by fuel type between 22 June and 5 July 2014,

which clearly shows that wind was the major fuel source over this period. On 27 June

2014 at 3 am, wind output was around 99% of native demand in South Australia and

around 71% of total South Australian generation. Here is latent how important are, and

further will be, stochastic production in the power systems.

Investing in new generation units and transmission lines may help to introduce stochas-

tic generation into the power system. There may be periods where ramp rates providers

of the power system are not quick enough to adapt their generation level to stochastic

generation changes. It is here where investment plays an important role with the intro-

duction of new types of generation, allowing to overcome fast changes in the equilibrium

generation-demand, or lines, distributing better the existent power.

Contingencies due to stochastic sources can be estimated using the predicted curve

demand - solar/wind generation. Generation investment, as a long-term issue, is related

with features as ramp rate, start-up time, minimum generation and the capacity to

adequacy of the resource. With line investment, the power transmission is improved. If

22

Chapter 4. Investment Planning 23

Figure 4.1: Generation in South Australia by fuel type between 22 June and 5 July2014.

power transmission system is limited, the ability to transport power from places where

the price of electricity is cheap to those where it is expensive is reduced. Thus, if huge

amount of stochastic generation exists in the system, it is necessary a good transmission

network for transporting this cheap generation to those areas where it is needed.

4.2 Investment Planning

The main difference between investment decisions and operation decisions is the actua-

tion period. Investment is with long-term expectations while the importance of operation

decisions increase when short dispatch intervals are succeeding. However, a combination

of both of them is desirable to overcome fast changes in the generation side.

In the past, numerous methods for power system planning have been proposed. With the

objective of achieve economic, technical and reliability issues, system planning have been

focused mainly on economic or technical aspects. However, modern planning trends to

optimize between the reliability of the power system and the economic efficiency of the

electricity market. Thus, studies to analyze how the changes in stochastic generation

affect the system are the main feature to research.

An efficiently operated power system would, in principle, ensure that all devices as gen-

erators, consumption, storage devices, etc. are used conveniently in the short-term and


there is efficient investment in such devices in the longer term. For studying operation

decisions issues, unit commitment conditions [15] are added in the final case and system

flexibility is measured.

4.3 Investment Planning Model

Notation for this chapter is similar to those found in Chapter 2. Ideally, ∆t should be

chosen as a period short enough to reflect the time interval over which the power system

would remain within its operating limits following any credible contingency. However,

in practice, ∆t is bounded below by computational limits. All assumptions are discussed

in detail in [12] and [16].

Generation and transmission investment decisions are the flexibility sources in this chap-

ter. Now, demand is inflexible and storage in one hydro power plant is considered. Four

different cases are designed. When no investment decisions are studied, α and β are

equal to 0, while if lines investment decisions are taken into account, α is equal to 1

and β remains 0. Generation units investment decisions are calculated with α equal

to 0 and β equal to 1. If the power system is expanded in generation and lines terms

coordinately, both α and β are set to 1.

Minimize α

(∑v

xvTICv

)+ β

(∑u

yuGICu

)+∑ω

πω

[(1−

∑k

pk)∑i

(cigi,ω

+ cSPi si,ω + cSDi wi,ω) + (1−∑k

pk)∑u

(cugu,ω + cSPu su,ω + cSDu wu,ω)

]+∑i,k,t

pkcigi,k,t,ω + cSPi si,k,t,ω + cSDi wi,k,t,ω

(1 + r)t(4.1)

+∑u,k,t

pkcugu,k,t,ω + cSPu su,k,t,ω + cSDu wu,k,t,ω

(1 + r)t

+1

(1 + r)Tr

∑i,k

pk(cigi,k,ω + cSPi si,k,ω + cSDi wi,k,ω)

+1

(1 + r)Tr

∑u,k

pk(cugu,k,ω + cSPu su,k,ω + cSDu wu,k,ω)


Subject to :

Energy balance constraints :

Initial :∑

(i,u):n

(gi,ω + gu,ω + pgi,ω + ppi,ω) +∑

(l,v):term(n)

(fl,ω + fv,ω)

= Dn,ω +∑

(l,v):org(n)

(fl,ω + fv,ω) (4.2)

Transition :∑

(i,u):n

(gi,k,t,ω + gi,k,t,ω + pgi,k,t,ω + ppi,k,t,ω)

+∑

(l,v):term(n)

(fl,k,t,ω + fl,k,t,ω) = Dn,k,t,ω +∑

(l,v):org(n)

(fl,k,t,ω + fl,k,t,ω)

(4.3)

Final :∑

(i,u):n

(gi,k,ω + gi,k,t,ω + pgi,k,ω + ppi,k,ω) +∑

(l,v):term(n)

(fl,k,ω + fl,k,ω)

= Dn,k,ω +∑

(l,v):org(n)

(fl,k,ω + fl,k,ω) (4.4)

Transmission limit constraints for existing lines :

− Fl ≤ Bl(θn,ω − θm,ω) ≤ Fl, (θn,ω, θm,ω)↔ l (4.5)

− Fl,k ≤ Bl(θn,k,t,ω − θm,k,t,ω) ≤ Fl,k, (θn,k,t,ω, θm,k,t,ω)↔ l (4.6)

− Fl,k ≤ Bl(θn,k,ω − θm,k,ω) ≤ Fl,k, (θn,k,ω, θm,k,ω)↔ l (4.7)

− Fl ≤ fl,ω ≤ Fl (4.8)

− Fl,k ≤ fl,k,t,ω ≤ Fl,k (4.9)

− Fl,k ≤ fl,k,ω ≤ Fl,k (4.10)


Linearised transmission flow constraints for candidates lines :

− Ξ(1− xv) ≤ fv,ω −Bl(θn,ω − θm,ω) ≤ Ξ(1− xv), (θn,ω, θm,ω)↔ v

(4.11)

− Ξ(1− xv) ≤ fv,k,t,ω −Bl(θn,k,t,ω − θm,k,t,ω) ≤

Ξ(1− xv), (θn,k,t,ω, θm,k,t,ω)↔ v (4.12)

− Ξ(1− xv) ≤ fv,k,ω −Bl(θn,k,ω − θm,k,ω) ≤ Ξ(1− xv), (θn,k,ω, θm,k,ω)↔ v

(4.13)

Transmission limit constraints for candidate lines :

− xvFv ≤ fv,ω ≤ xvFv (4.14)

− xvFv,k ≤ fv,k,t,ω ≤ xvFv,k (4.15)

− xvFv,k ≤ fv,k,ω ≤ xvFv,k (4.16)

Generation limit constraints for existing generators :

zi,ωGMi ≤ gi,ω ≤ zi,ωGi (4.17)

zi,k,t,ωGMi ≤ gi,k,t,ω ≤ zi,k,t,ωGi (4.18)

zi,k,ωGMi ≤ gi,k,ω ≤ zi,k,ωGi (4.19)

Linearised generation limit constraints for candidate generators :

hu,ωGMu ≤ gu,ω ≤ hu,ωGu (4.20)

hu,k,t,ωGMu ≤ gu,k,t,ω ≤ hu,k,t,ωGu (4.21)

hu,k,ωGMu ≤ gu,k,ω ≤ hu,k,ωGu (4.22)

Ramp− rate constraints for existing generators :

− zi,k,t,ωRDi − (zi,ω − zi,k,t,ω) ∗GMi ≤ gi,k,1,ω − gi,ω

zi,ωRUi + (zi,k,t,ω − zi,ω) ∗GMi ≥ gi,k,1,ω − gi,ω t = 1 (4.23)

− zi,k,t,ωRDi − (zi,k,t−1,ω − zi,k,t,ω) ∗GMi ≤ gi,k,t,ω − gi,k,t−1,ω


zi,k,t−1,ωRUi + (zi,k,t,ω − zi,k,t−1,ω) ∗GMi ≥ gi,k,t,ω − gi,k,t−1,ω t = 2, ..., T

(4.24)

− zi,k,T+1,ωRDi − (zi,k,T,ω − zi,k,T+1,ω) ∗GMi ≤ gi,k,ω − gi,k,T,ω

zi,k,T,ωRUi + (zi,k,T+1,ω − zi,k,T,ω) ∗GMi ≥ gi,k,ω − gi,k,T,ω t = T + 1

(4.25)

Linearised ramp− rate constraints for candidate generators :

− hu,ωRDu∆t ≤ gu,k,1,ω − gu,ω ≤ hu,ωRUu∆t t = 1 (4.26)

− hu,k,t,ωRDu∆t ≤ gu,k,t,ω − gu,k,t−1,ω ≤ hu,k,t,ωRUu∆t t = 2, ..., T

(4.27)

− hu,k,ωRDu∆t ≤ gu,k,ω − gu,k,T,ω ≤ hu,k,ωRUu∆t t = T + 1 (4.28)

Hydro generation constraints :

(gi,ω +∑t

gi,k,t,ω + gi,k,ω)∆t ≤ Ei (4.29)

Constraint for pump storage plants :

Qi,ω = Qi,0,ω + ppi,ωηp − pgi,ω/ηp (4.30)

Qi,k,t,ω = Qi,ω + ppi,k,t,ωηp − pgi,k,t,ω/ηp (4.31)

Qi,k,ω = Qi,k,T,ω + ppi,k,ωηp − pgi,k,ω/ηp (4.32)

0 ≤ Qi,ω ≤ Qmaxi , 0 ≤ Qi,k,t,ω ≤ Qmaxi , 0 ≤ Qi,k,ω ≤ Qmaxi

Linearised bilinear terms zu,ωyu, zu,k,t,ωyu, zu,k,ωyu :

hu,ω ≤ zu,ω , hu,ω ≤ yu , hu,ω ≥ zu,ω + yu + 1 , hu,k,t,ω ≤ zu,k,t,ω (4.33)

hu,k,t,ω ≤ yu , hu,k,t,ω ≥ zu,k,t,ω + yu + 1 , hu,k,ω ≤ zu,k,ω (4.34)

hu,k,ω ≤ yu , hu,k,ω ≥ zu,k,ω + yu + 1 (4.35)


Start− up constraints for existing and candidate generators :

si,ω = zi,ω , su,ω = zu,ω

si,k,t,ω ≤ zi,k,t,ω , su,k,t,ω ≤ zu,k,t,ω

si,k,t,ω ≤ 1− zi,k.t−1.ω , su,k,t,ω ≤ 1− zu,k.t−1.ω

si,k,t,ω ≥ zi,k,t,ω − zi,k,t−1,ω , su,k,t,ω ≥ zu,k,t,ω − zu,k,t−1,ω (4.36)

si,k,ω ≤ zi,k,ω , su,k,ω ≤ zu,k,ω

si,k,ω ≤ 1− zi,k,T,ω , su,k,ω ≤ 1− zu,k,T,ω

si,k,ω ≥ zi,k,t,ω − zi,k,T,ω , su,k,ω ≥ zu,k,t,ω − zu,k,T,ω

Shut− down constraints for existing and candidate generators :

wi,,k,t,ω = zi,k,t−1,ω , wu,,k,t,ω = zu,k,t−1,ω

wi,k,t,ω ≤ 1− zi,k,t,ω , wu,k,t,ω ≤ 1− zu,k,t,ω

wi,k,t,ω ≥ zi,k,t−1,ω − zi,k,t,ω , wu,k,t,ω ≥ zu,k,t−1,ω − zu,k,t,ω (4.37)

wi,k,ω ≤ zi,k,T,ω , wu,k,ω ≤ zu,k,T,ω

wi,k,ω ≤ 1− zi,k,ω , wu,k,ω ≤ 1− zu,k,ω

wi,k,ω ≥ zi,k,T,ω − zi,k,ω , wu,k,ω ≥ zu,k,T,ω − zu,k,ω

In this model, there are two types of non-linear terms. However, as a linear system is

being looking for, following linearization techniques are applied:

1. The transmission flow constraints for candidate lines fv − xvBv(θj − θi) = 0 have

non-linear terms xvθj and xvθi, where xv ∈ 0, 1 and θj , θi ∈ <. Using a large enough

positive number Ξ, this non-linear constraint can be written as the linear constraints:

−Ξ(1− xv) ≤ fv −Bl(θj − θi) ≤ Ξ(1− xv).

2. The flexibility investment constraint zuyuGMu ≤ gu ≤ zuyuGu has the non-linear

term zuyu, where zu ∈ 0, 1 and yu ∈ 0, 1. It is defined hu = zuyu and then replace

it with hu ≤ zu, hu ≤ yu and hu ≤ zu + yu + 1.

The optimisation problem (4.1)-(4.37) has two sets of linearised constraints: transmission

flow constraints for candidate lines (4.12) and flexibility investment constraint (4.34).

The resulting problem is a Mixed Integer Linear Programming (MILP), since non-linear

terms are removed. There are some power plants with energy-limited resources such as


energy-storage and hydro generators. The demand is considered fix and known unlike

the flexible demand studied before. It is also considered demand response in every load

node, expressed as penalty generators.

Ramp rate constraints 4.24 have binary terms (state of generator, 1-on, 0-off) for allowing

start or shut down the generator in any period of time. Minimum generation appears in

this constraint as dispatchable units start to produce with their minimum generation.

Chapter 5

Benders Decomposition

5.1 Introduction

Optimization plainly dominates the design, planning, operation, and control of engineer-

ing systems. In principle, one may think that the optimization of a system composed

by independent entities is simply to create a big mathematical model and solving it cen-

trally using currently available computing power and solution techniques. In practice,

however, this is usually impossible [17]. If it is desired to solve a problem centrally,

complete information on objective functions and constraints is need. These entities are

separated geographically and functionally, so this information may be unattainable or,

in terms of money, impossible to reach. More importantly, entities could be undesired to

share their own information, as it is not incentive compatible to do so, i.e., these entities

may have an incentive to misrepresent their true preferences. With the aim of optimize

certain benefits in those power systems, it is turned into the aspects of decomposition.

Specially with limited information, it is imperative to coordinate entities to reach an

optimal solution [18]. The goal is to coordinate the entities by optimizing a certain

objective (such as finding equilibrium resource price) while satisfying local and system

constraints.

All of these results into more convoluted calculations, being crucial the introduction of

decomposition techniques asDantzig−Wolfe [19], Benders [20], Lagrangian Relaxation

[21] or Augmented Lagrangian Decomposition [22] to reduce computational issues.

30

Chapter 5. Benders Decomposition 31

Several softwares have been developed to analyze those problems: Linear-Programming

(LP) if optimization has a linear objective function, subject to linear equality and linear

inequality constraints, Non-Linear Programming (NLP) if they have at least one non-

linear term, Mixed-Integer Linear Programming (MILP), a linear model in which some or

all of the variables are restricted to be integers, Mixed-Integer Non-Linear Programming

(MINLP), etc. In this research, MILP problems with unit commitment constraints and

investment decisions as binary variables are the matter to study.

5.2 Benders Decomposition for MILP problems

Benders decomposition is one of the most commonly used decomposition techniques

in power systems. Jaques. F. Benders (born 1925, Swalmen, Netherlands) introduced

the Benders decomposition algorithm for solving large-scale mixed-integer programming

(MIP) problems. Benders decomposition has been successfully applied to various op-

timization problems, such as power systems operation and planning, network design,

electronic packaging, transportation, manufacturing, logistics or military applications.

When applying Benders decomposition [23], the original problem is decomposed into

a master problem and several subproblems. Generally, master problem is integer and

subproblems are linear. This decomposition method is also used when exists a high-

level of priority in the model design which will form the master problem [24]. Master

problem is composed of the constraints that depend on the complicating variables while

the subproblems examine the solution of the master problem to check if it gratify the

remaining constraints. If subproblems are feasible, upper bound solution of the original

problem is calculated, while forming a new objective function for the further optimization

of the master problem solution. If any of the subproblems is infeasible, an infeasibility

cut representing the least satisfies constraint is added to the master problem. Then, a

new lower bound solution of the original problem is achieved by solving again the master

problem, adding more constraints (cuts). When the upper bound and the lower bound

are sufficiently close, the optimal solution of the original problem is achieved and the

iterative procedure stops.


5.2.1 Mathematical Formulation

The general MILP problem has the form [20]:

Minimize

n∑i

cixi +

m∑j

djyj (5.1)

Subject to :

n∑i

aixi +m∑j

ejyj = b (5.2)

xi ∈ 0, 1 (5.3)

ydownj ≤ yj ≤ yupj (5.4)

To solve the problem (5.1)-(5.4) with Benders decomposition, next steps must be fol-

lowed:

1. Solve complicated variables (integer) and fix them to given feasible integer values.

2. With those fixed integer variable values, solve the resulting continuous LP problem

(or Subproblems) and obtain the optimal objective function value and the sensitivities

(dual variables) associated with constraints that fix the integer variables to specific

values. Obtain an upper bound with the objective function optimal value.

3. Solve the MILP master problem to determine improved values of the integer variables.

Obtain a lower bound with the objective function optimal value.

4. If the gap between both bounds are smaller than a certain tolerance set by the user,

stop, the optimal solution has been reached; otherwise, the algorithm continues in Step 2.

Step 0. Initialization. Counter ν = 1 , Solve initial master problem:

Minimize

n∑i

cixi + α(xi) (5.5)

Subject to :

xi ∈ 0, 1 (5.6)

α ≥ αdown (5.7)


Step 1. Solve subproblem for values previously obtained:

Minimizen∑j

djyj (5.8)

Subject to :

m∑j

ejyj = b −n∑i

aixi (5.9)

ydownj ≤ yj ≤ yupj (5.10)

xi = xi : λi (5.11)

Where yj and dual variables λi associated to constraints fixing integer variables to

specific values are obtained.

Step 2. Convergence checking: Update upper and lower bounds and compare them.

zνup =n∑i

cixi +m∑j

dj yj (5.12)

zνdown =n∑i

cixi + α (5.13)

If zνup−zνdown is smaller than a small tolerance previously fixed (ε) the solution is optimal

and xi = xi , yj = yj . Otherwise go to step 3:

Step 3. Update master problem with cuts 5.15: Iteration ν → ν + 1.

Minimize

n∑i

cixi + α(xi) (5.14)

Subject to :

α ≥m∑j

dj yi +

n∑i

λi(xi − xi) (5.15)

xi ∈ 0, 1

α ≥ αdown (5.16)

This is solved again, obtaining new values for α and xi and then the iteration continues

to Step 1.

It may happen that one of the problem is infeasible. If master problem is infeasible,


Figure 5.1: Flowchart of the Benders Decomposition.

the whole problem is infeasible and no solution can be accomplished. If the infeasibility

happen in the subproblem, a feasibility cut is added to the master problem. These cuts

have the following form [25]:

0 ≥m∑j

dj yi +

n∑i

λi(xi − xi) (5.17)

Thus, the general form of the master problem is presented in:

Minimizen∑i

cixi + α(xi)

Subject to :

Optimality cuts :

α ≥m∑j

dj yi +

n∑i

λi(xi − xi)


Feasibility cuts :

0 ≥m∑j

dj yi,infeasible +

n∑i

λi(xi − xi)

xi ∈ 0, 1

α ≥ αdown

5.2.2 Benders Decomposition applied to a 2-node system with unit

commitment

Before applying Benders decomposition to a big and complex problem, its performance

is tested with a small example consisting in a system with two nodes connected with a

line (Figure 5.2).

Figure 5.2: 2-Node system, 1: Existent generator, 2: Existent generator, 3: Demandresponse, 4: Demand response.

Unit commitment decisions are modelled as binary variables. They are the complicating

variables to be decided in the master problem. To make this test study as close as

the problem studied later, same constraints as Section 4.3 are used, without those that

depends on candidate generators and lines. Three scenarios are modelled, setting 90%,

95% and 105% for final load regarding the initial one, 50MW for each node, affecting to

the node load as Table 5.1 shows.

Node 1 Node 2

Scenario 1 45 45

Scenario 2 47.5 47.5

Scenario 3 52.5 52.5

Table 5.1: Final load for three different scenarios.

Two dispatchable generators exist in each node with their respectively demand response,

modelled as penalty generators [26] (high variable cost) Table 5.2.


Max power Min power Ramping Rate Variable Cost Start-up Cost

G1 100 50 10 10 20

G2 100 50 10 30 35

P1 300 0 20 300 0

P2 300 0 20 300 0

Table 5.2: Data for Generation Units.

The line connecting both nodes has a transmission capacity of 100 MW and the fic-

titious cost for undelivered load is established to 9.999 kœ/MW . The probability for

contingencies is 0.1 and tolerance to control the convergence of Benders algorithm is

10−8.

For improving Benders decomposition method, one cut for each scenario is created [27],

[28] and [29]. The special decomposable structure of this form is suitable for Benders De-

composition because it takes advantage of subgradient information to construct convex

estimates of the recourse function and iteratively generates a Benders cut to be added

to the decomposed master problem. In the first step, a decomposed subproblem with

those constraints that do not include the second-stage variables is solved to obtain the

values of the first-stage decisions [30]. Then first-stage decisions are fixed and all the

scenario subproblems that include second-stage decisions are solved in order to obtain

the optimal values of the second-stage decisions (Figure 5.3).

Figure 5.3: Algorithm for Benders Decomposition with Multiple Scenarios.

Thus, master problem with multiple cuts (one cut for each scenario) depends on the


number of iterations and scenarios (ω) that compose the Benders decomposition prob-

lem, i.e. it is performed one cut for every scenario and every iteration until optimal

solution is achieved:

Minimize

n∑i

cixi +∑w

αω,it(xi) (5.18)

Subject to :

αω,it ≥m∑j

dj yi,ω +

n∑i

λi,ω(xi,ω − xi,ω)

0 ≥m∑j

dj yi,ω +n∑i

λi,ω(xi,ω − xi,ω)

If all integer variables are in the master problem, it is assured that α-function is convex,

so it is not necessary to implement heuristics. However, the advantage of decoupling in

scenarios that was achieved only by setting investment decisions in the master problem

is lost. At this point it is decided to continue with the first method [31], as solution is

more intuitive and no implementation for heuristics is mandatory, which could lead to a

increasing complexity of the model. Thus, both sets of binary variables associated with

investment as those associated with unit commitment belongs to master problem while

subproblem remains as LP.

The model is coded in GAMS environment using CPLEX solver in a computer Intel

Core i7-2600 CPU with a 3.40 GHz clocking frequency and 8 GB of RAM. The model

is simulated up to 9 transition periods. Table 5.3 shows that for some cases, lower

bound is bigger than the upper bound and the iterative Benders loop stop. This issue

is produced by GAMS default solver. It works with a MIP solution gap of 0.1, which

means that master problem allows ”upper” and ”lower” bound of his branch-and-bound

solving method to have a tolerance of 10%, so non optimal solution is found, leading to

incorrect results when applying Benders algorithm.

However, optimal solution can be achieved 1 without the need of complex methods. Re-

sults with optimal solution in master problem are presented in Table 5.5 and Table 5.6.

In this case, upper and lower bound are the same which means that correct optimal

solution is achieved.

1To solve this issue, GAMS has an easy sentence to adjust the MIP solution gap as the user desires.It can be done writing ”option optcr = 0.0; ” just before the solving statement of the MIP problem.


MIP solution (kœ) Upper Bound (kœ) Lower Bound (kœ)

T=t1 65.692 65.692 65.692

T=t1,t2 65.692 65.692 65.692

T=t1,...,t5 65.692 65.692 65.692

T=t1,...,t6 65.692 67.188 67.550

T=t1,...,t7 65.692 67.188 67.424

T=t1,...,t8 65.692 70.982 71.161

T=t1,...,t9 65.692 67.188 67.602

Table 5.3: Dispatch Cost with Non-Optimal Master Problem.

Time (seconds) Number of iterations

T=t1 1.744 17

T=t1,t2 2.877 27

T=t1,...,t5 5.203 65

T=t1,...,t6 13.297 132

T=t1,...,t7 21.515 276

T=t1,...,t8 29.528 314

T=t1,...,t9 38.194 402

Table 5.4: Calculation times with Non-Optimal Master Problem.

MIP solution (kœ) Upper Bound (kœ) Lower Bound (kœ)

T=t1 65.692 65.692 65,692

T=t1,t2 65.692 65.692 65.692

T=t1,...,t5 65.692 65.692 65.692

T=t1,...,t6 65.692 65.692 65.692

T=t1,...,t7 65.692 65.692 65.692

T=t1,...,t8 65.692 65.692 65.692

T=t1,...,t9 65.692 65.692 65.692

Table 5.5: Dispatch Cost with Optimal Master Problem.

Time Benders Number of iterations

T=t1 1.375 17

T=t1,t2 5.189 27

T=t1,...,t5 11.083 97

T=t1,...,t6 21.587 161

T=t1,...,t7 35.378 258

T=t1,...,t8 47.981 358

T=t1,...,t9 51.827 570

Table 5.6: Calculation times with Optimal Master Problem.

As seen, Benders Decomposition has been successfully applied to this simple 2-node

system without investment, in addition to apply an acceleration technique as it is the


separation of Benders cuts by scenarios.

5.3 Benders Decomposition for Investment Planning

When investment is added, some modifications have to be applied to Benders method

[32] to work with discrete variables. For this case, investment and unit commitment

decisions are the variables to fix in the MP while optimal operation belongs to SP.

Based on proposed in [33], if α function is continuous, it would be enough using a

suitable branch-and-cut technique. Continuity of α function is guaranteed as all binary

variables are fixed in the master problem, while subproblem remains as LP [16].

Conventional Benders decomposition has been applied in this paper, but further im-

provements can be carried out in order to require less computation time, since the slow

convergence of Benders method with conventional solvers may be unacceptable [34], [35]

and [36].

Khodaei et. al. [31] proposed to add an extra subproblem before solving optimal oper-

ation problem. The goal of this subproblem is to guarantee the feasibility of the system

and thus start from a point closer to the optimal solution. This problem is called Security

Check.

The Security Check subproblem (SP1) analyzes when the proposed plan satisfies the

operation constraints. This subproblem would satisfy the power balance at each node

while preserving base case and contingency constraints. If Security Check conditions (in

this case, sum of all demand response lower than a set value) are not fulfilled, a feasibility

cut is created and added to the master problem. Until a secure plan is achieved, the

loop continues.

The Optimal Operation subproblem (SP2) is used to compute the optimal solution. This

subproblem checks the optimality comparing the lower bound, already calculated in the

master problem, with the upper bound of the original MIP planning problem. If the

proposed plan is not optimal, optimality cuts will be formed and added to the master

problem for the next iteration.

Note that two different types of cuts are used. The first one, generated in the Feasibility

Check subproblem, is a feasibility cut. This cut indicates that the security violations can


be reduced by adjusting the investment plan as well as the unit commitment decisions.

The second one is an optimality cut, which is generated in the Optimal Operation

subproblem. The optimality cut limits the range of master problem objective to make

it closer to the objective function of the original MIP planning problem. The optimality

cut indicates that the objective value of the function of expansion planning can decrease

by modifying the investment plan together with the state of the dispatchable power

plants [37]. The dual variables in the optimality cut are the incremental reduction in

the objective function of the Optimal Operation subproblem.

Figure 5.4: Benders Decomposition Flowchart with Feasibility Cuts [31].

Figure 5.4 shows the flowchart of this technique when applying Benders decomposition.

5.3.1 Model of Benders Decomposition for Investment Planning

As developed in the previous section, two levels of subproblem are created: one Security

Check Subproblem to guarantee feasibility and one Optimal Operation Subproblem to

minimize the operational costs. In master problem, all integer variables are solved to

ensure the convergence of alpha-function:


*Master Problem (MIP):

Minimize α

(∑v

xvTICv

)+ β

(∑u

yuGICu

)+∑ω

πω

[(1−

∑k

pk)∑i

(cigi,ω

+ cSPi si,ω + cSDi wi,ω) +

[(1−

∑k

pk)∑u

(cugu,ω + cSPu su,ω + cSDu wu,ω)

+∑i,k,t

pkcigi,k,t,ω + cSPi si,k,t,ω + cSDi wi,k,t,ω

(1 + r)t

+∑u,k,t

pkcugu,k,t,ω + cSPu su,k,t,ω + cSDu wu,k,t,ω

(1 + r)t

+1

(1 + r)Tr

∑i,k

pk(cigi,k,ω + cSPi si,k,ω + cSDi wi,k,ω)

+1

(1 + r)Tr

∑u,k

pk(cugu,k,ω + cSPu su,k,ω + cSDu wu,k,ω) + αω,it

Subject to :

constraints(4.34)− (4.37)

Optimal Operation Benders cuts :

αω,it ≥ zω,it,OO +∑v

λv,ω,it(xv − xv) +∑u

λu,ω,it(yu − yu)

+∑i,k,t

λi,k,t,ω,it(si,k,t,ω − si,k,t,ω) +∑i,k,t

λi,k,t,ω,it(ωi,k,t,ω − ωi,k,t,ω)

+∑i,k,t

λi,k,t,ω,it(zi,k,t,ω − zi,k,t,ω) +∑i,k,t

λi,k,t,ω,it(hi,k,t,ω − hi,k,t,ω)

Security Check Benders cuts :

0 ≥ zω,it,SC +∑v

λv,ω,it(xv − xv) +∑u

λu,ω,it(yu − yu)

+∑i,k,t

λi,k,t,ω,it(si,k,t,ω − si,k,t,ω) +∑i,k,t

λi,k,t,ω,it(wi,k,t,ω − wi,k,t,ω)

+∑i,k,t

λi,k,t,ω,it(zi,k,t,ω − zi,k,t,ω) +∑i,k,t

λi,k,t,ω,it(hi,k,t,ω − hi,k,t,ω)

xv, yu, s, w, z, h ∈ 0, 1

αω,it ≥ αdown


Where zω,it,SC and zω,it,OO are the value of the objective function in security check

subproblem and optimal operation subproblem for each scenario and each iteration re-

spectively. Only start-up, shut-down, linearization for h and Benders cuts constraints

limit the solution of the master problem. When it is solved, information of integer

variables is sent to Security Check Subproblem:

*Security Check subproblem (LP):

Minimize∑ω

πω

[(1−

∑k

pk)∑p

cpgp,ω +∑p,k,t

pkcpgp,k,t,ω(1 + r)t

+1

(1 + r)Tr

∑p,k

pk(cpgp,k,ω)

](5.19)

Subject to :


xv = x : λx

yu = y : λy

s = s : λs

w = w : λw

z = z : λz

h = h : λh

Where λx, λy, λs, λw, λz, λh are the marginal values for the security check subproblem,

belonging to the binaries variables x y, s, w, z, h respectively. Subset p in 5.19 refers

to demand response. In this problem, marginal values for feasibility Benders cuts are

calculated and added to master problem, until security check conditions are fulfilled

(sum of cost of all demand response lower than certain specified value). When it is

accomplished, optimal operation subproblem can be solved:

*Optimal Operation Subproblem (LP):

Minimize∑ω

πω

[(1−

∑k

pk)∑i

cigi,ω + (1−∑k

pk)∑u

cugu,ω +∑i,k,t

pkcigi,k,t,ω(1 + r)t

+

∑i,k,t

pkcigi,k,t,ω(1 + r)t

+∑u,k,t

pkcugu,k,t,ω(1 + r)t

+1

(1 + r)Tr

∑i,k

pk(cigi,k,ω) +1

(1 + r)Tr

∑u,k

pk(cugu,k,ω)

]


Subject to :


xv = x : λx

yu = y : λy

s = s : λs

w = w : λw

z = z : λz

h = h : λh

Same constraints as Security Check Subproblem are used, with the difference that here

optimal operation marginal values are calculated. Objective function consists of opera-

tion cost terms. Iterative procedure is followed until upper bound and lower bound are

smaller than an specified value as it was explained in previous sections.

5.3.2 Benders Decomposition applied to a 2-node system with invest-

ment and unit commitment

The example of two nodes system (Section 5.2.2) is implemented again in both GAMS

and GLPK solvers for checking the time performed by adding the intermediate Feasibility

Subproblem. Investment has been added (see equations in Section 5.3.1) with one can-

didate generator in each node and a candidate line connecting both nodes (Figure 5.5).

Figure 5.5: 2-Node system with investment, 1: Existent generator, 2: Existent gener-ator, 3: Demand response, 4: Demand response, 5: Candidate generator, 6: Candidate

generator.

A single load of 400 MW is added to node 2 while node 1 remains unloaded. Data

for candidate generators are assumed to be the same as in the case discussed above for

no-investment (Table 5.7). The transmission capacity of both existent and candidate

line is 100 MW. Three different scenarios are developed, corresponding to different final


load levels (Table 5.8). Diverse security check bounds are studied, from the acting of all

demand response is permitted (i.e., as if there were not feasibility check subproblem) to

the case that no demand response is allowed.

Max power Min power Ramping Rate Variable Cost Start-up Cost

G1 100 50 10 10 20

G2 100 50 10 30 35

P1 300 0 20 300 0

P2 300 0 20 300 0

G3 100 50 10 10 20

G4 100 50 10 30 35

Table 5.7: Data for Generation Units in 2-node system with investment.

Node 1 Node 2

Scenario 1 0 400

Scenario 2 0 380

Scenario 3 0 360

Table 5.8: Final load for three different scenarios in 2-node system with investment.

Pen. Generation Objetive (kœ) Calculation No sol. SP2/Allowed Function Time (s) No sol. total

840000 (100%) 374.589 1:54.899 49/49

630000 (75%) 374.589 49.436 31/32

420000 (50%) 374.589 32.203 22/23

210000 (25%) 374.589 38.240 20/31

84000 (10%) 374.589 23.504 7/25

42000 (5%) 374.589 22.460 5/26

16800 (2%) 374.589 21.738 4/27

8400 (1%) 374.589 23.145 2/30

4200 (0,5%) 374.589 22.328 2/30

0 (0%) 374.589 22.633 2/30

Table 5.9: Dispatch cost solution for Feasibility Benders and Standard Benders.

Based on shown in Table 5.9, same objective function values are achieved, but less

computation time is needed for cases that are more strict when limiting the amount of

demand response available (MP parts closer to the final situation). In all cases, with

the introduction of both new generators and the line, demand response is no needed .

In the first column of Table 5.9, the condition to satisfy security check restrictions (sum

of the cost of all demand response smaller than this value) is indicated. Last column

shows the relation between the number of times that the optimal operation subproblem


(second level of the subproblem) is solved and the total iterations needed. When the

feasibility condition is more restrictive regarding demand response, master problem is

closer to the final solution, so only minimizing generation cost will remain and leading to

a faster convergence of α term. This term, that can take any value, is closer to its final

value that checks Benders decomposition concurrence. It is noteworthy that feasibility

condition should not be extremely restrictive, as it may lead to a non-optimal solution

if demand response is included in the optimal solution. Also, as reader can notice in

Table 5.9, the lowest calculation time is achieved when some demand response is allowed

before granting to solve the optimality subproblem.

Chapter 6

Case Study

6.1 Introduction

In this chapter, the mathematical models developed for short term (2.3-2.21 and 3.28-

3.35) and long term (4.1-4.37) are used in three case studies. Different power systems

are examined to give a broader view about how important are the ramp rate providers in

markets with high penetration of stochastic production. The 5-node National Electricity

Market of Australia (NEM), IEEE 6-node system and IEEE 30-node system are studied

with SRED model and numerical results are discussed.

In Section 6.2, the mathematical model exposed in Chapter 3 is tested. In Section 6.3,

investment and unit commitment decisions are studied applying Benders decomposition.

6.2 Study of Flexible Demand and Storage Availability

In this first part, demand and storage are considered as flexible resources. They are

introduced into a 5-node model, representing the five eastern states of Australia, and the

utility of those flexible resources is tested. As it was mentioned in Chapter 3, flexibility

from storage availability is studied increasing the features of pump-hydro storage plants.

These features are the initial and maximum water content, but it not considered the

necessary cost to implement these improvements. Flexible demand is modelled between

some margins. Nowadays, with the introduction of domestic devices or electric cars, it is

possible for the consumer choose not to consume energy when there is a high electricity

46

Chapter 6. Case Study 47

price, but consume it when this is low, leading to lower electricity price in the system

[38].

6.2.1 5-node model of the Australian National Electricity Market

The mathematical model explained in the previous chapters is applied to a real case and

conclusions are exposed. The 5-node model of the National Electricity Market (NEM) of

Australia [39],[40] is used because, as it was mentioned before, system is modelled with

SRED, which means small dispatch intervals, as those used in the Australian market

[41]. Five different zones, representing each single region of the NEM (five eastern states

of Australia), compose the system: Queensland (QLD), New South Wales (NSW), South

Australia (SA), Victoria (VIC) and Tasmania (TAS). Figure 6.1 illustrate this system.

Figure 6.1: The 5-node model of Australian National Electricity Market [39].

To study the effect of big quantity of wind in the NEM, wind farms under construction

has been taken as existent in the system, as the goal of the development of modern

electricity market models is to use them in a near future where there will be a high

penetration of stochastic generation. Nowadays around 80% of Australian power gener-

ation comes from fossil fuels [42], but the challenge is to reach around 25% of renewable

generation by the next ten years. With that, it is assumed that the zone with higher

wind generation is Victoria [43], with big wind production areas as Macarthur Wind


Farm or Darlington. On the other hand, Tasmania is characterized by the large number

of hydro power plants, but with low wind production [44]. Information for QLD, SA

and NSW is adopted from [45], [8] and [46].

Generation data for this study can be found in [39], following the assumptions explained

there. To analyze the decision of the system operator when choosing generation to

be dispatched, generation is decomposed into five different types: expensive dispatch-

able generators, dispatchable generators, hydro storage generators, pump-hydro storage

generators and wind farms.

Total generation capacity for each zone is set out in Table B.1 in AppendixB. To

condense these generation and thus reduce the number of variables, the existence of one

single type of generator in each node is supposed, according to the power installed in

every zone of the NEM. Generators are modelled to have some consistency with the

variable cost and ramping rate, i.e. thermal variable cost bigger than hydro, thermal

ramp rates smaller than hydro and zero wind variable cost [47].

Demand data [39] is deliberately modified to give its flexible character. In those regions

which important quantity of cities and intense industrial activity (QSD, NSW and VIC)

load is large [48], while those more inhospitable (SA and TAS), less power is demanded.

Same utility cost and minimum daily energy data is assumed for all loads (Table B.2).

Lines details are given in Table B.3. For simplicity of the model, equal maximum

transmission flow is considered in both directions and the value for susceptance and

capacitance is set to 0,1 p.u.

As it was previously mentioned, there is a distinction between conventional (expensive

thermal, thermal and hydro) and special (wind and hydro-pumped) power plants. Data

for pump-hydro storage plants is only contemplated if it is considerable enough to have

an important impact in system operation [49] and [50], which can be found essentially

in Queensland and New South Wales. It is estimated for the starting point the half of

initial water of maximum capacity for both upper and lower lake reservoir (Table B.4).

It is assumed that it can produce energy until there is no water in the reservoir. Variable

cost [51] are same as hydro power plants. σTp (Conversion factor water flow to produce

power < 1) and σPp (Conversion factor water flow to consumed power > 1) are set to

0.8 and 1.2 MW/(Hm3/h) respectively.


Data for wind power is illustrated in Table B.5. Notice that wind production is supposed

to be known and constant (only change due to contingencies that form the different

scenarios). Ramp rate is zero, as much as the variable cost.

Finally, five different equiprobable contingency scenarios are defined. Each scenario

represents lost of wind power (50% of the rated power) in each area and back recovery

in the final period (Table B.6).

There are some missing data which is estimated for getting a good approximation. The

price of lost of load is high enough (10000 œ/h) to avoid this situation, interest rate is

0.02 and contingencies are equiprobable (0.2). PTDF matrix 6.1 [52], [53] and [54] is

calculated, taking as reference the third node, as it has the better connections with the

other nodes.

PTDF (l, n) =

1 0 0 0 0

1 1 0 0 0

0 0 0 −1 0

0 0 0 0 −1

(6.1)

Flexibility is measured as the benefit obtained introducing the resources commented in

Chapter 3: flexible demand and storage availability. Simulations for just only one of

these resources and then both together is tested and results are compared.

Optimization problem (2.3)-(2.21) and (3.28)-(3.35) is solved with General Algebraic

Modeling System (GAMS) platform. To get a broader view about the impact of flexible

resources on the market when there is a high penetration of stochastic production,

different cases are studied.

For flexible demand, one unique and constant demand in each node is considered, but

different demand intervals, up to 50% regarding the average measure, are tested. This

means that Dmax and Dmin are progressively changed from having both same value

(inflexible demand) to 25% up and down value of existing load respectively.

Storage availability has a higher relation with the balance of the power system. If the

number of pumped hydro power plants is increased or the size of those existing increment,

storage can be an important feature to consider in the balance generation-demand. In


this document, flexibility from storage availability is treated as an increase in the initial

and maximum water content.

6.2.1.1 Flexible Demand

To start, inflexible load is considered and the operation cost is 47,542 Mœ. Figure 6.2

shows the evolution of dispatch cost while flexible load is progressively added into the

system, achieving lower operation cost.

% Flexible Demand Dispatch Cost (kœ)

0 % 47,542

5 % 44,267

10 % 40,993

15 % 37,972

20 % 35,365

25 % 32,901

30 % 30,771

35 % 28,797

40 % 26,940

45 % 25,187

50 % 23,475

Table 6.1: Dispatch Cost with Flexible Demand.

Figure 6.2: Dispatch Cost with Flexible Demand.


To measure flexibility, benefit is analyzed. This term is calculated as the substraction

of operational cost in the level of flexible demand from the case without flexibility:

FB = OC ′ −OC (6.2)

Where FB means flexibility benefit, OC’ is the operational cost for the original case

(inflexible system) and OC is the operational cost for the flexible system. Figure 6.3

shows that a significant amount of benefit is achieved if load is flexible enough to face

changes in stochastic production side.

% Flexible Demand Benefit (kœ)

0 % 0

5 % 3,274

10 % 6,548

15 % 9,570

20 % 12,176

25 % 14,640

30 % 16,770

35 % 18,745

40 % 20,602

45 % 22,355

50 % 24,066

Table 6.2: Benefit of Flexible Demand.

Generation schedule is showed with detail in the case when both flexibility resources are

available because, as it will be seen in the next section, storage availability has a small

impact in the system, so solution obtained here is close to those calculated in the last

section.

6.2.1.2 Storage Availability

Figure 6.4 shows the dispatch cost from the inflexible system case to cases with different

levels of initial (Vinit) and maximum water content (Vmax). In Table 6.3 and Table 6.4,

”% Storage Availability” means how is the increase of hydro resources. Up to doubling

initial data is studied, with an increasing interval of 0.1 times the initial size.


Figure 6.3: Benefit of Flexible Demand.

% Storage Availability Dispatch Cost (kœ)

(Vmax, Vinit) 47,542

1.1 (Vmax, Vinit) 47,345

1.2 (Vmax, Vinit) 47,148

1.3 (Vmax, Vinit) 46,951

1.4 (Vmax, Vinit) 46,753

1.5 (Vmax, Vinit) 46,556

1.6 (Vmax, Vinit) 46,359

1.7 (Vmax, Vinit) 46,162

1.8 (Vmax, Vinit) 45,965

1.9 (Vmax, Vinit) 45,768

2 (Vmax, Vinit) 45,571

Table 6.3: Dispatch Cost with Storage Availability.

It is observed that the effect in the flexibility of storage availability is smaller than the

flexible demand. The flexibility benefit obtained when pump hydro power plants are

improved (Figure 6.5) is smaller than for the previous case.


Figure 6.4: Dispatch Cost with Storage Availability.

% Storage Availability Benefit (kœ)

(Vmax, Vinit) 0

1.1 (Vmax, Vinit) 197





1.6 (Vmax, Vinit) 1,182

1.7 (Vmax, Vinit) 1,379

1.8 (Vmax, Vinit) 1,576

1.9 (Vmax, Vinit) 1,774

2 (Vmax, Vinit) 1,971

Table 6.4: Benefit of Storage Availability.

6.2.1.3 Flexible Demand and Storage Availability

Previous results show that flexible demand reduces more significantly operation cost

than storage availability in this specific case. However, as both resources may exist

in the system, the combination of both of them is performed. Different conditions for

flexible demand and storage availability are set, from case of inflexible system to a high

flexible system. Figure 6.6 shows the improvement when using both flexible resources.


Figure 6.5: Benefit of Storage Availability.

% Storage Availability and Flexible Demand Dispatch Cost (kœ)

(Vmax, Vinit) and 0 % 47,542

1.1 (Vmax, Vinit) and 5 % 44,070

1.2 (Vmax, Vinit) and 10 % 40,613

1.3 (Vmax, Vinit) and 15 % 37,440

1.4 (Vmax, Vinit) and 20 % 34,687

1.5 (Vmax, Vinit) and 25 % 32,143

1.6 (Vmax, Vinit) and 30 % 29,928

1.7 (Vmax, Vinit) and 35 % 27,816

1.8 (Vmax, Vinit) and 40 % 25,821

1.9 (Vmax, Vinit) and 45 % 23,931

2 (Vmax, Vinit) and 50 % 22,081

Table 6.5: Dispatch Cost with Storage Availability and Flexible Demand.

With this technique, up to more than the half of production cost is accomplished

(Figure 6.7, 25,461 Mœ of benefit versus 47,542 Mœ of production cost in the in-

flexible case), as expensive dispatch generators are less used and cheap generators as

hydro and wind, with the backup of cheap dispatchable ones, are enough to cover all

the demand. Note that it is assumed that wind production remains constant and it will

only change when the contingency happen (loose of 50 % of wind power for each zone

in the transition periods).


Figure 6.6: Dispatch Cost with Storage Availability and Flexible Demand.

% Storage Availability and Flexible Demand Benefit (kœ)

(Vmax, Vinit) and 0 % 0

1.1 (Vmax, Vinit) and 5 % 3,472

1.2 (Vmax, Vinit) and 10 % 6,929

1.3 (Vmax, Vinit) and 15 % 10,101

1.4 (Vmax, Vinit) and 20 % 12,855

1.5 (Vmax, Vinit) and 25 % 15,399

1.6 (Vmax, Vinit) and 30 % 17,614

1.7 (Vmax, Vinit) and 35 % 19,725

1.8 (Vmax, Vinit) and 40 % 21,720

1.9 (Vmax, Vinit) and 45 % 23,610

2 (Vmax, Vinit) and 50 % 25,461

Table 6.6: Benefit of Storage Availability and Flexible Demand.

Generation schedule is deeper analyzed in the following lines. Two scenarios are chosen

to be studied: Scenario 2, where lost of wind happen in New South Wales and scenario

5, where lost of find power appears in Tasmania. The selection of them is motivated

as they represent extreme cases, where a lot of of wind production is lost (Scenario 2)

and low wind generation is removed (Scenario 5), so the effect of expensive dispatch

generators is clearly noticed.

• Scenario 2: Lost of wind in New South Wales:


Figure 6.7: Benefit of Storage Availability and Flexible Demand.

Figure 6.8 shows generation schedule in scenario 2 for three different cases: in-

flexible system 6.8a, medium-flexible system 6.8b and high-flexible system 6.8c.

Generation is clustered in five types: expensive dispatchable generators (clear

blue), dispatchable generators (purple), hydro pumped (green), hydro (red) and

wind (dark blue). As the model has been developed with ten periods of transition

time, only even transition states, plus initial and final, are shown. This will be

enough for explaining how flexible resources improves the performance of expensive

dispatch generators, as in the odd transition states they have the same behaviour.

-Inflexible system in Scenario 2 6.8a: Second scenario means an important

reduction of wind generation. In the initial state, the system was consuming all

cheap resources, while a small amount of expensive generation is needed to satisfy

all the demand. However, in the next period, part of the wind power is lost,

which has to be covered with expensive dispatchable units, as they have enough

ramp rates to fulfill the demand. This condition endures for the remain transition

states until final period is achieved, where recovery of wind production leads to

a reduction of expensive dispatchable generation. It can be noted that for this

case of inflexible demand, it is supposed to remain constant so the sum of the

generation in all states is the same.


(a) Inflexible system

(b) Medium-flexible system

(c) High-flexible system

Figure 6.8: Generation for Storage Availability and Flexible Demand for three cases,scenario 2.

-Medium flexible system in Scenario 2 6.8b: It is considered the case where

there is 15% of interval for Flexible Demand and pumped-hydro power plants

water content is increased to achieve 1.3 of its initial value. In the initial state, no

expensive dispatchable generation is needed, as with big amount of pumped-hydro

generation all demand is fulfilled. However, as it was in the previous case, this


high amount of power can be only used once, as it would be necessary to pump

water in some transition states. While transition periods are happening and lost

of wind is most latent, expensive dispatchable generation is needed, but, unlike

the previous case, the quantity is reduced. Here is where flexible demand helps

the system to reduce load (as in this period, cost of production is bigger) and thus

leading to a lower optimal dispatch cost. In the final state, demand is satisfied

without expensive dispatch generation but load is lower than initial state. This

result is achieved because, as for this period not enough amount of cheap hydro-

pumped power can supply the system, it would be necessary to use dispatchable

generators, which pose a greater increase in total dispatch cost.

-High flexible system in Scenario 2 6.8c: In this case, there are big opportuni-

ties of producing cheap (hydro-pumped) generation and load is very flexible. Here

it is possible to eliminate completely the use of expensive dispatchable generators.

Case of 35% of interval for flexible demand and 1.7 times higher pumped hydro

water content is studied. In the initial state, pumped hydro generation replaces

expensive dispatchable generation. In the transition states, where wind power is

lost, load is flexible enough to reduce its level up to there is no need of using

expensive dispatchable generators. When final state is achieved, recovery of wind

power is set, so higher quantity of demand can be satisfied. In this case, the impact

of hydro-power plants is more important as higher amount of it are supplying the

system in both initial and final state, unlike the previous cases where it was just

acting in one of them.

• Scenario 5: Lost of wind in Tasmania.

Figure 6.9 shows generation schedule where small amount of wind power is lost

in Tasmania. In this case, it is possible to notice that less amount of expensive

dispatchable generation is needed, which can be totally eliminated even for less

flexible systems.

-Inflexible system in Scenario 5 6.9a: For initial and final state, when no

flexibility is added to the system, the behaviour is the same than in the previous

scenario studied. When contingency occurs in Tasmania, with a small reduction of

wind power (154 MW), expensive dispatchable generation has to supply the load to

compensate this change. Unlike previous case, this amount is not that considerable

as compared with the reduction of wind production (that is the base of cheap


(a) Inflexible system

(b) Medium-flexible system

(c) High-flexible system

Figure 6.9: Generation for Storage Availability and Flexible Demand for three cases,scenario 5.

generation). It will remain in all transition states until wind power is recovered in

the final state, where, thanks to the rehabilitation of wind and the performance of

big amount of pumped-hydro production, less dispatchable generation is needed.

-Medium flexible system in Scenario 5 6.9b: As it is possible to see in


Figure 6.9, there is no need at all of using expensive dispatchable generation in

all periods. Pumped hydro generation is enough to satisfy initial demand, where

there is no chance of reducing the load, and it also helps final state to reduce

dispatchable generation. During transition periods, flexible load is adjusted to be

able to be feeded by cheap generators, as lost of wind power is not high enough to

exceed the limit of minimum demand.

-High flexible system in Scenario 5: 6.9c If for the case of high flexible system

in the second scenario there were not expensive dispatchable generation, for this

scenario it is not less. However, as now there are more amount of wind power

in the transition states, supplied load is bigger than in the second scenario, even

when dispatchable generators are producing less amount of power. Pumped hydro

power plants are being used till their maximum capacity when they are specially

needed, i.e. in the initial state, where there is no chance of flexible demand, and

final state, where it allows to reduce the quantity of dispatchable generation. In

the second period of time, there is no need of pumped hydro generation. Moreover,

10.5 MW are being pumped to the upper reservoir in Queensland.

6.2.2 Sensitivity of the SRED to the Interest Rate

This section studies the influence of the interest rate in the system. This term, that

appears in SRED model, changes the result of the objective function regarding the basic

model explained in Chapter 2. Interest rate or discount rate can be defined as the

devaluation of price of the electricity market when it is happening, i.e. when transition

periods are appearing.

Interest rate in SRED may change the value of optimal dispatch, giving place to erro-

neous results if this term is not well set. It is considered that it should have a small

value, so up to 0.2 (20% of interest rate) is studied. The sensitivity of this term in the

objective function is analyzed. If important changes occur in the the objective function

(dc), interest rate has a significant importance in the development of the problem. For

that, the case discussed previously for inflexible system is used, taking values for the

interest rate from 0.01 to 0.2, as Table 6.7 shows.


Figure 6.10: Deviation from Dispatch Cost.

r dc (kœ) dc(0,01)dc (×10−6) dc(0, 01)− dc (œ)

0.01 47,542 1 0

0.015 47,542 2.958 140.64

0.02 47,542 5.916 281.28

0.025 47,541 8.875 421.92

0.03 47,541 11.833 562.56

0.035 47,541 14.791 703.19

0.04 47,541 17.749 843.83

0.045 47,541 20.707 984.46

0.05 47,541 23.666 1,125.12

0.055 47,541 26.624 1,265.73

0.06 47,540 29.582 1,406.36

0.065 47,540 32.543 1,547.56

0.07 47,540 35.499 1,687.63

0.075 47,540 38.457 1,828.26

0.08 47,540 41.415 1,968.89

0.085 47,540 44.373 2,109.52

0.09 47,540 47.332 2,250.15

0.095 47,539 50.295 2,390.78

0.1 47,539 53.248 2,531.41

0.125 47,539 68.045 3,234.54

0.15 47,538 82.831 3,937.65

0.175 47,537 97.623 4,640.76

0.2 47,537 112.414 5,343.84

Table 6.7: Variation Interest Rate.

Figure 6.10 shows the ratio of the dispatch cost for a given interest rate regarding the

dispatch cost for the smaller interest rate case (1%). Figure 6.11 shows the difference

between them.

As the function Dispatch Cost− Interest Rate does not presents any abrupt change, it


Figure 6.11: Difference from Dispatch Cost.

is confirmed that there is not a strong dependence of the interest rate on the outcome

of costs, besides which itself should have when interest is increased. Thus, it is demon-

strated that the election of this term will not have a critical influence in the problem

results.

6.3 Study of Investment Decisions

Investment decisions are represented as binary variables in the model. These binary

variables result in an integer problem and solution grows in complexity. Thus, investment

decisions, such as whether to build a power plant or a line, make the optimization

problem non-convex and therefore difficult to solve. Solution time and memory may

increase when more integer variables compose the problem.

Benders decomposition is applied to the developed model and it is used to test two cases:

IEEE 6-node system and IEEE 30-node system. The objective of the study is to measure

flexibility benefit when investment decisions are considered. Thus, dispatch cost of the

original problem is compared with those where investment in lines, generators and both

of them is suggested.

As it was mentioned in the first paragraph, solution time and memory may increase

when more binary variables are added. For this reason, first only investment decisions

are applied, while unit commitment decisions are included to the study the IEEE 6-node

system.


Constraints 4.34-4.37 of Section 4.3 are removed when only investment is considered

as binary variable. Thus, start up (son), shut down (soff ), state of the generator (z)

and state of the candidate generator (h) disappear, while investing in new lines (x) and

investing in new generators (y) remain as the only integer variables.

As reader may notice, master problem formulation changes too. This results in a sim-

plification of the objective function, where only binary variables related with generation

and lines investment decisions and α-function remain. Constraints are simplified too,

where only Benders cuts and limitation for the α-function appear, leading to a faster

convergence of Benders decomposition.

Feasibility check subproblem is included to decrease computational time, where different

levels of demand response is allowed for each case.

In the optimal operation subproblem, besides eliminating constraints commented above,

objective function is only composed by operational dispatch cost, while this cost related

with start-up is removed.

6.3.1 IEEE 6-node system with investment

6.3.1.1 Data

Benders Decomposition was successfully applied to a small example of 2-node system.

After that first step, it is applied to IEEE 6-node system [31]. IEEE 6-node system

is chosen as it is an intermediate move before analyzing a bigger and more complex

network as IEEE 30-node system. Some modifications are deliberately done to analyse

the need of investing in new lines and generators when there is a considerable amount

of stochastic generation in the system.

Some renewable generation services are added to the generation nodes (1, 2 and 6) and

candidates generators and lines are suggested for transmitting this power to load nodes

(3, 4 and 5), resulting the model presented in Figure 6.12.

Two dispatchable units, one hydro and three stochastic power plants exists, while three

dispatchable generators and one hydro power plant are available to invest. (Table C.1 ,

Table C.2).


Figure 6.12: The modified IEEE 6-Node example system, 1: Coal, 2: Hydro, 3: Gas,4: Coal 5: Hydro, 6: Gas, 7: Gas, 8: Wind, 9: Wind, 10: Wind.

Hydro power plants have low operation and start-up costs, with high ramp rates, while

dispatchable units have high operation and start-up costs with low ramp rates. There

are two different types of dispatchable units, with same maximum production capacity

but disparate ramp rates. Investing in hydro power plants is more expensive than

dispatchable as in the operation level a big reduction of cost is achieved. Between

thermal plants, those with lower operation costs have higher investment cost. Seven

lines exist in the system and four more are candidate to invest (Table C.3), linking the

nodes with generators to those with loads, to strengthen them and being able to send

all the power without congesting the existing lines.

Net load is modelled as a ”Duck Curve”. This term changes the conventional load models

(Figure 6.13), where there are three clearly difference operational factors that affect the

price: the morning ramp, the afternoon/evening peak, and the differences between the

hourly and five-minute peaks. In this case, not only wind power is the principal source

of stochastic production, but solar generation plays also an important role.

Duck Curve was first noticed in California in 2012 [56], but later it was observed in

another parts of the world [57]. This effect is produced due to cheap photovoltaic

modules reduce the demand during the sunny part of the day, so the morning ramp

flatters out and the evening ramp gets steeper. There are some predictions about how the

introduction of huge amount of solar generation may affect to the system (Figure 6.14),

reaching the conclusion that such effect may be much more pronounced. It means that

power systems have to be ready for stochastic and variable generation, which results in


Figure 6.13: Electric load curve: New England 22/10/2010 [55].

an increasing importance of investment decisions, as existent generators may not be able

to overcome the big lost of stochastic generation that is produced in the sunset times.

Figure 6.14: Duck Curve Prediction [56].

To simplify the calculation, wind and solar generation are joined as a single variable.

Different scenarios are created, fixing the difference between demand and stochastic

generation. As Duck curve represents this difference, it is modelled changing the value

of the demand in the transition states while stochastic generation remains as a variable.

With that, up to nine scenarios are studied.


Node Load (MW)

3 94

4 70.5

5 70.5

Table 6.8: Load data for IEEE 6-node system with investment.

If Duck curve is representing one day (24 hours), with just 5 time intervals is enough to

describe its behaviour, simplifying the calculation. Figure 6.15 shows up to 9 different

scenarios, as it is studied, the difference between demand and stochastic generation.

For modelling this feature, constant demand (Table 6.8) is set and it is multiplied in

the transition intervals by some coefficients (Table 6.9), resulting in this characteristic

shape. As reader can notice, it means that load changes evenly in all demand nodes,

without preferring any of them.

Figure 6.15: Duck Curve for 9 scenarios (ss) in IEEE 6-node system with investment.

time ss1 ss2 ss3 ss4 ss5 ss6 ss7 ss8 ss9

1 1.058 1.041 1.023 1.010 0.988 0.970 0.952 0.935 0.917

2 0.988 0.970 0.952 0.935 0.917 0.889 0.882 0.864 1.023

3 0.388 0.423 0.430 0.494 0.529 0.564 0.599 0.829 0.970

4 1.482 1.464 1.446 1.429 1.411 1.393 1.376 1.323 1.199

5 1.058 1.041 1.023 1.010 0.988 0.970 0.952 0.935 0.917

Table 6.9: Coefficients for Duck curve.

As it is possible to see in Figure 6.15, scenario 1 (dark blue) represent the future

case where there is high penetration of stochastic production in the system (2020) while


scenario 9 (green) represent past electricity market when it was composed in its majority

by dispatchable generators (2012), which means that there are not extreme changes in

the Duck curve. Big change of difference demand-stochastic generation is produced

between the 3th and the 4th period of time, and here it is where the need of investment

is important, as ramp rates of existing generators may not big enough to adapt their

levels to this new generation.

Finally, demand response is set in every load node (node 3, 4 and 5). If generation is

not enough to fulfill the demand, this demand could adapt to the new generation levels

and reduce its value, participating in the generation-demand balance. However, the cost

of disconnecting a consumer in the system is relatively large (4 Mœ). Scenarios are

equiprobable (probability equal to 1/9) and the interest rate is set to 0.02.

Modifications are made in the commented data in order to adjust mentioned changes in

the generation side with the load:

• Generation of existing dispatch generators are multiplied by 1.5 in order to have

a peak of demand of 417.924 MW, when the value of existing generation is 420

MW (including wind power plants).

• Ramp Rates of existing dispatch generators are multiplied by 1.5.

• Capacity of candidate lines is increased 1.5 times to reduce congestion appeared

in these lines.

6.3.1.2 Results

System is modelled in GAMS and convergence of Benders method is accomplished.

Comparative analysis requires the simulation of four cases: System without investment,

investment in lines, investment in generators and investment in both resources.

Table 6.10 and Table 6.11 show the investment decisions for all cases.

• For no investment case, Benders Decomposition is not necessary as there are not

binary variables in the model. Objective function is set to 166.008 Mœ, where

expensive generators have to supply the demand.


Case L8 L9 L10 L11

No-investment

Line investment x x

Generation investment

Both investment x

Table 6.10: Line invest decisions for IEEE 6-node system with investment.

Case G4 G5 G6 G7

No-investment

Line investment

Generation investment x x x

Both investment x x x x

Table 6.11: Generation invest decisions for IEEE 6-node system with investment.

• When line investment is simulated, lower cost is achieved (138.852 Mœ) as power

transmission is improved, so it can flow better from generation nodes to demand

nodes. However, more generation is still needed in the system.

• Generation investment shows a big improvement respect the previous case (3.895

Mœ), but still demand response (penalty generation) is required.

• Last, lines and generators investment decisions are tested, leading to a big im-

provement in the system (224,104.535 œ), as there is enough generation and power

transmission to feed the demand without using demand response.

Benders convergence is achieved in all cases, where upper and lower bound have exactly

the same value, which means that optimal solution has been achieved. However, as when

both investment decisions are simulated the problem is harder and the solver needs more

time to converge, only this one is reported. Figure 6.16 shows that 36 iterations are

needed to achieve the final solution.

Different measures are developed to analyze system flexibility. Two of them, related

with economic impact, are presented here:

1. First of them is to split the cost into four different types: investment cost, normal

operation cost and adjustment cost (Table 6.12):

Decreasing reduction of normal operation cost and adjustment cost are obtained

while investment decisions are introduced. Thus, lower total costs are achieved in


(a) Benders Convergence.

(b) Benders Convergence Expanded.

Figure 6.16: Benders Decomposition Convergence for IEEE 6-node system with in-vestment.

the investment cases, even where there are extra cost (new lines and generators),

regarding the given case.

Thus, benefit is decomposed in three different types: Efficiency benefit (EB), flex-

ibility benefit (FB) and total benefit (TB). These terms are calculated as:

EB = NC ′ −NC (6.3)

FB = AC ′ −AC (6.4)

TB = EB + FB (6.5)

Where NC’ and NC are the normal operation cost for non investment system and

investment system respectively, AC’ and AC are the adjustment costs and TB is


Cost No invest (kœ) L. invest (kœ) G. invest (kœ) B. invest (kœ)

GIC 0 0 168.378 202.625

TIC 0 20.776 0 9.589

NC 16.994 7,860.968 9.978 0.576

AC 1.659×105 1.387×105 3,717.085 11.313

Total 1.660×105 1.388×105 3,895.443 224.104

Table 6.12: Generation investment costs (GIC), transmission investment costs (TIC),normal- operation costs (NC) and adjustment costs (AC) for IEEE 6-node system with

investment.

the final benefit achieved. Table 6.13 shows these results.

Cost L. invest (kœ) G. invest (kœ) B. invest (kœ)

EB 9.133 7.015 16.418

FB 27,234 1.621×105 1.658×105

TB 27,235 1.622×105 1.658×105

Table 6.13: Efficiency Benefit (EB), Flexibility Benefit (FB) and Total Benefit (TB)for IEEE 6-node system with investment.

Thus, the most important benefit is the flexibility benefit, as it compose the ma-

jority of the total benefit achieved.

2. The second one, suggested in [12], is to use the System Flexibility Index (SFI):

SFI =DC(s)∑

k

pkAC(s→ s′k)(6.6)

where,

AC(s→ s′k) =

T∑t=1

(DC(stk)−DC(s′k))

(1 + r)t−1(6.7)

These equations give a way of calculating how flexible resources may affect to the

reduction of costs. (6.6) shows how is the difference between normal operation cost

(DC(s), initial dispatch cost) and the cost necessary to adjust the system due to

the appearance of contingencies. (6.7) presents the adjustment, which is expressed

as difference between cost in transition states DC(stk) and the final state achieved

DC(s′k), which belongs to the new steady state achieved.


Case SFI

No invest 0.001

Line invest 0.2082

Gen. invest 0.9613

Both invest 1.1216

Table 6.14: SFI for all cases in IEEE 6-node system with investment.

A power system with higher SFI can respond at low cost to variable and unpre-

dictable changes in power system and therefore it can operate more efficiently ex

ante. Table 6.14 shows SFI for no investment, only line investment, generation in-

vestment and the combination of both of them. When more flexible is the system,

higher is this value because lower cost for adjustment is needed.

It is possible to see that both methods report that the system is becoming more flexi-

ble when investment decisions are introduced. Lastly, generation schedule is analyzed.

Two scenarios are studied, those more extreme (1 and 9), as the rest of them have an

intermediate behaviour in the system. All different four cases of investment decisions

are drawn.

• Study of Scenario 1: This scenario belongs to the potential stochastic generation

that will exist in future electricity markets (2020). In this case, where exits a high

penetration of stochastic production, there is an important need of ramp rate

providers during the 3th and the 4th interval of time.

Figure 6.17 shows how the different types of generators produce power in each

period. For simplicity, units have been grouped in four different types, where

stochastic generators (wind and sun) are classified in the category ”wind”, dis-

patchable generators are differentiated with their prices in ”dispatchable cheap”

and ”dispatchable expensive” and hydro power plants have their own group. As it

is possible to see in all the figures, in this system there is a large amount of hydro

generation, as there was an existing one while another is offered to invest, which

results in an important amount of power coming from this resource.

The base for the system generation is the cheap stochastic power. Even knowing

that they vary a lot during the day, for this study this variation has been repre-

sented just changing the demand data that, as it is possible to see, is modified


(a) No investment. (b) Line investment.

(c) Generation investment. (d) Line and Generation investment.

Figure 6.17: Generation level in scenario 1 for IEEE 6-node system

in every period. Figure 6.17a shows that when there is no possibility to invest,

the system is no able to overcome the lost of stochastic generation and demand

response is needed in the 4th period. When lines decisions are taken into account

(Figure 6.17b), the better distribution of the power allows to reduce the amount

of demand response. However, it is still not enough, so investing in new generation

units is needed. Figure 6.17c shows that demand response has been significantly

reduced when units are invested. However, there is still a need of distribute better

the power. It is accomplished when both lines and generators are offered to invest

(Figure 6.17d), where demand response is completely eliminated.

Cheap dispatch generators are required in every moment because wind and hydro

are not high enough to supply all the load, while expensive generators are specially

need in the peak of demand, which provide a small amount of power to supply this

peak.

• Study of scenario 9: This scenario belongs to the past electricity market (2012),

where small amount of wind and sun generation exist. However, it may happen


that in those periods of the year where there are less amount of sun or wind, the

difference between load and stochastic generation have this shape.




As in the previous study, Figure 6.18 shows the producers generation level.

As in this case Duck Curve is flatter, ramp rate providers are less needed. In

Figure 6.18a, no investment case, demand response is required in the peak of

demand. For all other cases, this term is complectly eliminated, as power trans-

mission improvement is enough to supply the load in the 4th period (Figure 6.18b).

If generation investment is considered but lines remain the same, cheap dispatch-

able units have to be producing more than in the previous case. This effect may

happen because it is still needed to enhance the transmission network, to send

cheaper hydro power to the load nodes. This is achieved with the introduction

of lines investment (Figure 6.18d), where huge amount of hydro power is being

consumed in all the process.

The main difference with the previous case is the bigger use of cheap hydro genera-

tion. This phenomena happens because now ramp rates of hydro power plants are

enough to endure demand changes, while in scenario 1 they were not high enough


to adapt their generation levels so dispatchable units had to be producing power

before the extreme change of net load happen.

6.3.2 IEEE 30-node system with investment

6.3.2.1 Data

The IEEE 30 Node Test Case [58] represents a portion of the American Electric Power

System (in the Midwestern US) in December, 1961. This is the second normalized

system that is tested, where 30 nodes compose the model. Data is deliberately modified

for this study, where investment is introduced and huge amount of stochastic production

is supposed to exist in the system. Eleven different types of generators are included,

which are possible to classify in coal, peat, gas-A, gas-B, gas-C, distillate, waste, biogas,

hydro, pumped-hydro and wind (Table D.1). The location has been carefully chosen so

that the power is evenly distributed and there are not large areas with high generation

while other zones are isolated. As unit commitment is not included, minimum generation

for dispatchable generation is considered to be zero. Hydro power plant (G10) has 1000

MW of reserve and it is assumed that it cannot be run out water.

Distribution of existing lines is not modified from the given model, but their data variate

as it is explained below. Candidate lines are introduced in the system after compiling

the model without investment, locating which existing lines were congested. Thus,

Figure 6.19 shows how candidates lines are set.

Candidate generators are deliberately located to encompass a broader view of what that

kind of generators are needed in the system. Thus, three types of candidate generators

are offered to invest, which are divided in fast, medium and slow.

• Fast dispatchable generators: Having big ramp rates but also big variable

cost, these generators are designed to overcome fast changes in the equilibrium

generation-demand.

• Medium dispatchable generators: They are a combination of fast and slow

machines, which help the system stability in an intermediate manner.


Figure 6.19: The modified IEEE 30-Node example system, 1: Coal, 2: Gas-C, 3:Pump-storage hydro, 4: Waste 5: Peat, 6: Gas-A, 7: Distillate, 8: Biogas, 9: Gas-B,

10: Hydro , 11: Wind.

• Slow dispatchable generators: Small ramp rates and variable cost, they are

oriented to those moments where demand changes but not widely. They constitute

the base for generation.

Twenty one candidate generators in seven different nodes (4, 5, 6, 10, 13, 22 and 30) are

set. It is possible to see that most of them are located in the south part, where there are

more load, whereby said units may provide enough power to overcome the contingencies.

As in IEEE 6-node system with investment (6.3.1), demand is represented as Duck curve.

Thus, it is established a constant demand (Table 6.15) and scenarios are set multiplying


Node Load (MW) Node Load (MW)

2 21.7 17 9

3 2.4 18 3.2

4 7.6 19 9.5

5 94.2 20 2.2

7 22.8 21 17.5

8 30 23 3.2

10 5.8 24 8.7

12 11.2 26 3.5

14 6.2 29 2.4

15 8.2 30 10.6

16 3.5 - -

Table 6.15: Load data for IEEE 30-node system with investment.

this demand by same factors (Table 6.9), resulting in nine scenarios representing five

periods of time.

Finally demand response is introduced. For this case, three different levels of demand

response are introduced in every load node. It means that now demand can be adapted

to the generation levels, reducing its value if it is needed but resulting in a significant cost

for the system operator. Thus, three levels of demand response are set, incrementing

the cost in every level (10,000, 20,000 and 30,000 œ/MW), which will report how much

generation is needed to overcome peaks of demand.

Data left is assumed to be the same as in the previous case. However, some modifications

are made in the commented data in order to adjust mentioned changes in the generation

side to the load:

• Load is multiplied by 2.

• Generation capacity of existing dispatchable generators are multiplied by 0.4

in order to have a peak of demand of 892.196 MW when the value of existing

generation is 991 MW (including wind power plants).

• Capacity of lines is increased 15 times to reduce congestion appeared in these

lines.

• Capacity of lines 119, 120, 121, 127 and 131 is doubled.


• Generation investment prices are set for 20 years inversion. Thus, for adapting

them to the problem (5 periods of time in a day), this term is multiplied for 5

and divided by (20x365x24).

• Lines investment prices are set for one year inversion. Thus, for adapting them

to the problem (5 periods of time in a day), this term is multiplied for 5 and

divided by (365x24).

6.3.2.2 Results

Same reasoning is brought to this case. For a clear comparative analysis, four cases are

simulated for each investment decision. Table 6.16 and Table 6.17 show the investment

decisions for all cases.

Line Line investment Both investment

151 x

152 x x

153 x

154 x x

155

156

157 x

158 x x

159 x

160

161 x

162 x

163 x

164

165

166

167 x

Table 6.16: Line invest decisions for IEEE 30-node system with investment.

Candidate generators are gropued in the three types in every node (4, 5, 6, 10, 13, 22

and 30), representing fast, medium or slow units presented above. Thus, for the case

that only generation investment is considered, there are four slow machines in the nodes

4, 6, 10 and 22 and one fast in the 6th node. It is seen here that it is necessary to supply

much more generation than the existing one because the transfer of power is not ideal

(there are still more existing generation capacity than peak demand), resulting in the


Generator Generation investment Both investment

G11

G12

G13 x x

G14

G15

G16

G17 x

G18

G19 x

G20

G21

G22 x

G23

G24

G25

G26

G27

G28 x

G29 x

G30

G31

Table 6.17: Generation invest decisions for IEEE 30-node system with investment.

decision to invest in many cheap but slow dispatchable generators that will complement

hydro and wind production. The fast generator is needed to cover the peak of demand

as it does not constitute a cheap base for production.

When investment in generators and lines is applied, less candidate units are needed.

One slow dispatchable unit in the 4th node and one fast in the 30th is decided to

introduce. This happen because now with new lines existing in the system, power

transfer is optimized and existing plants produces enough power. However, for fulfilling

the peak of demand one fast dispatchable generator is needed.

Benders Decomposition loop returns a solution with a maximum gap of 1 %, achieving

the following results:

• If no investment decisions are considered, demand response is required. This results

in a considerable increase in costs, achieving a objective function of 36.915 Mœ.

• If lines are candidates to invest, objective function is set to 2.293 Mœ. It means

that the previous results has been improved so less demand response is needed,


just because the better power transmission, as generation levels remain the same.

298 iterations are required to achieve Benders convergence.

• Generation investment results in a cost of 360.122 kœ. Here the effect of increasing

generation is more important than enhancing transmission network, as no demand

response is used. 273 iterations are performed and the gap obtained between upper

and lower bound is 0,98 %.

• Last case is investing in both generators and lines, where objective function reaches

a value of 261.719 kœ. No demand response is activated in the system and Benders

converges in 843 iterations, with a gap of 0,998 %. Figure 6.20 shows Benders

convergence from the 39th iteration.

Figure 6.20: Benders Decomposition Convergence for IEEE 30-node system withinvestment.

Flexibility measures are calculated. As for the previous case, benefit and SFI are ana-

lyzed:

1. First, decomposition of cost is analyzed. In this case, same as in the previous one,

the introduction of investment decisions results into a decreasing cost (Table 6.18).

Thus, the adjustment cost, that can be explained as the cost required to pay be-

cause the appearance of contingencies respect the planned plan (day-ahead mar-

ket), has lower values when investment is considered, leading to an increasing


Cost No invest (kœ) L. invest(kœ) G. invest (kœ) B. invest (kœ)

GIC 0 0 98.914 126.288

TIC 0 330.615 0 61.526

NC 1,509.817 15.772 5.505 5.516

AC 35,405.231 1,946.850 255.702 68.389

Total 36,915.049 2,293.238 360.122 261.719

Table 6.18: Generation investment costs (GIC), transmission investment costs (TIC),normal- operation costs (NC) and adjustment costs (AC) for IEEE 30-node system with

investment.

system flexibility. As it is possible to see, in the last case where lines and gen-

erators are offered to be introduced in the system, very small adjustment cost is

required, resulting into a high flexible power system.

Benefits are calculated too. Table 6.19 shows this feature.

Cost L. invest (kœ) G. invest (kœ) B. invest (kœ)

EB 1,494.045 1,504.312 1,504.301

FB 33,458.381 35,149.529 35,336.842

TB 34,952.426 36,653.841 36,841.144

Table 6.19: Efficiency Benefit (EB), Flexibility Benefit (FB) and Total Benefit (TB)for IEEE 30-node system with investment.

Generation investment decisions case shows that efficiency benefit is bigger than

for both generation and lines investment decisions. However, flexibility benefit

is higher for this second case, leading to major total benefit. Lines investment

case has lower benefits, as expanding the transmission network helps the system

flexibility with lower strenght.

2. Second, System Flexibility Index is calculated. Table 6.20 shows these results,

where big flexibility is achieved with investment in generation and with the com-

bination of both resources.

Case SFI

No invest 0.1075

Line invest 0.1416

Gen. invest 0.2325

Both invest 0.3838

Table 6.20: SFI for all cases in IEEE 30-node system with investment.


In this case, higher SFI is achieved when investment decisions are taken. With

the coordination of investing in new lines and generators, around four times more

SFI is obtained, which gives an idea about how important improvement has been

achieved with the introduction of lines and generators in the system. As for the

IEEE 6-node system, line investment does not participate largely in the increment

of flexibility.

Thus, it is demonstrated that ramp rate providers play an important role in the system

stability. In order to analyze deeper their importance, generation schedule in the extreme

changing case (scenario 1) and in the flatter case (scenario 9) is shown. For that,

generation units are divided into six types depending on their characteristics: Renewable

(stochastic), hydro (conventional and pumped), slow generators (waste, peat and slow

candidate), medium generators (coal), fast generators (gas-A, gas-B, gas-C, distillate,

biogas and fast candidates) and Demand Response.

• Study of scenario 1: Scenario 1 represent the future electricity market where

huge amount of stochastic production is introduced, which leads to a extremely

changing curve conventional generation-demand. The 4th period of time involves

an important lack of stochastic generation just after the most stochastic generation

point, leading to a peak in the Duck Curve 6.15 after the valley.

Figure 6.21 shows generation levels for all four cases. As it is possible to see, the

base power is constituted by cheap stochastic generation (dark blue) with the help

of the slow but cheap dispatchable units (green). However, in the third period of

time the amount of renewable is significatively reduced while the others types of

generation remains in the system. This effect is caused by the lack of big enough

ramp rates in the generation side. This implies that some wind has to be spilled

in order to keep producing dispatchable generation and being able to cover the

peak of demand in the next period. Thus, between 3th and 4th period where

the difference between renewable and demand increases critically, ramp rates of

dispatchable generators have to act and compensate this change. Thus, as they

are not enough for keeping the same wind power in the previous period and feed

the load in the next one, they remain connected in the 3th period while stochastic

generation is reduced, allowing ramp rates to be big enough for keep the balance

in the next transition state.





When no investment is considered Figure 6.21a, huge amount of demand response

(red) is required. If lines are candidate to invest Figure 6.21b, demand response

is still needed in the peak of demand, fact that can be eliminated with the in-

troduction of candidate generation Figure 6.21c. For this case, the generation

is composed mostly by slow dispatchable units (around 200 MW during all the

process). When investing in both lines and generators Figure 6.21d, generation

is enough to cover the load without demand response. It is also possible to see

that expensive units (orange) as distillate and biogas are just needed in the peak

of demand. As it was mentioned before, three candidates units were suggested to

invest. Fast unit is needed in the peak of demand, as it has ramp rates big enough

to cover it. The cheap unit is introduced as base in the production, with low

variable cost that allows the system to use cheaper units and save the expensive

ones to cover those moments where stochastic generation is reduced.

• Study of scenario 9: In this case where few renewable production exists in the

system, Duck curve is flatter, which results in different results regarding the first


scenario. First of all, there is no need to reduce wind power in the third period.

As the change in conventional generation-demand is not that critical, ramp rates

of dispatchable generators are enough to increase their production and cover the

peak of demand.


(c) Generation investment. (d) Line and generation investment.


If no investment case is studied Figure 6.22a, a few amount of demand response

is required in the 4th period. The main difference with investment in only lines

Figure 6.22b and those cases where it is considered investing in new units is

the amount of slow generation, which indicate the important use of the slow but

cheap candidate units. Generation investment Figure 6.22c and both investment

Figure 6.22d differs in the utilization of fast generation. When line investment is

considered, the expansion of the transmission network allows a better distribution

of the resources that leads in a lower use of expensive generation.

Investment has been added in the SRED model for the IEEE 30-node system. Study

accomplished shows that ramp rate providers are needed to overcome fast changes in


stochastic production side. In the original system, the large amount of stochastic gener-

ation is an issue because, if it decreases rapidly, existing dispatchable generators are not

fast enough to cover it, leading to an unbalanced system and demand response has to

act. Introduction of investment resources allows the system to be balanced continuously,

releasing its dependence on other resources.

6.3.3 IEEE 6-node system with investment and unit commitment

6.3.3.1 Data

IEEE 6-node system model with investment is modified and unit commitment decisions

are introduced. Now all equations (4.1)-(4.37) are coded, while master problem grows in

complexity as Section 5.3.1 comments. Constraints that make a relation between start-

up, shut-down and the state of the unit either for existing as for candidates generators

belong to master problem and Benders cuts have more terms. Same data for the IEEE

6-node system with investment 6.3.1 is used but some other parameters have to be

arranged (Table C.4 and Table C.5), as start-up cost and minimum generation:

Hydro power plants have low operation and start-up costs, with an important flexibility

to produce energy (large difference between maximum and minimum capacity), while

dispatchable units have big operation and start-up costs and low difference between

its maximum and minimum production levels. Investing in the hydro power plants is

more expensive than dispatchables as in the operation level a big reduction of cost is

achieved. Between thermal plants, those with lower operation costs have bigger invest-

ment. Section 5.3.1

6.3.3.2 Results

Benders convergence for this case is very slow. Unlike 2-node system tested in Section 5.3.2

where unit commitment was successfully added, in this case no-convergence in a reason-

able time is achieved.

Case for both lines and generators investment with unit commitment is tested. After 16

hours, Benders convergence is still not achievable. To check how close upper and lower


bound have come in those 16 hours, the gap is measured in every iteration. This gap is

calculated as:

gap =UpperBound− LowerBound

| LowerBound |(6.8)

Figure 6.23 shows this feature. The closer to zero, the more accurate the calculation is.

If gap reaches exactly zero, optimal solution is found.

Figure 6.23: Gap for Benders Decomposition Convergence for IEEE 6-node systemwith investment and unit commitment.

As it is possible to see, more than 700 iterations are performed. After 16 hours, in the

iteration number 722, it is decided to stop. In this moment, the gap achieved is 0.062,

which means that there is still a difference of 6, 2% between upper and lower bound. An

acceptable solution may be if this gap is lower than 2-3%.

For very complex problems, Benders Decomposition may have some issues. Magnanti

and Wong (1981) commented that there is a major computational bottleneck in the

Benders algorithm [59] as Benders master problem has to be repeatedly solved and, if

it is an integer problem (IP), its solution is usually slow. In the case that concerns us,

it is observed that the solution of master problem becomes increasingly slower and each

cut needs more time in every iteration, resulting in unacceptable long calculation.

As Magnanti and Wong commented, an increased number of iterations can usually be

attributed to the following:


• 1- The algorithm starts with no initial information.

• 2.- Towards the end of the algorithm the values of the binary variables tend to be

similar to those of the previous iteration, resulting in similar cuts generated.

• 3.- A degenerate Benders Subproblem has a non-unique dual solution, and the

chosen one can correspond to a weak cut.

• 4.- The problem formulation might be inadequate.

There are several solutions to the above problems. Techniques as pareto-optimal cuts

[60], bundle method [61], multi-cut framework [28], valid inequalities [29] or selection

of initial solution [62] may accelerate Benders convergence and thus save computational

time and memory.

Chapter 7

Conclusions and Future Work

7.1 Conclusions

Flexibility resources are successfully introduced in the Short-run Economic Dispatch

model. As they have been studied separately, it is performed an individual analysis for

each of them.

-Flexible demand: The degree of flexibility is the ability of a load to vary in response

to an external signal with minimal disruption to consumer utility [38]. When flexible

demand is applied to the NEM model, a considerable amount of benefit is achieved.

Future electricity supply will include a higher fraction of stochastic generation, increasing

opportunities for demand side management to maintain the supply/demand balance.

Nowadays, important amount of domestic devices that can participate in the market

balance are being developed, as electric cars or heat pumps, which can warm water

when electricity price is low. Electric cars are an state-of-the-art technology which will

play an important paper in the future power systems balance, as they will help to keep

electricity price in a small interval, consuming energy when there is cheap electric price

and even, if it is necessary, sell energy when it is needed. For modelling this flexible

demand, constant load has been assumed, which can increase o reduce up to 25 % of

its nominal value. More detailed models for flexible demand can be developed, as they

depend on another parameters, as the season (in winter or summer flexible demand

changes a lot because of domestic devices as air-cooling or light)[38] or splitting the

flexible demand in their different types.

87

Chapter 7. Conclusions and Future Work 88

-Storage Availability: With the construction of hydro power plants with large capac-

ity, new forms of regulating the supply-demand balance have been opened. Pumped-

hydro power plants exist for more than a century, but until the development of reliable

techniques of efficient pump and storage of important amount of water, it was not taken

into consideration as much as it is nowadays. In the actual power systems, pumped-

hydro power plants are a good match for balancing the equilibrium generation-demand,

pumping water if there are cheap electricity price (low demand and nuclear power plant

production) and selling it when the electricity price is high. However, future electricity

markets will have high penetration of stochastic generation as wind or solar power, so

pumped-hydro generators will play a key role to hold a balanced power system when

there is a change in these stochastic generation levels. Storage availability is modelled

in this system increasing the size of the reservoir (initial and maximum), which means

investing in the expansion of such power plants. The introduction of these resources

increases the flexibility of the system as an improved balanced market is achieved.

-Investment decisions: When investment is mentioned, a deeper analysis about its

profitability is necessary to be done. Investment decisions involve large amount of money

with a long term profitability, so its implementation must be concluded with justified

reasons. However, future electricity markets have one important justification: the in-

troduction of huge amount of stochastic generation may increase the uncertainty of the

power system. Thus, benefits that on one hand are obtained because of their low vari-

able cost, may cause a problem when its stochastic nature leads to a critical changes in

the production. Expanding the transmission network, exchanging more efficiently exis-

tent power to those zones where stochastic generations has changed, or introducing new

types or generation units, for covering them, has been proved to helps system stability.

Investment in different machines is proposed, where slow units helps to cover the basic

demand while fast units are used to cover the peaks of demand.

Introduction of flexible sources leads the electricity market to a better distribution of

the available resources, while old and expensive dispatchable generators have a minor

importance in the generation-demand balance. All this implies a reduction of cost for all

participants in the system and, what is more important, improving the reliability. This

is really necessary in the near future, where stochastic sources will have a paramount

importance in the generation side and its changing behaviour may lead to problems in

the system. There is an important feature of dispatchable generators that may not allow

Chapter 7. Conclusions and Future Work 89

to supply all the demand required in a short period of time: ramp rates. As it is know,

dispatchable generators (coal, fossil fuel, gas, nuclear,...) need some time to achieve its

demanded generation. If suddenly part of stochastic cheap production stop (because

of lack of wind or sun, for example), the existence of ramp rate providers is needed

for being able to keep feeding the requested load. Thus, either from the point of view

of generation (investing in new units or lines) or the demand side, generation-demand

balance has to be satisfied to serious problems such under/overdelivery will not occur.

7.2 Future Work

• Developing a more realistic model [38] for Flexible Demand, as for this study this

term was just estimated.

• Include investment when Flexibility from the Storage Availability is modelled.

• Develop Benders Decomposition with the dual problem instead with the primal,

that may lead to a reduction of computational time.

• Add acceleration techniques for reducing computational time. [59], [60], [61], [28],

[29] or [62] suggest some implementation to accelerate it that have been tested

with good results.

• Add unit commitment for bigger system, as IEEE 30-node system, that was not

possible to do in this research due to convergence issues.

• Increasing the periods of time to 24 (one day), so ramp rate effect is less pro-

nounced. If SRED model is used, implement even shorter dispatch intervals (up

to 5 minutes).

Appendix A

Integrating Renewables in

Electricity Market

A.1 Initial Mathematical Model

In this appendix, a conventional model for electricity market is introduced. This model

is designed for those power systems that operate with balancing market. For that, using

hourly dispatch intervals, features as reserve acquire higher importance, as generation

units must be prepared to adapt changes in demand during these intervals. Extending

the model that can be found in Chapter 2 of [13] to a multi-period case, it is possible

to get the model below:

Nomenclature:

• Sets:

i sets of conventional production units

q set of stochastic production units

t periods of optimization

j loads

n set of nodes

ω sets of scenarios

` sets of transmission lines

90

Appendix A. Integrating Renewables in Electricity Market 91

• Parameters:

Ci marginal production cost of power plant i (day-ahead market)

CRUi cost of maintain upward reserve capacity in unit i (day-ahead market)

CRDi cost of maintain downward reserve capacity in unit i (day-ahead market)

Cq cost of stochastic production units

CUi cost of increment production in power plant i (at balancing time)

CDi cost of decrease production in power plant i (at balancing time)

πω probability of scenario ω

Wqω(t) power generation of stochastic producers in scenario ω

V LOLj cost of lost load

b` susceptance of the line `

Cmax` transmission capacity limits

Wmaxq maximum power output of the wind power scheduled in the day-ahead

market

Pmaxi maximum power output of the unit i

Rupmaxi maximum upward reserve capacity of unit i

Rdownmaxi maximum downward reserve capacity of unit i

Loadj(t) load in each period

Φin matrix with the production units in the nodes

Φjn matrix with the loads in the nodes

Φqn matrix with the stochastic production units in the nodes

GCM `n matrix with the lines linking the nodes

• Variables:

Pi(t) power output of the unit i

RUi (t) upward reserve capacity of unit i

RD1 (t) downward reserve capacity of unit i

rUiω(t) increment of production of power plant i

rDiω(t) decrease of production of power plant i


WSq (t) power output of the wind power scheduled in the day-ahead market

W spillqω (t) part of wind power that can be curtailed

Lshedjω (t) part of the load that can be curtailed

δ0` (t) initial power transmission between n nodes

δ`ω(t) transition power transmission between n nodes

Mathematical model:

The equation to minimize is the sum of all production costs, belonging to generators

and reservoirs, plus cost of lost of load. Systems have to be prepared to face all possible

demands and prevent load shedding, so high cost for lost of load is set.

MinimizeT∑t=1

∑i∈I

(CiPi(t) + CRUi RUi (t) + CRDi RDi (t)) +T∑t=1

∑q∈Q

CqWSq (t)+

T∑t=1

∑ω∈Ω

πω

∑i∈I

(CUi rUiω(t)− CDi rDiω(t)) +

∑q∈Q

Cq(Wqω(t)−WSq (t)−W spill

qω (t))+

(A.1)∑j∈J

V LOLj Lshedjω (t)

Subject to :

Energy balance constraints :∑i∈ΦI

n

Pi(t) +∑q∈ΦQ

n

WSq (t)−

∑j∈ΦJ

n

Lj(t)−∑

`∈Λ|o(`)=n

b`(δ0o(`)(t)− δ

0e(`)(t))+

∑`∈Λ|e(`)=n

b`(δ0o(`)(t)− δ

0e(`)(t)) = 0 : λDn (t),∀n ∈ N, ∀t = 1, ..., T (A.2)

∑i∈ΦI

n

(rUiω(t)− rDiω(t)) +∑j∈ΦJ

n

Lshedjω (t) +∑q∈ΦQ

n

(Wqω(t) −WSq (t)−W spill

qω (t))+

∑`∈Λ|o(`)=n

b`(δ0o(`)(t)− δo(`)ω(t)− δ0

e(`)(t) + δe(`)ω(t))− (A.3)


∑`∈Λ|e(`)=n

b`(δ0o(`)(t)− δo(`)ω(t)− δ0

e(`)(t) + δe(`)ω(t)) = 0 :

γnω(t),∀n ∈ N, ∀ω ∈ Ω,∀t = 1, ..., T

Maximum transmission flow :

b`(δ0o(`)(t)− δ

0e(`)(t)) ≤ C

max` , ∀` ∈ Λ,∀t = 1, ..., T (A.4)

b`(δ0e(`)(t)− δ

0o(`)(t)) ≤ C

max` , ∀` ∈ Λ,∀t = 1, ..., T (A.5)

b`(δo(`)ω(t)− δe(`)ω(t)) ≤ Cmax` , ∀` ∈ Λ,∀ω ∈ Ω,∀t = 1, ..., T (A.6)

b`(δe(`)ω(t)− δo(`)ω(t)) ≤ Cmax` , ∀` ∈ Λ,∀ω ∈ Ω,∀t = 1, ..., T (A.7)

Reference node :

δ01(t) = 0,∀t = 1, ..., T (A.8)

δ1ω(t) = 0, ∀ω ∈ Ω, ∀t = 1, ..., T (A.9)

Maximum wind production :

WSq (t) ≤Wmax

q , ∀q ∈ Q,∀t = 1, ..., T (A.10)

Maximum/minimum power :

Pi(t) +RUi (t) ≤ Pmaxi , ∀i ∈ I, ∀t = 1, ..., T (A.11)

Pi(t)−RDi (t) ≥ 0,∀i ∈ I, ∀t = 1, ..., T (A.12)

Maximum/minimum reserves :

RUi (t) ≤ RU,maxi , ∀i ∈ I, ∀t = 1, ..., T (A.13)

RDi (t) ≤ RD,maxi ,∀i ∈ I, ∀t = 1, ..., T (A.14)

Maximum/minimum ramping rates :

rUiω(t) ≤ RUi (t),∀i ∈ I, ∀ω ∈ Ω,∀t = 1, ..., T (A.15)

rDiω(t) ≤ RDi (t), ∀i ∈ I, ∀ω ∈ Ω,∀t = 1, ..., T (A.16)


Maximum load shedding :

Lshedjω (t) ≤ Lj(t), ∀j ∈ J, ∀ω ∈ Ω,∀t = 1, ..., T (A.17)

Maximum wind spilled :

W spillqω (t) ≤ Lqω(t),∀q ∈ Q,∀ω ∈ Ω,∀t = 1, ..., T (A.18)

Positive variables :

Pi(t), RUi (t), RDi (t) ≥ 0, ∀i ∈ I, ∀t = 1, ..., T (A.19)

rUiω(t), rDiω(t) ≥ 0,∀i ∈ I, ∀ω ∈ Ω,∀t = 1, ..., T (A.20)

WSq (t) ≥ 0, ∀q ∈ Q,∀t = 1, ..., T (A.21)

W spillqω (t) ≥ 0,∀q ∈ Q,∀ω ∈ Ω, ∀t = 1, ..., T (A.22)

Lshedjω (t) ≥ 0, ∀j ∈ J, ∀ω ∈ Ω,∀t = 1, ..., T (A.23)

Where Pi(t), RUi (t), RDi (t), rUiω(t), rDiω(t),WSq (t),W spill

qω (t), Lshedjω (t), δ0n(t), δnω(t),

∀i ∈ I, ∀q ∈ Q,∀j ∈ J, ∀n ∈ N, ∀ω ∈ Ω,∀t = 1, ..., T are the set of decision variables.

For this formulation, first node is established as reference, which means that δ01(t) = 0

and δ1ω(t) = 0. e(`) is the receiving node of line ` and o(`) is the sending node of line `.

The objective function is to minimize the cost of the day-ahead market plus the cost of

the balancing market. Different costs are defined, making distinctions between upward

and downward reserve capacity costs, day-ahead market cost or balancing costs, for

decreasing or increasing generation (CD and CU ). Stochastic production (Cq) is also

included along with fictitious cost of lost of load V LOL, that, as it was mentioned above,

have a very high value to prevent that undesirable effect. Further features, as restrictions

(A.2) - (A.23), can be found in the literature [13].

A.2 Mathematical Model for Flexible Demands

In this subsection, it is presented a simple model that will allow to understand how to

introduce flexible demand in the system [13].


First, there are defined the relevant constants needed to represent the behaviour of

flexible demands [63]:

• Constants:

Dmink (t) Minimum load required by flexible demand k during period t

Dmaxk (t) Maximum load that consumed by flexible demand k during period t

DUk Maximum load pickup rate of flexible demand k

DDk Maximum load drop rate of flexible demand k

Edayk Minimum daily energy consumption for flexible demand k

• Variables:

dk(t) Scheduled load for flexible demand k during period t

dkω(t) Load increase of flexible demand k during period t and scenario ω

cDkω(t) Actual load for flexible demand k during period t and scenario ω, where the

superscript D indicates that the demand is providing down-regulation

cUkω(t)Load Curtailment for flexible demand k during period t and scenario ω,

where the superscript U stands for up-regulation

With all of that, is possible to see that the actual load for flexible demand is expressed

as:

dkω(t) = dk(t) + cDkω(t)− cUkω(t) (A.24)

and

0 ≤ cDkω(t) ≤ Dmaxk (t)− dk(t) (A.25)

0 ≤ cUkω(t) ≤ dk(t)−Dmaxk (t) (A.26)

that represent the change in the load scheduled which is bounded by its maximum and

minimum levels. Maximum daily energy can be restricted with:

T∑t=1

dkω(t) ≥ Edayk , ∀ω (A.27)


Therefore, ”ramp rates” for the load can be constrained as:

dkω(t)− dkω(t− 1) ≤ DUk ,∀t,∀ω (A.28)

dkω(t− 1)− dkω(t) ≤ DDk ,∀t,∀ω (A.29)

And the load shedding constraint is modify as:

Lshedjω (t) ≤ dkω(t) + cDkω(t)− cUkω(t),∀t,∀ω (A.30)

Finally, the objective function is changed to add the flexible demand, which reason-

ably means a cost for the system. The following term is subtracted from the objective

function:

T∑t=1

∑k∈J

Uk(t)dk(t) +

∑ω∈Ω

π[UDk (t)cDkω(t)− UUk (t)cUkω(t)]

(A.31)

where Uk(t) is the utility (cost) for flexible demand at the time of the dispatching

stage, UDk (t) and UUk (t) the utilities of electricity purchase and sale at the balancing

stage respectively. Also balancing conditions are reformulated including these terms. It

is necessary to include scheduled load in the power balance on the day-ahead energy

dispatch (A.2) and the increment of real load (actual load) in the power balance on the

energy redispatch resulting from the real-time balancing (A.3).

A.3 Flexibility from Storage Availability

Analysis

These kinds of power plants allows shifting in time the demand, producing energy at

high-price periods and consuming it at low-price periods, getting greatly improve in

the system performance. When there is a valley in the demand curve and the electricity

prices are low, they consume power, pumping water to the upper reservoir, and in periods

with high demand and prices, turbine the water and generate power. It is important to

notice that these types of units does not actively participate in the profit of the market,

it is more a tool for making the system operation easier, as they can hardly participate


in the regulation. In this manner, storage units play more a role as the transmission

system than as a pure generation device.

The simple mathematical model that explains this feature is the following:

Mathematical Model for Storage Availability

First, constants and variables that define pumped-storage units p are:

• Constants:

Qmaxp Maximum water flow in production/pumping mode.

σTp Conversion factor water flow to produce power (σTp < 1)

σPp Conversion factor water flow to consumed power (σPp > 1)

V U,inip Initial water content of the upper reservoir

V U,minp Minimum water content of the upper reservoir

V U,maxp Maximum water content of the upper reservoir

V L,inip Initial water content of the lower reservoir

V L,minp Minimum water content of the lower reservoir

V L,maxp Maximumwater content of the lower reservoir

• variables:

qTpω(t) Water flow turbined during period t and scenario ω

qPpω(t) Water flow pumped during period t and scenario ω

P Tpω(t) Power production during period t and scenario ω

PPpω(t) Power consumption during period t and scenario ω

νUpω(t) Water content of the upper reservoir at the beginning of period t and sce-

nario ω

νLpω(t) Water content of the lower reservoir at the beginning of period t and scenario

ω

Equations that represent storage availability are:

P Tpω(t) = σTp × qTpω(t),∀p, t, ω (A.32)

PPpω(t) = σPp × qPpω(t), ∀p, t, ω (A.33)


Which define the power generated by turbine (A.32) or pump (A.33) water. It is possible

to see that power consumed to pump water is higher than the power produced turbining

it. It is due to the potential energy (force of gravity) that water has to overcome.

νUpω(t+ 1) = νUpω(t) + qPpω(t)− qTpω(t), ∀p,∀t,∀ω (A.34)

νLpω(t+ 1) = νLpω(t) + qTpω(t)− qPpω(t), ∀p, ∀t,∀ω (A.35)

(A.34) and (A.35) define water balance for upper and lower reservoir respectively. Con-

straints limits for water content in the upper (A.35) and lower (A.37) reservoir and the

minimum water restriction for both reservoirs (A.38),(A.39) are :

V U,minp ≤ νUpω(t) ≤ V U,max

p , ∀p,∀t,∀ω (A.36)

V L,minp ≤ νLpω(t) ≤ V L,max

p , ∀p,∀t,∀ω (A.37)

νUpω(t) ≥ V U,inip , t = T, ∀p,∀ω (A.38)

νLpω(t) ≥ V L,inip , t = T, ∀p,∀ω (A.39)

Finally, (A.40) is the limit of water extracted from the upper reservoir to turbine (gen-

erate) electricity and (A.41) is the limit of water extracted from the lower reservoir to

pump (consume) electricity and storage water:

0 ≤ qTpω(t) ≤ Qmaxp ,∀p,∀t,∀ω (A.40)

0 ≤ qPpω(t) ≤ Qmaxp ,∀p,∀t,∀ω (A.41)

Balancing conditions are reformulated, adding the difference of power produced and

power consumed in the real-time balance equation (A.3). Cost of variable generation is

added to objective function (A.1) but those belonging to consume power to pump water

is not considered, as it is a matter of the pump-hydro power plant owner (it could also

be considered as a load).

Appendix B

Data for 5-node model of the

Australian NEM

Zone and Type Max capacity (MW) Ramp Rate (MW/h) Variable Cost (œ/h)

QLD thermal 3,100 220 90

QLD exp. thermal 9,512 1,200 100

QLD hydro 600 540 10

NSW thermal 3,800 240 90

NSW exp. thermal 10,724 1,400 100

NSW hydro 1,000 900 10

VIC thermal 26,001 200 90

VIC exp. thermal 8,061 1,100 100

VIC hydro 837 750 10

SA thermal 1,050 500 90

SA exp. thermal 2,859 400 100

SA hydro 223 200 10

TAS thermal 269 10 90

TAS exp. thermal 539 50 100

TAS hydro 2,462 2,200 10

Table B.1: NEM Conventional Generation.

Zone Average Load Max Load Min Load Max Pickup Max Drop Min Daily Utility(MW) (MW) (MW) Rate (MW/h) Rate (MW/h) Energy (MW/h) (œ/h)

QSD 6,026.3 7,532.8 4,519.7 500 500 1,000 10

NSW 8,909.7 11,137.1 6,682.2 600 600 1,000 10

VIC 5,870.6 7,338.2 4,402.9 400 400 1,000 10

SA 1,536.9 1,921.1 1,152.6 250 250 1,000 10

TAS 1,131.2 1,414.2 848.4 200 200 1,000 10

Table B.2: NEM Flexible Demand.

99

Appendix B. Data for 5-node model of the Australian NEM 100

Interconnector Max transmission capacity (MW) Reactance X (p.u.) Susceptance B (p.u.)

QLD-NSW 1,078.4 0.1 0.1

NSW-VIC 1,685.7 0.1 0.1

VIC-SA 430.6 0.1 0.1

VIC-TAS 594.9 0.1 0.1

Table B.3: NEM Line Data.

Name Max Capacity (MW) Init Water Up Init Water Down Min Water Max Water Variable CostCont (Hm3) Cont (Hm3) Cont (Hm3) Cont (Hm3) (œ/h)

QLD (Wivenhoe) 500 1,000 0 0 2,000 10

NSW (Bendeela) 80 160 0 0 320 10

NSW (Kangaroo Valley) 160 320 0 0 640 10

NSW (Tumut 3) 1,500 3,000 0 0 6,000 10

Table B.4: NEM Hydro-Pumped Generators.

Zone Max Capacity (MW) Ramp Rate (MW/h) Variable Cost (œ/h)

QLD 1,802 0 0

NSW 1,940 0 0

VIC 2,678 0 0

SA 1,622 0 0

TAS 308 0 0

Table B.5: NEM Wind Generators.

Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5Area Low wind Low wind Low wind Low wind Low wind

in QSD (MW) in NSW (MW) in VIC (MW) in SA (MW) in TAS (MW)

QSD 901 1,802 1,802 1,802 1,802

NSW 1,940 970 1,940 1,940 1,940

VIC 2,678 2,678 1,339 2,678 2,678

SA 1,622 1,622 1,622 811 1,622

TAS 308 308 308 308 154

Table B.6: NEM Wind Contingencies.

Appendix C

Data for IEEE-6 Bus System

Gen. Type Bus Max. Gen. (MW) RR (MW/h)

1 coal 1 100 30

2 hydro 2 100 50

3 gas 6 50 10

4 coal 6 100 30

5 hydro 2 100 50

6 gas 2 50 10

7 gas 1 50 10

8 wind 1 10 -

9 wind 2 10 -

10 wind 6 10 -

Table C.1: Generators data for IEEE 6-node system.

Gen. Op. cost (œ/h) Investment cost (œ/year)

1 15 -

2 0.1 -

3 23 -

4 15 20×103

5 0.1 27×106

6 23 12×103

7 23 12×103

8 0 -

9 0 -

10 0 -

Table C.2: Generators costs for IEEE 6-node system.

101

Appendix C. Data for IEEE-6 Bus System 102

Line From To B(p.u.) Cap(MW) Invest. Cost (œ/year)

1 1 2 1/0.170 80 -

2 2 3 1/0.037 70 -

3 1 4 1/0,.258 140 -

4 2 4 1/0.197 100 -

5 4 5 1/0.037 50 -

6 5 6 1/0.140 140 -

7 3 6 1/0.018 130 -

8 1 2 1/0.170 80 1.6×106

9 2 3 1/0.037 70 1.68×106

10 1 4 1/0.258 140 4.2×106

11 5 6 1/0.140 140 1.96×106

Table C.3: Lines Data for IEEE 6-node system.

Gen. Type Bus Max. Gen. (MW) Min. Gen. (MW) RR (MW/h)

1 coal 1 100 65 3

2 hydro 2 100 15 5

3 gas 6 50 20 10

4 coal 6 100 65 3

5 hydro 2 100 15 5

6 gas 2 50 20 10

7 gas 1 50 20 10

8 wind 1 10 - -

9 wind 2 10 - -

10 wind 6 10 - -

Table C.4: Generators data for IEEE 6-node system with unit commitment.

Gen. Op. cost (œ/h) Start-up cost (œ) Investment cost (œ/year)

1 15 30×103 -

2 0.1 0 -

3 23 40×103 -

4 15 30×103 20×103

5 0.1 0 27×106

6 23 40×103 12×103

7 23 40×103 12×103

8 0 0 -

9 0 0 -

10 0 0 -

Table C.5: Generators costs for IEEE 6-node system with unit commitment.

Appendix D

Data for IEEE-30 Bus System

Gen. Type Bus Max. Gen. (MW) RR (MW/h) Var. cost (œ/h)

1 Coal 1 280 36 7.378

2 Gas-C 2 200 30 13.045

3 Pump-hydro 6 14.75 4.66 0.003

4 Waste 29 21 5 3.757

5 Peat 13 115 14 2.687

6 Gas-A 22 355 30 9.299

7 Distillate 23 100 26 12.572

8 Biogas 25 22 5 14.066

9 Gas-B 27 355 30 9.299

10 Hydro 28 14.75 4.66 0.003

11 Wind 14 100 - 0

11 Wind 16 100 - 0

11 Wind 19 100 - 0

11 Wind 21 100 - 0

Table D.1: The locations, types, data and costs of existing generators for the modifiedIEEE 30-node system.

Type Max. Gen. (MW) RR (MW/h) Var. cost (œ/h) Investment cost (œ/year)

Fast 100 55 9.519 2.78×108

Medium 280 20 7.378 4.59×108

Slow 115.67 10 2.687 1.64×108

Table D.2: Types, data and costs of candidates generators for the modified IEEE30-node system.

103

Appendix D. Data for IEEE-30 Bus System 104

Line From To B Cap Line From To B Cap(p.u.) (MW) (p.u.) (MW)

101 1 2 17.391 13 122 15 18 4.577 1.6

102 1 3 6.053 13 123 22 24 7.740 1.6

103 2 4 5.757 6.5 124 19 20 14.706 1.6

104 3 4 26.385 13 125 10 20 4.785 3.2

105 2 5 5.043 13 126 10 17 11.834 3.2

106 2 6 5.672 6.5 127 10 21 13.351 3.2

107 4 6 24.155 9 128 6 22 6.671 3.2

108 5 7 8.621 7 129 21 22 42.373 3.2

109 6 7 12.195 13 130 15 23 4.950 1.6

110 6 8 23.810 5.2 131 22 24 5.587 1.6

111 6 9 4.808 6.5 132 23 24 3.704 1.6

112 6 10 1.799 3.2 133 24 25 3.038 1.6

113 9 11 4.808 6.5 134 25 26 2.632 1.6

114 9 10 9.091 6.5 135 25 27 4.792 1.6

115 4 12 3.906 6.5 136 27 28 2.525 6.5

116 12 13 7.143 6.5 137 27 29 2.408 1.6

117 12 14 3.908 3.2 138 27 30 1.659 1.6

118 12 20 7.669 3.2 139 29 30 2.206 2.6

119 12 16 5.033 1.6 140 8 28 5.000 3.2

120 14 15 5.008 1.6 141 6 28 16.694 3.2

121 16 17 5.200 1.6 - - - - -

Table D.3: Data of existing lines for the modified IEEE 30-node system.

Line From To B Cap Investment cost(p.u.) (MW) (œ)

151 2 3 5.587 1.6 3.14×106

152 3 13 3.908 3.2 4.99×106

153 4 11 23.810 3.2 7.21×105

154 14 18 5.757 6.5 2.86×106

155 16 18 7.669 3.2 2.34×106

156 18 24 3.704 1.6 5.35×106

157 19 24 3.038 1.6 5.84×106

158 19 22 5.757 6.5 2.93×106

159 21 27 1.659 1.6 10.13×106

160 21 28 2.632 1.6 7.13×106

161 4 16 3.038 1.6 5.94×106

162 4 17 2.632 1.6 6.82×106

163 3 12 3.704 1.6 4.62×106

164 4 5 3.704 1.6 4.72×106

165 5 6 9.091 6.5 1.61×106

166 7 8 9.091 6.5 1.61×106

167 21 28 1.799 3.2 9.94×106

Table D.4: Data of candidate lines for the modified IEEE 30-node system.

Bibliography

[1] M. R. Hesamzadeh, N. Hosseinzadeh, and P. J. Wolfs. Transmission system augmen-

tation based on the concepts of quantity withheld and monopoly rent for reducing

market power. IEEE Transactions on Power Systems, 25(1):167–180, Feb 2010.

[2] Renewable energy capacity statistics 2015. International Renewable Energy Agency,

2015.

[3] M. R. Hesamzadeh D. R. Biggar. The economics of electricity markets. IEEE-Wiley

Press, August 2014.

[4] Y. Tohidi and M. R. Hesamzadeh. Multi-regional transmission planning as a non-

cooperative decision-making. IEEE Transactions on Power Systems, 29(6):2662–

2671, Nov 2014.

[5] M. R. Hesamzadeh and M. Yazdani. Transmission capacity expansion in imperfectly

competitive power markets. IEEE Transactions on Power Systems, 29(1):62–71,

Jan 2014.

[6] Dr. G. Papaefthymiou, K. Grave, and K. Dragoon. Flexibility options in electricity

systems. Ecofys, 2014.

[7] E. Moiseeva, M. R. Hesamzadeh, and D. R. Biggar. Exercise of market power on

ramp rate in wind integrated power systems. IEEE Transactions on Power Systems,

30(3):1614–1623, May 2015.

[8] Water energy and South Australia enviroment. Web-site: https:

//www.sa.gov.au/topics/water-energy-and-environment/energy/

energy-supply-and-sources/renewable-energy-sources/wind-energy/

wind-energy-in-sa.

105

https://www.sa.gov.au/topics/water-energy-and-environment/energy/energy-supply-and-sources/renewable-energy-sources/wind-energy/wind-energy-in-sa




Bibliography 106

[9] M. R. Hesamzadeh, D. R. Biggar, N. Hosseinzadeh, and P. J. Wolfs. Economic

transmission augmentation with explicit modeling of the competition benefit. IEEE

Transactions on Power Systems, 25(3):1714–1721, Aug 2010.

[10] GAMS. Web-site:. http://www.gams.com. .

[11] GLPK. Web-site:. https://www.gnu.org/software/glpk. .

[12] M.R. Hesamzadeh, O. Galland, and D.R. Biggar. Short-run economic dispatch with

mathematical modelling of the adjustment cost. International Journal of Electrical

Power and Energy Systems, 58:9 – 18, 2014.

[13] J. Miguel M. Gonzalez, A. J. Conejo, H. Madsen, P. Pinson, and M. Zugno. Inte-

grating Renewables in Electricity Markets: Operational Problems. Springer, 2014.

ISBN 978-1-4614-9410-2. doi: 10.1007/978-1-4614-9411-9.

[14] P. Seok Kwon and P. Østergaard. Assessment and evaluation of flexible demand in

a danish future energy scenario. Applied Energy, 134:309 – 320, 2014.

[15] N. Menemenlis, M. Huneault, and A. Robitaille. Thoughts on power system flexi-

bility quantification for the short-term horizon.

[16] D. R. Biggar and M. R. Hesamzadeh. Towards a theory of optimal dispatch in the

short run. 2011.

[17] M. R. Hesamzadeh and D. R. Biggar. Computation of extremal-nash equilibria in

a wholesale power market using a single-stage milp. IEEE Transactions on Power

Systems, 27(3):1706–1707, Aug 2012.

[18] M. Baradar, M. R. Hesamzadeh, and M. Ghandhari. Second-order cone program-

ming for optimal power flow in vsc-type ac-dc grids. IEEE Transactions on Power

Systems, 28(4):4282–4291, Nov 2013.

[19] M. Aganagic and S. Mokhtari. Security constrained economic dispatch using non-

linear dantzig-wolfe decomposition. Power Systems, IEEE Transactions on, 12(1):

105–112, Feb 1997.

[20] A. J. Conejo and R. Minguez. Decomposition techniques in mathematical program-

ming : engineering and science applications. Springer, Berlin, New York, 2006.

http://www.gams.com

https://www.gnu.org/software/glpk

Bibliography 107

[21] S. Jiang and L. Tang. Lagrangian relaxation algorithms for re-entrant hybrid flow-

shop scheduling. In Information Management, Innovation Management and Indus-

trial Engineering, 2008. ICIII ’08. International Conference on, volume 1, pages

78–81, Dec 2008.

[22] M. Jabarnejad and J. Valenzuela. Optimal investment plan for dynamic thermal

rating using benders decomposition. European Journal of Operational Research,

248(3):917 – 929, 2016.

[23] E. Keyvanshokooh, S. M. Ryan, and E. Kabir. Hybrid robust and stochastic op-

timization for closed-loop supply chain network design using accelerated benders

decomposition. European Journal of Operational Research, 249(1):76 – 92, 2016.

[24] M. R. Hesamzadeh, D. R. Biggar, N. Hosseinzadeh, and P. J. Wolfs. Transmission

augmentation with mathematical modeling of market power and strategic genera-

tion expansion , part ii. IEEE Transactions on Power Systems, 26(4):2040–2048,

Nov 2011.

[25] S. Cvijic and J. Xiong. Security constrained unit commitment and economic dis-

patch through benders decomposition: A comparative study. 2011.

[26] M. Parsa Moghaddam, A. Abdollahi, and M. Rashidinejad. Flexible demand re-

sponse programs modeling in competitive electricity markets. Applied Energy, 88

(9):3257 – 3269, 2011.

[27] F. You and I. E. Grossmann. Multicut benders decomposition algorithm for process

supply chain planning under uncertainty, 2013.

[28] F. de Oliveira Santini, I. E. Grossmann, and S. Hamacher. Accelerating benders

stochastic decomposition for the optimization under uncertainty of the petroleum

product supply chain, 2014.

[29] L. Tang, W. Jiang, and G. K. D. Saharidis. An improved benders decomposition

algorithm for the logistics facility location problem with capacity expansions, 2013.

[30] J. Verstichel, J. Kinable, P. De Causmaecker, and G. Vanden Berghe. A combi-

natorial benders decomposition for the lock scheduling problem. Computers and

Operations Research, 54:117 – 128, 2015.

Bibliography 108

[31] A. Khodaei, M. Shahidehpour, and S. Kamalinia. Transmission switching in ex-

pansion planning. Power Systems, IEEE Transactions on, 25(3):1722–1733, Aug

2010.

[32] O. Alizadeh-Mousavi and M. Zima-Bockarjova. Efficient benders cuts for transmis-

sion expansion planning. Electric Power Systems Research, 131:275 – 284, 2016.

[33] A. J. Conejo S. de la Torre R. Garcıa-Bertrand S. A. Gabriel, Y. Shim. A ben-

ders decomposition method for discretely-constrained mathematical programs with

equilibrium constraints. The Journal of the Operational Research Society, 61(9):

1404–1419, 2010.

[34] X. Li. Parallel nonconvex generalized benders decomposition for natural gas pro-

duction network planning under uncertainty. Computers and Chemical Engineering,

55:97 – 108, 2013.

[35] S. M. Frank and S. Rebennack. Optimal design of mixed ac–dc distribution sys-

tems for commercial buildings: A nonconvex generalized benders decomposition

approach. European Journal of Operational Research, 242(3):710 – 729, 2015.

[36] W. Zhu and G. Lin. A dynamic convexized method for nonconvex mixed integer

nonlinear programming. Computers and Operations Research, 38(12):1792 – 1804,

2011.

[37] A. Kumar Vatsa and S. Jayaswal. A new formulation and benders decomposition for

the multi-period maximal covering facility location problem with server uncertainty.

European Journal of Operational Research, pages –, 2015.

[38] B. Drysdale, J. Wu, and N. Jenkins. Flexible demand in the GB domestic elec-

tricity sector in 2030. Applied Energy, 139:281 – 290, 2015.

[39] M. R. Hesamzadeh, D. R. Biggar, and N. Hosseinzadeh. The tc-psi indicator for

forecasting the potential for market power in wholesale electricity markets. Energy

Policy, 39(10):5988–5998, 2011.

[40] W. P. Bell, P. Wild, J. Foster, and M. Hewson. Wind speed and electricity demand

correlation analysis in the australian national electricity market: Determining wind

turbine generators’ ability to meet electricity demand without energy storage. Eco-

nomic Analysis and Policy, 48:182 – 191, 2015. Energy.

Bibliography 109

[41] C. Nguyen, C. Ma, A. Hailu, and M. Chalak. Factors influencing calculation of

capacity value of wind power: A case study of the australian national electricity

market (nem). Renewable Energy, 90:319 – 328, 2016.

[42] Clean energy australia report 2013. Clean Energy Council, 2013.

[43] Victorian Wind Alliance. Web-site:. http://ramblingsdc.net/Australia/

WindVic.html. .

[44] Tasmania. Web-site:. http://ramblingsdc.net/Australia/WindTas.html. .

[45] Queensland Goverment. Business and industry portal. Web-site:. https:

//www.business.qld.gov.au/industry/energy/electricity-industry/

electricity-queensland/electricity-generation.

[46] New South Wales Goverment. Department industry. Web-site:. http://www.

resourcesandenergy.nsw.gov.au/energy-consumers/sustainable-energy/

wind.

[47] N. Mahmoudi, T. K. Saha, and M. Eghbal. Wind offering strategy in the australian

national electricity market: A two-step plan considering demand response. Electric

Power Systems Research, 119:187 – 198, 2015.

[48] M. R. Hesamzadeh and D. R. Biggar. Merger analysis in wholesale power market

using the equilibria band methodology. 2013.

[49] Global Energy Obervatory. Web-site:. http://globalenergyobservatory.org/

list.php?db=PowerPlants&type=Hydro. .

[50] R. Brooks and E. Harris. Price leadership and information transmission in aus-

tralian water allocation markets. Agricultural Water Management, 145:83 – 91,

2014. Exploring some of the socio-economic realities of sustainable water manage-

ment in irrigation.

[51] D. Cotton and L. De Mello. Econometric analysis of australian emissions markets

and electricity prices. Energy Policy, 74:475 – 485, 2014.

[52] A. Kumar and J. Kumar. ATC determination with FACTS devices using

PTDFs approach for multi-transactions in competitive electricity markets. Inter-

national Journal of Electrical Power and Energy Systems, 44(1):308 – 317, 2013.

http://ramblingsdc.net/Australia/WindVic.html

http://ramblingsdc.net/Australia/WindVic.html

http://ramblingsdc.net/Australia/WindTas.html

https://www.business.qld.gov.au/industry/energy/electricity-industry/electricity-queensland/electricity-generation



http://www.resourcesandenergy.nsw.gov.au/energy-consumers/sustainable-energy/wind



http://globalenergyobservatory.org/list.php?db=PowerPlants&type=Hydro

http://globalenergyobservatory.org/list.php?db=PowerPlants&type=Hydro

Bibliography 110

[53] D. Huang. Methodology of dynamic ptdf for the zonal model. In North American

Power Symposium (NAPS), 2012, pages 1–6, Sept 2012.

[54] R. Srinu Naik, K. Vaisakh, and K. Anand. Determination of atc with ptdf using

linear methods in presence of tcsc. In Computer and Automation Engineering (IC-

CAE), 2010 The 2nd International Conference on, volume 5, pages 146–151, Feb

2010.

[55] U.S. Energy Information Administration. New england load curve. 2010.

[56] CaliforniaISO. Web-site:. http://www.caiso.com/. .

[57] U.S. Energy Information Administration. Web-site:. http://www.eia.gov/

todayinenergy/detail.cfm?id=19731. .

[58] B. Divya and R. Devarapalli. Estimation of sensitive node for ieee-30 bus system

by load variation. In Green Computing Communication and Electrical Engineering

(ICGCCEE), 2014 International Conference on, pages 1–4, March 2014.

[59] N. Papadakos. Integrated airline scheduling: Decomposition and acceleration tech-

niques, 2006.

[60] L. Kuznia, B. Zeng, G. Centeno, and Z. Miao. Stochastic optimization for power

system configuration with renewable energy in remote areas. 2012.

[61] A. J. Rubialesa, P. A. Lotitoa, M. A. Rissoa, F. J. Mayoranoa, and L. A. Par-

entea. Estabilizacion de la decomposicion generalizada de benders aplicada a la

coordinacion hidrotermica a corto plazo. 2011.

[62] X. Peng and P. Jirutitijaroen. Convergence acceleration techniques for the stochas-

tic unit commitment problem. 2010.

[63] N. Szarka, F. Scholwin, M. Trommler, H. F. Jacobi, M. Eichhorn, A. Ortwein, and

D. Thran. A novel role for bioenergy: A flexible, demand-oriented power supply.

Energy, 61:18 – 26, 2013.

http://www.caiso.com/

http://www.eia.gov/todayinenergy/detail.cfm?id=19731

http://www.eia.gov/todayinenergy/detail.cfm?id=19731

TRITA EE 2016:052

www.kth.se

stochastic investment in power system flexibility: a ...930844/fulltext01.pdf · yaser tohidi a...

Documents