Master’s Thesis Financial Storage Rights for Integration

Technische Universitaumlt MuumlnchenDepartment of Electrical and Computer EngineeringInstitute for Renewable and Sustainable Energy Systems

Politecnico di TorinoDipartimento Energia Galileo FerrarisCorso di Laurea Magistrale in Ingegneria Elettrica

Masterrsquos Thesis

Financial Storage Rights for Integrationof Battery Energy Storage Systems Us-ing AC-OPF

written by

Hasan Gega

Advisors Prof Maurizio RepettoProf Thomas HamacherMSc Andrej Trpovski Dr Arif Ahmed

Abstract

As renewable energy is gaining an increasing attention due to global warm-ing traditional grids are facing numerous challenges A combination of dif-ferent technologies is essential to ensure energy security and equity whilemaintaining an acceptable level of environmental sustainability Energy stor-age devices will play a key role at different levels of the electrical grids formaintaining a stable supply of energy both in a short- and long-term contextAs storage can provide a wide variety of services to the grid an accurateregulatory framework is necessary for a detailed consideration of its bene-fitsIn this thesis after introducing the basics of mathematical optimization andpower system analysis a discussion on the role and consideration of stor-age devices in electrical grids is presented After defining an optimal powerflow model in which storage is integrated as a transmission asset finan-cial instruments called Financial Storage Rights are reviewed consideringa nodal pricing scheme Since manuscripts and reports concerning thistopic have been considering Direct Current (DC) approximations of the op-timal power flow equations a full Alternating Current (AC) formulation ispresented without neglecting losses reactive power flows and voltage lev-els Consequently the mathematical computation of Financial TransmissionRights and Financial Storage Rights is performed using the same proce-dures as in the approximated model presented in [1] The simulations havebeen carried out using the General Algebraic Modeling System (GAMS)software in order to evaluate how approximations affect results quality andto provide an overview on the potential use of Financial Storage Rights inelectricity markets Last this thesis is concluded with a discussion of thepotential future research scope

Statement of Academic Integrity

Last name GegaFirst name HasanID No 03739935

hereby confirm that the attached thesis

Financial Storage Rights for Integration of Battery Energy Storage SystemsUsing AC-OPF

was written independently by me without the use of any sources or aidsbeyond those cited and all passages and ideas taken from other sourcesare indicated in the text and given the corresponding citations appropriatelyTools provided by the institute and its staff such as models or programs arealso listed

I agree to the further use of my work and its results (including programsproduced and methods used) for research and instructional purposes

I have not previously submitted this thesis for academic credit

Munich June 2021

Acknowledgements

I would like to express my gratitude to Andrej Trpovski and Arif Ahmed fortheir constant support during the preparation of my thesis I would also liketo thank Prof Maurizio Repetto and Prof Thomas Hamacher for their pro-fessional contribution and assistance throughout this workI would like to thank all the colleagues met in Turin and in Trondheim dur-ing my studies it has been a great pleasure working with you I am verygrateful to my friends for their support and affection I will never forget themoments we shared and how our friendship is important to me My deepestappreciation goes to my beloved family their patience and support duringthis journey Last but not least I would like to thank Selene for her trust herlove and these wonderful years togetherAll of this would not have been possible without all of you

Contents

Statement of Academic Integrity 3

Contents 6

List of Figures 8

List of Tables 9

1 Introduction 11

2 Optimization in Electrical Engineering 1521 Primal and Dual Problem 1822 Karush-Kuhn-Tucker Conditions 22

3 Power System Analysis 2531 The pu System 2832 Conventional Power Flow 2933 Optimal Power Flow 3134 Applications 37

341 DC-OPF 37342 Other Formulations 39

35 OPF and Dual Variables as Prices 39

4 Energy Storage 4341 Storage Characteristics 4542 Technologies 47

421 Mechanical Energy Storage 47422 Electrical and Magnetic Energy Storage 47423 Thermal Energy Storage 48424 Chemical Energy Storage 48425 Electro-chemical Energy Storage 48

43 Storage Role and Advantages 50

5 Financial Rights 5551 Grid Congestions 5552 Storage Congestions 58

CONTENTS CONTENTS

6 Financial Storage Rights Calculation 6361 DC-OPF 6362 AC-OPF 68

7 Simulations 7771 General Considerations 7772 DC - AC Results Comparison 7873 Storage Role 8474 Financial Transmission Rights and Financial Storage Rights 86

8 Conclusion 91

9 Future Work 9391 Model Extensions 9392 V2G 94

Bibliography 97

List of Figures

11 Percentage regional electricity generation by fuel (2019) [2] 1112 A Ragone plot for energy storage devices comparison [5] 13

21 Classification of optimization problems 1822 Primal and Dual problem linear case 2023 Example 1 LP Primal and Dual problem 2124 Example 1 LP Primal and Dual problem matrix form 2125 Example 1 Duality between decision variables and Lagrangian

variables (1) 2126 Example 1 Duality between decision variables and Lagrangian

variables (2) 22

31 A generic representation of a bus i with multiple connection tothe grid 26

32 π-equivalent circuit for a generic branch 2733 Approximated voltage profiles in a transmission line with respect

to the SIL 3634 Loadability curve for a transmission line [17] 3635 The analogy for DC-OPF 3836 Two bus system example 4037 Two bus non-congested system example 4138 LMP in case of congestion in a two bus system 41

41 Technologies for energy storage 4442 Storage benefits according to the location in the supply chain [40] 4543 Power-Energy view of storage benefits [59] 5144 Electrical Energy Storage (ESS) role for Peak shaving (a) and

Load Levelling (b) [64] 53

51 LMP in case of congestion in a two bus system 57

71 5 bus system single line diagram 7972 Relative errors on phase angles and net active powers 10 time

periods 8273 Relative errors on Locational Marginal Prices (LMPs) 8374 Phase angle at bus 5 85

75 Power generated from the slack bus and the most expensive gen-erator 86

76 LMPs [ k$pu

] in incrementing load simulation 8877 Financial Transmission Rights and congestion surplus match [ k$

pu] 88

78 Financial Transmission Rights congestion surplus and loss com-ponent in case of AC results [ k$

pu] 89

79 FSR computation first approach [1] 90710 FSR computation second approach [71] 90

91 New passenger car registrations by fuel type in the EuropeanUnion Q1 2021 [83] 95

92 New passenger car registrations in the EU by alternative fuel type[83] 96

List of Tables

71 Generators data [18] 7972 Bus and Load data [18] 7973 Lines data [18] 7974 DC-OPFAC-OPF results comparison Voltages and phase angles 8075 DC-OPFAC-OPF results comparison Generated and net active

and reactive powers 8076 Relative error between DC-OPFAC-OPF 81

Chapter 1

Introduction

Contemporary transmission and distribution grids are nowadays facing achallenging modernization process in order to use energy in more effec-tively and efficiently concepts such as Smart Grids renewable genera-tion and energy storage systems have increasingly been used in the lastdecades in order to better explain what modern electric networks are ex-periencing A crucial aspect is the growing integration of solar and windrenewable sources considering their intrinsic intermittency and need for anefficient way of storing the energy produced Electrical consumption is goingto increase especially after a widespread uptake of Electric Vehicles (EVs)This will bring to higher energy demand but also opportunities to use EVsbatteries to improve the flexibility of the grid Also after a first period ofuse as on-board storage systems second-life stationary applications maybring a huge benefit both in terms of capacity and battery recycling Section82 gives a view of the potential widespread diffusion of EV technologiesand second-life battery possibilities The first step towards a lowering of

Figure 11 Percentage regional electricity generation by fuel (2019) [2]

the CO2 footprint is the phasing-out of coal-fired power plants that in Euro-

1 Introduction

pean countries will occur in the next few years (2025 is the target for someEuropean countries) while as a worldwide perspective might take a longerperiod Figure 11 shows the different portfolios according to the locationsAs traditional plants will give way to local renewable sources active powerflows in high voltage lines could see a decrease in magnitude While onthe one hand it will represent an advantage in terms of savings in electricitybills for the final consumer and reduced CO2 footprint on the other hand thiswill test the network stability and resilienceAs discussed in chapter 3 events in which there are low load profiles andlow power flowing in a transmission line could lead to a reactive power pro-duction at the transmission side This is due to the capacitive effects of thelines that could bring voltage levels to a higher level than their nominal val-ues (this can be overcome via inductors in parallel) However transmissiongrids will still play a crucial role to maintain a reliable energy supply anda stable network Frequency is also another fundamental parameter to betaken into account since in future grids there will be less conventional powerplants and therefore less available synchronous generators for frequencysupportAlso renewable energy sources (RES) bring a high level of unpredictabilityand uncertainty of production In this framework storage is important forabsorbing the surplus generated from RES when their output is higher thanthe actual demand of the grid avoiding curtailment processes The energystored is then released when this aleatory generation drops resulting in thepossibility of active power control Additionally the presence of storage sys-tems is essential if coupled with high power devices in the framework ofmodern smart grids eg in case of a widespread of fast charging technolo-gies the peak demand required could be unsustainable for the distributionsystemsStorage along with other solutions can be a flexibility resource also for thetransmission side of the grid Due to environmental issues and economicreasons it might be difficult to build new lines Therefore it is common tosee alternatives that match in other ways the new power requirements [3]

ndash Storage devices that charge and discharge over a multi-period timeframe allowing a redistribution of high peak loads over the day

ndash Demand response strategy aiming at utilizing electrical energy in asmart way by the modulation of the consumption to avoid congestionsin the grid and better redistribute the load in a larger timescale

ndash An increase of energy efficiency brings advantages in terms of overallconsumptions [3]

ndash Distributed generation sources helps alleviating transmission lines con-gestion because power production is closer to consumers but the vari-

1 Introduction

ability in production arises new challenges both in transmission anddistribution grids

Many storage technologies are currently available and in use while othersare still in development some comprehensive overviews have been pre-sented see for example [4] and [5] Each of these devices is more suitablein different circumstances since their characteristics are extremely differentThese dissimilarities can be efficiently outlined using a Ragone plot in termof specific energy (Whkg) and specific power (Wkg) The use of logarith-mic axes shows how these values differ according to the technology usedconsidering also physical characteristics related to the volume or weight ofthe devices (Figure 12) See chapter 4 for a discussion about storage tech-nologies for grid applications

Considering their technological maturity and good compromise betweenenergy density and power density in this work we will focus on simulationsusing Battery Energy Storage Systems (BESS) their modelling in powersystems and their consideration within the financial framework of powersystem economics Nevertheless some of these considerations may bemore broadly extended to the other storage technologies Since the in-terface between BESS and the grid is based on converters ancillary ser-vices such as reactive power and voltagefrequency control are also possi-ble This scenario is particularly significant in weak grids wherein voltageand frequency excursions make BESS more valuable than in strong grids[6]

Figure 12 A Ragone plot for energy storage devices comparison [5]

Future networks will see a widespread of decentralized producers resultingin a less-predictable power flow To get the most out of the existing network

1 Introduction

a good autonomy in terms of energy storage will be essential Storage helpsalso preventing congestions that would rise the prices in determinate areasavoiding meanwhile high-cost infrastructure upgrades BESS as other stor-age technologies can be considered as investments for the purpose of pro-ducing an income for example they can be charged when the electricityprice is low and discharged when the cost rises Whether they are used formaximizing energy penetration from renewable sources or for load powerlevelling BESS can improve power system stability and efficiency

A storage device is an attractive investment when profits exceed overallcosts for this reason initial investment cost and operational expenses mustbe taken into account also considering their durability and the cost of thepower electronics conversion stage Some of these ancillary services canbe difficultly valued [7] provides a description about non-technical issuesrelated to BESS integration One example is the inefficient pricing of someservices provided by energy storage systems that can result in unproduc-tive amount of storage investmentsFurthermore the regulatory treatment of storage brings to two opportuni-ties either providing regulated services or getting back their costs throughthe market These and other methods such as a hybrid treatment or openaccess approach are discussed in [7]System complexity of course increases if many storage devices are con-nected to the grid instead of building new transmission lines but an ap-propriate treatment of storage could become a solid competitor to TampD up-grades (or at least could help with the deferral of new infrastructures)

Since storage can be categorized as a generation transmission or dis-tribution asset due to its characteristics every treatment can undervalue itspotential revenue because the services offered overlap among these sec-tions [7] Moreover while they are usually classified as generation assetstransmission asset usage can be worth investigating Thus the battery doesnot buy or sell at the wholesale market but gains just through rate payments[1] This brings to considering storage as a passive and price insensitive de-vice aiming at maximizing its utility to the grid For this reason an efficientmodel is critical for not making storage operation uneconomicIn this work we will consider utility-scale storage systems as a transmissionasset for different purposes Financial storage rights similar as financialtransmission rights are introduced to add a revenue source for the batteryowner while leaving the task of running the device to the system operatorto maximize social welfareFirst a general overview of mathematical optimization (chapter 2) and powersystems operation (chapter 3) is presented for understanding the basics ofoptimal power flow with storage Then the role of storage is discussed inchapter 4 and the concept of financial rights is reviewed in chapter 5 Themathematical model is presented in chapter 6 and software simulations aresummarized in chapter 7 Then the final sections conclude the thesis

Chapter 2

Optimization in ElectricalEngineering

Mathematical Optimization is a branch of applied mathematics that studiesmethods and algorithms in order to find a solution to a maximization or min-imization problem Optimization is applied in every subfield of engineeringscience and industry Many methods can be used for solving an optimizationproblem but some of them may be more suitable in certain cases thereforea basic knowledge and the classification of the methods must be clear whensolving these problemsWhen choosing the most appropriate method the computational complexityis an important factor even with modern computer performances In somecases it is useful to choose linear approximation of more complex problemsbecause computational time could be excessive If an optimization problemhas a complex structure the user can also decide to ldquorelaxrdquo some or all theentities that bring complexity to the problem As an example non-linear pro-gramming problems can be relaxed to apply linear programming algorithmsacting on the ldquonon-linearitiesrdquo via piecewise linear approximations

In a general framework an optimization problem consists of an objectivefunction to be minimized or maximized given a domain for the variablessubject to constraints

minmax f(x) (201)subject to ii(x) le 0 i = 1 (202)

ej(x) = 0 j = 1 (203)(204)

ndash x is a vector containing all the primal decision variables (or degrees offreedom)

ndash f(x) is the objective function to be minimized or maximized

2 Optimization in Electrical Engineering

ndash ii(x) are the inequality constraints

ndash ej(x) are the equality constraints

When no constraints are considered in a problem it is called unconstrainedoptimization Since a general term can be fixed some variables may besubject to an equality constraint Almost every variable or combination ofthem is subject to inequality constraints because domain boundaries areimportant to maintain the solution physically feasible The output of thisproblem is a vector containing the values to be assigned to the decision vari-ables in order to obtain the smallest objective function (largest in the case ofmaximization problems) As previously stated when dealing with non-linearproblems the search of an optimal point can be hard because of compu-tational reasons Therefore to avoid an excessive complexity sometimesit is good to choose a ldquoneighbourhoodrdquo that is part of the entire domain ofthe solution It is possible that the optimal point found in the neighbourhooddoes not correspond to the global minimum of the solution domain in thiscase it is called ldquolocal optimumrdquo In most cases the best option is a trade-off choosing a neighbourhood wide enough to have a good quality solutionbut not too large to avoid a problem extremely difficult to be computed [8]With this clear it is immediate to state that functions that have a uniqueminimum are simpler to be treated Thus convex optimization problems areeasier to be solved rather than non-convex ones A convex problem has anobjective function and all the constraints that must be convex The percep-tion of convexity is straightforward in the case of a simple one real variablefunction the function must be below any lines connecting two points of thefunction For convex sets any lines connecting two points of the set mustrely entirely inside the set Mathematically this can be verified using the firstorder or the second order conditions [9]

Let f Rn rarr R be a function whose first and second derivative (gradientnablaf and Hessian nabla2f ) exist in every point of the domain dom(f) If f is con-vex then [9]

minusrarr f(y) ge f(x) +nablaf(x)T (y minus x) for all x y isin dom(f)

minusrarr nabla2f(x) ge 0 for all x isin dom(f)

Again relaxation methods are possible for simplifying studies of non-convexproblems (convex relaxation)Another important classification consists of Linear and Non-Linear Opti-mization problems (also called Linear Programming and Non-Linear Pro-gramming LP and NLP) It is important to highlight that one single non-linearity in the objective function or one of the constraints results in a Non-Linear Programming problemIn case of minimization in a general linear problem

min f(xi) = cixi

st Ax le b

x ge 0

or in matrix form

min cTx

st Ax le b

x ge 0

In this way we defined the ldquoPrimal Problemrdquo Another ldquoDual Problemrdquocan be defined from these data see subsection (21) for this important ap-plicationIn general the simplex method introduced by Dantzig [10] is one of the mostwell-known methods for solving a Linear Programming problem It consistsof an algorithm that solves LP problems by considering a feasible regiondefined by a geometric entity termed polytope The polytope is drawn fromthe inequality constraints and the optimal value is found evaluating the ob-jective function in the feasibility region and will occur at the vertex of thisarea Unlike the simplex method that follows the path on the boundaries ofthe polytope interior point methods cross the feasible regionCountless other methods have been developed and applied for solving bothLP and NLP problems In figure (21) a not comprehensive list is presented

The distinction between deterministic and stochastic methods is impor-tant because both of them are widely used in many subfields Generally ifan algorithm has no random components and seeks to find an exact solu-tion is called deterministic If somehow randomness is necessary becauseof the uncertain nature or challenging structure of the problem randomizedstochastic methods are used In many applications deterministic methodscan bring to a computationally expensive result therefore exact algorithmsare impossible to be implemented in some cases This results in an increas-ing success of meta-heuristics stochastic methods Nevertheless since thesize of the grids considered in the simulations of this work is not too widedeterministic methods have been preferred thanks to their specific proce-duresHybrid approaches that use both exact and meta-heuristics methods arealso possible resulting in good quality of solutions with an acceptable timeof computation They can be classified into integrative combinations andcollaborative combinations [11]The solutions of the problems are different in case of integer variables in

Figure 21 Classification of optimization problems

this case we talk about Integer Programming (IP) If only some of the vari-ables are integer it is the case of Mixed Integer Programming (MIP) Inconclusion dealing with discrete variables rather than continuous variablesbrings to an even more complex structure of the problem This chapter andin particular the next sections contains relevant mathematical backgroundthat will be useful when introducing financial storage rights

21 Primal and Dual Problem

Given an optimization problem that we call the ldquoPrimal Problemrdquo we canalso define the ldquoDual Problemrdquo that is strictly related to the primal and insome cases might be very useful for finding the optimal solution of the pri-mal We can define the duality gap as the discrepancy between the solu-tions of these two problems Only in some cases the duality gap is zeroie the two problems have the same objective solution (eg in the caseof a linear programming problem) Strong duality and weak duality can bedefined see [9] for this and other aspects of optimization Given a problemin the form of eqs (20) we can define the Lagrangian function associated

L(x λ micro) = f(x) +sumi

λiii(x) +sumj

microjej(x) (211)

ndash λi ge 0 is called ldquoLagrangian dual multiplierrdquo or dual variable of the i-thinequality constraint

ndash microj is called ldquoLagrangian dual multiplierrdquo or dual variable of the j-thequality constraint (note that in chapter 6 a Lagrangian function is de-fined but the micro are replaced by λ also for equality constraints)

In this way a weighted sum of the constraints increases the objective func-tion Then the Lagrangian dual function can be defined

Ldual(λ micro) = inf (L(x λ micro) = inf (f(x) +sumi

λiii(x) +sumj

microjej(x)) (212)

We can therefore state that

ndash the dual function gives a lower bound to the optimal objective value

ndash since we are looking for the best lower bound computable from thedual function the dual problem is a maximization problem

ndash the constraints for this maximization problem are the inequality con-straints ensuring the non-negativity of the dual variables (λi ge 0)

ndash the dual maximization problem is always a convex optimization prob-lem regardless of the nature of the primal problem

ndash the number of constraints in the primal problem is equal to the num-ber of variables in the dual problem while the number of variables inthe primal problem is equal to the number of constraints of the dualproblem

The introduction of dual variables is extremely important They are oftencalled Lagrange multipliers or dual multipliers but when they represent acost they are sometimes termed shadow prices implicit prices or dualprices [12] When using optimization modeling systems it sometimes takethe acronym of marginal value This will give relevant insights in chapters 5and 6 where dual multipliers are considered as representing the cost of pro-ducing one extra MWh of energy after a supposed optimized re-dispatch ina specific location of the network [13] From the definition of the Lagrangianfunction the value researched (minimum or maximum) can be found with

the so-called Lagrange multipliers method However since this method onlyconsiders the presence of equality constraints we are more interested onthe Karush-Kuhn-Tucker approach that comprise also inequality constraints(see section 22 for more details) In case of linear programming problemsthe dual problem can be easily found using the following notation

Figure 22 Primal and Dual problem linear case

The General Algebraic Modeling System (GAMS) language [by GAMS De-velopment Corporation] is useful in this framework After stating that thereis a duality in the number of constraints and decision variables in the primaland dual problems when solving an optimization problem it can be foundthat the values of the variables of the dual problem are equal to the valuesof the marginal costs associated to the inequality constraints of the primalproblem (MARGINAL in GAMS) Vice versa if solving the dual problem thelevels of the primal variables match with the marginals of the dual problemThe example in figure 23 (that is solved in [14] using dual simplex method)shows a simple linear programming problem and the results have been im-plemented using GAMS Duality is even clearer in matrix form (figure 24)In this case since in LP strong duality occurs the objective functions of pri-mal and dual problems take the same value

The results from GAMS simulation is reported here It can be seen thatapart from having found the same solution for the primal and dual problemthe dual variables value can be found under the column MARGINAL of theprimal problem This can be very useful in more complex problems

Figure 23 Example 1 LP Primal and Dual problem

Figure 24 Example 1 LP Primal and Dual problem matrix form

Figure 25 Example 1 Duality between decision variables and Lagrangianvariables (1)

22 Karush-Kuhn-Tucker ConditionsTo prove if a solution is optimal the Karush-Kuhn-Tucker conditions (KKT)can be computed [15] They represent a more general approach with re-spect to the Lagrange multipliers method since it also considers inequalityconstraints After the definition of the dual problem the Karush-Kuhn-Tuckerconditions can be computed considering the gradient of the Lagrangian (theobjective and constraints must be differentiable) The gradient (derivativewith respect to the primal and dual variables) is set to zero because this sta-tionary condition brings to a maximum or minimum optimal point The KKTconditions can be a way to solve simple optimization problems but it is notrealistically useful in the case of problems including several variables andconstraints Yet these conditions are useful to approach the solutions andin our cases to help finding a connection between the constraints throughthe use of the Lagrangian (see financial storage rights calculations in chap-ter 6)Setting the gradient of the Lagrangian to zero gives the stationary condi-tion

nablaL(xlowast λ micro) = nablaf(xlowast) +sumi

λinablaii(xlowast) +sumj

microjnablaej(xlowast) = 0 (221)

From the primal problem constraints we can define the primal feasibilityconditions

ii(xlowast) le 0 i = 1 (222)

ej(xlowast) = 0 j = 1 (223)

From the dual multipliers associated with the inequality constraints we candefine the dual feasibility conditions

λi ge 0 (225)

The KKT conditions stated above [16] do not give information about whichinequality constraint is binding or not in the optimal solution found There-fore the following additional conditions labelled as complementary slack-ness can be added to the model

λi ii(xlowast) = 0 (226)

A constraint is active (binding) if the corresponding dual multiplier is positivewhile it is not binding if the corresponding multiplier is zero We can alsostate that

ndash Given a problem and its dual with strong duality the solutions form asaddle point

ndash The KKT optimality conditions are necessary and also sufficient con-ditions in case of convex objective functions

ndash Since several optimization problems related to power systems are non-linear and non-convex it is relevant to remark that the KKT conditionsare not sufficient in those cases

Chapter 3

Power System Analysis

A generic electrical grid can be modelled as a graph consisting of busesand branches representing every component of the power system (trans-formers lines physical points of connection etc) A system with n busescan be described by

ndash 4n variables comprising voltages voltage angles active and reactivenet powers

ndash 2n equations representing the power flow equations for each bus

Therefore 2n variables are calculated through computation of the 2n powerflow equations whereas the remaining 2n variables must be specified ac-cording to the following criteria

ndash PV nodes active power and voltage magnitude in case of a generatingunit connected to the bus

ndash PQ nodes active and reactive power in case of load buses or transitbuses

ndash Slack bus (V δ node) voltage magnitude and phase angle for only onebus

In some cases during particular operating conditions some nodes canchange the specified variables The active power cannot be assigned toevery bus because in quite large systems the total amount of power lossesis non-negligible Then it is necessary to select one bus in which the activepower is not assigned while phase angle and voltage magnitude are pre-specified called as slack bus Apart from the key-role played in the contextof balancing the active power the slack bus works also as a phase anglereference (usually set to 0deg)

Computing voltages power flows and other variables of interest is rel-evant for planning and operation of power systems Therefore an appro-priate mathematical model is necessary to describe the system ensuring

3 Power System Analysis

that electrical energy is transferred from supply buses to loads Genera-tors loads and the grid are represented using mathematical approximationsEach bus has a complex voltage Vi with respect to a reference and the phys-ical characteristics of branches and shunt elements can be described usingthe admittance matrix A generic bus can be denoted with the notation infigure (31)

Figure 31 A generic representation of a bus i with multiple connection tothe grid

Where S = P + jQ denotes the complex power and Y shi is the shunt ele-ment for the bus in consideration The currents and reactive powers can berepresented using the same sign convention as the active power We talkabout injected power (or current) considering it positive if it is injected tothe gridThe grid is described using the admittance matrix Ybus

[Y ] =

Y 11 Y 1n

Y n1 Y nn

The definition of the admittance matrix depends on the physical character-istics of the grid in term of resistances and reactances

A generic branch connecting two nodes can be described with its pi-equivalentcircuit considering a linear and passive behaviour of the lines

ZL = (r + jωl)a (302)

Y s = (g + jωc)a

2(303)

in which the physical and electrical characteristics of the lines are describedusing the following coefficients

Figure 32 π-equivalent circuit for a generic branch

ndash a is the length of the line [m]

ndash ra represents the resistance of the line [Ω]

ndash ωla is the reactance of the line [H]

ndash ga is the conductance of the line [S]

ndash ωca is the susceptance of the line [S]

It is relevant to highlight that

This is a lumped-element model that approximates the physical phe-nomena occurring along the lines

These approximations are valid for not too long lines (See [17] for aclassification of line length and a discussion on when approximationscan be considered valid)

Typically the power losses in the insulating material can be neglectedresulting in a capacitive shunt component (that can also be neglectedfor short overhead lines)

Sometimes power losses can be neglected with respect to the activepower flowing through the system (eg in high voltage grids) Conse-quently in some cases the resistances are neglected

The two shunt equivalent elements are equal for lines consideringa symmetrical behaviour but they generally can be different (eg intransformers)

The bus admittance matrix can be constructed from the pi-equivalent circuitof the line using the current nodal balance

ndash The diagonal elements depend on all the admittances connected toa specific node In case of shunt elements on the bus they can beconsidered by adding its admittance value to the corresponding diag-onal term (this is possible thanks to the use of admittances instead of

impedances)

Y ii = Y shunti +sumj

(Y sij + Y Lij) (304)

ndash Off-diagonal elements consist of the admittances connecting the twocorresponding nodes (buses not connected result in a bus admittanceelement equal to 0)

Y ij = minusY Lij (305)

The main goal of these mathematical problems as previously stated is tofind a solution to the system equations using numerical methods to com-pute voltages currents and powers in each point of the network Voltagesare considered as independent decision variables while net currents are de-pendent variables that can be computed through the bus admittance matrixVoltages cause current flows in proportion to the corresponding admittanceI1

= [Y bus]

V 1V n

The use of a bus impedance matrix Zbus is also possible selecting net in-jected currents as independent variables and voltages as dependent vari-ables This approach is useful for faults analysis but unlike the bus ad-mittance matrix Zbus is not a sparse matrix and typically results in highcomputational demandThe bus admittance matrix can be divided into conductance matrix Gbus andsusceptance matrix Bbus This approach can be useful when writing the op-timization code using GAMS In our cases we will consider the values ofGbus and Bbus as constants avoiding therefore a computationally demand-ing extended general approach comprising active transmission componentssuch as PST (phase-shifting-transformers) tap-changing transformers orpower electronics-based FACTS (flexible alternating current transmissionsystems) These components as well as the status of switched reactors orcapacitors bring also complexity to the nature of the variables since someof them are discrete

31 The pu SystemWhen dealing with power systems it is usually convenient the use of perunit dimensionless quantities (pu) that represent a value of an electricalquantity with respect to a base quantity expressed with the SI

Xpu =X

As base quantities it is common to use

Sbase = a unique threeminus phase power for the systemVbase = line to line voltages

The other base values can be found

Ibase =Sbaseradic3Vbase

Zbase =Vbaseradic3Ibase

=Vbaseradic3 Sbaseradic

3Vbase

=V 2base

Ybase =1

Unless otherwise specified all the quantities will be expressed in pu Sev-eral advantages arise from this choice

3 andradic

3 factors typical in three-phase systems are not present whenusing pu formulas

Most transformers have a voltage ratio equal to 1 because the volt-ages at the two ends are both expressed in pu (11)

Magnitudes are quite close to 1 pu resulting in improved numericalstability [18]

32 Conventional Power FlowIn power system analysis the computation of active and reactive power flowsis fundamental It can be done through the definition of the complex power

P + jQ = S = V Ilowast

= V (Y V )lowast (321)

where denotes the complex conjugation Then we can write down the netcurrent at a generic bus i as

I i = Y i1V 1 + + Y iiV i + + Y inV n =sumj

Y ijV j (322)

Considering rectangular coordinates for the admittance matrix entries and

polar coordinates for complex bus voltages

Y ij = Gij + jBij (323)

V i = |Vi|angδi (324)

From Eulerrsquos formula

V i = |Vi|ejδi = |Vi|(cosδi + jsinδi) (325)

Then we can find the expression of active and reactive power flows usingthe so-called rsquobus injection modelrsquo from eqs (321) (323) (324) and(325)

Pi = Visumj

Vj [Gij cos(δi minus δj) +Bij sin(δi minus δj)] (326)

Qi = Visumj

Vj [Gij sin(δi minus δj)minusBij cos(δi minus δj)] (327)

Where the symbol Vi = |V i| denotes the magnitude of voltage at bus i Itis important to note that these powers are the net values resulting from thedual operation of generating supply and loading demand in a transit nodewithout any loads or generation the injected power is equal to zero

Si = SGi minus SLi (328)

Pi = PGi minus PLi (329)

Qi = QGi minus QLi (3210)

An alternative notation of the power flow equations is characterized by theselection of different coordinates for voltages and admittancesIn the previous case (eqs 326 and 327) voltages are expressed in po-lar coordinates (V i = Viangδi) while admittances in rectangular coordinates(Y ij = Gij + jBij) It can also be useful to use polar coordinates for bothvoltages and admittances (Y ij = Yijangθij)

Pi = Visumj

VjY ijcos(δi minus δj minus θij) (3211)

Qi = Visumj

VjY ijsin(δi minus δj minus θij) (3212)

The other representations are less useful in our context and can be foundin the literature (see [18])The main idea of the conventional power flow is to compute a deterministicsolution without specifying any objective functions [18] The loads are char-acterized using their powers that can be considered as constant even ifmost of the loads have voltage-dependent powers

33 Optimal Power FlowThe OPF formulation is an extension of the conventional power flow Tra-ditionally the first literature works are from Carpentier [19] Dommel andTinney [20] from lsquo60s In the first paper the power flow equations had beenincluded for the first time in the Economic Dispatch classical formulation interms of equality constraints According to Frank and Rebennack [18] plan-ning of power systems operation have different time scales from real-timesimulations to a planning horizon of months or years In this framework theOptimal Power Flow formulation can be used for simulations in almost everytime scaleThe main idea is to minimize an objective function that represents the costsassociated to the production of electrical energy seen as the summation ofthe costs in all generation nodes G

minsumiisinG

CG (331)

These costs are typically described using linear or convex functions (quadraticcost function) It is not unusual to choose different objective functions ac-cording to which quantity one may want to minimize Some examples aregiven at the end of this chapterThe constraints represent the power flow equations and the operational lim-its in terms of electrical quantities powers voltages currents and phaseangles

Pi = PG minus PL = Visumj

Vj [Gij cos(δi minus δj) + [Bij sin(δi minus δj)] (332)

Qi = QG minusQL = Visumj

Vj [Gij sin(δi minus δj)minus [Bij cos(δi minus δj)] (333)

PminGi le PGi le Pmax

Gi (334)

QminGi le QGi le Qmax

Gi (335)

V mini le Vi le V max

i (336)

δmini le δi le δmaxi (337)

Modern formulations include also line limits in terms of maximum current ormaximum apparent power The capability of the line has to be accuratelyassessed in order to prevent damages on electrical systems componentsbut also to avoid underestimation of the power that can flow that may resultin an inefficient operation of the grid The capability of a line depends on thevoltage level on stability limits and thermal limits (see figure 34)

ndash The voltage level is typically fixed under an operating point of view Asa general rule in order to increase the power transfer that can flow inlong lines voltage has to be increased This is one of the reasons thatbrought to the growing development of HVDC technologies

ndash Stability limits are extremely important they are referred to voltage andangle stability They are related to the generators and the dynamicsof the loads [21] When planning power systems reactive power com-pensation methods can be designed in order to increase the maximumpower that can flow in quite long lines

ndash Thermal limits are related to the conductor loss of strength and espe-cially the line sagging (they also avoid that lines protections activate)[17] Wind velocity external temperature and solar radiation are fac-tors that affect the current limits of a line that are consequently relatedto the thermal limits

In terms of equations the latter limit will be considered as a constant valueof maximum power that can flow without compromising the thermal limits ofthe lines The inequality constraint in the case of AC power flow studiesmay be considered in terms of maximum current that can flow

|Vi minus Vj| yij le IijMAX (338)

Or maximum apparent power

P 2ij + Q2

ij le S2ijMAX (339)

Optimal power flow studies that comprise a constraint for line limits flow havean increased computational time (between 2 and 20 times according to [22]and an increase in the objective solution (up to 25 according to [22]) de-pending on the network size and characteristics However congestion is animportant aspect to be considered when dealing with financial rights Theconsideration of lines limits is therefore essential in our caseFor our calculations we will consider the power limit (339) So the expres-sion for the power flowing in a transmission line is required as well as for

the reactive power For this purpose we consider the complex power flow-ing through a line as

Pij + Qij = Sij = V i Ilowastij (3310)

The current flowing through the line from bus i to bus j (and the shunt path)can be computed using Kirchhoffrsquos Current Law in the pi-equivalent circuit(32)

I ij =ysij

2V i + yLij(V i minus V j) (3311)

Computing the complex power

Sij = V iIlowastij = V i

(Ylowastsij

2Vlowasti + Y

lowastLij(V

lowasti minus V

lowastj)

= |V |2i(Ylowastsij

lowastLij

)minus |V i||V j|ang(δi minus δj)Y

lowastLij

Using Eulerrsquos Formula

Sij = |V i|2(Ylowastsij

lowastLij

)minus |V i||V j| (cos(δi minus δj) + j sin(δi minus δj)Y

lowastLij =

(3312)

= |V i|2(Ylowastsij

lowastLij

)minus [|V i||V j| cos(δi minus δj) + |V i||V j| j sin(δi minus δj)]Y

lowastLij

(3313)

We introduce now the series and shunt admittances and neglect the re-sistive behaviour of the insulating material (neglecting therefore the shuntconductance GSij)

Y Lij = Gij + jBij

YlowastLij = Gij minus jBij

YlowastSij

2= minusjBSij

From (3312) we can calculate the active and reactive power flowing through

the line computing the real and imaginary parts of the complex power

Pij = ltSij (3314)

Qij = =Sij (3315)

Consequently

Sij = |V i|2(minusjBSij

2+Gij minus jBij

)minus [|V i||V j| cos(δi minus δj) + |V i||V j| j sin(δi minus δj)] [Gij minus jBij]

And then we can finally find the expression for the calculation of active andreactive powers through a generic transmission line

Pij = |V i|2(Gij)minusGij|V i||V j| cos(δiminus δj)minusBij |V i||V j| sin(δiminus δj)] (3316)

Qij = minus|V i|2(BSij

)minusGij |V i||V j| sin(δiminus δj)] +Bij|V i||V j| cos(δiminus δj)

(3317)

At this point it can be relevant to calculate the power losses that occurthrough the line This can be done simply via summation of the power flow-ing from node i to j with the power from j to i [23]

PLOSSij =Pij + Pji =

|V i|2(Gij)minusGij|V i||V j| cos(δi minus δj)minusBij |V i||V j| sin(δi minus δj)]+|V j|2(Gji)minusGji|V j||V i| cos(δj minus δi)minusBji |V j||V i| sin(δj minus δi)]

Therefore using trigonometric proprieties we can consider sin(minusx) = minussin(x)and cos(minusx) = cos(x)

PLOSSij = Gij (|V i|2 + |V j|2)minus 2Gij |V i||V j| cos(δi minus δj) (3318)

We will not consider how the maximum value of current is calculated be-cause as previously stated it depends on many factors such as wind speedand temperature requiring solution of the non-linear conductor heat balancemodel An approximated way of calculating the maximum apparent powerflowing can be calculated considering the maximum current provided by the

manufacturer in standard environment conditions according to [23]

Smax = 3Vradic

3Imax (3319)

Where V represents the phase to phase voltage Formula (3319) is not inpuIt is important to highlight that the maximum power we are considering hereis related to thermal limits For long lines the limits are set by voltage dropand stability limitations of the system [21] Reactive power compensationis useful in this framework for increasing the stability limits allowing longerlines to be builtAlso the reactive power flow in a generic line can be computed as well

QLINEij = minus(BSij

)(|V i|2+ |V j|2)+2Bij |V i||V j| cos(δiminusδj) (3320)

For the reactive power we can make a few comments

ndash If the line needs reactive power for its operation QLINEij gt 0 whileQLINEij lt 0 in the opposite case

ndash The reactive power in a transmission system depends on the activepower flowing through the line with respect to the natural power (orSIL Surge Impedance Loading)

ndash As a general rule reactive power is generated by the line in low loadconditions while it is needed in high load conditions

ndash This depends on the capacitive and inductive behaviours of the linesif no reactive power is needed or produced then these two effectscompensate each other

ndash Voltage profile through the line can be considered as constant - orflat - at the SIL (if we consider the same voltage at the two ends ofthe line) it increases at the middle of the line in low power conditionswhile it decreases in high power conditions (see figure 33)

These aspects are important along with the effective reactive power flowthrough the line A transmission system can be considered approximatelyas a constant voltage system not suitable for reactive power transfer [21]The previous considerations are valid also when the voltage at the two endsof a generic line are not too far from their nominal values This is one of thereasons that bring to simplifications of the OPF formulation to a DC-OPF

Figure 33 Approximated voltage profiles in a transmission line with respectto the SIL

Figure 34 Loadability curve for a transmission line [17]

The inequality constraint (339) can be simplified as well as the other con-straints of this model to an approximated model called DC-OPF that will bediscussed in subsection (341)Finally a sensitivity factor can be introduced if we want to avoid comput-ing the voltage angles in order to calculate the power flowing through aline It is called Power Transfer Distribution Factor (PTDF) and it is equal tothe marginal increase of power in a line considering a marginal increase ofpower in a specified bus See [24] for a detailed description

34 ApplicationsIn the following discussion a short introduction about other models similarto the classical OPF formulation is listed

ndash Economic Dispatch and Security-constrained ED OPF was first intro-duced from the Economic Dispatch (ED) formulation that did not con-sider any flows and transmission constraints The objective functioncan be defined as in the OPF case and we only have the inequal-ity constraints for generation limits and an equality constraint that en-sures the supply of the loads from the generating units

sumi PGi = PL

The security-constraint extensions comprises also the contingenciesthat may occur during the power system operation Contingencies areusually related to N-1 security state that considers the possibility thatone component of the system (eg a line) can exhibit an outage

ndash Unit Commitment and Security-constrained UC this refers to a formu-lation that schedules the generators considering that they can also beturned off during a multi-period time scheme [18] Start-up and shutdown costs have to be considered in the objective function

ndash Optimal Reactive Power Flow and Reactive Power Planning the ob-jective function minimizes the losses in the network while consideringin the second case also the insertion of new reactive power devices

Finally the DC-OPF approximation will be introduced in the next subsectionFor a detailed discussion about those topics see [18]

341 DC-OPFOne of the most well-known and still widely used approximations of the fullAC-OPF formulation comes out from the following assumptions

ndash Resistances and therefore power losses are neglected based on thefact that transmission networks have low losses compared to the power

flowing through the lines and resistances are small if compared to thelines reactances

ndash Voltages are considered as constants at their nominal value Vi =1 pu In power systems they are typically bounded to their nomi-nal value with good approximation but this also means that reactivepower generation is needed to keep the voltage stable [18]

ndash All reactive powers are neglected because small if compared to activepower flows In a voltage-constant transmission system the reactivepower flow is related to the magnitude difference between the volt-ages that in the first approximation were assumed to be equal for allthe nodes

ndash The voltage angles differences between connected buses is quite smalltherefore cos (δi minus δj) asymp 1 and sin (δi minus δj) asymp (δi minus δj)

It is called DC-OPF because the formulation is similar in form to a dc circuit(figure 35)

Figure 35 The analogy for DC-OPF

These can be quite good approximations for power systems not excessivelystressed But it is not uncommon to see voltages under their nominal valuesand voltage angle differences might be non-negligible [18] Therefore sincein this work there is a focus on congestion management stressed systemssimulations might not be accurate using the DC-OPFIn many papers DC inaccuracies have been addressed using a power lossesincorporation [25] [26] Some of these approximations use a quadraticfunction to represent system losses that can be relaxed using a piecewise-defined function to obtain a simpler problem [27] and [28] give a compre-hensive discussion about this topic as well as convex relaxation methodsfor AC-OPF problems with storage integration

342 Other FormulationsOptimal Storage Scheduling is another approach that can be used In thiscase described in [29] simulated with wind production the profit of storagedevices can be exploited through the maximization of the following function

max (Rminus C) (341)

Where R represents the total revenue of providing energy to the loads andC are the costs of energy from the transmission network In this case thenetwork losses are also decreasedThe voltage profile enhancement is another example of optimization prob-lem seeking to minimize the variation of the voltage from 1 pu at the loadbuses [30]Multi-objective optimization is also a viable alternative when the objectivefunctions to be optimized are more than one

35 OPF and Dual Variables as PricesThe following section contains relevant links between power systems andelectricity markets An important distinction has to be made when dealingwith marginal prices in power systems Nodal pricing and zonal pricing arethe most common choices that are used worldwide

ndash Many countries (eg the European countries) use a zonal pricingscheme in which the concept of zone or area depends on two differ-ent approaches a price zone might correspond to a country or withinthe same country several price zones may be defined

ndash Several countries (US Singapore New Zealand Argentina and Chile[13]) adopt a nodal pricing scheme or Locational Marginal Pricing(LMP) method

The nodal marginal price is the cost for the supply of an extra unit of power(typically 1 MW) at a specific node This price reflects the cheapest solutionfor re-dispatching this extra unit of power to that specific node consider-ing transmission constraints and capabilities [3] Since this nodal price (orLocational Marginal Price) reflects also costs associated to the transmissionsystem operation it might be higher than the most expensive generator cost[31]As a general rule due to system losses and possible congestions the LMPor nodal price is different in all the nodes Only in the case of a losslessapproximation without congestion ongoing we can consider the price forelectrical energy to be equal in all the system buses

In this work we will deal with the nodal pricing scheme but some con-siderations might be extended also to zonal pricing This latter alternativeconsiders the fact that only the interconnections between the zones can becongested Sometimes a zone is a whole country that have such a highlymeshed grid that congestion might hardly occur [32] In a nodal pricingscheme costumers pay the energy according to the price of that specificlocation [33] [34] as said before accounting for losses and congestion sur-charges

LMPi = LMPE + LMPL + LMPC (351)

Where the three terms are referred respectively to energy losses and con-gestion marginal prices [31] [33]We present now a simple example to show the price differences in a simple2-bus system (figure 36) These considerations can be extended to morecomplex systems for LMP and other relevant topics about power systemeconomics see [3] Note that the lines losses are neglected in order to fo-cus on the energy economics of this problem

Figure 36 Two bus system example

Let us suppose that generator i has a marginal price of πi = 25 $MWh

for sup-plying electrical energy while generator j has a lower cost of πj = 20 $

The transmission line limit is set to 100MW while the load demands are atfirst both equal to 60 MW If we suppose that generator j has enough poweroutput to supply both loads then it will provide 120MW equally distributedto the two loads and the line power flow would be 60MW

The nodal price in this first example is equal for both buses and it cor-responds to the marginal price of generator j 20 $

MWh This is because the

merit order brings on the market the generators with the least marginal costs[31] [35]

If in a second case we consider the load at the i bus to be 120MW gen-erator j will provide a 100MW supply to load i while maintaining the 60MWload at bus j because line has a power limit of 100MW This is a simple case that brings to a marginal price difference between

Figure 37 Two bus non-congested system example

these two nodes caused by transmission congestion In this simple casethe calculation of the nodal prices is straightforward since an additionalmegawatt of load at bus i must come necessarily from generator i while atbus j from generator j

Figure 38 LMP in case of congestion in a two bus system

Considering a 3 bus system is a more complex problem because the pathof the power flow follows the physical rule of power systems according tolines admittances [31] In this perspective we can see how can be useful toimplement an OPF formulation with dual variables interpreted as prices be-cause in much more complex systems involving many buses and branchesthey reflect the marginal costs for supplying energy to that specific nodeAnother crucial aspect that arises after a congestion event is that a mer-chandising surplus (or congestion surplus in this case) results from the sys-tem operation This is caused by the higher price paid by the customerswith respect to the revenue gained by generators [3] This surplus can be

calculated as

Payments minus Revenues = (LMPi minus LMPj)Pji = (352)

= (25$

MWhminus 20

MWh)100MW = 500

In chapter 5 we introduce a hedging financial instrument named Finan-cial Transmission Right (FTR) that redistributes the congestion surplus col-lected by the system operator when the lines are congestedThe DC-OPF formulation previously introduced is popular when dealingwith locational marginal pricing However since in markets and nodal pricesschemes the accuracy of the result of simulations is important under theeconomical point of view approximations may bring to results that do notreflect precisely the physical operation of the system [36] AC-OPF is there-fore getting more attention in the framework of energy markets [24]

Chapter 4

Energy Storage

The changes brought by the energy transition will have huge impacts onthe whole electrical grid Distributed energy sources will lead to a radi-cal change in the flow direction in distribution grids and the transmissionlevel will also see relevant effects Aleatory power production from PV andwind plants will also bring to an increased uncertainty in energy supply Re-silience and flexibility are some of the main features that we are looking forfrom power grids qualities that traditional fuels-based plants can bring inefficiently in terms of voltage stability frequency regulation and flexibility inproduction In order to maintain the grid stable as it has been so far weneed to integrate new technologies and bring innovationIn this framework energy storage will be highly needed and will play a crit-ical role For this reason many ways of storing energy have been testeddeveloped and deployed in distribution networks and also for utility-scaletransmission grids An important role will also be played by the residential-scale and microgrid storage that will lead to an increasing level of self-consumption However in this work we will deal with utility-scale energystorage systems their role consideration and integration in transmissionsystemsThe benefits brought by storage systems can comprise different servicesfrom multi-period time shifting (also known as arbitrage) to short-term an-cillary services This multi-source income framework brings interesting in-sights in terms of return of investment However several regulatory difficul-ties arise when considering the benefits brought by storage devices to theelectrical system [7]The electrical grid is an efficient carrier for transporting energy but whenenergy is produced asynchronously from the required demand it has to bestored over a wide range of time from milliseconds in case of supercapac-itors to days months or seasonal energy balance in case of hydroelectricenergy storageStraightforward ways to store electrical energy directly are not present in na-ture apart from the energy in capacitors that is accumulated in an electricfield A conversion stage is therefore necessary for obtaining another form

4 Energy Storage

of energy and this brings to conversion losses and lack of flexibility

Figure 41 Technologies for energy storage

The services delivered by storage devices are very different according tothe type of technology considered Large-scale pumped hydro storage sys-tems mainly help in terms of energy time shifting while electro-chemicalstorage can help with voltage support and frequency regulation but have alimited role in a long-term energy time shifting As a general rule multi-ple services can be provided by a single storage plant In this work we willinvestigate some aspects of the integration of Battery Energy Storage Sys-tems (BESS) in power systems because of their potential benefit both interms of power (in a short-term scale) and of energy (load shifting or peakshaving) Nevertheless several features are also valid for the other types oftechnologies A categorization of storage technologies is provided in figure(41)As mentioned in the introduction storage can be considered also under thegrid planning point of view especially as an alternative to new expensivelines (at least for their deferral) This aspect along with the considera-tion of storage as a transmission asset have a crucial role in the modelintroduced in the following sections Transmission lines and storage deviceshave some similarities that can bring to a similar consideration in a marketand regulatory framework according to [1] In particular

ndash Storage moves electric power ahead in time while lines physicallycarry electric power

ndash Both have high investment costs and low operating expenses

ndash Both have constraints related to their power capabilities storage hasalso energy constraints

4 Energy Storage

Storage benefits can be categorized according to the placement location(see figure 42) that is a critical aspect when deploying storage devicesboth in transmission (eg [37]) or distribution grids (eg [38] [39]) Alongwith services related to the resilience of the grid and the flexibility providedit is also significant to highlight that it helps the decarbonization of the elec-tricity sector and the reduction of emission when coupled with a growingpenetration of green technologies In addition non-technical benefits canbe listed as storage can provide market-related flexibility We will further as-sess how a market participant can hedge against nodal price volatility usingfinancial instruments called Financial Storage Rights in section 5

Figure 42 Storage benefits according to the location in the supply chain [40]

41 Storage CharacteristicsDepending on the different devices or technology storage can exhibit differ-ent physical operating parameters Power and energy are key factors thatcan be compared using a Ragone Plot see figure (12) Energy densityand power density take into account also weight and volume aspects thatare crucial in some applications of storage The efficiency has to be takeninto account because both power losses and energy leakage coefficientscan affect the quality of the benefits Moreover technology-related aspectssuch as the response time [41] have to be taken into account especially for

4 Energy Storage

applications in the electricity sector that might need great peaks of power ina very short time or a huge amount of energy over timeAs we are extending our optimal power flow formulation to a system com-prising storage technologies we are interested in the constraints related toits operation

0 le pchit le pchMAXit (411)

pdchMAXit le pdchit le 0 (412)

EMINstorage le SOCit le EMAX

storage (413)

SOCit = SOCitminus1 + ηchpchit∆t +pdchitηdch

∆t (414)

Where pch and pdch denote the charging and discharging rates while theSOCit denotes the State of Charge of the device at bus i at the time stept Expression (411) limits the charging power to its maximum value andits minimum is set to zero because the discharging process is governedby another variable The inequality constraint (412) is therefore neededto limit the power also when discharging Inequality (413) represents themaximum and minimum energy that can be stored in the device Emin

storage

can also be set to zero but it is known from the literature that especiallyfor Li-ion-based technologies the Depth of Discharge (that is how much astorage device is discharged DOD = 100 minus SOC) as well as the maxi-mum State of Charge (SOC) affect the aging of the battery It is thereforeconvenient to consider both energy limits Equation (414) relates the multi-period SOC value as a function of the previous state of charge and thechargingdischarging rates Storage losses are modelled using efficienciesηch and ηdch while the leakage coefficient of the batteries can be consideredwith an efficiency coefficient that multiplies SOCitminus1 The presence of twodifferent efficiencies for charging and discharging powers justifies the uti-lization of two different variables representing the charging and dischargingprocessesThe state of charge can also be described by different equation consider-ing SOCit+1 but our approach is more convenient when using the GAMSenvironment Finally sometimes it is also required to set the initial and finalstate of charge

SOCi0 = SOCiEND (415)

where END indicates the final state of charge at the end of the simulationSome storage integration approaches consider also the fact that batteriescan provide reactive power to the grid A quadratic constraint is necessaryto consider the capability curve of the inverter that connects the device to

4 Energy Storage

the grid [38]

P 2it +Q2

it le S2MAXit (416)

In all those constraints the multi-period multi-bus approach has been adoptedconsidering i as the set of nodes and t the set of time steps

42 Technologies

421 Mechanical Energy Storage

So far most of the grid-scale storage is realized using pumped-storage hy-dro plants (in 2017 accounted about 97 of total storage [41]) It is a well-known technology that allows to store water in form of potential gravitationalenergy to an upper reservoir It can provide energy and start a storage modeoperation in a matter of seconds resulting in a quite good response time[42] Since the capital costs of such an investment are massive buildingnew pumped hydro-plants is a prohibitive task considering also the envi-ronmental aspects of such huge plants Only in Asia Pacific during the lastyears new hydroelectricity plants have been built while the situation in therest of the world is almost stable [2]Flywheel devices store kinetic rotational energy that is exchanged usingan electric motor as bidirectional converter The use of magnetic bearingsbrings high advantages in terms of friction reduction that is also lowered bythe vacuum enclosure in which the rotating parts are placed It is typicallyuseful for frequency regulation but there are also benefits in terms of activepower shifting [43]Compressed Air Storage Systems uses motors and compressors to storeair in high-pressure chambers that is released to produce electrical energyusing an air turbine [44]

422 Electrical and Magnetic Energy Storage

The use of capacitors and in particular ultrasuper capacitors is based ondirect storage of electrical energy in an electric field It is relevant when highpower requirements are necessary in very short periods However new ca-pacitors allow to store much more energy than traditional technologies butstill small compared to the other grid-connected devicesOn the other hand superconducting magnetic storage is based on the mag-netic field energy produced by a DC (direct current) in a superconductingcoil Criogenically cooling the conductor brings the resistance to a very lowvalue resulting in high overall efficiency Again the power quality enhance-

4 Energy Storage

ments are good but the costs are in this case a limiting factor [45]

423 Thermal Energy StorageElectrical energy can also be converted to thermal energy and if not di-rectly utilized for heating or cooling can be reconverted back into electricalpower within a large time-frame from hours to months or years [46] Theimplementation of heat pumps for different purposes such as ancillary ser-vices has received some attention [47] There are several possibilities whendealing with thermal storage

ndash Sensible heat energy is stored increasing or decreasing the temper-ature of a material without any phase changes

ndash Latent heat energy is stored or released when a material is changingphase at constant temperature

ndash Thermochemical energy in this case reversible chemical reactionscan require or emit heat and store energy changing the chemical bondsof the materials involved in the reaction [41]

424 Chemical Energy StorageThe conversion of electrical energy into chemical energy is possible thanksto the so-called water electrolysis process The operation of an electrolyserbrings to formation of hydrogen and oxygen from water The hydrogen canbe stored an then converted back to electricity using fuel cells or convertedto another type of gas or chemical (Power-to-Gas) [41]Water electrolysis can be obtained using different technologies polymerelectrolyte membranes (PEM) alkaline electrolysis cells (AEC) Researchesare also going towards solid oxide technology [48]Several researcher projects suggest also that hydrogen will help the widespreaduptake of renewable sources by seasonal storage of gas in the network [49]but it is still an emerging technology that so far may not be profitable dueto the costs and the regulatory framework [48]Nevertheless hydrogen-based technologies can potentially bring many ad-vantages in many fields helping the society lowering the carbon emissionsSee [50] for a comparison analysis between hydrogen and BESS

425 Electro-chemical Energy StorageCost indices such as the Levelized Cost of Electricity (LCOE) dropped dur-ing the last years for Battery Energy Storage Systems (BESS) PV and windinstallations [51] Since the cost for lithium-ion batteries (LIB) has seen an

4 Energy Storage

87 price drop in the last decade [52] and thanks to their maturity and goodenergy density as well as power density BESS are receiving attentions forbringing benefits to transmission systems distribution grids and also micro-gridsA list of battery technologies according to the chemical components is listed

ndash Lead-Acid Batteries it is historically the most used and cheapest so-lution therefore has seen a wide diffusion (eg in the automotive sec-tor) Due to the fact that the lead is in solid state the material leakage isvery limited However due to its heavy weight has a low energy den-sity but with quite good power density It has a limited role in electricalsystems but can be used when a high-power low amount of energyis required eg for Uninterruptible Power Supply (UPS) applications[53]

ndash Nickel-based batteries they present higher energy densities with re-spect to the lead-acid technology NiCd (nickel cadmium) has not greatrelevance nowadays but MiMeH (Nickel metal hydride) batteries havestill some applications in the industry and in the automotive world [53]

ndash ZEBRA and Sodium-Sulphur batteries they are molten-salt batteriesthat are characterized by very high temperature operating conditionsThey show good characteristics in terms of energy density and havebeen utilized for both arbitrage voltage regulation [54] and renewableenergy sources integration [55]

ndash Lithium-ion batteries they have the biggest market share and thanksto a reduction in prices it is the most used battery technology in manyapplications They are common in portable applications and electricvehicles but also for grid-scale applications such as time shifting orfrequency regulation [41]

ndash Flow batteries it is an early technology that is characterized by the factthat the reactants are outside of the battery and are stored in tanks Itis used for stationary applications because of the high volume requiredfor all its components

Together with power energy efficiency and costs many other factors arepractically important when designing a battery storage power station Thetemperatures of operation the cycle frequency the minimum and maximumstate of charge (SOC) are other aspects that affect the so-called aging ofthe battery pack resulting in a natural degradation of the performance [53]Due to the fact that batteries loose their original capacity as they age it canbe introduced the factor State of Health (SoH) of the battery it refers tothe number of Coulombs available at a given charging rate compared tothe original manufacturing conditions (Itrsquos about 100 when the battery isnew and decreases aging) Then the charging power is important when

4 Energy Storage

calculating aging consequences It is also known that the SoH has differentvalues if calculated with different charging rates for this purpose it is nec-essary to specify at which C-rate 1 is the SoH referred to [53]The SoH is an important factor when dealing with end-of-life applicationsof batteries that cannot provide an adequate performance in certain frame-works The most relevant example is the reuse of old batteries from electricvehicles to stationary applications there is an ongoing debate on how thisstorage devices can have a second life application and which difficultiesmay arise In general caused by the expected wide spread of an enormousamount of EVs the battery could arise problems related to the recycling af-ter their use (that depends also on the annual mileage of the vehicle 2 andhow V2G applications will affect battery aging [57]) This problem can turninto an opportunity for stationary applications but research on both reuseand recycle must bring efficient ideas on both these paradigms [56]

When dealing with Electric Vehicles the most relevant technology is theLi-ion Battery (LIB) thanks to its capability to provide good energy and powerdensities in quite small volumes but when considering grid storage some ofthe above-mentioned alternatives may compete in terms of suitability Thearticle [58] gives an overview on the competitors to LIB-based technologywhen dealing with grid-scale storage Weight and volume have less impor-tance for stationary applications if compared to the automotive world Tech-nologies that can bring a high amount of maximum energy will be needed toincrease the storage time-frame without increasing too much the expenseseg flow batteries use quite cheap materials and if compared to LIB have alower increasing cost when rising in volume [58]

43 Storage Role and AdvantagesIn the classification section it was clear that many technologies are availableand consolidated while some of them are undergoing a research processthat will hopefully bring relevant solutions for the future issues of electricalgrids Operational limits are also key factors when choosing the most appro-priate storage technologies there is no storage device that can be suitablein every circumstance Figure 43 shows how different services can be pro-vided in different time scalesParameters such as frequency and voltage are important in power systemsand must remain within their operating limits Frequency is a crucial pa-rameter and is strictly related to the supply-demand balance Moment bymoment loads change their behaviour and power supply must follow thesechanges It is however a change in the paradigm because so far the tra-ditional power plants could provide an efficient balance by controlling their

1The C-rate factor describes the power required for a full-charge of the battery in onehour eg if C minus rate = 2C the battery is charged in half an hour [53]

2A typical SoH for EVs at the end of their use is 07-08 [56]

4 Energy Storage

input power according to the needs In the framework of renewable energysources the grid must be reinforced with flexible devices that can help over-come these challenges

Figure 43 Power-Energy view of storage benefits [59]

If we analyze the different time scales

ndash In the short term it is not uncommon to see voltage dips caused byoutages or the insertion of high loads The reactive power that asstated in chapter 3 is strictly related to the voltage absolute valuehas to be supplied by devices such as synchronous compensatorsin order to offset the inductive effect of most loads Battery energystorage systems can also provide a reactive power regulation sincetheir connection to the grid is based on a DCAC converter (inverter)that can operate in 4 quadrants of the P-Q characteristics Frequencymust also be kept under control because it is strictly related to therotating speed of generators When a contingency a loss of a largepower plant or a steep change in the load occurs the electrical systemhave been traditionally providing the power required through the rotorinertia of the generators connected to the grid

ndash Since storage does not require a start-up time such as traditional gen-erators they can provide an operating reserve very quickly (withinsome milliseconds [60]) Power systems have been traditionally pro-viding spinning reserve thanks to the automatic speed controller of thesynchronous generators connected to the grid A decrease in the sys-tem frequency means also a decrease in the rotor speed The speedcontroller (governor) then acts providing more torque to the generator

4 Energy Storage

eg by supplying more fuel and the balance between power generatedand loads is then re-established This is called primary frequency reg-ulation [61] [62]

ndash The secondary regulation then brings the frequency to its nominalvalue (eg 50 Hz in Europe) in a longer time-frame through an au-tomatic centralized control

In all these time periods BESS and other storage devices can help the gridovercome temporal issues and transients Other benefits comprise linescongestions reduction back-up power in case of outages black start capa-bilities RES curtailment avoidance deferral of new lines For our purposesregardless on the typology of frequency regulation we can state that storagein general and Battery Storage Systems in particular can provide activepower support that is strictly related to the frequency regulation (see [63] fora discussion on both primary and secondary regulation using BESS) Wewill focus on what is commonly referred as load shifting peak shaving or ar-bitrage That is charging the battery (or the storage device in general) whenelectrical energy prices are cheap and discharge it when the price rise Weare reasonably considering high-load periods to be more expensive thanlow-load conditions

Several problems arise when trying to evaluate the benefits brought bystorage devices to the electrical grid We discussed above that they com-prise active and reactive power balance but also voltage and frequencyregulation Not all these services are efficiently priced and looking at stor-age devices as an investment it might happen that they could bring to non-profitable results In [7] a wide discussion about the non-technical issuesand barriers for storage deployment is summarized Starting from its con-sideration it is common to set storage as a generation asset that buy andsell energy according to nodal prices (arbitrage) In this work we will how-ever assess which benefits could bring the consideration of storage as atransmission asset that consists on leaving the operation task to the sys-tem operator that manages it according to its physical constraints in orderto maximize social welfare This consideration is facilitated by the fact thatstorage has some similarities with transmission lines that have previouslybeen outlined in this chapterHowever it is relevant to point out that the differences are important Whenlines are congested we will consider the prices for energy different in thenodes of the system (using nodal pricing) Storage instead arbitrages inthe same node but between two different time instants This will bring todifferent results when dealing with financial rights in chapters 5-7Regarding ancillary services and grid services it is important to provide awelcoming market for storage deployment and efficient consideration of itsbenefits that in the past were hardly valued or not valued at all [7]The contribution of this thesis is to try to help building a welcoming mar-ket framework under the arbitrage point of view (see figure 44) Several

4 Energy Storage

other contributions are essential to evaluate as much as possible the bene-fits brought by storage devices to electrical grids

Figure 44 Electrical Energy Storage (ESS) role for Peak shaving (a) andLoad Levelling (b) [64]

Chapter 5

Financial Rights

51 Grid Congestions

Nodal pricing scheme described in section (35) is strictly related to the con-cept of lines congestion It has been explained how nodal prices can befound in simple 2 or 3 bus systems but a mathematical model is neededwhen dealing with real systems comprising hundreds or thousands of buses(and an average of 3 lines connected to each bus)One of the main results from such simulations is the cost of energy at eachbus also referred as LMP Unpredictability in spot market can bring to veryhigh costs and may encourage the consumers to find a way to hedge fromprice volatilityAn approach presented in many works comprises the use of financial rightswhat have already been adopted by some Independent System Operators(ISOs) is the concept of Financial Transmission Rights (eg PJM Intercon-nection LLC [65]) It is also possible when zonal pricing scheme is appliedconsidering only the lines interconnecting the different zones An exampleis Europersquos Forward Capacity Allocation (FCA) regulation that schedules theoperation of cross-zonal interconnections and capacities in a forward marketframework using long-term transmission rights (LTTRs) [66] [67] Howeverwe will consider a nodal pricing scheme with different LMPs for each bus ofthe system

A first approach can be the definition of Physical Transmission Rights(PTRs) They are bought by suppliers and consumers at auctions to havethe right to physically use a transmission line for supplying or receivingpower This approach however has the problem that the flow follows thephysical rules of the power system and not the intended path because ofparallel flows [68] Also a market participant might want to buy physicalrights and not use them only for increasing their profits A use them or losethem condition can be binded to PTRs to avoid these circumstances [3]A transmission right provides to its owner the property of a certain amount of

5 Financial Rights

transmission capacity giving the permission to use it Furthermore insteadof considering this physical approach Financial Transmission Rights (FTRs)can be introduced their owners receive a profit that is proportional to theamount of FTR purchased and to the price for each FTR This allows to theirholder to gain a revenue in case of congestions The value of these trans-mission rights is therefore zero if the system has no congestion eventsFinancial Transmission Rights can be introduced using two different ap-proaches

ndash Point-to-Point Transmission Rights they are defined between any twonodes of the grid and the value of the right is the difference betweenthe two nodal prices in (eg in $

MWh) times the quantity of rights bought

[69] The rights are assumed to be simultaneously possible for thegrid and their value must be less than the merchandising surplus of thesystem operator under congestion (also known as congestion surplus)[70]

ndash Flowgate or Flow-based Transmission Rights they are defined usingthe definition of a flowgate that can be described as a line that canbe congested Since they are related to a specific line their feasibilityis more related to the physical capacity of the line Therefore it can becalculated using the dual variable of the capacity constraint of the line[3]

Other approaches are compared in [68] Also simultaneous markets forboth point-to-point and flowgate rights have been proposed [69]Since we are more interested in the flowgate approach for the definition offinancial storage rights we will consider only this type of instruments eventhough many system operators chose the other approachThen we can extend the example introduced in subsection (35) for practi-cally understanding the value of FTRLet us consider the congested case of fig 38Here the price difference is quite low but in some cases it can take high

valuesThe consumer PiL may want to pay for hedging instruments together withenergy prices That is paying for Financial Transmission Rights and gain arevenue that depends on the congesting conditionsAssuming that the generators revenues and the consumers expenses arebased on the nodal price we decouple the production and consumption lo-cations [3] We can therefore find the two costs associated to the productionat generators i and j

ProductionCosti = πi middot PiG = 25$

MWhmiddot 20MW = 500

h(511)

5 Financial Rights

ProductionCostj = πj middot PjG = 20$

h(512)

While the total cost for the consumers is

ConsumtpionCosti = πi middot PiG = 25$

h(513)

ConsumptionCostj = πj middot PjG = 20$

h(514)

If we calculate the difference between the total costs for the consumers andthe total revenues of the generators [3]

(ConsumptionCosti + ConsumptionCostj)minus(ProductionCosti + ProductionCostj) =

= (πi minus πj)PijMAX = 4200minus 3700 = 500$

We obtained the value of the merchandising surplus (or congestion surplus)It can be profitable (or hedgeable) now for consumer at node i to recovertheir additional costs using Financial Transmission Rights and own the prof-its from the congestion between nodes j and i The system operator canthen decide to sell these financial instruments to market participants andreallocate the surplus coming from the system operation Since the totalsurplus is 500 $

h it is not profitable for consumer at bus i to spend more than

this amount The maximum price that brings to a zero-hedge position istherefore

5 Financial Rights

FTRjiMAX = πi minus πj = 5$

MWh(515)

It is important to highlight that

ndash If consumer at bus i would be able to pay 0 $MWh

for the FTRs he wouldbe perfectly hedged It is equivalent to buying the energy according tothe production location

ndash What a market participant can purchase is a portion of the total amountof FTRs available

ndash This two bus example can explain the basics around Flowgate Trans-mission Rights while it does not comprehensively outline the wholeframework also comprising the point-to-point approach

ndash Realistic situations are much more complex and a mathematical modelis necessary to calculate the nodal prices and congestion surplus (thiswill be done in chapter 6 using dual variables or shadow prices)

52 Storage Congestions

We already discussed about storage integration into electricity markets andthe possibility of avoiding nodal price transactions considering them astransmission assets [1] Extending what we introduced in the previous sec-tion storage can be used to protect against the unpredictability of nodalinter-temporal prices In this case we can consider or not the presence oftransmission congestion because storage provides a hedge against tem-poral price fluctuationsThere are two different approaches for Financial Storage Rights definition[1] and [71] Both of them propose the joint sale of both financial storageand transmission rightsIn the first model [1] similarly as with FTRs financial storage rights lead to again only if they are used during a storage congestion Furthermore stor-age rights cannot be negative while transmission rights can in some casestake negative value bringing to an obligation for its holderUnderstanding storage congestion might not be as intuitive as the trans-mission lines congestion is Defining then this situation mathematically wesuppose that without the presence of storage the power demand has thefollowing trend

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T2)

Generators and loads are perfectly balanced in terms of power With theintroduction of storage letrsquos suppose that the demand will increase by Cduring the first period (02) in order to charge the storage device Then itwill be discharged during the second period (24) in order to supply part ofthe load and avoid for example congestion in transmission lines

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

We can see that a storage congestion is ongoing since the device is ex-ploited to the fullest capacity This situation occurs as soon as [1]

C ltP (T2)minus P (T1)

2(521)

In case the difference between the powers required in these periods divided

5 Financial Rights

by 2 equals the value of the power capacity the storage unit would perfectlysmooth the demand curve

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

If the storage has a capacity higher than the above mentioned value itshould not be congested in our example but it will use just a part of its totalcapacity In case of congestion the system operator would earn a revenuethat is the exact amount that would be paid if the storage device acquiredand sold power at nodal prices [1] Merchandising surpluses are then re-distributed to financial storage rights owners Reasonably this occurs if theprice in the second period is higher than the first one This situation mightencourage consumers to protect against price unpredictability and financialstorage rights can hedge them in case of a congestion event Since theyare defined as financial rights do not affect the physical operation of thesystemA multi-period Optimal Power Flow formulation is presented in order to eval-uate the effects of the storage presence Moreover a lossless DC powerflow is an approximation widely used in power systems However DC PFdoes not consider voltage limits and stability and can exhibit significant in-accuracy for heavily loaded systems [18] Therefore an AC Power Flow canbe computed in order to overcome these issues and better evaluate storageeffects within electric grids Yet this solution is difficult for the presence ofnon-convex constraints (voltages) and non-linear constraints (real and reac-tive powers) [72] In order to incorporate energy storage technologies withinmarkets a multi-period OPF is necessary due to the time-coupled behaviourof power system dynamic components

While considering storage as a generation asset would bring the owner abenefit related to arbitrage activities the system operator could manage

5 Financial Rights

storage bringing to an increased overall benefit to the system Reasonablywhen seeking to maximize social welfare the overall costs for producing anddispatching energy would be lower than in other cases Also if a detailedconsideration of ancillary services would be available storage would resultin much more cost-effective investments Under the system operator pointof view the approaches in the literature propose that an auction would bethe best way to allocate financial rights to potential buyers Also both [71]and [73] state that this auction methodology is revenue adequate that isthe budget surplus is greater than the overall value of financial rights Thisconditions ensures that the system operator does not incur in a financialdebtHaving assessed the advantages for hedging consumers and the positionof the system operator we must evaluate if the storage ownerrsquos perspectiveis profitable Instead of gaining from arbitrage transactions at nodal pricesbring to a revenue that is redistributed by the system operator to financialstorage rights buyers that have to pay for those financial instruments If theFSR buyer partially hedges then a portion of the arbitrage revenue remainsto the system operator Then the storage owner receives rate paymentfrom the system operator from the sale of FSRs and thanks to the overallcost benefits both in terms of diminished objective value that for ancillaryservices provision to the grid It is clear that an accurate benefits assess-ment is necessary for storage integrationAs an alternative to this method based on the definition of dual variables forstorage congestion [71] proposes an approach that calculates FSRs usingnodal prices at the different time steps

Chapter 6

Financial Storage RightsCalculation

61 DC-OPFThe definition of Financial Storage Rights (FSR) divided into Power Ca-pacity Rights and Energy Capacity Rights comes from the Lagrangian dualproblem of a multi-period Optimal Power Flow linked by storage with lin-earized approximation This model had been introduced in [1] and will beextended to a full AC approach in the next section The following mathemat-ical model represents the DC Optimal Power Flow formulation in a generictransmission system

minsumit

fit(pit) (611)

st pit = pchit + pdchit +sumk

bik (δi minus δk)t (612)

pMINit le pit le pMAX

it (613)

bik (δi minus δk)t le pMAXik (614)

storage (617)

∆t (618)

Where i and t denote the set of nodes and time periods From the OPFformulation the Lagrangian dual problem can be expressed using dual mul-tipliers λi for each constraint Since some variables are limited with lower

6 Financial Storage Rights Calculation

and upper bounds two dual variables have to be assigned

pit = pchit + pdchit +sumk

bik (δi minus δk)t larrrarr λ(1)it

it larrrarr λ(2L)it λ

(2U)it

bik (δi minus δk)t le pMAXik larrrarr λ

(3)ikt

0 le pchit le pchMAXit larrrarr λ

(4L)it λ

pdchMAXit le pdchit le 0 larrrarr λ

(5)it λ

(5U)it

storage larrrarr λ(6L)it λ

∆t larrrarr λ(7)it

SOCi0 = 0 larrrarr λ(7)i0

The Lagrangian for this optimization problem can be constructed from theobjective function and the constraints

L(pi δi δk pchi p

dchi SOCi λi)t =

(fit(pit))

+ λ(1)it (pchit + pdchit +

(bik(δit minus δkt))minus pit)

+ λ(2L)it (pMIN

it minus pit) + λ(2)it (pit minus pMAX

+ λ(3)ikt(bik(δit minus δkt)minus p

MAXik )

+ λ(4L)it (minuspchit) + λ

(4)it (pchit minus p

chMAXit )

+ λ(5)it (pdchMAX

it minus pdchit ) + λ(5U)it (pdchit )

+ λ(6L)it (EMIN

i minus SOCit) + λ(6)it (SOCit minus EMAX

+ λ(7)it (SOCitminus1 + ηchpchit∆t+

pdchitηdch

∆tminus SOCit) (619)

Setting the derivatives of the Lagrangian with respect to the primal variablesto zero we get an optimal solution

partpit=partOF

partpitminus λ(1)it minus λ

(2L)it + λ

(2U)it = 0 (6110)

partδit= λ

(bik) + λ(3)ikt

(bik) = 0 (6111)

partlkpartδkt

= λ(1)kt

(bik) + λ(3)kit

(bik) = 0 (6112)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6113)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6114)

partSOCit= minusλ(6L)it + λ

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6115)

The solution satisfies also the complementary slackness

λ(3)ikt(bik(δit minus δkt)minus p

MAXik ) = 0 (6116)

λ(4)it (pchit minus p

chMAXit ) = 0 (6117)

λ(5)it (pdchMAX

it minus pdchit ) = 0 (6118)

λ(6)it (SOCit minus EMAX

i ) = 0 (6119)

Note that only the conditions related to the lines limits and storage energyand power constraints have been written down Assuming that each in-equality constraint is binding it behaves like an equality constraint and thecorresponding dual variable is positive and equal to the shadow price of thatconstraint Considering all the dual variables as non-zero might be physi-cally unfeasible but our purpose is to find an expression between the dualvariables In case some of the inequality constraints will not be binding inour simulations we will consider the corresponding dual variable as zeroand it will not affect our calculationFurthermore multiplying λ

(1)i and (612) the merchandising surplus is ob-

tained because it is like computing the net surplus between injections ofpayments and withdrawals

λ(1)it (pchit + pdchit︸︷︷︸

Storage term

bik (δi minus δk)t︸︷︷︸Flowgate term

minusSurplus term︷︸︸︷

pit ) = 0 (6120)

From (6111) (and similarly for (6112)) follows that

partδit= (λ

(1)it + λ

(3)ikt)

(bik) = 0 (6121)

Consequently from (6121) and the complementary slackness condition(6116) we set the power flowing through a generic line at its maximum ca-pacity pMAX

ik and then we can therefore replace the Flowgate termsumit

λ(1)it

bik (δi minus δk)t minusrarr minussumikt

λ(3)iktp

MAXik (6122)

Since we are considering the possibility that storage has a congestion wecan assume the corresponding inequality constraint as binding For this rea-son

pchit = pchMAXit (6123)

pdchit = pdchMAXit (6124)

SOCit = EMAXi (6125)

For our purposes in (6113) (6114) and (6115) we can consider only thelambdas relevant to our constraints since in a double inequality only one ofthe limits can be binding

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6126)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6127)

minusλ(6L)it + λ

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6128)

We can therefore replace the the storage term in (6120)sumit

λ(1)it (pchit + pdchit ) =

λ(1)it (pchit) +

λ(1)it (pdchit ) (6129)

From (6126) and (6127)sumit

λ(1)it (pchit) =

(minusλ(4)it minus ηchλ(7)it )(pchit) (6130)

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

ηdch)(pdchit ) (6131)

And therefore

(minusλ(4)it minus ηchλ(7)it )(pchit) +

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

(minusλ(4)it (pchit) + λ(5)it (pdchit ))minus

(ηchλ(7)it (pchit)minus

λ(7)it

ηdch(pdchit ))

(6132)

This last term is equal to

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6133)

That can be substituted using (6125) and (618) with

minussumit

λ(7)it (EMAX

i minus SOCitminus1) (6134)

From (6128)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

MAXi minus SOCitminus1) (6135)

Writing down the calculation

minussumit

(minusλ(7)itminus1EMAXi + λ

(7)itminus1SOCitminus1 + λ

(6)it E

MAXi + λ

(6)it SOCitminus1) (6136)

Using the following assumption

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6137)

According to (6128)

minussumit

(minusλ(7)itminus1EMAX

i +λ(7)itminus1SOCitminus1 + λ

(6)it E

MAXi +

λ(6)it SOCitminus1) (6138)

And therefore

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6139)

Consequently from (6132)sumit

(minusλ(4)it (pchit) + λ(5)it (pdchit )minus λ(6)it EMAX

i ) (6140)

And finally from (6122) and (6140) we can replace the terms in (6120)

SOrsquos Budget Surplus︷︸︸︷minussumit

λ(1)it (pit)minus

λ(3)iktp

MAXik︸︷︷︸

+sumit

(minusλ(4)it pchit + λ(5)it p

dchit︸︷︷︸

minusλ(6)it EMAXi︸︷︷︸

(6141)

lowast FTR = Flowgate Transmission Rights

lowast PCR = Power Capacity Rights

lowast ECR = Energy Capacity Rights

lowast The sum of PCR and ECR is the definition of Financial Storage Rights

62 AC-OPFA similar approach can be used starting from the following mathematicalmodel that represents the Optimal Power Flow formulation in a generictransmission system with a full AC representation

minsumit

fit(pit) (621)

st pit = pchit + pdchit + Vitsumk

Vkt[gikcos (δi minus δk)t + biksin (δi minus δk)t] (622)

qit = Visumk

Vk[giksin (δi minus δk)t minus bikcos (δi minus δk)t] (623)

it (624)

qMINit le qit le qMAX

it (625)

P 2ik + Q2

ik le S2ikMAX (626)

V MINit le Vit le V MAX

it (627)

δMINit le δit le δMAX

it (628)

storage (6211)

∆t (6212)

Pik = V 2i (gik)minus gik|Vi||Vk| cos(δi minus δk)minus bik |Vi||Vk| sin(δi minus δk)] (6213)

Qik = minusV 2i

(bSik2

)minus gik |Vi||Vk| sin(δi minus δk)] + bik|Vi||Vk| cos(δi minus δk)

(6214)

However since the consideration of a quadratic constraint for the maximumpower brings to a complex structure of the financial storage rights formula-tion we will consider the maximum power as a maximum active power Amargin can be taken from that value in order to compensate possible devia-tions due to the reactive power flow through the line Therefore

gikV2it minus VitVkt[gikcos (δi minus δk)t + biksin (δi minus δk)t] le pMAX

ik (6215)

From the OPF formulation the Lagrangian dual problem can be expressedusing dual multipliers λi for each constraint

pit = pchit + pdchit + Vitsumk

Vkt[gikcos (δi minus δk)t + biksin (δi minus δk)t] larrrarr λ(1)it

qit = Visumk

Vk[giksin (δi minus δk)t minus bikcos (δi minus δk)t] larrrarr λ(8)it

(2U)it

(9U)it

(10U)it

(11U)it

ik larrrarr λ(3)ikt

(4L)it λ

(5)it λ

(5U)it

l(pi δi δk pchi p

dchi SOCi qi Vi λi)t =

(fit(pit))

+ λ(1)it (pchit + pdchit + Vit

Vkt[gikcos (δi minus δk)t + biksin (δi minus δk)t]minus pit)

+ λ(2L)it (pMIN

it minus pit) + λ(2U)it (pit minus pMAX

+ λ(3)ikt(gikV

2it minus VitVkt[gikcos (δi minus δk)t + biksin (δi minus δk)t]minus p

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

+ λ(7)it (SOCit + ηchpchit∆t+

pdchitηdch

∆tminus SOCit+1)

+ λ(8)it (Vi

Vk[giksin (δi minus δk)t minus bikcos (δi minus δk)t]minus qit)

+ λ(9L)it (qMIN

it minus qit) + λ(9U)it (qit minus qMAX

+ λ(10L)it (V MIN

it minus Vit) + λ(10U)it (Vit minus V MAX

+ λ(11L)it (δMIN

it minus δit) + λ(11U)it (δit minus δMAX

(6216)

partpit=partOF

(2L)it + λ

(2U)it = 0 (6217)

partδit= λ

(1)it (Vit

Vkt[minusgiksin (δi minus δk)t + bikcos (δi minus δk)t])

+ λ(3)ikt(VitVkt[giksin (δi minus δk)t minus bikcos (δi minus δk)t])

+ λ(8)ikt(Vit

Vkt[gikcos (δi minus δk)t + biksin (δi minus δk)t])

minus λ(11L)it + λ(11U)it = 0 (6218)

partδkt= λ

(1)kt(Vkt

Vit[giksin (δk minus δi)t minus bikcos (δk minus δi)t])

+ λ(3)kit(VitVkt[minusgiksin (δk minus δi)t + bikcos (δk minus δi)t])

+ λ(8)kt(Vkt

Vit[minusgikcos (δk minus δi)t minus biksin (δk minus δi)t])

minus λ(11L)kt + λ(11U)kt = 0 (6219)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6220)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6221)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6222)

The solution satisfies also the complementary slackness conditions

λ(3)ikt(gikV

2itminusVitVkt[gikcos (δi minus δk)t + biksin (δi minus δk)t]minus p

MAXik ) = 0 (6223)

chMAXit ) = 0 (6224)

λ(5)it (pdchMAX

i ) = 0 (6226)

Note that only the conditions related to the lines limits and storage energyand power constraints have been written down Assuming that each in-equality constraint is binding it behaves like an equality constraint and thecorresponding dual variable is positive and equal to the shadow price ofthat constraint Equations (6220) to (6222) and (6224) to (6226) didnot change from the previous DC approximation since the storage con-straints are not influenced by the linearized approximation Multiplying λ

and (622) gives the merchandising surplus

Storage term

+Vitsumk

Vkt[gikcos (δi minus δk)t + biksin (δi minus δk)t]︸︷︷︸Flowgate term

pit ) = 0

(6227)

It is reasonable that the flowgate term has now a different value since weare considering effects that have been neglected before In particular λ(11L)it

and λ(11U)it are terms referred to voltage angles that are limited due to sta-

bility reasons However because we are considering in our approach the

flowgate to be limited according to the thermal limits of the lines we canfind a similar expression for FTRs and FSRs considering the operation ofthe system far from these two stability limitsAlso the following terms

(Vitsumk

Vkt[gikcos (δi minus δk)t + biksin (δi minus δk)t])(λ(8)it ) (6228)

(Vktsumi

Vit[gikcos (δk minus δi)t + biksin (δk minus δi)t])(λ(8)kt) (6229)

can be considered as negligible Indeed λ(8)it is the term reflecting the dualvariable of the net reactive power balance at each bus The active powercounterpart is described by λ(1)it that reflects the costs of the active energyand re-dispatch (as already discussed in chapter 3) Simulations show re-sults for λ(8)it very close to zero This term can be considered as reflectingthe costs for additional reactive power requested to the system Howeverwe are not considering any reactive power costs because the transmissiongrid is not built for providing reactive power to the consumers We will there-fore neglect dual variables associated to the reactive power balance λ(8) aswell as the stability-associated dual variables λ(11L) and λ(11U)

partδit= λ

(1)it (Vit

+ λ(3)ikt(VitVkt[giksin (δi minus δk)t minus bikcos (δi minus δk)t]) = 0 (6230)

partδkt= λ

(1)kt(Vkt

Vit[giksin (δi minus δk)t minus bikcos (δi minus δk)t])

+ λ(3)kit(VitVkt[minusgiksin (δi minus δk)t + bikcos (δi minus δk)t]) = 0

Consequently from the complementary slackness condition (6223) we setthe power flowing through a generic line at its maximum capacity pMAX

gikV2it minus VitVkt[gikcos (δi minus δk)t + biksin (δi minus δk)t] = pMAX

ik (6231)

and then we can therefore replace the Flowgate term using equation (6230)sumit

λ(1)it

(VitVkt[gikcos (δi minus δk)t + biksin (δi minus δk)t]) (6232)

with sumit

λ(3)iktp

MAXij minus

λ(1)it gikV

2it (6233)

It is clear how the solution is different from the DC case since nodal priceshave a losses-related component and cannot bring to the same results asin that case Furthermore the final expression will have a FTR componentbut also a loss coefficient λ(1)it gikV

Since we are considering the possibility that storage has a congestion wecan assume the corresponding inequality constraint as binding for this rea-son

For our purposes in (6220) (6221) and (6222) we can consider only thelambdas relevant to our constraints since in a double inequality only one ofthe limits is binding

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6237)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6238)

minusλ(6L)it + λ(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6239)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6240)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6243)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6244)

From (6236) and (6212)

minussumit

λ(7)it (EMAX

From (6239)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6248)

We can neglect the following terms according to (6239)

minussumit

(6)it E

MAXi +

Consequently

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6250)

Then from (6243)sumit

i ) (6251)

λ(1)it (pit)minus

λ(3)iktp

MAXij︸︷︷︸

FTR NEW

+sumit

dchit︸︷︷︸

LOSS︷︸︸︷sumit

λ(1)it gikV

2it = 0

(6252)

lowast FTR NEW = Flowgate Transmission Rights (neglecting λ(11L)it and λ(11U)it

and assuming λ(8)it = λ(8)kt )

lowast LOSS is the component that brings to a LMP difference due to lineslosses

It is highly important to mention that the quantities pMAXik pchit pchit and EMAX

are the quantities that bring to the whole financial rights value That valuehas to be redistributed to multiple participants since they are allowed to buya percentage of the financial right (transmission or storage)In this model we made the assumption that voltage angles are within theirlimits If we want to consider stability limits what is going to change in theprevious expression is the flowgate term Storage-related terms were intro-duced in the same way and bring to the same results

Chapter 7

Simulations

71 General ConsiderationsFor the purpose of testing and simulation the software GAMS (General Al-gebraic Modeling System) have been used and different algorithms havebeen chosen from GAMS solvers database The two different approacheschosen are relative to a DC approximation of the optimal power flow formu-lation and the AC-OPFThe main problem related to AC formulations is the computational complex-ity that in real systems can be a key factor because of the size of powersystems Therefore DC approximations are widely used both in power sys-tem analysis and in market-related simulationsFor the implementation of the codes it has been considered that

ndash Grid data as well as load and generation characteristics are loadedinto GAMS from an external Excel file In this way different grids canbe studied without changing the code

ndash The sets are defined for nodes branches time periods PV nodesand slack bus nodes with batteries

ndash Tap changing transformers and phase shifting transformers have notbeen considered in our model because they can increase heavily thecomplexity of the system see [18] for their integration in a OPF model

ndash The decision variables and parameters depend on the model In gen-eral the unknown variables depend on the choice of each bus asstated in chapter 3

ndash Ramp rates have also been neglected but they can be easily incorpo-rated ensuring that eg PGt+1 minus PGt le Pmax

rampup

ndash Network characteristics load data and generator limits have been takenfrom a 5-bus example Results for the AC-OPF are very close to the

7 Simulations

original model presented in [18] except for very small numerical er-rors Also lines limits are introduced for giving a more realistic studyof the grids as well as a computation of financial transmission rights

ndash the solvers used for the linear cases is CPLEX and for NLP prob-lems is CONOPT Most of the simulations were non-linear even for theDC-approximation because of the non-linear structure of the objectivefunction

The model starts with the definition of the physical characteristics of the gridin terms of resistances and reactances It is also possible to consider par-allel admittance both for lines features that in terms of shunt admittancesAfter building the bus susceptance matrix and in the case of AC-OPF alsothe bus conductance matrix the bus admittance matrix can be found Theadmittance angle matrix have not been computed because of the choice ofrectangular coordinates Gik and Bik Given the resistance and reactancethe series conductance and susceptance can be found through

G = ltY =R

R2 +X2(711)

B = =Y = minus X

R2 +X2(712)

Then the conductance and susceptance matrices are found using the rulesexplained in chapter 3 for diagonal and off-diagonal entriesFor having a specific overview of the storage advantages the State of Chargeat the beginning and at the end of the simulations are set to be equalSOCit0 = SOCitfinal

All quantities unless otherwise specified are expressed in per unit

72 DC - AC Results ComparisonIn the following tables all the input data for the simple test case can befound they are taken from an analogous example in [18] For this simplecase congestion is not considered but can simply be valued setting the ca-pacity of some lines to 06 pu or less (or increasing the load)

Table (71) shows generating limits both for active and reactive power rsquoLin-earrsquo and rsquoquadraticrsquo are the coefficient that multiply the power and the squaredpower considering a classical quadratic cost function Table (72) shows thevoltage limits for each bus considering the slack bus as the connection to

7 Simulations

Figure 71 5 bus system single line diagram

P_Gmin P_Gmax Q_Gmin Q_Gmax linear quadraticbus1 -20 20 -2 2 035bus2bus3 01 04 -02 03 02 04bus4 005 04 -02 02 03 05bus5

Table 71 Generators data [18]

V_min V_max g_shunt b_shunt P_Load Q_Loadbus1 1 1bus2 095 105 03bus3 095 105 005bus4 095 105 09 04bus5 095 105 0239 0129

Table 72 Bus and Load data [18]

R X gS2 bS2 Limitbus1 bus2 03 1bus1 bus3 0023 0145 004 1bus2 bus4 0006 0032 001 1bus3 bus4 002 026 1bus3 bus5 032 1bus4 bus5 05 1

Table 73 Lines data [18]

7 Simulations

the rsquomain gridrsquo The other columns represent the shunt elements and loaddataTable (73) lists the grid data in terms of resistances and reactances Herethe shunt elements represent the physical behaviour of the line rsquoLimitrsquo is themaximum power that can flow complying with the thermal limits of each lineThe DC approximation neglects the reactive power flows resistances andvoltage limits Therefore the corresponding columns are not introduced inthe modelResults show a good approximation in terms of phase angles and the volt-age value is not too far between the two cases (about 5-6) When com-puting active power differences can be quite high because they are com-puted using voltage values trigonometric functions and comprise the resis-tive part Therefore relative deviations on active power is more significantwhile the reactive power has a maximum error since it is neglected in theDC approximation see Table (76)

V [pu] Angle [degree]DC AC DC AC

bus1 1 1 0 0bus2 1 099 -8077537 -8435bus3 1 09795 -3585418 -3803835bus4 1 09776 -8939141 -9246847bus5 1 095 -8346616 -8802888

Table 74 DC-OPFAC-OPF results comparison Voltages and phase an-gles

P [pu] P_G [pu] Q [pu] Q_G [pu]DC AC DC AC DC AC DC AC

bus1 09015 094541 0901 0945 0 0131914 0 0132bus2 0 0 0 0 0 0 0 0bus3 01875 019421 0187 0194 0 002777 0 0028bus4 -085 -084309 005 0057 0 -02 0 02bus5 -0239 -0239 0 0 0 -0129 0 0

Table 75 DC-OPFAC-OPF results comparison Generated and net activeand reactive powers

It can be relevant now to consider an increasing load framework becausewe know that DC approximations suffer inaccuracy when the system isstressed and electrical quantities are closer to their operating limits Thiscan happen in real systems and is really relevant when computing financial

7 Simulations

Relative Error []Voltages up to 526

Voltage Angles up to 574Active Power up to 464

Generated Active Power up to 1228Reactive Power maximum

Generated Reactive Power maximum

Table 76 Relative error between DC-OPFAC-OPF

transmission rights since they take value only in the case of transmissioncongestionThe same models have been investigated with an increase of load (+40)and a multi-period increasing load (up to 100) While inaccuracies relativeto voltages and phase angles might be acceptable (up to 6-7) generatedand net active power errors increase more significantly Having consideredrelative errors it results that phase angles are even more accurate when thesystem is highly loaded (but still not congested)Finally when the load is increased too much it has been seen that the DC-OPF is still feasible in this 5 bus example while the AC-OPF is unfeasibleusing GAMS

In figure (72) an incremental load has been applied in a 10 time stepsframework The reactive power has been chosen as decreasing to accom-modate the increase in real power In the last time step the power at bus 3hits the operating limits of its generating unit resulting in a zero error whilethe power generated at bus 4 has a high error (sim 30) From t7 onwardscongestions on the lines play a key role in the results accuracy

It is more important for our purposes considering the errors in the nodalprices The computation of marginal values in general has been done us-ing the MARGINAL feature of GAMS In order to find the nodal prices (orshadow prices) we must calculate the marginal of the nodal balance equa-tion at each node Figure (73) shows how in percentage these nodal pricesdiffer when comparing the DC approximation with full AC-OPFLMPs as expected have a small deviation in low-load conditions This iscaused by the LMP factor due to losses that is small if compared to theLMP factor due to congestions (see equation (351)) As the load increasesleading to congestions in the grid the LMP increases in value and lacks inaccuracy LMP at bus 1 has no error since the slack bus is the cheapestgenerating supply and an increase in load would not lead to a re-dispatchcalculationLocational marginal prices as well as other marginal values are used for cal-culating financial transmission rights and financial storage rights (see equa-tion (6252)) Marginal values computation is often realized using DC-OPF

7 Simulations

Figure 72 Relative errors on phase angles and net active powers 10 timeperiods

in real cases and this might bring to non-negligible errors on LMPs calcula-tion [74]Computational complexity might still be a key factor when ignoring the useof AC-OPF for certain purposes Therefore it can be good to choose atrade-off between the two models to get closer in terms of results accuracyThis can be done by including real behaviours of variables into the DC-OPFor via relaxation methods of the AC-OPFSeveral ways have been introduced to improve the accuracy of the DC ap-proximation [75] proposes a loss compensation method and an α-matchingDC-PF formulation [27] assesses how line losses the reactive power inclu-sion in the apparent power and phase angle consideration improves theaccuracy while [76] focuses on the network capabilities and introduces areactive power flow integration into the DC-OPFOne simple known way to integrate losses in the DC formulation is replac-

7 Simulations

Figure 73 Relative errors on Locational Marginal Prices (LMPs)

ing the power flow balance equation [26] [77] with an equation consideringa power losses approximation The power flow in a line from nodes i to kcan be computed as

Pij = V 2i (Gij)minusGij|Vi||Vj| cos(δi minus δj)minusBij |Vi||Vj| sin(δi minus δj)] (721)

And the losses are (see chapter 3 for the calculation)

PLOSSij = Pij + Pji = Gij (V 2i + V 2

j )minus 2Gij |Vi||Vj| cos(δi minus δj) (722)

Neglecting the voltages level (that is set to 1 pu)

PLOSSij asymp 2 middotGij minus 2 middotGij cos(δi minus δj) = 2Gij(1minus cos(δi minus δj)) (723)

Using small angle approximations

cos(δi minus δj) sim 1minus (δi minus δj)2

2(724)

And therefore

1minus cos(δi minus δj) sim 1minus 1 +(δi minus δj)2

2(725)

PLOSSij asymp 2 middotGij(δi minus δj)2

2(726)

7 Simulations

And finally

PLOSSij asymp Gij(δi minus δj)2 (727)

We can now replace the balance power flow equation in the DC approxima-tion with

PG minus PL minussum

Bij(δi minus δj) = PLOSSij asymp1

2Gij(δi minus δj)2 (728)

The 12

factor is necessary for not including the losses of a single line in boththe nodal balance at bus i and j Furthermore [77] shows that this quadraticfunction can be approximated via piece-wise linearization while [26] showsan alternative approximation as a conic quadratic problemFinally [28] covers also the role of storage integration while computing powerlosses and together with [78] state that DC-approximations play a criticalrole while assessing storage siting and sizing problem This can lead toa sub-optimal energy storage systems integration decisions according to[28]

73 Storage RoleThe storage integration brings several advantages that have been discussedin the previous chapters As reactive power-related benefits are not eval-uated in these simulations storage devices are not allowed to exchangereactive power with the grid This can be done as with power balanceequations by adding charging and discharging terms to the reactive powerbalance equations In that case a quadratic inequality constraint has to limitthe apparent power provided by the inverterbattery

The similarities outlined in the previous chapters between lines and storagedevices bring those devices to a positive consideration within a congestionframework Transmission expansion planning are typically realized with thedeployment of new lines but as storage got cheaper and becoming wellregarded a deferral of new lines can be possible deploying large-scale stor-age systemsTypically energy prices are cheaper in low-load conditions and increase inpeak periods This increases the profit of storage devices and allows gridsto redistribute the load to a larger time-frame The following simulations rununder the increasing load conditions of the previous example show howelectrical quantities are affected by the presence of BESSA storage device is introduced at bus 5 since it is the only bus with a loadwithout a generating unit Several other choices are possible and would lead

7 Simulations

to different results However it is not relevant for our purposes the place-ment choices of the storage device

Figure 74 Phase angle at bus 5

The phase angle at bus 5 follows a more stable trend when consideringstorage (brown line) The negativity of the angles means that power is go-ing from the slack bus (δ1t = 0) to the other nodes because of the cheapercost for energyRegarding the generated powers figure (75) refer to the generated powerfrom the slack bus and the most expensive generator in the system Thestorage system is charging in the first periods and discharging when loadis higher (and then the costs are higher because of the quadratic objectivefunction) More power is required from the (cheap) slack bus in the first timeperiods in order to charge the storage device and less power is requiredfrom the (expensive) generator 4 that was asked to produce quite a lot ofpower at t9 and t10 when storage was not yet introduced and congestionswere increasing the costs of the systemHowever the single equations increased from 441 to 451 and the singlevariables from 441 to 471 10 equations for the update of the SOC and 3variables representing Pch Pdch and SOC This increase in variables andequations is linearly greater if considering more than one storage device

7 Simulations

Figure 75 Power generated from the slack bus and the most expensivegenerator

74 Financial Transmission Rights andFinancial Storage Rights

Storage characteristics have not been mentioned in the previous sectionsThe storage device introduced has a great storable energy and high powerrate limits In the 10-time-periods simulations it charged up to 1 pu and dis-charged back to the initial state of charge Having introduced a multi-periodmonotonic power it is reasonably charged a lot in the very first time stepsand follows a dual behaviour discharging when the load and the costs arevery high However the maximum charging and discharging powers are notreached and according to eq (6252) only the Energy Capacity Right hasvalue (for 3 time periods) If for example we can introduce a high-powerhigh-energy storage device in our system we might be uncertain about theutility of Financial Storage Rights as well as storage arbitrage possibilitiesIn other words if the overall storage capability of the grid allows the systemoperator to perfectly balance the supply levelling the load (figure 44) thenthe investments in storage would be uneconomic because of the small pricedifferences between the time periods investigated According to [79] theprice difference in a time framework acts as price signals for investors inorder to realize if investing in storage would be profitable or not If in the fu-ture there will be a great amount of storage devices in the grid the benefitswould be less concrete and the investments would have a lower return

Financial Storage Rights have a hedging role in electricity markets and

7 Simulations

allow system operator to redistribute the surplus resulting from storage con-gestion events If the capacity of storage devices will be very high thecongestion will hardly occur and the rights value would be zero just likethe case in which a highly meshed grid will hardly have frequent congestionevents due to multiple interconnections In that case the market partici-pants would have less interests in hedging against price volatility that hasbeen reduced by the presence of storage On the other hand the storageowner would not benefit from selling Financial Storage Rights and wouldonly earn rate-based payments from the system operator Therefore thereturn of investment would be exclusively related to these payments andthe usefulness of storage for the gridFor those reasons in these section the storage device has been set to anenergy capacity of 1 pu per hour charging and discharging rates at 02pu and -02 pu respectively while maintaining the efficiencies to the unityfor simplicity

The financial rights calculation comprise both flowgate and storage rightsbut for the sake of simplicity we consider them in separate cases We re-fer the reader to the papers previously cited for a comprehensive view ofsimultaneous rights allocations under simultaneous feasibility conditionsAs previously discussed it can be useful for a market participant to hedgeagainst spatio-temporal price volatilities buying FTRs and FSRs [1]

We begin with a simple example in which we show how using DC approxi-mation results in an exact match between merchandising surplus and finan-cial transmission rights value Summing up

ndash In case of a DC approximation nodal prices have no losses compo-nents and completely depend on the additional costs of energy andcongestion re-dispatch costs

ndash If no congestion occurs the LMP is the same at every node and equalsthe linear costs of the slack bus (035 k$

pu) that is the cheapest gener-

ating unit (no quadratic components have been considered so far)

ndash Figure (76) shows that increasing the load from t1 to t10 no conges-tions occur up to t7 Nodal prices difference at t8 t9 t10 is caused bycongestion

The line interested by the congestion event is branch 1-3 According toequation (6252) we can calculate the congestion surplus (CS) or mer-chandising surplus as sum

λ(1)it pit = MS (741)

7 Simulations

The total Financial Transmission Rights value (FTR) can be computed fromthe line shadow price (or dual variable)sum

λ(3)ijtpijMAX = FTR (742)

Figure 76 LMPs [ k$pu

] in incrementing load simulation

Figure 77 Financial Transmission Rights and congestion surplus match[ k$pu

As expected the FTR result in a zero value when there is no congestionwhile they perfectly match with the congestion surplus when congestion oc-curs

When computing the same simulation case with the AC approach it is clearthat the surplus cannot be equal to the FTR term due to a component thatwe called LOSS in the calculations in chapter 6 Trying to compute λ(1)gikV 2

gives an approximation on LMP component due to line losses Howeverthis approximation gave rough results in terms of accuracy Several differ-ent computation possibilities have been presented for calculating the losscomponent of locational marginal prices we refer the reader to [80] [81]and [82] for this topic Typically the loss component is found using penaltyfactors (or delivery factors) for each bus [82] For the sake of simplicity sinceour formulation is not directly referring to LMPs but relate shadow prices withpower injections using (6252) we can directly compute the LOSS compo-nent from the congestion surplus and financial transmission rights value

7 Simulations

Figure 78 Financial Transmission Rights congestion surplus and loss com-ponent in case of AC results [ k$

When comparing the two results tables we show that the approximationsbring to a non-negligible errorAs shown in figure (78) any loss component considered would not matchperfectly the expression (6252) in terms of congestion surplus and finan-cial transmission rights It has been demonstrated that under non-convexityconditions the revenue adequacy might not be obtained [80] Thus as a re-sult for our purposes the problem needs to be convex to ensure that thesystem operator does not incur in a financial deficit debt (even if quite low)Convex relaxation or losses incorporation methods are worth investigatingin the framework of locational marginal prices and financial rights

Having assessed the importance of precise mathematical models for thedescription of market-related FTRs and LMP calculations we introduce nowstorage in our simulations and see how financial storage rights can be com-puted As in the previous case the marginal feature in GAMS can providethe computation of dual variables at each time period We use here thesame notation as in chapter 6 and calculate Energy Capacity Rights (ECRs)and Power Capacity Rights (PCRs) Charging and discharging rates areconsidered smaller than in section 73 for showing the role of PCRs Mean-while a quadratic cost function is considered also for the slack busAs a first result the Financial Storage Rights computation match with thenodal inter-temporal transaction scheme In the previous chapter we dis-cussed about the possibility of implementing FSR either as a sequence ofnodal price transaction over the time periods [71] or using a constraint-based approach [1] In figure (710) we can see how Financial StorageRights can be computed through the summation over all the time periods ofthe different components of eq (6252)

ndash CS is the congestion surplus table calculated with nodal dual pricesand power net balance

ndash PCRch PCRdch and ECR are respectively the power capacity rightsfor charging discharging and energy capacity rights

7 Simulations

Figure 79 FSR computation first approach [1]

ndash The values in red boxes must not be considered when computing FSRbecause they refer to dual variables not relevant in the congestionevents (eg when the storage device has a SOC = 0 or pch = 0 orpdch = 0)

Financial Storage Rights total value can be computed now as

PCRch + PCRdch + ECR = minus003549minus 003725minus 019517 = minus02679k$(743)

Following now another approach by calculating the arbitraging sequence ofnodal transactions [71] sum

λ1setCtpsetCt (744)

Where setC denotes the set of nodes with batteries The following figureshows how the same result is computed with this second approach

Figure 710 FSR computation second approach [71]

The two results match for the value of Financial Storage Rights but thereis a remaining term necessary to compensate the full congestion surplusthat corresponds to the loss component and to inaccuracies due to the sim-plification of the branch limit inequality constraint from (626) to (6215)

Chapter 8

Conclusion

In this thesis after introducing the main topics around the integration ofstorage devices in future grids an AC-OPF model is presented FinancialTransmission Rights and Financial Storage Rights are calculated from theKKT conditions following a similar approach as in [1] but another method isalso presented using nodal price transactions [71] So far to the best of ourknowledge all the papers dealing with FSRs focus on a DC approximationcommonly used in many fields of power systems However it is shown thatas expected the approximations of DC models can yield imprecise resultsYet when dealing with pricing and markets results accuracy is an importantfactorIt has been highlighted that dealing with an AC-OPF could result in problemsif the structure of the problem is non-convex [80] The most reasonablemodel for accurate financial rights calculation seems to be a convex AC-OPF or an enhanced DC-OPFThis thesis is framed in the context of storage integration in transmissiongrids Financial rights could help investors decouple the storage operationfrom any wholesale market mechanisms leaving the system operator thetask of utilizing it to maximize social welfareSince storage technologies will be fundamental to guarantee a good level ofautonomy and flexibility a detailed consideration of their benefits is essentialfor ensuring an appropriate return of investment

Chapter 9

Future Work

91 Model Extensions

As we have seen several problems may occur when choosing a DC approx-imation of Optimal Power Flow or using a full-AC approachAlso coupled with the deployment of a huge amount of distributed gen-eration devices relatively small sized storage systems can help avoid con-gestion in grids where the configuration is radial or weakly meshed Further-more several other services can be supplied by storage systems in generalAncillary services can be provided by a variety of storage technologiesAdding such evaluations to the mathematical model would lead to a moreaccurate assessment of potential benefits Eg as discussed in chapter3 optimal reactive power flow seeks to minimize power transfer losses re-questing reactive power from storage devices If an increased penetrationof renewable energy sources is expected the merit order curve would leadto decreased overall prices because PV and wind technologies have verylow operating costs In such framework it may be reasonable to minimizenetwork losses instead of production costsThe incorporation of ancillary services can be a viable way However sinceAC-OPF is already computationally demanding adding new complexitiescould lead to an infeasible problem Probabilistic approaches can thereforesee a variety of applications due to the nature of such uncertain problemsFurthermore due to AC-OPF complexity it can be reasonable to get closerto its results without considering the full AC approach relaxation methodsfor the AC model or losses integration for the DC approximation can lead toquite good results with reasonable computational complexityConceptually the model previously presented can be applied to meshedtransmission grids However a similar approach can be used to extend theutilization of storage systems at the distribution level However pricing indistribution grids needs to follow the nodal scheme as in the transmissioncounterpartAn accurate and detailed Financial Storage Rights framework would need

9 Future Work

the incorporation of such instruments in auctions as shown in [73]Finally a more precise model with characteristics of storage devices thatalso depend on aging of the batteries would lead to more accurate resultswhen dealing with return on investment calculations

92 V2G

Road vehicles and automobiles are the most common means of transportin the world Since future power grids will be highly dependent on renew-able energy systems EVs contain a battery and can provide an affordablehigh-power density supply for this grid support this is named V2G (Vehicle-To-Grid) mode operationOnly a few countries have nowadays a discrete number of plug-in electricvehicles but in the near future there will be an exponential growth in salesthat could bring to a potential widespread of V2G systems Storage systemsspread throughout the territory will lead to a direct instantaneous exchangeof electric power with the grid improving its stability reliability and efficiencyAs a potential forecast millions of new EVs (Electric Vehicles) are expectedto be manufactured in the upcoming years (figures (91) and (92)) leadingto a non-negligible contribution brought by EVs connected to the grid con-sidering also that they are typically used just for a short time over the daySince the grid operates in AC and the battery is charged using DC a powerelectronic converter is required High-power charging systems are decreas-ing the time required to a full charge of the battery resulting in a high avail-ability for V2G purposes Since EVs have been considered as loads for thegrid in the past a new infrastructure is needed for the double purpose ofcharging the vehicle when needed (G2V operation mode) and dischargingEVs batteries in V2G operation mode This role is mainly covered by a bidi-rectional power electronic converter and other devices in order to optimizethe connection to the grid If connected to the grid EVs also offer other an-cillary advantages voltage and frequency regulation reactive power com-pensation active power regulation current harmonic filtering load balanc-ing and peak load shaving (rearranging power required by the loads over awide range of hours) V2G costs include batteries degradation power elec-tronics maintenance additional devices for grid optimal connection and forchargingdischarging the batteries according to the needs of the grid or theownerrsquos specific claimsDespite the huge negative impact brought by Covid-19 pandemic to the au-tomobile industry the share of hybrid and electric vehicles is growing allow-ing the transition to an electrical energy-based transportation sector

It is expected that the market share of EVs will keep growing and will be oneof the key technologies for power grid autonomous operation As we can

9 Future Work

Figure 91 New passenger car registrations by fuel type in the EuropeanUnion Q1 2021 [83]

see from data publicly available on ACEA website [83] the market share ishighly dependent on the country and policies play a critical role Concern-ing charging points most of the stations are located in a few countries butin the near future a widespread of both EVs and charging stations will bepossible Since power electronics is used for these applications accuratecontrol techniques are necessary for matching the requirements in terms ofvoltages and currents Many other details and issues must be discussedwhen introducing V2G mode operation power quality and connection to thedistribution grid SOC SOH and EVrsquos owner availability It is essential thatEVs widespread will be combined with a fewer fossil fuels-dependent soci-ety for reduced overall emissions There are many other issues related toEVs market (like batteries waste management or bigger demand for newmaterials in automotive sector) but there are several advantages introduc-ing V2G technology

Therefore some research projects have been established in order to betterassess the benefits brought by V2G mode operation to the grid One ofthe first pilot projects have been launched in September 2020 inside FCArsquosMirafiori industrial area in Turin Italy The main idea of these projects is tolsquoaggregatersquo a great number of vehicles in the same point of connection asflexibility resources that can provide services to the grid Along with thesepilot projects it is important to fairly recognize the benefits brought by V2G[84] Again the regulatory framework has a great importance because boththe EV owner and the companies involved need to gain a revenue from V2Goperation mode The former expects an income because of the use of itsbattery the latter ones need to recover the cost related to the use of innova-tive bidirectional converters and measurement devices providing ancillaryservices [84]Financial storage rights could be an interesting way to help integrate EVs

9 Future Work

Figure 92 New passenger car registrations in the EU by alternative fuel type[83]

into the grid in countries in which nodal pricing is applied Several difficul-ties may arise from such an application since research is still focused onbringing the right revenues to the owners of EVs when operating in V2Gmode However large hubs connecting EVs in an aggregate way mightbe considered as a flexibility resource for the grid similar to a utility-scalestorage plant In this framework the consideration of these hubs as trans-mission assets could bring to an additional source of revenue for both theowner of the vehicle and the company providing the charging stations be-cause the overall social welfare would be enhancedThe participation in wholesale markets is unfeasible for small-scale partic-ipants as well as distributed resources because the market would be in-tractable with a huge number of participants and because the regulationsof the market is too complex for a single small contributor The introductionof aggregators will help these distributed energy resources to manage theinteraction with the wholesale markets [3]All these considerations are placed in the framework of integrating new flex-ibility sources in the grid as discussed in the previous chapters of this the-sis

Bibliography

[1] J A Taylor ldquoFinancial Storage Rightsrdquo In IEEE Transactions on PowerSystems 302 (2015) pp 997ndash1005 DOI 10 1109 TPWRS 2014 2339016 (cit on pp 1 14 44 58ndash60 63 87 89ndash91)

[2] bp Statistical Review of World Energy 69th edition 2020 URL httpswwwbpcomcontentdambpbusiness- sitesenglobalcorporate pdfs energy - economics statistical - review bp -stats-review-2020-full-reportpdf (cit on pp 11 47)

[3] Daniel S Kirschen and Goran Strbac Fundamentals on Power Sys-tem Economics second edition 2019 (cit on pp 12 39ndash41 55ndash5796)

[4] H Ibrahim A Ilinca and J Perron ldquoEnergy storage systems mdash Char-acteristics and comparisonsrdquo In Renewable and Sustainable EnergyReviews 125 (2008) pp 1221 ndash1250 ISSN 1364-0321 DOI httpsdoiorg101016jrser200701023 URL httpwwwsciencedirectcomsciencearticlepiiS1364032107000238 (citon p 13)

[5] Xing Luo et al ldquoOverview of current development in electrical energystorage technologies and the application potential in power systemoperationrdquo In Applied Energy 137 (2015) pp 511 ndash536 ISSN 0306-2619 DOI https doi org 10 1016 j apenergy 2014 09 081 URL httpwwwsciencedirectcomsciencearticlepiiS0306261914010290 (cit on p 13)

[6] L Rouco and L Sigrist ldquoActive and reactive power control of batteryenergy storage systems in weak gridsrdquo In 2013 IREP SymposiumBulk Power System Dynamics and Control - IX Optimization Securityand Control of the Emerging Power Grid 2013 pp 1ndash7 DOI 101109IREP20136629422 (cit on p 13)

[7] Ramteen Sioshansi Paul Denholm and Thomas Jenkin ldquoMarket andPolicy Barriers to Deployment of Energy Storagerdquo In Economics ofEnergy amp Environmental Policy 12 (2012) pp 47ndash64 ISSN 2160588221605890 URL httpwwwjstororgstable26189491 (cit onpp 14 43 52)

BIBLIOGRAPHY BIBLIOGRAPHY

[8] Elisa Pappalardo Panos Pardalos and Giovanni Stracquadanio Opti-mization Approaches for Solving String Selection Problems Jan 2013pp 13ndash25 ISBN 978-1-4614-9052-4 DOI 101007978- 1- 4614-9053-1 (cit on p 16)

[9] Stephen Boyd and Lieven Vandenberghe Convex Optimization Cam-bridge University Press 2004 DOI 101017CBO9780511804441 (citon pp 16 18)

[10] George Bernard Dantzig Linear Programming and Extensions SantaMonica CA RAND Corporation 1963 DOI 107249R366 (cit onp 17)

[11] Jakob Puchinger and Guumlnther R Raidl ldquoCombining Metaheuristicsand Exact Algorithms in Combinatorial Optimization A Survey andClassificationrdquo In Artificial Intelligence and Knowledge EngineeringApplications A Bioinspired Approach Ed by Joseacute Mira and Joseacute RAacutelvarez Berlin Heidelberg Springer Berlin Heidelberg 2005 pp 41ndash53 (cit on p 17)

[12] Joel Sobel Linear Programming Notes VI Duality and ComplementarySlackness URL httpseconwebucsdedu~jsobel172aw02notes6pdf (cit on p 19)

[13] URL httpsenwikipediaorgwikiElectricity_market (cit onpp 19 39)

[14] Bernard Kolman and Robert E Beck ldquo3 - Further Topics in LinearProgrammingrdquo In Elementary Linear Programming with Applications(Second Edition) Ed by Bernard Kolman and Robert E Beck SecondEdition San Diego Academic Press 1995 pp 155ndash247 ISBN 978-0-12-417910-3 DOI httpsdoiorg101016B978-012417910-350006-1 URL httpswwwsciencedirectcomsciencearticlepiiB9780124179103500061 (cit on p 20)

[15] Harold W Kuhn and Albert W Tucker ldquoNonlinear programmingrdquo InProceedings of the Second Berkeley Symposium on MathematicalStatistics and Probability Berkeley CA USA University of CaliforniaPress 1951 pp 481ndash492 (cit on p 22)

[16] URL enwikipediaorgwikiKarushE28093KuhnE28093Tucker_conditions (cit on p 23)

[17] P Kundur NJ Balu and MG Lauby Power System Stability andControl EPRI power system engineering series McGraw-Hill Educa-tion 1994 ISBN 9780070359581 URL httpsbooksgoogleitbooksid=wOlSAAAAMAAJ (cit on pp 27 32 36)

[18] Stephen Frank and Steffen Rebennack ldquoAn introduction to optimalpower flow Theory formulation and examplesrdquo In IIE Transactions4812 (2016) pp 1172ndash1197 DOI 1010800740817X20161189626eprint httpsdoiorg1010800740817X20161189626 URLhttpsdoiorg1010800740817X20161189626 (cit on pp 2931 37 38 60 77ndash79)

[19] J Carpentier ldquoContribution arsquo lrsquoetude du Dispatching EconomiquerdquoIn 3 (1962) pp 431ndash447 (cit on p 31)

[20] Hermann W Dommel and William F Tinney ldquoOptimal Power FlowSolutionsrdquo In IEEE Transactions on Power Apparatus and SystemsPAS-8710 (1968) pp 1866ndash1876 DOI 101109TPAS1968292150(cit on p 31)

[21] E Carpaneto Sistemi Elettrici di Potenza Lecture Notes 2020 (citon pp 32 35)

[22] P Lipka R OrsquoNeill and S Oren Developing line current magnitudeconstraints for IEEE test problems Optimal Power Flow Paper 7 Staffpaper 2013 URL httpswwwfercgovsitesdefaultfiles2020-04acopf-7-line-constraintspdf (cit on p 32)

[23] SA Daza Electric Power System Fundamentals Artech House powerengineering series Artech House Publishers 2016 ISBN 9781630814328URL httpsbooksgoogleitbooksid=c2uwDgAAQBAJ (cit onpp 34 35)

[24] Spyros Chatzivasileiadis Optimization in Modern Power Systems DTUCourse 31765 Lecture Notes Technical University of Denmark (DTU)2018 URL httpsarxivorgpdf181100943pdf (cit on pp 3742)

[25] AJ Conejo and JA Aguado ldquoMulti-area coordinated decentralizedDC optimal power flowrdquo In IEEE Transactions on Power Systems134 (1998) pp 1272ndash1278 DOI 10110959736264 (cit on p 38)

[26] RA Jabr ldquoModeling network losses using quadratic conesrdquo In IEEETransactions on Power Systems 201 (2005) pp 505ndash506 DOI 101109TPWRS2004841157 (cit on pp 38 83 84)

[27] Carleton Coffrin Pascal Van Hentenryck and Russell Bent ldquoApprox-imating line losses and apparent power in AC power flow lineariza-tionsrdquo In 2012 IEEE Power and Energy Society General Meeting2012 pp 1ndash8 DOI 101109PESGM20126345342 (cit on pp 38 82)

[28] Anya Castillo and Dennice F Gayme ldquoEvaluating the Effects of RealPower Losses in Optimal Power Flow-Based Storage Integrationrdquo InIEEE Transactions on Control of Network Systems 53 (2018) pp 1132ndash1145 DOI 101109TCNS20172687819 (cit on pp 38 84)

[29] Ke Qing et al ldquoOptimized operating strategy for a distribution networkcontaining BESS and renewable energyrdquo In 2019 IEEE InnovativeSmart Grid Technologies - Asia (ISGT Asia) 2019 pp 1593ndash1597DOI 101109ISGT-Asia20198881304 (cit on p 39)

[30] Provas Kumar Roy and Susanta Dutta IGI Global 2019 Chap Eco-nomic Load Dispatch Optimal Power Flow and Optimal Reactive PowerDispatch Concept pp 46ndash64 URL httpdoi104018978-1-5225-6971-8ch002 (cit on p 39)

[31] Santosh Raikar and Seabron Adamson ldquo8 - Renewable energy inpower marketsrdquo In Renewable Energy Finance Ed by Santosh Raikarand Seabron Adamson Academic Press 2020 pp 115ndash129 ISBN978-0-12-816441-9 DOI https doi org 10 1016 B978 - 0 -12- 816441- 900008- 8 URL httpswwwsciencedirectcomsciencearticlepiiB9780128164419000088 (cit on pp 39ndash41)

[32] Ross Baldick Seams ancillary services and congestion managementUS versus EU Electricity Markets URL httpsmediumcomflorence-school-of-regulationseams-ancillary-services-and-congestion-managementus- versus- eu- electricity- markets- ab9ad53a16df(cit on p 40)

[33] Piotr F Borowski ldquoZonal and Nodal Models of Energy Market in Eu-ropean Unionrdquo In Energies 1316 (2020) ISSN 1996-1073 DOI 103390en13164182 URL httpswwwmdpicom1996-107313164182 (cit on p 40)

[34] Mette Bjoslashrndal and Kurt Joslashrnsten ldquoZonal Pricing in a DeregulatedElectricity Marketrdquo In The Energy Journal 221 (2001) pp 51ndash73ISSN 01956574 19449089 URL httpwwwjstororgstable41322907 (cit on p 40)

[35] URL httpsenwikipediaorgwikiMerit_order (cit on p 40)

[36] Brent Eldridge Richard P OrsquoNeill and Andrea R Castillo ldquoMarginalLoss Calculations for the DCOPFrdquo In (Dec 2016) DOI 1021721340633 URL httpswwwostigovbiblio1340633 (cit on p 42)

[37] Nayeem Chowdhury Giuditta Pisano and Fabrizio Pilo ldquoEnergy Stor-age Placement in the Transmission Network A Robust OptimizationApproachrdquo In 2019 AEIT International Annual Conference (AEIT)2019 pp 1ndash6 DOI 1023919AEIT20198893299 (cit on p 45)

[38] P Lazzeroni and M Repetto ldquoOptimal planning of battery systemsfor power losses reduction in distribution gridsrdquo In Electric PowerSystems Research 167 (2019) pp 94ndash112 ISSN 0378-7796 DOIhttpsdoiorg101016jepsr201810027 URL httpswwwsciencedirectcomsciencearticlepiiS0378779618303432(cit on pp 45 47)

[39] Hamidreza Mirtaheri et al ldquoOptimal Planning and Operation Schedul-ing of Battery Storage Units in Distribution Systemsrdquo In 2019 IEEEMilan PowerTech 2019 pp 1ndash6 DOI 101109PTC20198810421(cit on p 45)

[40] Benedikt Battke and Tobias S Schmidt ldquoCost-efficient demand-pullpolicies for multi-purpose technologies ndash The case of stationary elec-tricity storagerdquo In Applied Energy 155 (2015) pp 334ndash348 ISSN0306-2619 DOI httpsdoiorg101016japenergy201506010 URL httpswwwsciencedirectcomsciencearticlepiiS0306261915007680 (cit on p 45)

[41] European Commission Energy storage ndash the role of electricity URLhttpseceuropaeuenergysitesenerfilesdocumentsswd2017 _ 61 _ document _ travail _ service _ part1 _ v6 pdf (cit onpp 45 47ndash49)

[42] European Association for Storage of Energy (EASE) Pumped HydroStorage URL httpsease- storageeuwp- contentuploads201607EASE_TD_Mechanical_PHSpdf (cit on p 47)

[43] Flywheel energy storage URL httpsenwikipediaorgwikiFlywheel_energy_storage (cit on p 47)

[44] Paul Breeze ldquoChapter 3 - Compressed Air Energy Storagerdquo In PowerSystem Energy Storage Technologies Ed by Paul Breeze AcademicPress 2018 pp 23ndash31 ISBN 978-0-12-812902-9 DOI httpsdoiorg101016B978-0-12-812902-900003-1 URL httpswwwsciencedirectcomsciencearticlepiiB9780128129029000031(cit on p 47)

[45] Paul Breeze ldquoChapter 5 - Superconducting Magnetic Energy Stor-agerdquo In Power System Energy Storage Technologies Ed by PaulBreeze Academic Press 2018 pp 47ndash52 ISBN 978-0-12-812902-9DOI httpsdoiorg101016B978-0-12-812902-900005-5URL https www sciencedirect com science article pii B9780128129029000055 (cit on p 48)

[46] Bruno Pollet Energy Storage Lecture Notes Norwegian University ofScience and Technology (NTNU) 2019 (cit on p 48)

[47] Z E Lee et al ldquoProviding Grid Services With Heat Pumps A ReviewrdquoIn ASME Journal of Engineering for Sustainable Buildings and Cities(2020) URL httpsdoiorg10111514045819 (cit on p 48)

[48] Iain Staffell et al ldquoThe role of hydrogen and fuel cells in the globalenergy systemrdquo In Energy Environ Sci 12 (2 2019) pp 463ndash491DOI 101039C8EE01157E URL httpdxdoiorg101039C8EE01157E (cit on p 48)

[49] Stephen Clegg and Pierluigi Mancarella ldquoStoring renewables in thegas network modelling of power-to-gas seasonal storage flexibility inlow-carbon power systemsrdquo In IET Generation Transmission amp Dis-tribution 103 (2016) pp 566ndash575 DOI httpsdoiorg101049iet-gtd20150439 eprint httpsietresearchonlinelibrarywileycomdoipdf101049iet- gtd20150439 URL httpsietresearchonlinelibrarywileycomdoiabs101049iet-gtd20150439 (cit on p 48)

[50] Matthew A Pellow et al ldquoHydrogen or batteries for grid storage Anet energy analysisrdquo In Energy Environ Sci 8 (7 2015) pp 1938ndash1952 DOI 101039C4EE04041D URL httpdxdoiorg101039C4EE04041D (cit on p 48)

[51] BloombergNEF lsquoAlready cheaper to install new-build battery storagethan peaking plantsrsquo URL httpswwwenergy- storagenewsnews bloombergnef - lcoe - of - battery - storage - has - fallen -faster-than-solar-or-wind-i (cit on p 48)

[52] Andrej Trpovski et al ldquoA Hybrid Optimization Method for DistributionSystem Expansion Planning with Lithium-ion Battery Energy StorageSystemsrdquo In 2020 IEEE Sustainable Power and Energy Conference(iSPEC) 2020 pp 2015ndash2021 DOI 10 1109 iSPEC50848 2020 9351208 (cit on p 49)

[53] Odne Stokke Burheim ldquoChapter 7 - Secondary Batteriesrdquo In En-gineering Energy Storage Ed by Odne Stokke Burheim AcademicPress 2017 pp 111ndash145 ISBN 978-0-12-814100-7 DOI httpsdoiorg101016B978-0-12-814100-700007-9 URL httpswwwsciencedirectcomsciencearticlepiiB9780128141007000079(cit on pp 49 50)

[54] Rahul Walawalkar Jay Apt and Rick Mancini ldquoEconomics of elec-tric energy storage for energy arbitrage and regulation in New YorkrdquoIn Energy Policy 354 (2007) pp 2558ndash2568 ISSN 0301-4215 DOIhttpsdoiorg101016jenpol200609005 URL httpswwwsciencedirectcomsciencearticlepiiS0301421506003545(cit on p 49)

[55] Terna Rete Elettrica Nazionale IL RUOLO DELLO STORAGE NELLAGESTIONE DELLE RETI URL httpswwwternaititmedianews-eventiruolo-storage-gestione-reti (cit on p 49)

[56] Yash Kotak et al ldquoEnd of Electric Vehicle Batteries Reuse vs Re-cyclerdquo In Energies 148 (2021) ISSN 1996-1073 DOI 10 3390 en14082217 URL httpswwwmdpicom1996-10731482217(cit on p 50)

[57] Catherine Heymans et al ldquoEconomic analysis of second use electricvehicle batteries for residential energy storage and load-levellingrdquo InEnergy Policy 71 (2014) pp 22ndash30 ISSN 0301-4215 DOI httpsdoiorg101016jenpol201404016 URL httpswwwsciencedirectcomsciencearticlepiiS0301421514002328 (citon p 50)

[58] David Roberts Competitors to lithium-ion batteries in the grid stor-age market URL https www volts wtf p battery - week -competitors-to-lithium (cit on p 50)

[59] Maria C Argyrou Paul Christodoulides and Soteris A Kalogirou ldquoEn-ergy storage for electricity generation and related processes Tech-nologies appraisal and grid scale applicationsrdquo In Renewable andSustainable Energy Reviews 94 (2018) pp 804ndash821 ISSN 1364-0321 DOI https doi org 10 1016 j rser 2018 06 044URL https www sciencedirect com science article pii S1364032118304817 (cit on p 51)

[60] CD Parker ldquoAPPLICATIONS ndash STATIONARY | Energy Storage Sys-tems Batteriesrdquo In Encyclopedia of Electrochemical Power SourcesEd by Juumlrgen Garche Amsterdam Elsevier 2009 pp 53ndash64 ISBN978-0-444-52745-5 DOI httpsdoiorg101016B978-044452745-5 00382 - 8 URL https www sciencedirect com science articlepiiB9780444527455003828 (cit on p 51)

[61] Claudio Brivio Stefano Mandelli and Marco Merlo ldquoBattery energystorage system for primary control reserve and energy arbitragerdquo InSustainable Energy Grids and Networks 6 (2016) pp 152ndash165 ISSN2352-4677 DOI httpsdoiorg101016jsegan201603004URL https www sciencedirect com science article pii S2352467716300017 (cit on p 52)

[62] Philip C Kjaeligr and Rasmus Laeligrke ldquoExperience with primary reservesupplied from energy storage systemrdquo In 2015 17th European Con-ference on Power Electronics and Applications (EPErsquo15 ECCE-Europe)2015 pp 1ndash6 DOI 101109EPE20157311788 (cit on p 52)

[63] M Benini et al ldquoBattery energy storage systems for the provision ofprimary and secondary frequency regulation in Italyrdquo In 2016 IEEE16th International Conference on Environment and Electrical Engi-neering (EEEIC) 2016 pp 1ndash6 DOI 101109EEEIC20167555748(cit on p 52)

[64] Haisheng Chen et al ldquoProgress in electrical energy storage system Acritical reviewrdquo In Progress in Natural Science 193 (2009) pp 291ndash312 ISSN 1002-0071 DOI httpsdoiorg101016jpnsc200807014 URL httpswwwsciencedirectcomsciencearticlepiiS100200710800381X (cit on p 53)

[65] URL httpslearnpjmcomthree- prioritiesbuying- and-selling-energyftr-faqswhat-are-ftrsaspxfaq-box-text0(cit on p 55)

[66] URL httpsacereuropaeuenElectricityMARKET-CODESFORWARD-CAPACITY-ALLOCATIONPagesdefaultaspx (cit on p 55)

[67] Petr Spodniak Mari Makkonen and Samuli Honkapuro ldquoLong-termtransmission rights in the Nordic electricity markets TSO perspec-tivesrdquo In 2016 13th International Conference on the European En-ergy Market (EEM) 2016 pp 1ndash5 DOI 101109EEM20167521212(cit on p 55)

[68] Hung po Chao et al ldquoFlow-Based Transmission Rights and Conges-tion Managementrdquo In The Electricity Journal 138 (2000) pp 38ndash58ISSN 1040-6190 DOI httpsdoiorg101016S1040-6190(00)00146-9 URL httpswwwsciencedirectcomsciencearticlepiiS1040619000001469 (cit on pp 55 56)

[69] R P OrsquoNeill et al ldquoA joint energy and transmission rights auctionproposal and propertiesrdquo In IEEE Transactions on Power Systems174 (2002) pp 1058ndash1067 DOI 101109TPWRS2002804978 (citon p 56)

[70] William W Hogan ldquoContract networks for electric power transmis-sionrdquo In Journal of Regulatory Economics 4 (1992) pp 211ndash242ISSN 1573-0468 DOI httpsdoiorg101007BF00133621 (citon p 56)

[71] Daniel Munoz-Alvarez and Eilyan Bitar Financial Storage Rights inElectric Power Networks 2017 arXiv 14107822 [mathOC] (cit onpp 58 61 89ndash91)

[72] J Warrington et al ldquoA market mechanism for solving multi-period op-timal power flow exactly on AC networks with mixed participantsrdquoIn 2012 American Control Conference (ACC) 2012 pp 3101ndash3107DOI 101109ACC20126315477 (cit on p 60)

[73] Abu Alam and Joshua A Taylor ldquoAn auction for financial storage rightsrdquoIn Mediterranean Conference on Power Generation TransmissionDistribution and Energy Conversion (MEDPOWER 2018) 2018 pp 1ndash5 DOI 101049cp20181858 (cit on pp 61 94)

[74] Kyri Baker Solutions of DC OPF are Never AC Feasible 2020 arXiv191200319 [mathOC] (cit on p 82)

[75] Yingying Qi Di Shi and Daniel Tylavsky ldquoImpact of assumptions onDC power flow model accuracyrdquo In 2012 North American Power Sym-posium (NAPS) 2012 pp 1ndash6 DOI 101109NAPS20126336395(cit on p 82)

[76] S Grijalva PW Sauer and JD Weber ldquoEnhancement of linear ATCcalculations by the incorporation of reactive power flowsrdquo In IEEETransactions on Power Systems 182 (2003) pp 619ndash624 DOI 101109TPWRS2003810902 (cit on p 82)

[77] AL Motto et al ldquoNetwork-constrained multiperiod auction for a pool-based electricity marketrdquo In IEEE Transactions on Power Systems173 (2002) pp 646ndash653 DOI 101109TPWRS2002800909 (cit onpp 83 84)

[78] Anya Castillo and Dennice F Gayme ldquoProfit maximizing storage al-location in power gridsrdquo In 52nd IEEE Conference on Decision andControl 2013 pp 429ndash435 DOI 101109CDC20136759919 (cit onp 84)

[79] Josh Taylor Financial Transmission and Storage Rights presented byJosh Taylor (webinar) URL httpsresourcecentersmartgridieee org education webinar - videos SGWEB0056 html (cit onp 86)

[80] Andy Philpott and Geoffrey Pritchard ldquoFinancial transmission rights inconvex pool marketsrdquo In Operations Research Letters 322 (2004)pp 109ndash113 ISSN 0167-6377 DOI httpsdoiorg101016jorl200306002 URL httpswwwsciencedirectcomsciencearticlepiiS0167637703001032 (cit on pp 88 89 91)

[81] James B Bushnell and Steven E Stoft ldquoElectric grid investment un-der a contract network regimerdquo In Journal of Regulatory Economics(1996) DOI httpsdoiorg101007BF00133358 (cit on p 88)

[82] Fangxing Li Jiuping Pan and H Chao ldquoMarginal loss calculation incompetitive electrical energy marketsrdquo In 2004 IEEE InternationalConference on Electric Utility Deregulation Restructuring and PowerTechnologies Proceedings Vol 1 2004 205ndash209 Vol1 DOI 10 1109DRPT20041338494 (cit on p 88)

[83] ACEA NEW PASSENGER CAR REGISTRATIONS BY FUEL TYPEIN THE EUROPEAN UNION Q1 2021 URL httpswwwaceaautofiles20210423_PRPC_fuel_Q1_2021_FINALpdf (cit on pp 9596)

[84] URL https www terna it it media comunicati - stampa dettaglioInaugurato-a-Mirafiori-il-progetto-pilota-Vehicle-to-Grid (cit on p 95)

Contents

List of Figures

List of Tables

1 Introduction


22 Karush-Kuhn-Tucker Conditions


31 The pu System

32 Conventional Power Flow

33 Optimal Power Flow

34 Applications

341 DC-OPF

342 Other Formulations

35 OPF and Dual Variables as Prices

4 Energy Storage

41 Storage Characteristics

42 Technologies



423 Thermal Energy Storage

424 Chemical Energy Storage

425 Electro-chemical Energy Storage

43 Storage Role and Advantages

5 Financial Rights

51 Grid Congestions



61 DC-OPF

62 AC-OPF

7 Simulations

71 General Considerations

72 DC - AC Results Comparison

73 Storage Role

74 Financial Transmission Rights and Financial Storage Rights

8 Conclusion

9 Future Work

91 Model Extensions

92 V2G

Bibliography

Abstract

As renewable energy is gaining an increasing attention due to global warm-ing traditional grids are facing numerous challenges A combination of dif-ferent technologies is essential to ensure energy security and equity whilemaintaining an acceptable level of environmental sustainability Energy stor-age devices will play a key role at different levels of the electrical grids formaintaining a stable supply of energy both in a short- and long-term contextAs storage can provide a wide variety of services to the grid an accurateregulatory framework is necessary for a detailed consideration of its bene-fitsIn this thesis after introducing the basics of mathematical optimization andpower system analysis a discussion on the role and consideration of stor-age devices in electrical grids is presented After defining an optimal powerflow model in which storage is integrated as a transmission asset finan-cial instruments called Financial Storage Rights are reviewed consideringa nodal pricing scheme Since manuscripts and reports concerning thistopic have been considering Direct Current (DC) approximations of the op-timal power flow equations a full Alternating Current (AC) formulation ispresented without neglecting losses reactive power flows and voltage lev-els Consequently the mathematical computation of Financial TransmissionRights and Financial Storage Rights is performed using the same proce-dures as in the approximated model presented in [1] The simulations havebeen carried out using the General Algebraic Modeling System (GAMS)software in order to evaluate how approximations affect results quality andto provide an overview on the potential use of Financial Storage Rights inelectricity markets Last this thesis is concluded with a discussion of thepotential future research scope

Munich June 2021

Acknowledgements

Contents

Contents 6

List of Figures 8

List of Tables 9

1 Introduction 11

CONTENTS CONTENTS

8 Conclusion 91

Bibliography 97

List of Figures

variables (2) 22

76 LMPs [ k$pu

pu] 88

pu] 89

List of Tables

Chapter 1

Introduction

1 Introduction

Chapter 2

ej(x) = 0 j = 1 (203)(204)

min f(xi) = cixi

st Ax le b

x ge 0

or in matrix form

min cTx

st Ax le b

x ge 0

λiii(x) +sumj

microjej(x) (211)

λiii(x) +sumj

microjej(x)) (212)

ej(xlowast) = 0 j = 1 (223)

λi ge 0 (225)

Chapter 3

[Y ] =

Y 11 Y 1n

Y n1 Y nn

ZL = (r + jωl)a (302)

Y s = (g + jωc)a

2(303)

impedances)

(Y sij + Y Lij) (304)

= [Y bus]

V 1V n

Xpu =X

3Vbase

=V 2base

Ybase =1

3 andradic

Y ijV j (322)

Pi = Visumj

Qi = Visumj

Pi = Visumj

Qi = Visumj

minsumiisinG

CG (331)

Gi (334)

Gi (335)

i (336)

P 2ij + Q2

ij le S2ijMAX (339)

I ij =ysij

(Ylowastsij

2Vlowasti + Y

lowastLij(V

lowasti minus V

lowastj)

= |V |2i(Ylowastsij

lowastLij

lowastLij =

(3312)

= |V i|2(Ylowastsij

lowastLij

(3313)

Y Lij = Gij + jBij

YlowastSij

2= minusjBSij

Pij = ltSij (3314)

Qij = =Sij (3315)

Consequently

2+Gij minus jBij

(3317)

Smax = 3Vradic

3Imax (3319)

sumi PGi = PL

calculated as

= (25$

MWhminus 20

MWh)100MW = 500

Chapter 4

Energy Storage

4 Energy Storage

storage (413)

∆t (414)

storage

4 Energy Storage

the grid [38]

P 2it +Q2

it le S2MAXit (416)

42 Technologies

4 Energy Storage

Chapter 5

Financial Rights

51 Grid Congestions

5 Financial Rights

h(511)

5 Financial Rights

h(512)

h(513)

h(514)

5 Financial Rights

MWh(515)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T2)

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

2(521)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

5 Financial Rights

Chapter 6

minsumit

fit(pit) (611)

it (613)

storage (617)

∆t (618)

(2U)it

(3)ikt

(4L)it λ

(5)it λ

(5U)it

L(pi δi δk pchi p

dchi SOCi λi)t =

(fit(pit))

+ λ(2L)it (pMIN

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

partpit=partOF

(2L)it + λ

(2U)it = 0 (6110)

partδit= λ

(bik) + λ(3)ikt

(bik) = 0 (6111)

partlkpartδkt

= λ(1)kt

(bik) + λ(3)kit

(bik) = 0 (6112)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6113)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6114)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6115)

MAXik ) = 0 (6116)

chMAXit ) = 0 (6117)

λ(5)it (pdchMAX

i ) = 0 (6119)

Storage term

pit ) = 0 (6120)

partδit= (λ

(1)it + λ

(3)ikt)

(bik) = 0 (6121)

λ(1)it

λ(3)iktp

MAXik (6122)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6126)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6127)

minusλ(6L)it + λ

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6128)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6129)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6132)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6133)

minussumit

λ(7)it (EMAX

From (6128)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6137)

According to (6128)

minussumit

(6)it E

MAXi +

And therefore

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6139)

i ) (6140)

λ(1)it (pit)minus

λ(3)iktp

MAXik︸︷︷︸

+sumit

dchit︸︷︷︸

(6141)

minsumit

fit(pit) (621)

qit = Visumk

it (624)

it (625)

P 2ik + Q2

ik le S2ikMAX (626)

it (627)

it (628)

storage (6211)

∆t (6212)

Qik = minusV 2i

(bSik2

(6214)

ik (6215)

qit = Visumk

(2U)it

(9U)it

(10U)it

(11U)it

(4L)it λ

(5)it λ

(5U)it

l(pi δi δk pchi p

(fit(pit))

+ λ(2L)it (pMIN

+ λ(3)ikt(gikV

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

∆tminus SOCit+1)

+ λ(8)it (Vi

+ λ(9L)it (qMIN

+ λ(10L)it (V MIN

+ λ(11L)it (δMIN

(6216)

partpit=partOF

(2L)it + λ

(2U)it = 0 (6217)

partδit= λ

(1)it (Vit

+ λ(8)ikt(Vit

minus λ(11L)it + λ(11U)it = 0 (6218)

partδkt= λ

(1)kt(Vkt

+ λ(8)kt(Vkt

minus λ(11L)kt + λ(11U)kt = 0 (6219)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6220)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6221)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6222)

λ(3)ikt(gikV

MAXik ) = 0 (6223)

chMAXit ) = 0 (6224)

λ(5)it (pdchMAX

i ) = 0 (6226)

Storage term

+Vitsumk

pit ) = 0

(6227)

(Vitsumk

(Vktsumi

partδit= λ

(1)it (Vit

partδkt= λ

(1)kt(Vkt

ik (6231)

λ(1)it

with sumit

λ(3)iktp

MAXij minus

λ(1)it gikV

2it (6233)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6237)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6238)

(7)it minus λ

(7)itminus1 = 0 (6239)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6240)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6243)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6244)

From (6236) and (6212)

minussumit

λ(7)it (EMAX

From (6239)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6248)

minussumit

(6)it E

MAXi +

Consequently

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6250)

i ) (6251)

λ(1)it (pit)minus

λ(3)iktp

MAXij︸︷︷︸

FTR NEW

+sumit

dchit︸︷︷︸

λ(1)it gikV

2it = 0

(6252)

Chapter 7

Simulations

rampup

7 Simulations

G = ltY =R

R2 +X2(711)

B = =Y = minus X

R2 +X2(712)

7 Simulations

2(724)

And therefore

2(725)

2(726)

7 Simulations

And finally

The 12

7 Simulations

λ(1)it pit = MS (741)

7 Simulations

Chapter 8

Conclusion

Chapter 9

Future Work

91 Model Extensions

9 Future Work

92 V2G

9 Future Work

Bibliography

Contents

List of Figures

List of Tables

1 Introduction




31 The pu System



34 Applications

341 DC-OPF



4 Energy Storage


42 Technologies







5 Financial Rights

51 Grid Congestions



61 DC-OPF

62 AC-OPF

7 Simulations



73 Storage Role


8 Conclusion

9 Future Work

91 Model Extensions

92 V2G

Bibliography

Munich June 2021

Acknowledgements

Contents

Contents 6

List of Figures 8

List of Tables 9

1 Introduction 11

CONTENTS CONTENTS

8 Conclusion 91

Bibliography 97

List of Figures

variables (2) 22

76 LMPs [ k$pu

pu] 88

pu] 89

List of Tables

Chapter 1

Introduction

1 Introduction

Chapter 2

ej(x) = 0 j = 1 (203)(204)

min f(xi) = cixi

st Ax le b

x ge 0

or in matrix form

min cTx

st Ax le b

x ge 0

λiii(x) +sumj

microjej(x) (211)

λiii(x) +sumj

microjej(x)) (212)

ej(xlowast) = 0 j = 1 (223)

λi ge 0 (225)

Chapter 3

[Y ] =

Y 11 Y 1n

Y n1 Y nn

ZL = (r + jωl)a (302)

Y s = (g + jωc)a

2(303)

impedances)

(Y sij + Y Lij) (304)

= [Y bus]

V 1V n

Xpu =X

3Vbase

=V 2base

Ybase =1

3 andradic

Y ijV j (322)

Pi = Visumj

Qi = Visumj

Pi = Visumj

Qi = Visumj

minsumiisinG

CG (331)

Gi (334)

Gi (335)

i (336)

P 2ij + Q2

ij le S2ijMAX (339)

I ij =ysij

(Ylowastsij

2Vlowasti + Y

lowastLij(V

lowasti minus V

lowastj)

= |V |2i(Ylowastsij

lowastLij

lowastLij =

(3312)

= |V i|2(Ylowastsij

lowastLij

(3313)

Y Lij = Gij + jBij

YlowastSij

2= minusjBSij

Pij = ltSij (3314)

Qij = =Sij (3315)

Consequently

2+Gij minus jBij

(3317)

Smax = 3Vradic

3Imax (3319)

sumi PGi = PL

calculated as

= (25$

MWhminus 20

MWh)100MW = 500

Chapter 4

Energy Storage

4 Energy Storage

storage (413)

∆t (414)

storage

4 Energy Storage

the grid [38]

P 2it +Q2

it le S2MAXit (416)

42 Technologies

4 Energy Storage

Chapter 5

Financial Rights

51 Grid Congestions

5 Financial Rights

h(511)

5 Financial Rights

h(512)

h(513)

h(514)

5 Financial Rights

MWh(515)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T2)

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

2(521)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

5 Financial Rights

Chapter 6

minsumit

fit(pit) (611)

it (613)

storage (617)

∆t (618)

(2U)it

(3)ikt

(4L)it λ

(5)it λ

(5U)it

L(pi δi δk pchi p

dchi SOCi λi)t =

(fit(pit))

+ λ(2L)it (pMIN

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

partpit=partOF

(2L)it + λ

(2U)it = 0 (6110)

partδit= λ

(bik) + λ(3)ikt

(bik) = 0 (6111)

partlkpartδkt

= λ(1)kt

(bik) + λ(3)kit

(bik) = 0 (6112)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6113)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6114)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6115)

MAXik ) = 0 (6116)

chMAXit ) = 0 (6117)

λ(5)it (pdchMAX

i ) = 0 (6119)

Storage term

pit ) = 0 (6120)

partδit= (λ

(1)it + λ

(3)ikt)

(bik) = 0 (6121)

λ(1)it

λ(3)iktp

MAXik (6122)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6126)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6127)

minusλ(6L)it + λ

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6128)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6129)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6132)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6133)

minussumit

λ(7)it (EMAX

From (6128)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6137)

According to (6128)

minussumit

(6)it E

MAXi +

And therefore

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6139)

i ) (6140)

λ(1)it (pit)minus

λ(3)iktp

MAXik︸︷︷︸

+sumit

dchit︸︷︷︸

(6141)

minsumit

fit(pit) (621)

qit = Visumk

it (624)

it (625)

P 2ik + Q2

ik le S2ikMAX (626)

it (627)

it (628)

storage (6211)

∆t (6212)

Qik = minusV 2i

(bSik2

(6214)

ik (6215)

qit = Visumk

(2U)it

(9U)it

(10U)it

(11U)it

(4L)it λ

(5)it λ

(5U)it

l(pi δi δk pchi p

(fit(pit))

+ λ(2L)it (pMIN

+ λ(3)ikt(gikV

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

∆tminus SOCit+1)

+ λ(8)it (Vi

+ λ(9L)it (qMIN

+ λ(10L)it (V MIN

+ λ(11L)it (δMIN

(6216)

partpit=partOF

(2L)it + λ

(2U)it = 0 (6217)

partδit= λ

(1)it (Vit

+ λ(8)ikt(Vit

minus λ(11L)it + λ(11U)it = 0 (6218)

partδkt= λ

(1)kt(Vkt

+ λ(8)kt(Vkt

minus λ(11L)kt + λ(11U)kt = 0 (6219)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6220)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6221)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6222)

λ(3)ikt(gikV

MAXik ) = 0 (6223)

chMAXit ) = 0 (6224)

λ(5)it (pdchMAX

i ) = 0 (6226)

Storage term

+Vitsumk

pit ) = 0

(6227)

(Vitsumk

(Vktsumi

partδit= λ

(1)it (Vit

partδkt= λ

(1)kt(Vkt

ik (6231)

λ(1)it

with sumit

λ(3)iktp

MAXij minus

λ(1)it gikV

2it (6233)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6237)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6238)

(7)it minus λ

(7)itminus1 = 0 (6239)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6240)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6243)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6244)

From (6236) and (6212)

minussumit

λ(7)it (EMAX

From (6239)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6248)

minussumit

(6)it E

MAXi +

Consequently

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6250)

i ) (6251)

λ(1)it (pit)minus

λ(3)iktp

MAXij︸︷︷︸

FTR NEW

+sumit

dchit︸︷︷︸

λ(1)it gikV

2it = 0

(6252)

Chapter 7

Simulations

rampup

7 Simulations

G = ltY =R

R2 +X2(711)

B = =Y = minus X

R2 +X2(712)

7 Simulations

2(724)

And therefore

2(725)

2(726)

7 Simulations

And finally

The 12

7 Simulations

λ(1)it pit = MS (741)

7 Simulations

Chapter 8

Conclusion

Chapter 9

Future Work

91 Model Extensions

9 Future Work

92 V2G

9 Future Work

Bibliography

Contents

List of Figures

List of Tables

1 Introduction




31 The pu System



34 Applications

341 DC-OPF



4 Energy Storage


42 Technologies







5 Financial Rights

51 Grid Congestions



61 DC-OPF

62 AC-OPF

7 Simulations



73 Storage Role


8 Conclusion

9 Future Work

91 Model Extensions

92 V2G

Bibliography

Acknowledgements

Contents

Contents 6

List of Figures 8

List of Tables 9

1 Introduction 11

CONTENTS CONTENTS

8 Conclusion 91

Bibliography 97

List of Figures

variables (2) 22

76 LMPs [ k$pu

pu] 88

pu] 89

List of Tables

Chapter 1

Introduction

1 Introduction

Chapter 2

ej(x) = 0 j = 1 (203)(204)

min f(xi) = cixi

st Ax le b

x ge 0

or in matrix form

min cTx

st Ax le b

x ge 0

λiii(x) +sumj

microjej(x) (211)

λiii(x) +sumj

microjej(x)) (212)

ej(xlowast) = 0 j = 1 (223)

λi ge 0 (225)

Chapter 3

[Y ] =

Y 11 Y 1n

Y n1 Y nn

ZL = (r + jωl)a (302)

Y s = (g + jωc)a

2(303)

impedances)

(Y sij + Y Lij) (304)

= [Y bus]

V 1V n

Xpu =X

3Vbase

=V 2base

Ybase =1

3 andradic

Y ijV j (322)

Pi = Visumj

Qi = Visumj

Pi = Visumj

Qi = Visumj

minsumiisinG

CG (331)

Gi (334)

Gi (335)

i (336)

P 2ij + Q2

ij le S2ijMAX (339)

I ij =ysij

(Ylowastsij

2Vlowasti + Y

lowastLij(V

lowasti minus V

lowastj)

= |V |2i(Ylowastsij

lowastLij

lowastLij =

(3312)

= |V i|2(Ylowastsij

lowastLij

(3313)

Y Lij = Gij + jBij

YlowastSij

2= minusjBSij

Pij = ltSij (3314)

Qij = =Sij (3315)

Consequently

2+Gij minus jBij

(3317)

Smax = 3Vradic

3Imax (3319)

sumi PGi = PL

calculated as

= (25$

MWhminus 20

MWh)100MW = 500

Chapter 4

Energy Storage

4 Energy Storage

storage (413)

∆t (414)

storage

4 Energy Storage

the grid [38]

P 2it +Q2

it le S2MAXit (416)

42 Technologies

4 Energy Storage

Chapter 5

Financial Rights

51 Grid Congestions

5 Financial Rights

h(511)

5 Financial Rights

h(512)

h(513)

h(514)

5 Financial Rights

MWh(515)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T2)

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

2(521)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

5 Financial Rights

Chapter 6

minsumit

fit(pit) (611)

it (613)

storage (617)

∆t (618)

(2U)it

(3)ikt

(4L)it λ

(5)it λ

(5U)it

L(pi δi δk pchi p

dchi SOCi λi)t =

(fit(pit))

+ λ(2L)it (pMIN

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

partpit=partOF

(2L)it + λ

(2U)it = 0 (6110)

partδit= λ

(bik) + λ(3)ikt

(bik) = 0 (6111)

partlkpartδkt

= λ(1)kt

(bik) + λ(3)kit

(bik) = 0 (6112)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6113)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6114)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6115)

MAXik ) = 0 (6116)

chMAXit ) = 0 (6117)

λ(5)it (pdchMAX

i ) = 0 (6119)

Storage term

pit ) = 0 (6120)

partδit= (λ

(1)it + λ

(3)ikt)

(bik) = 0 (6121)

λ(1)it

λ(3)iktp

MAXik (6122)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6126)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6127)

minusλ(6L)it + λ

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6128)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6129)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6132)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6133)

minussumit

λ(7)it (EMAX

From (6128)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6137)

According to (6128)

minussumit

(6)it E

MAXi +

And therefore

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6139)

i ) (6140)

λ(1)it (pit)minus

λ(3)iktp

MAXik︸︷︷︸

+sumit

dchit︸︷︷︸

(6141)

minsumit

fit(pit) (621)

qit = Visumk

it (624)

it (625)

P 2ik + Q2

ik le S2ikMAX (626)

it (627)

it (628)

storage (6211)

∆t (6212)

Qik = minusV 2i

(bSik2

(6214)

ik (6215)

qit = Visumk

(2U)it

(9U)it

(10U)it

(11U)it

(4L)it λ

(5)it λ

(5U)it

l(pi δi δk pchi p

(fit(pit))

+ λ(2L)it (pMIN

+ λ(3)ikt(gikV

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

∆tminus SOCit+1)

+ λ(8)it (Vi

+ λ(9L)it (qMIN

+ λ(10L)it (V MIN

+ λ(11L)it (δMIN

(6216)

partpit=partOF

(2L)it + λ

(2U)it = 0 (6217)

partδit= λ

(1)it (Vit

+ λ(8)ikt(Vit

minus λ(11L)it + λ(11U)it = 0 (6218)

partδkt= λ

(1)kt(Vkt

+ λ(8)kt(Vkt

minus λ(11L)kt + λ(11U)kt = 0 (6219)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6220)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6221)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6222)

λ(3)ikt(gikV

MAXik ) = 0 (6223)

chMAXit ) = 0 (6224)

λ(5)it (pdchMAX

i ) = 0 (6226)

Storage term

+Vitsumk

pit ) = 0

(6227)

(Vitsumk

(Vktsumi

partδit= λ

(1)it (Vit

partδkt= λ

(1)kt(Vkt

ik (6231)

λ(1)it

with sumit

λ(3)iktp

MAXij minus

λ(1)it gikV

2it (6233)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6237)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6238)

(7)it minus λ

(7)itminus1 = 0 (6239)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6240)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6243)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6244)

From (6236) and (6212)

minussumit

λ(7)it (EMAX

From (6239)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6248)

minussumit

(6)it E

MAXi +

Consequently

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6250)

i ) (6251)

λ(1)it (pit)minus

λ(3)iktp

MAXij︸︷︷︸

FTR NEW

+sumit

dchit︸︷︷︸

λ(1)it gikV

2it = 0

(6252)

Chapter 7

Simulations

rampup

7 Simulations

G = ltY =R

R2 +X2(711)

B = =Y = minus X

R2 +X2(712)

7 Simulations

2(724)

And therefore

2(725)

2(726)

7 Simulations

And finally

The 12

7 Simulations

λ(1)it pit = MS (741)

7 Simulations

Chapter 8

Conclusion

Chapter 9

Future Work

91 Model Extensions

9 Future Work

92 V2G

9 Future Work

Bibliography

Contents

List of Figures

List of Tables

1 Introduction




31 The pu System



34 Applications

341 DC-OPF



4 Energy Storage


42 Technologies







5 Financial Rights

51 Grid Congestions



61 DC-OPF

62 AC-OPF

7 Simulations



73 Storage Role


8 Conclusion

9 Future Work

91 Model Extensions

92 V2G

Bibliography

Contents

Contents 6

List of Figures 8

List of Tables 9

1 Introduction 11

CONTENTS CONTENTS

8 Conclusion 91

Bibliography 97

List of Figures

variables (2) 22

76 LMPs [ k$pu

pu] 88

pu] 89

List of Tables

Chapter 1

Introduction

1 Introduction

Chapter 2

ej(x) = 0 j = 1 (203)(204)

min f(xi) = cixi

st Ax le b

x ge 0

or in matrix form

min cTx

st Ax le b

x ge 0

λiii(x) +sumj

microjej(x) (211)

λiii(x) +sumj

microjej(x)) (212)

ej(xlowast) = 0 j = 1 (223)

λi ge 0 (225)

Chapter 3

[Y ] =

Y 11 Y 1n

Y n1 Y nn

ZL = (r + jωl)a (302)

Y s = (g + jωc)a

2(303)

impedances)

(Y sij + Y Lij) (304)

= [Y bus]

V 1V n

Xpu =X

3Vbase

=V 2base

Ybase =1

3 andradic

Y ijV j (322)

Pi = Visumj

Qi = Visumj

Pi = Visumj

Qi = Visumj

minsumiisinG

CG (331)

Gi (334)

Gi (335)

i (336)

P 2ij + Q2

ij le S2ijMAX (339)

I ij =ysij

(Ylowastsij

2Vlowasti + Y

lowastLij(V

lowasti minus V

lowastj)

= |V |2i(Ylowastsij

lowastLij

lowastLij =

(3312)

= |V i|2(Ylowastsij

lowastLij

(3313)

Y Lij = Gij + jBij

YlowastSij

2= minusjBSij

Pij = ltSij (3314)

Qij = =Sij (3315)

Consequently

2+Gij minus jBij

(3317)

Smax = 3Vradic

3Imax (3319)

sumi PGi = PL

calculated as

= (25$

MWhminus 20

MWh)100MW = 500

Chapter 4

Energy Storage

4 Energy Storage

storage (413)

∆t (414)

storage

4 Energy Storage

the grid [38]

P 2it +Q2

it le S2MAXit (416)

42 Technologies

4 Energy Storage

Chapter 5

Financial Rights

51 Grid Congestions

5 Financial Rights

h(511)

5 Financial Rights

h(512)

h(513)

h(514)

5 Financial Rights

MWh(515)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T2)

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

2(521)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

5 Financial Rights

Chapter 6

minsumit

fit(pit) (611)

it (613)

storage (617)

∆t (618)

(2U)it

(3)ikt

(4L)it λ

(5)it λ

(5U)it

L(pi δi δk pchi p

dchi SOCi λi)t =

(fit(pit))

+ λ(2L)it (pMIN

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

partpit=partOF

(2L)it + λ

(2U)it = 0 (6110)

partδit= λ

(bik) + λ(3)ikt

(bik) = 0 (6111)

partlkpartδkt

= λ(1)kt

(bik) + λ(3)kit

(bik) = 0 (6112)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6113)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6114)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6115)

MAXik ) = 0 (6116)

chMAXit ) = 0 (6117)

λ(5)it (pdchMAX

i ) = 0 (6119)

Storage term

pit ) = 0 (6120)

partδit= (λ

(1)it + λ

(3)ikt)

(bik) = 0 (6121)

λ(1)it

λ(3)iktp

MAXik (6122)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6126)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6127)

minusλ(6L)it + λ

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6128)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6129)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6132)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6133)

minussumit

λ(7)it (EMAX

From (6128)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6137)

According to (6128)

minussumit

(6)it E

MAXi +

And therefore

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6139)

i ) (6140)

λ(1)it (pit)minus

λ(3)iktp

MAXik︸︷︷︸

+sumit

dchit︸︷︷︸

(6141)

minsumit

fit(pit) (621)

qit = Visumk

it (624)

it (625)

P 2ik + Q2

ik le S2ikMAX (626)

it (627)

it (628)

storage (6211)

∆t (6212)

Qik = minusV 2i

(bSik2

(6214)

ik (6215)

qit = Visumk

(2U)it

(9U)it

(10U)it

(11U)it

(4L)it λ

(5)it λ

(5U)it

l(pi δi δk pchi p

(fit(pit))

+ λ(2L)it (pMIN

+ λ(3)ikt(gikV

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

∆tminus SOCit+1)

+ λ(8)it (Vi

+ λ(9L)it (qMIN

+ λ(10L)it (V MIN

+ λ(11L)it (δMIN

(6216)

partpit=partOF

(2L)it + λ

(2U)it = 0 (6217)

partδit= λ

(1)it (Vit

+ λ(8)ikt(Vit

minus λ(11L)it + λ(11U)it = 0 (6218)

partδkt= λ

(1)kt(Vkt

+ λ(8)kt(Vkt

minus λ(11L)kt + λ(11U)kt = 0 (6219)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6220)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6221)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6222)

λ(3)ikt(gikV

MAXik ) = 0 (6223)

chMAXit ) = 0 (6224)

λ(5)it (pdchMAX

i ) = 0 (6226)

Storage term

+Vitsumk

pit ) = 0

(6227)

(Vitsumk

(Vktsumi

partδit= λ

(1)it (Vit

partδkt= λ

(1)kt(Vkt

ik (6231)

λ(1)it

with sumit

λ(3)iktp

MAXij minus

λ(1)it gikV

2it (6233)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6237)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6238)

(7)it minus λ

(7)itminus1 = 0 (6239)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6240)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6243)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6244)

From (6236) and (6212)

minussumit

λ(7)it (EMAX

From (6239)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6248)

minussumit

(6)it E

MAXi +

Consequently

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6250)

i ) (6251)

λ(1)it (pit)minus

λ(3)iktp

MAXij︸︷︷︸

FTR NEW

+sumit

dchit︸︷︷︸

λ(1)it gikV

2it = 0

(6252)

Chapter 7

Simulations

rampup

7 Simulations

G = ltY =R

R2 +X2(711)

B = =Y = minus X

R2 +X2(712)

7 Simulations

2(724)

And therefore

2(725)

2(726)

7 Simulations

And finally

The 12

7 Simulations

λ(1)it pit = MS (741)

7 Simulations

Chapter 8

Conclusion

Chapter 9

Future Work

91 Model Extensions

9 Future Work

92 V2G

9 Future Work

Bibliography

Contents

List of Figures

List of Tables

1 Introduction




31 The pu System



34 Applications

341 DC-OPF



4 Energy Storage


42 Technologies







5 Financial Rights

51 Grid Congestions



61 DC-OPF

62 AC-OPF

7 Simulations



73 Storage Role


8 Conclusion

9 Future Work

91 Model Extensions

92 V2G

Bibliography

CONTENTS CONTENTS

8 Conclusion 91

Bibliography 97

List of Figures

variables (2) 22

76 LMPs [ k$pu

pu] 88

pu] 89

List of Tables

Chapter 1

Introduction

1 Introduction

Chapter 2

ej(x) = 0 j = 1 (203)(204)

min f(xi) = cixi

st Ax le b

x ge 0

or in matrix form

min cTx

st Ax le b

x ge 0

λiii(x) +sumj

microjej(x) (211)

λiii(x) +sumj

microjej(x)) (212)

ej(xlowast) = 0 j = 1 (223)

λi ge 0 (225)

Chapter 3

[Y ] =

Y 11 Y 1n

Y n1 Y nn

ZL = (r + jωl)a (302)

Y s = (g + jωc)a

2(303)

impedances)

(Y sij + Y Lij) (304)

= [Y bus]

V 1V n

Xpu =X

3Vbase

=V 2base

Ybase =1

3 andradic

Y ijV j (322)

Pi = Visumj

Qi = Visumj

Pi = Visumj

Qi = Visumj

minsumiisinG

CG (331)

Gi (334)

Gi (335)

i (336)

P 2ij + Q2

ij le S2ijMAX (339)

I ij =ysij

(Ylowastsij

2Vlowasti + Y

lowastLij(V

lowasti minus V

lowastj)

= |V |2i(Ylowastsij

lowastLij

lowastLij =

(3312)

= |V i|2(Ylowastsij

lowastLij

(3313)

Y Lij = Gij + jBij

YlowastSij

2= minusjBSij

Pij = ltSij (3314)

Qij = =Sij (3315)

Consequently

2+Gij minus jBij

(3317)

Smax = 3Vradic

3Imax (3319)

sumi PGi = PL

calculated as

= (25$

MWhminus 20

MWh)100MW = 500

Chapter 4

Energy Storage

4 Energy Storage

storage (413)

∆t (414)

storage

4 Energy Storage

the grid [38]

P 2it +Q2

it le S2MAXit (416)

42 Technologies

4 Energy Storage

Chapter 5

Financial Rights

51 Grid Congestions

5 Financial Rights

h(511)

5 Financial Rights

h(512)

h(513)

h(514)

5 Financial Rights

MWh(515)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T2)

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

2(521)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

5 Financial Rights

Chapter 6

minsumit

fit(pit) (611)

it (613)

storage (617)

∆t (618)

(2U)it

(3)ikt

(4L)it λ

(5)it λ

(5U)it

L(pi δi δk pchi p

dchi SOCi λi)t =

(fit(pit))

+ λ(2L)it (pMIN

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

partpit=partOF

(2L)it + λ

(2U)it = 0 (6110)

partδit= λ

(bik) + λ(3)ikt

(bik) = 0 (6111)

partlkpartδkt

= λ(1)kt

(bik) + λ(3)kit

(bik) = 0 (6112)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6113)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6114)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6115)

MAXik ) = 0 (6116)

chMAXit ) = 0 (6117)

λ(5)it (pdchMAX

i ) = 0 (6119)

Storage term

pit ) = 0 (6120)

partδit= (λ

(1)it + λ

(3)ikt)

(bik) = 0 (6121)

λ(1)it

λ(3)iktp

MAXik (6122)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6126)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6127)

minusλ(6L)it + λ

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6128)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6129)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6132)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6133)

minussumit

λ(7)it (EMAX

From (6128)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6137)

According to (6128)

minussumit

(6)it E

MAXi +

And therefore

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6139)

i ) (6140)

λ(1)it (pit)minus

λ(3)iktp

MAXik︸︷︷︸

+sumit

dchit︸︷︷︸

(6141)

minsumit

fit(pit) (621)

qit = Visumk

it (624)

it (625)

P 2ik + Q2

ik le S2ikMAX (626)

it (627)

it (628)

storage (6211)

∆t (6212)

Qik = minusV 2i

(bSik2

(6214)

ik (6215)

qit = Visumk

(2U)it

(9U)it

(10U)it

(11U)it

(4L)it λ

(5)it λ

(5U)it

l(pi δi δk pchi p

(fit(pit))

+ λ(2L)it (pMIN

+ λ(3)ikt(gikV

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

∆tminus SOCit+1)

+ λ(8)it (Vi

+ λ(9L)it (qMIN

+ λ(10L)it (V MIN

+ λ(11L)it (δMIN

(6216)

partpit=partOF

(2L)it + λ

(2U)it = 0 (6217)

partδit= λ

(1)it (Vit

+ λ(8)ikt(Vit

minus λ(11L)it + λ(11U)it = 0 (6218)

partδkt= λ

(1)kt(Vkt

+ λ(8)kt(Vkt

minus λ(11L)kt + λ(11U)kt = 0 (6219)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6220)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6221)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6222)

λ(3)ikt(gikV

MAXik ) = 0 (6223)

chMAXit ) = 0 (6224)

λ(5)it (pdchMAX

i ) = 0 (6226)

Storage term

+Vitsumk

pit ) = 0

(6227)

(Vitsumk

(Vktsumi

partδit= λ

(1)it (Vit

partδkt= λ

(1)kt(Vkt

ik (6231)

λ(1)it

with sumit

λ(3)iktp

MAXij minus

λ(1)it gikV

2it (6233)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6237)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6238)

(7)it minus λ

(7)itminus1 = 0 (6239)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6240)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6243)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6244)

From (6236) and (6212)

minussumit

λ(7)it (EMAX

From (6239)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6248)

minussumit

(6)it E

MAXi +

Consequently

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6250)

i ) (6251)

λ(1)it (pit)minus

λ(3)iktp

MAXij︸︷︷︸

FTR NEW

+sumit

dchit︸︷︷︸

λ(1)it gikV

2it = 0

(6252)

Chapter 7

Simulations

rampup

7 Simulations

G = ltY =R

R2 +X2(711)

B = =Y = minus X

R2 +X2(712)

7 Simulations

2(724)

And therefore

2(725)

2(726)

7 Simulations

And finally

The 12

7 Simulations

λ(1)it pit = MS (741)

7 Simulations

Chapter 8

Conclusion

Chapter 9

Future Work

91 Model Extensions

9 Future Work

92 V2G

9 Future Work

Bibliography

Contents

List of Figures

List of Tables

1 Introduction




31 The pu System



34 Applications

341 DC-OPF



4 Energy Storage


42 Technologies







5 Financial Rights

51 Grid Congestions



61 DC-OPF

62 AC-OPF

7 Simulations



73 Storage Role


8 Conclusion

9 Future Work

91 Model Extensions

92 V2G

Bibliography

List of Figures

variables (2) 22

76 LMPs [ k$pu

pu] 88

pu] 89

List of Tables

Chapter 1

Introduction

1 Introduction

Chapter 2

ej(x) = 0 j = 1 (203)(204)

min f(xi) = cixi

st Ax le b

x ge 0

or in matrix form

min cTx

st Ax le b

x ge 0

λiii(x) +sumj

microjej(x) (211)

λiii(x) +sumj

microjej(x)) (212)

ej(xlowast) = 0 j = 1 (223)

λi ge 0 (225)

Chapter 3

[Y ] =

Y 11 Y 1n

Y n1 Y nn

ZL = (r + jωl)a (302)

Y s = (g + jωc)a

2(303)

impedances)

(Y sij + Y Lij) (304)

= [Y bus]

V 1V n

Xpu =X

3Vbase

=V 2base

Ybase =1

3 andradic

Y ijV j (322)

Pi = Visumj

Qi = Visumj

Pi = Visumj

Qi = Visumj

minsumiisinG

CG (331)

Gi (334)

Gi (335)

i (336)

P 2ij + Q2

ij le S2ijMAX (339)

I ij =ysij

(Ylowastsij

2Vlowasti + Y

lowastLij(V

lowasti minus V

lowastj)

= |V |2i(Ylowastsij

lowastLij

lowastLij =

(3312)

= |V i|2(Ylowastsij

lowastLij

(3313)

Y Lij = Gij + jBij

YlowastSij

2= minusjBSij

Pij = ltSij (3314)

Qij = =Sij (3315)

Consequently

2+Gij minus jBij

(3317)

Smax = 3Vradic

3Imax (3319)

sumi PGi = PL

calculated as

= (25$

MWhminus 20

MWh)100MW = 500

Chapter 4

Energy Storage

4 Energy Storage

storage (413)

∆t (414)

storage

4 Energy Storage

the grid [38]

P 2it +Q2

it le S2MAXit (416)

42 Technologies

4 Energy Storage

Chapter 5

Financial Rights

51 Grid Congestions

5 Financial Rights

h(511)

5 Financial Rights

h(512)

h(513)

h(514)

5 Financial Rights

MWh(515)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T2)

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

2(521)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

5 Financial Rights

Chapter 6

minsumit

fit(pit) (611)

it (613)

storage (617)

∆t (618)

(2U)it

(3)ikt

(4L)it λ

(5)it λ

(5U)it

L(pi δi δk pchi p

dchi SOCi λi)t =

(fit(pit))

+ λ(2L)it (pMIN

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

partpit=partOF

(2L)it + λ

(2U)it = 0 (6110)

partδit= λ

(bik) + λ(3)ikt

(bik) = 0 (6111)

partlkpartδkt

= λ(1)kt

(bik) + λ(3)kit

(bik) = 0 (6112)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6113)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6114)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6115)

MAXik ) = 0 (6116)

chMAXit ) = 0 (6117)

λ(5)it (pdchMAX

i ) = 0 (6119)

Storage term

pit ) = 0 (6120)

partδit= (λ

(1)it + λ

(3)ikt)

(bik) = 0 (6121)

λ(1)it

λ(3)iktp

MAXik (6122)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6126)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6127)

minusλ(6L)it + λ

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6128)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6129)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6132)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6133)

minussumit

λ(7)it (EMAX

From (6128)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6137)

According to (6128)

minussumit

(6)it E

MAXi +

And therefore

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6139)

i ) (6140)

λ(1)it (pit)minus

λ(3)iktp

MAXik︸︷︷︸

+sumit

dchit︸︷︷︸

(6141)

minsumit

fit(pit) (621)

qit = Visumk

it (624)

it (625)

P 2ik + Q2

ik le S2ikMAX (626)

it (627)

it (628)

storage (6211)

∆t (6212)

Qik = minusV 2i

(bSik2

(6214)

ik (6215)

qit = Visumk

(2U)it

(9U)it

(10U)it

(11U)it

(4L)it λ

(5)it λ

(5U)it

l(pi δi δk pchi p

(fit(pit))

+ λ(2L)it (pMIN

+ λ(3)ikt(gikV

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

∆tminus SOCit+1)

+ λ(8)it (Vi

+ λ(9L)it (qMIN

+ λ(10L)it (V MIN

+ λ(11L)it (δMIN

(6216)

partpit=partOF

(2L)it + λ

(2U)it = 0 (6217)

partδit= λ

(1)it (Vit

+ λ(8)ikt(Vit

minus λ(11L)it + λ(11U)it = 0 (6218)

partδkt= λ

(1)kt(Vkt

+ λ(8)kt(Vkt

minus λ(11L)kt + λ(11U)kt = 0 (6219)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6220)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6221)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6222)

λ(3)ikt(gikV

MAXik ) = 0 (6223)

chMAXit ) = 0 (6224)

λ(5)it (pdchMAX

i ) = 0 (6226)

Storage term

+Vitsumk

pit ) = 0

(6227)

(Vitsumk

(Vktsumi

partδit= λ

(1)it (Vit

partδkt= λ

(1)kt(Vkt

ik (6231)

λ(1)it

with sumit

λ(3)iktp

MAXij minus

λ(1)it gikV

2it (6233)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6237)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6238)

(7)it minus λ

(7)itminus1 = 0 (6239)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6240)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6243)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6244)

From (6236) and (6212)

minussumit

λ(7)it (EMAX

From (6239)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6248)

minussumit

(6)it E

MAXi +

Consequently

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6250)

i ) (6251)

λ(1)it (pit)minus

λ(3)iktp

MAXij︸︷︷︸

FTR NEW

+sumit

dchit︸︷︷︸

λ(1)it gikV

2it = 0

(6252)

Chapter 7

Simulations

rampup

7 Simulations

G = ltY =R

R2 +X2(711)

B = =Y = minus X

R2 +X2(712)

7 Simulations

2(724)

And therefore

2(725)

2(726)

7 Simulations

And finally

The 12

7 Simulations

λ(1)it pit = MS (741)

7 Simulations

Chapter 8

Conclusion

Chapter 9

Future Work

91 Model Extensions

9 Future Work

92 V2G

9 Future Work

Bibliography

Contents

List of Figures

List of Tables

1 Introduction




31 The pu System



34 Applications

341 DC-OPF



4 Energy Storage


42 Technologies







5 Financial Rights

51 Grid Congestions



61 DC-OPF

62 AC-OPF

7 Simulations



73 Storage Role


8 Conclusion

9 Future Work

91 Model Extensions

92 V2G

Bibliography

76 LMPs [ k$pu

pu] 88

pu] 89

List of Tables

Chapter 1

Introduction

1 Introduction

Chapter 2

ej(x) = 0 j = 1 (203)(204)

min f(xi) = cixi

st Ax le b

x ge 0

or in matrix form

min cTx

st Ax le b

x ge 0

λiii(x) +sumj

microjej(x) (211)

λiii(x) +sumj

microjej(x)) (212)

ej(xlowast) = 0 j = 1 (223)

λi ge 0 (225)

Chapter 3

[Y ] =

Y 11 Y 1n

Y n1 Y nn

ZL = (r + jωl)a (302)

Y s = (g + jωc)a

2(303)

impedances)

(Y sij + Y Lij) (304)

= [Y bus]

V 1V n

Xpu =X

3Vbase

=V 2base

Ybase =1

3 andradic

Y ijV j (322)

Pi = Visumj

Qi = Visumj

Pi = Visumj

Qi = Visumj

minsumiisinG

CG (331)

Gi (334)

Gi (335)

i (336)

P 2ij + Q2

ij le S2ijMAX (339)

I ij =ysij

(Ylowastsij

2Vlowasti + Y

lowastLij(V

lowasti minus V

lowastj)

= |V |2i(Ylowastsij

lowastLij

lowastLij =

(3312)

= |V i|2(Ylowastsij

lowastLij

(3313)

Y Lij = Gij + jBij

YlowastSij

2= minusjBSij

Pij = ltSij (3314)

Qij = =Sij (3315)

Consequently

2+Gij minus jBij

(3317)

Smax = 3Vradic

3Imax (3319)

sumi PGi = PL

calculated as

= (25$

MWhminus 20

MWh)100MW = 500

Chapter 4

Energy Storage

4 Energy Storage

storage (413)

∆t (414)

storage

4 Energy Storage

the grid [38]

P 2it +Q2

it le S2MAXit (416)

42 Technologies

4 Energy Storage

Chapter 5

Financial Rights

51 Grid Congestions

5 Financial Rights

h(511)

5 Financial Rights

h(512)

h(513)

h(514)

5 Financial Rights

MWh(515)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T2)

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

2(521)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

5 Financial Rights

Chapter 6

minsumit

fit(pit) (611)

it (613)

storage (617)

∆t (618)

(2U)it

(3)ikt

(4L)it λ

(5)it λ

(5U)it

L(pi δi δk pchi p

dchi SOCi λi)t =

(fit(pit))

+ λ(2L)it (pMIN

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

partpit=partOF

(2L)it + λ

(2U)it = 0 (6110)

partδit= λ

(bik) + λ(3)ikt

(bik) = 0 (6111)

partlkpartδkt

= λ(1)kt

(bik) + λ(3)kit

(bik) = 0 (6112)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6113)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6114)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6115)

MAXik ) = 0 (6116)

chMAXit ) = 0 (6117)

λ(5)it (pdchMAX

i ) = 0 (6119)

Storage term

pit ) = 0 (6120)

partδit= (λ

(1)it + λ

(3)ikt)

(bik) = 0 (6121)

λ(1)it

λ(3)iktp

MAXik (6122)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6126)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6127)

minusλ(6L)it + λ

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6128)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6129)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6132)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6133)

minussumit

λ(7)it (EMAX

From (6128)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6137)

According to (6128)

minussumit

(6)it E

MAXi +

And therefore

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6139)

i ) (6140)

λ(1)it (pit)minus

λ(3)iktp

MAXik︸︷︷︸

+sumit

dchit︸︷︷︸

(6141)

minsumit

fit(pit) (621)

qit = Visumk

it (624)

it (625)

P 2ik + Q2

ik le S2ikMAX (626)

it (627)

it (628)

storage (6211)

∆t (6212)

Qik = minusV 2i

(bSik2

(6214)

ik (6215)

qit = Visumk

(2U)it

(9U)it

(10U)it

(11U)it

(4L)it λ

(5)it λ

(5U)it

l(pi δi δk pchi p

(fit(pit))

+ λ(2L)it (pMIN

+ λ(3)ikt(gikV

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

∆tminus SOCit+1)

+ λ(8)it (Vi

+ λ(9L)it (qMIN

+ λ(10L)it (V MIN

+ λ(11L)it (δMIN

(6216)

partpit=partOF

(2L)it + λ

(2U)it = 0 (6217)

partδit= λ

(1)it (Vit

+ λ(8)ikt(Vit

minus λ(11L)it + λ(11U)it = 0 (6218)

partδkt= λ

(1)kt(Vkt

+ λ(8)kt(Vkt

minus λ(11L)kt + λ(11U)kt = 0 (6219)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6220)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6221)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6222)

λ(3)ikt(gikV

MAXik ) = 0 (6223)

chMAXit ) = 0 (6224)

λ(5)it (pdchMAX

i ) = 0 (6226)

Storage term

+Vitsumk

pit ) = 0

(6227)

(Vitsumk

(Vktsumi

partδit= λ

(1)it (Vit

partδkt= λ

(1)kt(Vkt

ik (6231)

λ(1)it

with sumit

λ(3)iktp

MAXij minus

λ(1)it gikV

2it (6233)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6237)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6238)

(7)it minus λ

(7)itminus1 = 0 (6239)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6240)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6243)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6244)

From (6236) and (6212)

minussumit

λ(7)it (EMAX

From (6239)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6248)

minussumit

(6)it E

MAXi +

Consequently

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6250)

i ) (6251)

λ(1)it (pit)minus

λ(3)iktp

MAXij︸︷︷︸

FTR NEW

+sumit

dchit︸︷︷︸

λ(1)it gikV

2it = 0

(6252)

Chapter 7

Simulations

rampup

7 Simulations

G = ltY =R

R2 +X2(711)

B = =Y = minus X

R2 +X2(712)

7 Simulations

2(724)

And therefore

2(725)

2(726)

7 Simulations

And finally

The 12

7 Simulations

λ(1)it pit = MS (741)

7 Simulations

Chapter 8

Conclusion

Chapter 9

Future Work

91 Model Extensions

9 Future Work

92 V2G

9 Future Work

Bibliography

Contents

List of Figures

List of Tables

1 Introduction




31 The pu System



34 Applications

341 DC-OPF



4 Energy Storage


42 Technologies







5 Financial Rights

51 Grid Congestions



61 DC-OPF

62 AC-OPF

7 Simulations



73 Storage Role


8 Conclusion

9 Future Work

91 Model Extensions

92 V2G

Bibliography

Chapter 1

Introduction

1 Introduction

Chapter 2

ej(x) = 0 j = 1 (203)(204)

min f(xi) = cixi

st Ax le b

x ge 0

or in matrix form

min cTx

st Ax le b

x ge 0

λiii(x) +sumj

microjej(x) (211)

λiii(x) +sumj

microjej(x)) (212)

ej(xlowast) = 0 j = 1 (223)

λi ge 0 (225)

Chapter 3

[Y ] =

Y 11 Y 1n

Y n1 Y nn

ZL = (r + jωl)a (302)

Y s = (g + jωc)a

2(303)

impedances)

(Y sij + Y Lij) (304)

= [Y bus]

V 1V n

Xpu =X

3Vbase

=V 2base

Ybase =1

3 andradic

Y ijV j (322)

Pi = Visumj

Qi = Visumj

Pi = Visumj

Qi = Visumj

minsumiisinG

CG (331)

Gi (334)

Gi (335)

i (336)

P 2ij + Q2

ij le S2ijMAX (339)

I ij =ysij

(Ylowastsij

2Vlowasti + Y

lowastLij(V

lowasti minus V

lowastj)

= |V |2i(Ylowastsij

lowastLij

lowastLij =

(3312)

= |V i|2(Ylowastsij

lowastLij

(3313)

Y Lij = Gij + jBij

YlowastSij

2= minusjBSij

Pij = ltSij (3314)

Qij = =Sij (3315)

Consequently

2+Gij minus jBij

(3317)

Smax = 3Vradic

3Imax (3319)

sumi PGi = PL

calculated as

= (25$

MWhminus 20

MWh)100MW = 500

Chapter 4

Energy Storage

4 Energy Storage

storage (413)

∆t (414)

storage

4 Energy Storage

the grid [38]

P 2it +Q2

it le S2MAXit (416)

42 Technologies

4 Energy Storage

Chapter 5

Financial Rights

51 Grid Congestions

5 Financial Rights

h(511)

5 Financial Rights

h(512)

h(513)

h(514)

5 Financial Rights

MWh(515)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T2)

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

2(521)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

5 Financial Rights

Chapter 6

minsumit

fit(pit) (611)

it (613)

storage (617)

∆t (618)

(2U)it

(3)ikt

(4L)it λ

(5)it λ

(5U)it

L(pi δi δk pchi p

dchi SOCi λi)t =

(fit(pit))

+ λ(2L)it (pMIN

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

partpit=partOF

(2L)it + λ

(2U)it = 0 (6110)

partδit= λ

(bik) + λ(3)ikt

(bik) = 0 (6111)

partlkpartδkt

= λ(1)kt

(bik) + λ(3)kit

(bik) = 0 (6112)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6113)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6114)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6115)

MAXik ) = 0 (6116)

chMAXit ) = 0 (6117)

λ(5)it (pdchMAX

i ) = 0 (6119)

Storage term

pit ) = 0 (6120)

partδit= (λ

(1)it + λ

(3)ikt)

(bik) = 0 (6121)

λ(1)it

λ(3)iktp

MAXik (6122)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6126)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6127)

minusλ(6L)it + λ

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6128)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6129)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6132)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6133)

minussumit

λ(7)it (EMAX

From (6128)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6137)

According to (6128)

minussumit

(6)it E

MAXi +

And therefore

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6139)

i ) (6140)

λ(1)it (pit)minus

λ(3)iktp

MAXik︸︷︷︸

+sumit

dchit︸︷︷︸

(6141)

minsumit

fit(pit) (621)

qit = Visumk

it (624)

it (625)

P 2ik + Q2

ik le S2ikMAX (626)

it (627)

it (628)

storage (6211)

∆t (6212)

Qik = minusV 2i

(bSik2

(6214)

ik (6215)

qit = Visumk

(2U)it

(9U)it

(10U)it

(11U)it

(4L)it λ

(5)it λ

(5U)it

l(pi δi δk pchi p

(fit(pit))

+ λ(2L)it (pMIN

+ λ(3)ikt(gikV

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

∆tminus SOCit+1)

+ λ(8)it (Vi

+ λ(9L)it (qMIN

+ λ(10L)it (V MIN

+ λ(11L)it (δMIN

(6216)

partpit=partOF

(2L)it + λ

(2U)it = 0 (6217)

partδit= λ

(1)it (Vit

+ λ(8)ikt(Vit

minus λ(11L)it + λ(11U)it = 0 (6218)

partδkt= λ

(1)kt(Vkt

+ λ(8)kt(Vkt

minus λ(11L)kt + λ(11U)kt = 0 (6219)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6220)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6221)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6222)

λ(3)ikt(gikV

MAXik ) = 0 (6223)

chMAXit ) = 0 (6224)

λ(5)it (pdchMAX

i ) = 0 (6226)

Storage term

+Vitsumk

pit ) = 0

(6227)

(Vitsumk

(Vktsumi

partδit= λ

(1)it (Vit

partδkt= λ

(1)kt(Vkt

ik (6231)

λ(1)it

with sumit

λ(3)iktp

MAXij minus

λ(1)it gikV

2it (6233)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6237)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6238)

(7)it minus λ

(7)itminus1 = 0 (6239)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6240)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6243)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6244)

From (6236) and (6212)

minussumit

λ(7)it (EMAX

From (6239)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6248)

minussumit

(6)it E

MAXi +

Consequently

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6250)

i ) (6251)

λ(1)it (pit)minus

λ(3)iktp

MAXij︸︷︷︸

FTR NEW

+sumit

dchit︸︷︷︸

λ(1)it gikV

2it = 0

(6252)

Chapter 7

Simulations

rampup

7 Simulations

G = ltY =R

R2 +X2(711)

B = =Y = minus X

R2 +X2(712)

7 Simulations

2(724)

And therefore

2(725)

2(726)

7 Simulations

And finally

The 12

7 Simulations

λ(1)it pit = MS (741)

7 Simulations

Chapter 8

Conclusion

Chapter 9

Future Work

91 Model Extensions

9 Future Work

92 V2G

9 Future Work

Bibliography

Contents

List of Figures

List of Tables

1 Introduction




31 The pu System



34 Applications

341 DC-OPF



4 Energy Storage


42 Technologies







5 Financial Rights

51 Grid Congestions



61 DC-OPF

62 AC-OPF

7 Simulations



73 Storage Role


8 Conclusion

9 Future Work

91 Model Extensions

92 V2G

Bibliography

1 Introduction

Chapter 2

ej(x) = 0 j = 1 (203)(204)

min f(xi) = cixi

st Ax le b

x ge 0

or in matrix form

min cTx

st Ax le b

x ge 0

λiii(x) +sumj

microjej(x) (211)

λiii(x) +sumj

microjej(x)) (212)

ej(xlowast) = 0 j = 1 (223)

λi ge 0 (225)

Chapter 3

[Y ] =

Y 11 Y 1n

Y n1 Y nn

ZL = (r + jωl)a (302)

Y s = (g + jωc)a

2(303)

impedances)

(Y sij + Y Lij) (304)

= [Y bus]

V 1V n

Xpu =X

3Vbase

=V 2base

Ybase =1

3 andradic

Y ijV j (322)

Pi = Visumj

Qi = Visumj

Pi = Visumj

Qi = Visumj

minsumiisinG

CG (331)

Gi (334)

Gi (335)

i (336)

P 2ij + Q2

ij le S2ijMAX (339)

I ij =ysij

(Ylowastsij

2Vlowasti + Y

lowastLij(V

lowasti minus V

lowastj)

= |V |2i(Ylowastsij

lowastLij

lowastLij =

(3312)

= |V i|2(Ylowastsij

lowastLij

(3313)

Y Lij = Gij + jBij

YlowastSij

2= minusjBSij

Pij = ltSij (3314)

Qij = =Sij (3315)

Consequently

2+Gij minus jBij

(3317)

Smax = 3Vradic

3Imax (3319)

sumi PGi = PL

calculated as

= (25$

MWhminus 20

MWh)100MW = 500

Chapter 4

Energy Storage

4 Energy Storage

storage (413)

∆t (414)

storage

4 Energy Storage

the grid [38]

P 2it +Q2

it le S2MAXit (416)

42 Technologies

4 Energy Storage

Chapter 5

Financial Rights

51 Grid Congestions

5 Financial Rights

h(511)

5 Financial Rights

h(512)

h(513)

h(514)

5 Financial Rights

MWh(515)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T2)

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

2(521)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

5 Financial Rights

Chapter 6

minsumit

fit(pit) (611)

it (613)

storage (617)

∆t (618)

(2U)it

(3)ikt

(4L)it λ

(5)it λ

(5U)it

L(pi δi δk pchi p

dchi SOCi λi)t =

(fit(pit))

+ λ(2L)it (pMIN

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

partpit=partOF

(2L)it + λ

(2U)it = 0 (6110)

partδit= λ

(bik) + λ(3)ikt

(bik) = 0 (6111)

partlkpartδkt

= λ(1)kt

(bik) + λ(3)kit

(bik) = 0 (6112)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6113)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6114)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6115)

MAXik ) = 0 (6116)

chMAXit ) = 0 (6117)

λ(5)it (pdchMAX

i ) = 0 (6119)

Storage term

pit ) = 0 (6120)

partδit= (λ

(1)it + λ

(3)ikt)

(bik) = 0 (6121)

λ(1)it

λ(3)iktp

MAXik (6122)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6126)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6127)

minusλ(6L)it + λ

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6128)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6129)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6132)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6133)

minussumit

λ(7)it (EMAX

From (6128)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6137)

According to (6128)

minussumit

(6)it E

MAXi +

And therefore

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6139)

i ) (6140)

λ(1)it (pit)minus

λ(3)iktp

MAXik︸︷︷︸

+sumit

dchit︸︷︷︸

(6141)

minsumit

fit(pit) (621)

qit = Visumk

it (624)

it (625)

P 2ik + Q2

ik le S2ikMAX (626)

it (627)

it (628)

storage (6211)

∆t (6212)

Qik = minusV 2i

(bSik2

(6214)

ik (6215)

qit = Visumk

(2U)it

(9U)it

(10U)it

(11U)it

(4L)it λ

(5)it λ

(5U)it

l(pi δi δk pchi p

(fit(pit))

+ λ(2L)it (pMIN

+ λ(3)ikt(gikV

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

∆tminus SOCit+1)

+ λ(8)it (Vi

+ λ(9L)it (qMIN

+ λ(10L)it (V MIN

+ λ(11L)it (δMIN

(6216)

partpit=partOF

(2L)it + λ

(2U)it = 0 (6217)

partδit= λ

(1)it (Vit

+ λ(8)ikt(Vit

minus λ(11L)it + λ(11U)it = 0 (6218)

partδkt= λ

(1)kt(Vkt

+ λ(8)kt(Vkt

minus λ(11L)kt + λ(11U)kt = 0 (6219)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6220)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6221)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6222)

λ(3)ikt(gikV

MAXik ) = 0 (6223)

chMAXit ) = 0 (6224)

λ(5)it (pdchMAX

i ) = 0 (6226)

Storage term

+Vitsumk

pit ) = 0

(6227)

(Vitsumk

(Vktsumi

partδit= λ

(1)it (Vit

partδkt= λ

(1)kt(Vkt

ik (6231)

λ(1)it

with sumit

λ(3)iktp

MAXij minus

λ(1)it gikV

2it (6233)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6237)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6238)

(7)it minus λ

(7)itminus1 = 0 (6239)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6240)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6243)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6244)

From (6236) and (6212)

minussumit

λ(7)it (EMAX

From (6239)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6248)

minussumit

(6)it E

MAXi +

Consequently

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6250)

i ) (6251)

λ(1)it (pit)minus

λ(3)iktp

MAXij︸︷︷︸

FTR NEW

+sumit

dchit︸︷︷︸

λ(1)it gikV

2it = 0

(6252)

Chapter 7

Simulations

rampup

7 Simulations

G = ltY =R

R2 +X2(711)

B = =Y = minus X

R2 +X2(712)

7 Simulations

2(724)

And therefore

2(725)

2(726)

7 Simulations

And finally

The 12

7 Simulations

λ(1)it pit = MS (741)

7 Simulations

Chapter 8

Conclusion

Chapter 9

Future Work

91 Model Extensions

9 Future Work

92 V2G

9 Future Work

Bibliography

Contents

List of Figures

List of Tables

1 Introduction




31 The pu System



34 Applications

341 DC-OPF



4 Energy Storage


42 Technologies







5 Financial Rights

51 Grid Congestions



61 DC-OPF

62 AC-OPF

7 Simulations



73 Storage Role


8 Conclusion

9 Future Work

91 Model Extensions

92 V2G

Bibliography

1 Introduction

Chapter 2

ej(x) = 0 j = 1 (203)(204)

min f(xi) = cixi

st Ax le b

x ge 0

or in matrix form

min cTx

st Ax le b

x ge 0

λiii(x) +sumj

microjej(x) (211)

λiii(x) +sumj

microjej(x)) (212)

ej(xlowast) = 0 j = 1 (223)

λi ge 0 (225)

Chapter 3

[Y ] =

Y 11 Y 1n

Y n1 Y nn

ZL = (r + jωl)a (302)

Y s = (g + jωc)a

2(303)

impedances)

(Y sij + Y Lij) (304)

= [Y bus]

V 1V n

Xpu =X

3Vbase

=V 2base

Ybase =1

3 andradic

Y ijV j (322)

Pi = Visumj

Qi = Visumj

Pi = Visumj

Qi = Visumj

minsumiisinG

CG (331)

Gi (334)

Gi (335)

i (336)

P 2ij + Q2

ij le S2ijMAX (339)

I ij =ysij

(Ylowastsij

2Vlowasti + Y

lowastLij(V

lowasti minus V

lowastj)

= |V |2i(Ylowastsij

lowastLij

lowastLij =

(3312)

= |V i|2(Ylowastsij

lowastLij

(3313)

Y Lij = Gij + jBij

YlowastSij

2= minusjBSij

Pij = ltSij (3314)

Qij = =Sij (3315)

Consequently

2+Gij minus jBij

(3317)

Smax = 3Vradic

3Imax (3319)

sumi PGi = PL

calculated as

= (25$

MWhminus 20

MWh)100MW = 500

Chapter 4

Energy Storage

4 Energy Storage

storage (413)

∆t (414)

storage

4 Energy Storage

the grid [38]

P 2it +Q2

it le S2MAXit (416)

42 Technologies

4 Energy Storage

Chapter 5

Financial Rights

51 Grid Congestions

5 Financial Rights

h(511)

5 Financial Rights

h(512)

h(513)

h(514)

5 Financial Rights

MWh(515)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T2)

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

2(521)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

5 Financial Rights

Chapter 6

minsumit

fit(pit) (611)

it (613)

storage (617)

∆t (618)

(2U)it

(3)ikt

(4L)it λ

(5)it λ

(5U)it

L(pi δi δk pchi p

dchi SOCi λi)t =

(fit(pit))

+ λ(2L)it (pMIN

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

partpit=partOF

(2L)it + λ

(2U)it = 0 (6110)

partδit= λ

(bik) + λ(3)ikt

(bik) = 0 (6111)

partlkpartδkt

= λ(1)kt

(bik) + λ(3)kit

(bik) = 0 (6112)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6113)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6114)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6115)

MAXik ) = 0 (6116)

chMAXit ) = 0 (6117)

λ(5)it (pdchMAX

i ) = 0 (6119)

Storage term

pit ) = 0 (6120)

partδit= (λ

(1)it + λ

(3)ikt)

(bik) = 0 (6121)

λ(1)it

λ(3)iktp

MAXik (6122)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6126)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6127)

minusλ(6L)it + λ

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6128)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6129)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6132)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6133)

minussumit

λ(7)it (EMAX

From (6128)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6137)

According to (6128)

minussumit

(6)it E

MAXi +

And therefore

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6139)

i ) (6140)

λ(1)it (pit)minus

λ(3)iktp

MAXik︸︷︷︸

+sumit

dchit︸︷︷︸

(6141)

minsumit

fit(pit) (621)

qit = Visumk

it (624)

it (625)

P 2ik + Q2

ik le S2ikMAX (626)

it (627)

it (628)

storage (6211)

∆t (6212)

Qik = minusV 2i

(bSik2

(6214)

ik (6215)

qit = Visumk

(2U)it

(9U)it

(10U)it

(11U)it

(4L)it λ

(5)it λ

(5U)it

l(pi δi δk pchi p

(fit(pit))

+ λ(2L)it (pMIN

+ λ(3)ikt(gikV

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

∆tminus SOCit+1)

+ λ(8)it (Vi

+ λ(9L)it (qMIN

+ λ(10L)it (V MIN

+ λ(11L)it (δMIN

(6216)

partpit=partOF

(2L)it + λ

(2U)it = 0 (6217)

partδit= λ

(1)it (Vit

+ λ(8)ikt(Vit

minus λ(11L)it + λ(11U)it = 0 (6218)

partδkt= λ

(1)kt(Vkt

+ λ(8)kt(Vkt

minus λ(11L)kt + λ(11U)kt = 0 (6219)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6220)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6221)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6222)

λ(3)ikt(gikV

MAXik ) = 0 (6223)

chMAXit ) = 0 (6224)

λ(5)it (pdchMAX

i ) = 0 (6226)

Storage term

+Vitsumk

pit ) = 0

(6227)

(Vitsumk

(Vktsumi

partδit= λ

(1)it (Vit

partδkt= λ

(1)kt(Vkt

ik (6231)

λ(1)it

with sumit

λ(3)iktp

MAXij minus

λ(1)it gikV

2it (6233)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6237)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6238)

(7)it minus λ

(7)itminus1 = 0 (6239)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6240)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6243)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6244)

From (6236) and (6212)

minussumit

λ(7)it (EMAX

From (6239)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6248)

minussumit

(6)it E

MAXi +

Consequently

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6250)

i ) (6251)

λ(1)it (pit)minus

λ(3)iktp

MAXij︸︷︷︸

FTR NEW

+sumit

dchit︸︷︷︸

λ(1)it gikV

2it = 0

(6252)

Chapter 7

Simulations

rampup

7 Simulations

G = ltY =R

R2 +X2(711)

B = =Y = minus X

R2 +X2(712)

7 Simulations

2(724)

And therefore

2(725)

2(726)

7 Simulations

And finally

The 12

7 Simulations

λ(1)it pit = MS (741)

7 Simulations

Chapter 8

Conclusion

Chapter 9

Future Work

91 Model Extensions

9 Future Work

92 V2G

9 Future Work

Bibliography

Contents

List of Figures

List of Tables

1 Introduction




31 The pu System



34 Applications

341 DC-OPF



4 Energy Storage


42 Technologies







5 Financial Rights

51 Grid Congestions



61 DC-OPF

62 AC-OPF

7 Simulations



73 Storage Role


8 Conclusion

9 Future Work

91 Model Extensions

92 V2G

Bibliography

1 Introduction

Chapter 2

ej(x) = 0 j = 1 (203)(204)

min f(xi) = cixi

st Ax le b

x ge 0

or in matrix form

min cTx

st Ax le b

x ge 0

λiii(x) +sumj

microjej(x) (211)

λiii(x) +sumj

microjej(x)) (212)

ej(xlowast) = 0 j = 1 (223)

λi ge 0 (225)

Chapter 3

[Y ] =

Y 11 Y 1n

Y n1 Y nn

ZL = (r + jωl)a (302)

Y s = (g + jωc)a

2(303)

impedances)

(Y sij + Y Lij) (304)

= [Y bus]

V 1V n

Xpu =X

3Vbase

=V 2base

Ybase =1

3 andradic

Y ijV j (322)

Pi = Visumj

Qi = Visumj

Pi = Visumj

Qi = Visumj

minsumiisinG

CG (331)

Gi (334)

Gi (335)

i (336)

P 2ij + Q2

ij le S2ijMAX (339)

I ij =ysij

(Ylowastsij

2Vlowasti + Y

lowastLij(V

lowasti minus V

lowastj)

= |V |2i(Ylowastsij

lowastLij

lowastLij =

(3312)

= |V i|2(Ylowastsij

lowastLij

(3313)

Y Lij = Gij + jBij

YlowastSij

2= minusjBSij

Pij = ltSij (3314)

Qij = =Sij (3315)

Consequently

2+Gij minus jBij

(3317)

Smax = 3Vradic

3Imax (3319)

sumi PGi = PL

calculated as

= (25$

MWhminus 20

MWh)100MW = 500

Chapter 4

Energy Storage

4 Energy Storage

storage (413)

∆t (414)

storage

4 Energy Storage

the grid [38]

P 2it +Q2

it le S2MAXit (416)

42 Technologies

4 Energy Storage

Chapter 5

Financial Rights

51 Grid Congestions

5 Financial Rights

h(511)

5 Financial Rights

h(512)

h(513)

h(514)

5 Financial Rights

MWh(515)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T2)

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

2(521)

5 Financial Rights

0 1 2 3 4 50

P (T1)P (T1) + CP (T2)

P (T2)minus C

5 Financial Rights

Chapter 6

minsumit

fit(pit) (611)

it (613)

storage (617)

∆t (618)

(2U)it

(3)ikt

(4L)it λ

(5)it λ

(5U)it

L(pi δi δk pchi p

dchi SOCi λi)t =

(fit(pit))

+ λ(2L)it (pMIN

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

partpit=partOF

(2L)it + λ

(2U)it = 0 (6110)

partδit= λ

(bik) + λ(3)ikt

(bik) = 0 (6111)

partlkpartδkt

= λ(1)kt

(bik) + λ(3)kit

(bik) = 0 (6112)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6113)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6114)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6115)

MAXik ) = 0 (6116)

chMAXit ) = 0 (6117)

λ(5)it (pdchMAX

i ) = 0 (6119)

Storage term

pit ) = 0 (6120)

partδit= (λ

(1)it + λ

(3)ikt)

(bik) = 0 (6121)

λ(1)it

λ(3)iktp

MAXik (6122)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6126)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6127)

minusλ(6L)it + λ

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6128)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6129)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6132)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6133)

minussumit

λ(7)it (EMAX

From (6128)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6137)

According to (6128)

minussumit

(6)it E

MAXi +

And therefore

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6139)

i ) (6140)

λ(1)it (pit)minus

λ(3)iktp

MAXik︸︷︷︸

+sumit

dchit︸︷︷︸

(6141)

minsumit

fit(pit) (621)

qit = Visumk

it (624)

it (625)

P 2ik + Q2

ik le S2ikMAX (626)

it (627)

it (628)

storage (6211)

∆t (6212)

Qik = minusV 2i

(bSik2

(6214)

ik (6215)

qit = Visumk

(2U)it

(9U)it

(10U)it

(11U)it

(4L)it λ

(5)it λ

(5U)it

l(pi δi δk pchi p

(fit(pit))

+ λ(2L)it (pMIN

+ λ(3)ikt(gikV

MAXik )

chMAXit )

+ λ(5)it (pdchMAX

+ λ(6L)it (EMIN

pdchitηdch

∆tminus SOCit+1)

+ λ(8)it (Vi

+ λ(9L)it (qMIN

+ λ(10L)it (V MIN

+ λ(11L)it (δMIN

(6216)

partpit=partOF

(2L)it + λ

(2U)it = 0 (6217)

partδit= λ

(1)it (Vit

+ λ(8)ikt(Vit

minus λ(11L)it + λ(11U)it = 0 (6218)

partδkt= λ

(1)kt(Vkt

+ λ(8)kt(Vkt

minus λ(11L)kt + λ(11U)kt = 0 (6219)

partpchit= λ

(1)it minus λ

(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6220)

partpdchit= λ

(1)it minus λ

(5)it + λ

(5U)it +

λ(7)it

ηdch= 0 (6221)

(6)it + λ

(7)it minus λ

(7)itminus1 = 0 (6222)

λ(3)ikt(gikV

MAXik ) = 0 (6223)

chMAXit ) = 0 (6224)

λ(5)it (pdchMAX

i ) = 0 (6226)

Storage term

+Vitsumk

pit ) = 0

(6227)

(Vitsumk

(Vktsumi

partδit= λ

(1)it (Vit

partδkt= λ

(1)kt(Vkt

ik (6231)

λ(1)it

with sumit

λ(3)iktp

MAXij minus

λ(1)it gikV

2it (6233)

λ(1)it minus

λ(4L)it + λ

(4)it + ηchλ

(7)it = 0 (6237)

λ(1)it minus λ

(5)it +

λ(5U)it +

λ(7)it

ηdch= 0 (6238)

(7)it minus λ

(7)itminus1 = 0 (6239)

λ(1)it (pchit) +

λ(1)it (pdchit ) (6240)

λ(1)it (pchit) =

λ(1)it (pdchit ) =

(λ(5)it minus

λ(7)it

And therefore

(λ(5)it minus

λ(7)it

ηdch)(pdchit )

=sumit

λ(7)it

ηdch(pdchit ))

(6243)

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) (6244)

From (6236) and (6212)

minussumit

λ(7)it (EMAX

From (6239)

minussumit

(λ(6)it minus λ

(7)itminus1)(E

minussumit

(6)it E

MAXi + λ

λ(7)it

λ(7)itminus1

=SOCitSOCitminus1

(6248)

minussumit

(6)it E

MAXi +

Consequently

minussumit

λ(7)it (ηchp

chit minus

pdchitηdch

) = λ(6)it E

MAXi (6250)

i ) (6251)

λ(1)it (pit)minus

λ(3)iktp

MAXij︸︷︷︸

FTR NEW

+sumit

dchit︸︷︷︸

λ(1)it gikV

2it = 0

(6252)

Chapter 7

Simulations

rampup

7 Simulations

G = ltY =R

R2 +X2(711)

B = =Y = minus X

R2 +X2(712)

7 Simulations

2(724)

And therefore

2(725)

2(726)

7 Simulations

And finally

The 12

7 Simulations

λ(1)it pit = MS (741)

7 Simulations

Chapter 8

Conclusion

Chapter 9

Future Work

91 Model Extensions

9 Future Work

92 V2G

9 Future Work

Bibliography

Contents

List of Figures

List of Tables

1 Introduction




31 The pu System



34 Applications

341 DC-OPF



4 Energy Storage


42 Technologies







5 Financial Rights

51 Grid Congestions



61 DC-OPF

62 AC-OPF

7 Simulations



73 Storage Role


8 Conclusion

9 Future Work

91 Model Extensions

92 V2G

Bibliography

Documents

Master’s Thesis Financial Storage Rights for Integration