Stochastic Investment in Power System Flexibility:A Benders Decomposition Approach

Stochastic Investment in Power System Flexibility:

A Benders Decomposition Approach

FERNANDO GARCIA MARIÑO

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL ENGINEERING

Master Thesis

STOCHASTIC INVESTMENT IN POWER SYSTEM FLEXIBILITY: A

BENDERS DECOMPOSITION APPROACH.

Author:

Fernando Garcia Mari˜no

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

Albert Einstein

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 Mari˜no

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.

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. Perell´o, 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

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

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

Bibliography 105

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 July 2014. . . 23

5.1 Flowchart of the Benders Decomposition. . . 34

5.2 2-Node system, 1: Existent generator, 2: Existent generator, 3: Demand response, 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

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 system

with investment and unit commitment.. . . 85

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-node system 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

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 modified IEEE 30-node system. . . 103

D.2 Types, data and costs of candidates generators for the modified IEEE 30-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

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

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,

π_ω Probability of realization of scenario ω,

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

RD_i(u) Ramp down rate of generator i (u), RU_i(u) Ramp up rate of generator i (u),

GM_i(u) Minimum stable generation of generator i (u), G_i(u),ω Capacity of generator i (u) in scenario ω,

G_i(u),k,ω Capacity of generator i (u) in contingency k in scenario ω, c_i(u) Production cost of generator i (u),

c^SP_i(u) Start-up cost of generator i (u),

xii

c^SD_i(u) Shut-down cost of generator i (u), B_l(v) Susceptance of transmission line l (v ), F_l(v) Capacity of transmission line l (v ),

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

T ICv Transmission investment cost for candidate line v, GIC_u Generation investment cost for candidate generator u, Υi,n Matrix linking generators i and nodes n ,

Φ_l,n Matrix linking lines l and nodes n , Hl,n PTDF matrix,

Ξ Very big number,

r Short term interest rate, p_k Probability of contingency k, Mi Energy limit of hydro plant i,

Q_i,0 Amount of energy stored before the operation in the reservoir of hydro generator i,

Q^max_i Capacity of reservoir of hydro generator i, η_i^g Efficiency of pump-storage generator i, η_i^p Efficiency of pump-storage motor i, α_down Minimum value of α_it,

xcont Fixed value for candidate line decision, y_cont Fixed value for candidate generator decision.

Variables

xv Binary variable of transmission line v, y_u Binary variable of generator u,

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

g_i(u),t,k,ω Dispatch of generator i (u) at time t for contingency k in scenario ω,

g⁰_i(u),k,ω Dispatch of generator i (u) after clearing of contingency k in scenario ω,

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

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

pp⁰_i(u),k,ω Power pumped (consumed) by i (u) after clearing of contingency k in scenario ω,

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

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

pg_i(u),k,ω⁰ Power turbinated (produced) by i (u) after clearing of contingency k in scenario ω,

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

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

O_i(u),k,ω⁰ Over produced energy by unit i (u) after clearing of contingency k in scenario ω,

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

LLˆ _n,t,k,ω Lost of Load in node n at time t for contingency k in scenario ω,

LL⁰_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

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

U_n,k,ω⁰ Underdelivery in node n after clearing of contingency k in scenario ω,

D_n,ω Demand in node n , in scenario ω

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

D⁰_n,k,ω Demand in node n after clearing of contingency k in scenario ω,

f_l(v),ω Power flow of line l (v ) under normal operation in scenario ω, fˆ_l(v),t,k,ω Power flow of line l (v ) at time t for contingency k

in scenario ω,

f_l(v),k,ω⁰ Power flow of line l (v ) after clearing of contingency k in scenario ω,

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

θˆn,t,k,ω Voltage angle of node n at time t for contingency k in scenario ω,

θ⁰_n,k,ω Voltage angle of node n after clearing of contingency k in scenario ω,

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

ˆ

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

s⁰_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,

ˆ

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

w⁰_i,k,ω Shut-down binary variable of generator i(u) after clearing of contingency k in scenario ω,

z_i(u),ω On-line/off-line binary variable of generator i(u) under normal operation in scenario ω,

z⁰_i(u),t,k,ω On-line/off-line binary variable of generator i(u) at time t contingency k in scenario ω,

ˆ

z_i(u),k,ω On-line/off-line binary variable of generator i(u) after clearing of contingency k in scenario ω,

h_i(u),ω On-line/off-line binary variable of generator u under normal operation in scenario ω,

h⁰_i(u),t,k,ω On-line/off-line binary variable of generator u at time t contingency k in scenario ω,

ˆh_i(u),k,ω On-line/off-line binary variable of generator u after clearing of contingency k in scenario ω,

Q_i,ω Stored water of hydro generator i in scenario ω, Q⁰_i,t,k,ω Stored water of hydro generator i

for contingency k in scenario ω,

Qˆi,k,ω Stored water of hydro generator i after clearing of contingency k, in scenario ω,

αit Auxiliary term for Benders decomposition,

z_it,SC Objective function of Security Check Subproblem in iteration it, zit,OO Objective function of Optimal Operation Subproblem in iteration it, λˆ_v,it Marginal value for candidate line in iteration it,

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

Introduction

1.1 Background

In the wholesale electricity market, demand is satisfied by generation suppliers that results in a given electricity price. F igure 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 (F igure 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

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].

F igure 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

Chapter2summarizes Short-Run Economic Dispatch, the electricity market model that is used in the simulations. In Chapter3, flexibility from Demand Side and from Storage Availability is introduced into SRED. Chapter4is 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, Chapter7 summarizes the conclusions achieved and some actions to be taken for future work are suggested.

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 AppendixA 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 (F igure2.1).

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

8

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(s⁰) the dispatch cost in the final state reached s⁰. 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 → s⁰)[12]

of:

AC(s → s⁰) =

Nt

X

t=1

(DC(s_t) − DC(s⁰))

(1 + r)^t−1 (2.1)

Where Ntis 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:

P V (DC) = 1 + p 1 + r



(1 − p)DC(s) + p



AC(s → s⁰) + DC(s⁰) 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:

M inimize (1 −

N k

X

k=1

p_k)DC(Gi, LLn) +

N k

X

k=1 N t

X

t=1

p_k

(1 + r)^t−1DC(G_n,t,k, LL_n,t,k)+

γ

N k

X

k=1

p_kDC(G_i,k, LL_i,k) (2.3)

Subject to :

Energy balance constraints :

N g

X

i=1

(G_i− O_i)Υ_i,n+

N l

X

l=1

f_lΦ_l,n+ U_n− D_n= 0 (2.4)

N g

X

i=1

(G_i,t,k− O_i,t,k)Υ_i,n+

N l

X

l=1

f_l,t,kΦ_l,n+ U_n,t,k− D_n,t,k = 0 (2.5)

N g

X

i=1

(G_i,k− O_i,k)Υ_i,n+

N l

X

l=1

f_l,kΦ_l,n+ U_n,k− D_n,k= 0 (2.6)

T ransmission f low balance constraints :

N n

X

n=1

"

H_l,n

N g

X

i=1

(G_i− O_i)Υ_i,n− (D_n− U_n)

!#

= f_l (2.7)

N n

X

n=1

"

Hl,n N g

X

i=1

(Gi,t,k− O_i,t,k)Υi,n− (D_n,t,k− U_n,t,k)

!#

= fl,t,k (2.8)

N n

X

n=1

"

H_l,n

N g

X

i=1

(G_i,k− O_i,k)Υ_i,n− (D_n,k− U_n,k)

!#

= f_l,k (2.9)

Lost − load constraints : LLn= Un+

N g

X

i=1

Oi∗ Υ_i,n (2.10)

LL_n,t,k= U_n,t,k+

N g

X

i=1

O_i,t,k∗ Υ_i,n (2.11)

LLn,k= Un,k+

N g

X

i=1

Oi,k∗ Υ_i,n (2.12)

T ransmission f low limits :

− ¯f_i≤ f_i≤ ¯f_i (2.13)

− ¯f_i≤ f_l,t,k≤ ¯f_i (2.14)

− ¯f_i≤ f_l,k≤ ¯f_i (2.15)

Generation production limits :

0 ≤ G_i ≤ ¯G_i (2.16)

0 ≤ G_i,t,k ≤ ¯Gi (2.17)

0 ≤ G_i,k ≤ ¯Gi (2.18)

Ramp − rate limits :

0 ≤ |Gi,t,k− G_i,t−1,k| ≤ RR_i (2.19)

0 ≤ |Gi,k− G_i,T0,k| ≤ RR_i (2.20)

Hydro energy production limits :

∀i ∈ N h,

N t

X

t=1

G_i,t,k· ∆t ≤ M_i (2.21)

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

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 AppendixA, 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.2have 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:

d_n,t,k = d_n,t+ c^D_n,t,k− c^U_n,t,k (3.1)

0 ≤ c^D_n,t,k≤ D_n,t^max− d_n,t (3.2)

0 ≤ c^U_n,t,k≤ d_n,t− D_n,t^max (3.3)

N t

X

t=1

dn,t,k ≥ E_n^day (3.4)

d_n,t,k− d_n,t−1,k ≤ D_n^U (3.5)

d_n,t−1,k− d_n,t,k ≤ D_n^D (3.6)

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

Un≤ L_initial (3.7)

U_n,t,k≤ d_n,t,k+ c^D_n,t,k− c^U_n,t,k (3.8)

Un,k≤ d_{n,t=N t,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_p^T = σ^T_p × q^T_p (3.10)

P_p,t,k^T = σ_p^T × q_p,t,k^T (3.11)

P_p,k^T = σ_p^T × q_p,k^T (3.12)

P_p^P = σ_p^P × q_p^P (3.13)

P_p,t,k^P = σ_p^P × q^P_p,t,k (3.14)

P_p,k^P = σ_p^P × q^P_p,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:

ν_p,t,k^U = ν_p,t−1,k^U + q^P_p,t−1,k− q_p,t−1,k^T (3.16)

ν_p,t,k^L = ν_p,t−1,k^L + q^T_p,t−1,k− q_p,t−1,k^P (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_p^U,min≤ ν_p^U ≤ V_p^U,max (3.18)

V_p^U,min≤ ν_p,t,k^U ≤ V_p^U,max (3.19)

V_p^L,min ≤ ν_p^L≤ V_p^L,max (3.20)

V_p^L,min ≤ ν_p,t,k^L ≤ V_p^L,max (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 ≤ q_p^T ≤ Q^max_p (3.22)

0 ≤ q_p,t,k^T ≤ Q^max_p (3.23)

0 ≤ q_p,k^T ≤ Q^max_p (3.24)

0 ≤ q^P_p ≤ Q^max_p (3.25)

0 ≤ q^P_p,t,k≤ Q^max_p (3.26)

0 ≤ q^P_p,k≤ Q^max_p (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(G_i, LL_n) =

N g

X

i=1



C_iG_i+ C_ruR^U_i + C_rdR^D_i

 +

N n

X

n=1

V^LOLU_n+ (3.28)

N p

X

p=1

CpP_p^T +

N q

X

q=1

CqWq−

N n

X

n=1

UnL_initial

• Transition Dispatch Cost

DC(G_i,t,k, LL_n,t,k) =

N t

X

t=1 N k

X

k=1

( _{N g} X

i=1



CiG_i,t,k+ CruR^U_i,t,k+ C_rdR^D_i,t,k

 +

N n

X

n=1

V^LOLUn,t,k+

N p

X

p=1

CpP_p,t,k^T +

N q

X

q=1

CqWq,t,k− (3.29)

N n

X

n=1

U_n(d_n,t+ 1.05 · c^D_n,t,k− 0.95 · c^U_n,t,k) )

• Final Dispatch Cost

DC(G_i,k, LL_n,k) =

N k

X

k=1

( _{N g} X

i=1



C_iG_i,k+ C_ruR^U_i,k+ C_rdR^D_i,k

 +

N n

X

n=1

V^LOLU_n,k+

N p

X

p=1

CpP_p,k^T +

N q

X

q=1

CqW_q,k−

N n

X

n=1

Un(dn,t=N t+ 1.05 · c^D_{n,t=N t,k}− 0.95 · c^U_{n,t=N t,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

N g

X

i=1

(Gi− O_i)Υi,n+

N l

X

l=1

P_lΦ_l,n+

N q

X

q=1

WqΨq,n+

N p

X

p=1

(P_p^T − P_p^P)χp,n+ Un− L_initial= 0

(3.31)

N g

X

i=1

(G_i,t,k− O_i,t,k)Υi,n+

N l

X

l=1

P_l,t,kΦ_l,n+

N q

X

q=1

W_q,t,kΨq,n+

N p

X

p=1

(P_p,t,k^T − P_p,t,k^P )χp,n

+ Un,t,k− d_n,t,k= 0 (3.32)

N g

X

i=1

(G_i,k− O_i,k)Υi,n+

N l

X

l=1

P_l,kΦ_l,n+

N q

X

q=1

W_q,kΨq,n+

N p

X

p=1

(P_p,k^T − P_p,k^P )χp,n

+ U_n,k− d_{n,t=N t,k}= 0 (3.33)

• Transmission Flow Balance

N n

X

n=1

"

Hl,n N g

X

i=1

(Gi− O_i)Υi,n+

N q

X

q=1

WqΨq,n+

N p

X

p=1

(P_p^T − P_p^P)χp,n− (L_initial− U_n)

!#

= Pl

(3.34)

N n

X

n=1

"

H_l,n

N g

X

i=1

(G_i,t,k− O_i,t,k)Υ_i,n+

N q

X

q=1

W_q,t,kΨ_q,n

+

N p

X

p=1

(P_p,t,k^T − P_p,t,k^P )χp,n− (d_n,t,k− U_n,t,k)

!#

= P_l,t,k (3.35)

N n

X

n=1

"

H_l,n

N g

X

i=1

(G_i,k− O_i,k)Υi,n+

N q

X

q=1

W_q,kΨq,n+

N p

X

p=1

(P_p,k^T − P_p,k^P )χp,n

− (d_{n,t=N t,k}− U_n,k)

!#

= P_l,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.

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, F igure 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

Figure 4.1: Generation in South Australia by fuel type between 22 June and 5 July 2014.

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 Chapter2. 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.

M inimize α

 X

v

x_vT IC_v

 + β

 X

u

y_uGIC_u



+ X

ω

π_ω



(1 −X

k

p_k)X

i

(c_ig_i,ω

+ c^SP_i s_i,ω+ c^SD_i w_i,ω) + (1 −X

k

p_k)X

u

(c_ug_u,ω+ c^SP_u s_u,ω+ c^SD_u w_u,ω)



+ X

i,k,t

pk

c_ig_i,k,t,ω+ c^SP_i s_i,k,t,ω+ c^SD_i w_i,k,t,ω

(1 + r)^t (4.1)

+ X

u,k,t

p_kcugu,k,t,ω+ c^SP_u su,k,t,ω+ c^SD_u wu,k,t,ω

(1 + r)^t

+ 1

(1 + r)^Tr X

i,k

pk(cigi,k,ω+ c^SP_i si,k,ω+ c^SD_i wi,k,ω)

+ 1

(1 + r)^Tr X

u,k

p_k(c_ug_u,k,ω+ c^SP_u s_u,k,ω+ c^SD_u w_u,k,ω)