eeh
power systems
laboratory
Haoyuan Qu
Three-Stage Stochastic Market-Clearing
Model for the Swiss Reserve Market
Master Thesis
PSL 1519
Department:
EEH – Power Systems Laboratory, ETH Zürich
In collaboration with Swissgrid Ltd
Examiner:
Prof. Dr. Göran Andersson, ETH Zürich
Supervisor:
Farzaneh Abbaspourtorbati, Swissgrid
Line Roald, ETH Zürich
Dr. Marek Zima, Swissgrid
Zürich, December 16, 2015
ii
Abstract
The topic of this master thesis originates from the reserve procurement process in Switzerland. Currently, a two-stage reserve market has been operated where secondary control reserves are procured in a weekly auction and
tertiary control reserves are split between weekly and daily auctions. In order to make use of additional available power from producers and to allow
the participation of Renewable Energy Sources (RES), a third market stage
which is closer to real-time operation is likely to be established in the future,
converting the reserve procurement process into a three-stage problem.
The chief objective of this master thesis is to develop a three-stage
stochastic market-clearing model for the Swiss reserve market. Within the
framework of this thesis, scenarios for daily market are scrutinized and improved, which can be readily appended to the current two-stage stochastic
market-clearing model. Scenarios for the third stage are generated based on
reference data in the current market and various cases are simulated. Simulation results show that both improvements on the two-stage model and
the incorporation of an additional third stage could lead to cost savings for
Transmission System Operator (TSO).
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iv
Acknowledgements
This thesis is the outcome of research work done in cooperation between
Power Systems Laboratory (PSL) at ETH and Swissgrid. After six months
of efforts, it concludes my master’s studies in Energy Science and Technology at ETH and is one of the most important milestones in my life.
First, I would like to express my gratitude towards Prof. Dr. Göran Andersson for being my tutor and enabling this collaboration with Swissgrid.
My profound thanks go to Dr. Marek Zima, who has provided me with
the opportunity of conducting research at Swissgrid and introduced me to
the fascinating world of ancillary services market.
Furthermore, I would like to give my deepest appreciation to my supervisor
at PSL, Line Roald and my supervisor at Swissgrid, Farzaneh Abbaspourtorbati. Without their continuous support and valuable input, I would not
have managed to come to this final stage.
I should not forget to thank my colleagues from Swissgrid who provided
me with all sorts of support, be it technically or spiritually.
Last but not least, I would like to thank and share this piece of work with
my beloved family and friends for their lasting love, patience and support.
Zürich, December 2015
Haoyuan
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Contents
List of Figures
x
List of Tables
xi
List of Acronyms
xiii
List of Symbols
1 Introduction
1.1 Balancing Reserves . . .
1.2 Reserve Market . . . . .
1.3 Stochastic Programming
1.4 Structure of the Thesis .
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2 Reserve Market in Switzerland
2.1 Self-scheduling Market . . . . . . . . . . . . . . .
2.2 Overview of Ancillary Services . . . . . . . . . .
2.3 Structure of Reserve Market . . . . . . . . . . . .
2.3.1 Primary Control Reserves . . . . . . . . .
2.3.2 Secondary and Tertiary Control Reserves
2.4 Bid Structure . . . . . . . . . . . . . . . . . . . .
2.4.1 Indivisible Bids . . . . . . . . . . . . . . .
2.4.2 Conditional Bids . . . . . . . . . . . . . .
2.5 Dimensioning Criteria . . . . . . . . . . . . . . .
2.5.1 Probabilistic Approach . . . . . . . . . . .
2.5.2 Deterministic Approach . . . . . . . . . .
2.6 Remuneration Scheme . . . . . . . . . . . . . . .
2.6.1 Remuneration of Capacity . . . . . . . . .
2.6.2 Remuneration of Energy . . . . . . . . . .
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3 Two-Stage Market-Clearing Model
19
3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Stochastic Market-Clearing Model . . . . . . . . . . . . . . . 20
3.2.1 Decision Variables . . . . . . . . . . . . . . . . . . . . 21
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CONTENTS
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4 Three-Stage Market-Clearing Model
4.1 Introduction . . . . . . . . . . . . . . . . . . .
4.2 Stochastic Market-Clearing Model . . . . . .
4.2.1 Decision Variables . . . . . . . . . . .
4.2.2 Objective Function . . . . . . . . . . .
4.2.3 Non-anticipativity Matrix . . . . . . .
4.2.4 Constraints . . . . . . . . . . . . . . .
4.2.5 Formulation . . . . . . . . . . . . . . .
4.3 Scenarios for Hourly Market . . . . . . . . . .
4.3.1 Modelling of Hourly Bid Curves . . .
4.3.2 Scenario Construction . . . . . . . . .
4.4 Simulation Results . . . . . . . . . . . . . . .
4.4.1 Case Study: Impact of Hourly Market
4.4.2 Complete Scenario Simulations . . . .
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3.3
3.2.2 Objective Function . . . . .
3.2.3 Constraints . . . . . . . . .
3.2.4 Formulation . . . . . . . . .
Improvements of Two-Stage Model
3.3.1 Linearization of Bid Curves
3.3.2 Selection of Scenarios . . .
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5 Conclusions and Outlook
67
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
A Hourly Discretizing Factors
69
Bibliography
71
List of Figures
1.1
1.2
Real-time electricity consumption and demand forecast of
November 16, 2015 [1] . . . . . . . . . . . . . . . . . . . . . .
Example of a scenario tree for three-stage problems . . . . . .
2
4
2.1
2.2
2.3
2.4
2.5
Temporal structure of frequency control after a disturbance [2]
Simplified diagram of Swiss ancillary services market [3] . . .
Scheme of a two-stage reserve market in Switzerland [4] . . .
Example of SCR and TCR provision . . . . . . . . . . . . . .
Deficit curves for dimensioning reserves in Switzerland [4] . .
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12
14
16
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
Two-stage stochastic market-clearing scheme . . . . . . . . .
Example of a bid curve (before and after linearization) . . . .
Example of piecewise linearized deficit curve . . . . . . . . . .
Overview of bid curve linearization methods . . . . . . . . . .
Example of bid curve linearization by four methods . . . . . .
Residual of fitted bid curve . . . . . . . . . . . . . . . . . . .
Amount of procured reserves using four fitting methods . . .
Total procurement cost of reserves using four fitting methods
Procurement cost in daily market using four fitting methods
after fixing weekly decision . . . . . . . . . . . . . . . . . . .
3.10 Overview of scenario selection methods . . . . . . . . . . . . .
3.11 Cost difference w.r.t. perfect information scenario . . . . . . .
3.12 Cost difference w.r.t. perfect information scenario (Methods
2, 5 and 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
Three-stage stochastic market-clearing scheme .
Hypothetical hourly bid curve . . . . . . . . . .
Free TCE+ volume in 2015 (Week 01−35) . . .
Free TCE− volume in 2015 (Week 01−35) . . .
Scenario construction process . . . . . . . . . .
Definition of cases . . . . . . . . . . . . . . . .
Amount of reserves procured for Week 35 2015
Amount of reserves procured for Week 04 2015
Estimated cost savings w.r.t two-stage model .
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x
LIST OF FIGURES
4.10
4.11
4.12
4.13
Scenario tree of complete model . . . . . . . . . . .
Reserve amount of complete model . . . . . . . . .
Substitution between TCR+ and TCR− products
Substitution between daily and hourly products . .
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66
66
List of Tables
2.1
2.2
2.3
2.4
2.5
Volume of PCR Cooperation in 2015 [5] . . . .
Bid structure of the Swiss reserve market [5] . .
Example bids for demonstration of indivisibility
Example of conditional bids . . . . . . . . . . .
Remuneration of activated SCR [5] . . . . . . .
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18
3.1
3.2
Performance of linearization methods . . . . . . . . . . . . . .
Estimation of cost savings by improved linearization method
(Week 02−35, 2015) . . . . . . . . . . . . . . . . . . . . . . .
Explanation of week names . . . . . . . . . . . . . . . . . . .
Correlation coefficients between weeks . . . . . . . . . . . . .
RMSE between weeks [CHF/MW] . . . . . . . . . . . . . . .
Information of selected weeks [6] . . . . . . . . . . . . . . . .
Comparison of Methods 2, 5 and 6 w.r.t. perfect information
Estimation of savings by improved scenario selection method
(Week 02−35, 2015) . . . . . . . . . . . . . . . . . . . . . . .
Estimation of savings by implementing both improvements
(Week 02−35, 2015) . . . . . . . . . . . . . . . . . . . . . . .
35
Probability factors of hourly scenarios . . . . .
Potential cost savings w.r.t. two-stage model .
Scenario notations . . . . . . . . . . . . . . . .
Problem size . . . . . . . . . . . . . . . . . . .
Amount of reserves procured in complete model
Total cost of procurement of complete model .
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65
A.1 Hourly discretizing factors . . . . . . . . . . . . . . . . . . . .
70
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.1
4.2
4.3
4.4
4.5
4.6
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xii
LIST OF TABLES
List of Acronyms
TSO
ENTSO-E
AS
BG
PCR
SCR
TCR
FCR
FRR
aFRR
mFRR
RR
AGC
LFC
ACE
RES
MEAS
TCE
MOL
CDF
MILP
RMSE
Transmission System Operator
European Network of Transmission System Operators for Electricity
Ancillary Services
Balance Group
Primary Control Reserves
Secondary Control Reserves
Tertiary Control Reserves
Frequency Containment Reserves
Frequency Restoration Reserves
automatic Frequency Restoration Reserves
manual Frequency Restoration Reserves
Replacement Reserves
Automatic Generation Control
Load Frequency Control
Area Control Error
Renewable Energy Sources
Mutual Emergency Assistance Service
Tertiary Control Energy
Merit Order List
Cumulative Distribution Function
Mixed Integer Linear Programming
Root Mean Square Error
xiii
xiv
LIST OF ACRONYMS
List of Symbols
xw
xw
S
xw
T+
xw
T−
xd
xdT +
xdT −
xh
xhT +
xhT −
Decision variable vector of bids in weekly market
Decision variable vector of SCR bids in weekly market
Decision variable vector of TCR+ bids in weekly market
Decision variable vector of TCR− bids in weekly market
Decision variable vector of reserve procurement in daily market
Vector of TCR+ procurement amount in daily market
Vector of TCR− procurement amount in daily market
Decision variable vector of reserve procurement in hourly market
Vector of TCR+ procurement amount in hourly market
Vector of TCR− procurement amount in hourly market
ωd
ωh
Ωd
Ωh
πd (ω d )
πh (ω h )
Index of scenarios for daily market
Index of scenarios for hourly market
Scenario set for daily market
Scenario set for hourly market
Probability of daily scenario ω d
Probability of hourly scenario ω h
cw
λT + (ω d )
λT − (ω d )
ζT + (ω d , ω h )
ζT − (ω d , ω h )
Cost vector of bids in weekly market
Cost vector of TCR+ in daily market for scenario ω d
Cost vector of TCR− in daily market for scenario ω d
Cost vector of TCR+ in hourly market for scenario combination (ω d , ω h )
Cost vector of TCR− in hourly market for scenario combination (ω d , ω h )
α+
i
α−
i
βi+
βi−
Slope of ith piece of linearized daily bid curve for TCR+
Slope of ith piece of linearized daily bid curve for TCR−
Intercept of ith piece of linearized daily bid curve for TCR+
Intercept of ith piece of linearized daily bid curve for TCR−
xv
xvi
LIST OF SYMBOLS
ρ+
i
ρ−
i
ϕ+
i
ϕ−
i
(i)
as+
(i)
as−
(i)
ao+
(i)
ao−
(i)
bs+
(i)
bs−
(i)
bo+
(i)
bo−
εs+
εs−
εo+
εo−
d
xmin
d,T + (ω )
d
xmax
d,T + (ω )
d
xmin
d,T − (ω )
d
xmax
d,T − (ω )
h
xmin
h,T + (ω )
h
xmax
h,T + (ω )
h
xmin
h,T − (ω )
h
xmax
h,T − (ω )
Slope of ith piece of linearized hourly bid curve for TCR+
Slope of ith piece of linearized hourly bid curve for TCR−
Intercept of ith piece of linearized hourly bid curve for
TCR+
Intercept of ith piece of linearized hourly bid curve for
TCR−
Slope of ith piece of linearized deficit curve for secondary
positive reserves
Slope of ith piece of linearized deficit curve for secondary
negative reserves
Slope of ith piece of linearized deficit curve for overall positive reserves
Slope of ith piece of linearized deficit curve for overall negative reserves
Intercept of ith piece of linearized deficit curve for secondary
positive reserves
Intercept of ith piece of linearized deficit curve for secondary
negative reserves
Intercept of ith piece of linearized deficit curve for overall
positive reserves
Intercept of ith piece of linearized deficit curve for overall
negative reserves
Probability of deficit of secondary positive reserves
Probability of deficit of secondary negative reserves
Probability of deficit of overall positive reserves
Probability of deficit of overall negative reserves
Lower bound of daily TCR+ procurement amount for daily
scenario ω d
Upper bound of daily TCR+ procurement amount for daily
scenario ω d
Lower bound of daily TCR− procurement amount for daily
scenario ω d
Upper bound of daily TCR− procurement amount for daily
scenario ω d
Lower bound of hourly TCR+ procurement amount for
hourly scenario ω h
Upper bound of hourly TCR+ procurement amount for
hourly scenario ω h
Lower bound of hourly TCR− procurement amount for
hourly scenario ω h
Upper bound of hourly TCR− procurement amount for
hourly scenario ω h
Chapter 1
Introduction
This chapter unveils the background of this thesis. Key concepts involved are
balancing reserves and market for reserve procurement. Stochastic programming, the core methodology applied in this thesis, will be briefly introduced
in Section 1.3. The framework of this thesis is then exhibited in Section 1.4.
1.1
Balancing Reserves
Power systems operate at a certain frequency, e.g. 50 Hz in Europe and a
majority of countries in the world, and 60 Hz in the Americas and part of
Asia. It is important to maintain system frequency within a small range of
deviation. On the one hand, large frequency deviations can damage equipment and affect load performance. On the other hand, if frequency drops
too much, generation units might be disconnected from the grid by protection devices, further enlarging frequency deviation. In worst cases, large
frequency fluctuations could possibly lead to interruption of power supply
and system collapse [7].
In most European countries, Transmission System Operators (TSOs)
are responsible for the security of transmission system and coordinating the
supply of and demand for electrical energy to avoid frequency deviations.
Generally, a surplus in generation shifts system frequency upwards, while
a deficit depresses it [7]. Therefore, in order to maintain constant system
frequency, production and consumption of electricity have to be balanced
instantaneously and continuously.
However, demand forecast can never yield 100% precision. Forecast errors, sudden load changes and unforeseen generation incidents can cause
power imbalances between supply and demand in the system. Under the
circumstances, TSOs have to deploy balancing energy to “fill the gap” between supply and demand. Yet one of the most unique features of electrical
energy is that it cannot be stored. In order to ensure that there is sufficient
energy in real-time operation for the purpose of balancing, TSOs usually
1
2
CHAPTER 1. INTRODUCTION
Figure 1.1: Real-time electricity consumption and demand forecast of
November 16, 2015 [1]
“reserve” some generation capacity in advance, which is also referred to as
the procurement of balancing reserves.
1.2
Reserve Market
Reserves are often categorized as a type of Ancillary Services (AS), as they
help to maintain system stability and support normal electricity trading and
delivery. The provision of balancing reserves can be either compulsory or
market-based. A market-based reserve procurement scheme can be advantageous over the mandatory provision in many ways. It avoids unnecessary
investment, optimizes allocation of resources, and fosters technological and
economical innovation. From the economic point of view, a market for reserves provides an incentive for participants to offer their services more cost
effectively.
According to the definitions in microeconomics [8], public goods refer to
goods that are neither excludable nor rival in consumption. From this perspective, power system security can be viewed as a type of public goods.
To avoid the “free-rider” problem, cost of maintaining system security is
usually covered by electricity consumers as part of their electricity bill. In
most countries, electricity tariffs are under the supervision of national electricity regulator and should be controlled within a certain range. Within
this setting, a market for reserves is highly desirable for their economical
efficiency.
Following the electricity reform at the beginning of this century, a market
for reserves has arisen in most European countries. Participants in this
1.3. STOCHASTIC PROGRAMMING
3
market can be pre-qualified power plants, industrial consumers or possibly
someone providing demand side response. TSOs are normally the sole buyer
in this market. The demand is determined by dimensioning criteria set by
ENTSO-E and TSO. Depending on different market setup, providers with
accepted bids are reimbursed either at their bid price (pay-as-bid ) or the
price of the last accepted bid (market-clearing price).
The operation of reserve market is usually separated into two steps: dimensioning and procurement. Traditionally, TSOs complete these two steps
in a sequential order, i.e. first determine the demand using dimensioning
criteria and then procure the fixed amount in the market. In Germany,
for example, reserves are dimensioned using probabilistic criteria and the
necessary amount is published every three months for the next quarter [9].
Afterwards, a tender auction process for the fixed demand will take place,
in which bids with the most attractive offering price will be awarded.
While this traditional approach obeys the security criteria, it neglects
the temporal coupling between different market stages and potential substitution between different types of reserves by unlinking dimensioning and
procurement [4], which results in overly conservative procurement decisions.
In [4], a new market-clearing approach for reserve market is proposed. This
new approach is a two-stage stochastic market-clearing model based on the
structure of the Swiss reserve market and has been implemented since January 2014. The economic saving of this new approach is estimated to be
around 12 million Swiss francs (CHF) in 2014.
1.3
Stochastic Programming
Stochastic programming is a model for optimization under uncertainty. In
general, stochastic programming incorporates a wide range of problems: twostage recourse problems, multi-stage stochastic problems, stochastic integer
programs, chance-constrained programs, etc. [10], of which the first one is
most widely studied and applied. This thesis is based on a two-stage recourse
problem, and extends it to a three-stage stochastic problem.
In a two-stage or three-stage stochastic programming problem, multiple
decisions are made as we progress in time, with more information on the
unknown parameter being disclosed. Here, uncertainty is represented by a
finite number of input data, which can be random variables or stochastic
processes. The objective function is formulated as the sum of individual
solutions of each set of input data weighted by the corresponding probability
factor. Hence, instead of optimizing deterministic objective functions, the
expected value of objective function (if not risk-averse) is optimized. The
final solution is therefore the best solution for all sets of input data, but not
for each individual scenario particularly [11].
Some common terminologies in stochastic programming are explained
4
CHAPTER 1. INTRODUCTION
below [10, 11]:
• Stage: a point in time where a new decision has to be made with a
change in uncertainty.
• Here-and-now decisions: decisions made without any realization of
uncertain information.
• Wait-and-see decisions: decisions made after uncertainty has totally
or partially unfolded.
• Recourse action: additional actions possible in second or further stages
when uncertainty reveals.
• Scenarios: a set of data representing stochastic processes spanning
a given time horizon. For instance, if λ is the hourly electricity
price of next week (168 hours), λ can be represented by NΩ scenarios
λ(1), · · · , λ(ω), · · · , λ(NΩ ), each with a length of 168×1 and occurring
with a probability of π(ω), where ω is the scenario index.
• Scenario tree: a graphical demonstration of scenarios. A node is a
point where a decision has to be made. A node can be succeeded by
multiple nodes and can be traced back to only one node. The number
of nodes at each stage is equal to the number of scenarios at this stage.
There is only one node at first stage, which is named root. A branch
corresponds to a “path” from the first-stage node to a final-stage node,
which represents a realization of the random variables.
Figure 1.2: Example of a scenario tree for three-stage problems
1.4. STRUCTURE OF THE THESIS
5
In recent years, there has been a growing trend of real-world applications
of stochastic programming. Examples are especially concentrated in the
fields of finance and energy [12]. The application of three-stage stochastic
programming in electricity market is generally focused on maximizing the
profits of power producers in multi-stage trading activities [13–15]. Research
on reserve market-clearing using stochastic programming models is mostly
conducted under the setting of a centralized market where reserve and energy
are jointly cleared [16,17]. These approaches are, however, not applicable for
the market design in Europe, where energy trading is separated from reserve
market operation. To our knowledge, Swiss reserve market is one of the only
real-world implementations of a stochastic market-clearing model reported
so far [4]. The idea of further extending it to a three-stage stochastic marketclearing model is thus novel as well.
1.4
Structure of the Thesis
The main objective of this thesis is to develop a three-stage stochastic
market-clearing model for the Swiss reserve market. This market-clearing
model can be adapted to future reserve market design in Switzerland, where
power plants may be requested to offer all or part of their remaining generation capacity close to real-time for the purpose of balancing. An alternative
application can be based on current market design in situations where plenty
of balancing energy bids are received without a priori accepted reserve bids.
The contributions of this thesis are twofold:
1. Scenarios for current two-stage stochastic market-clearing model are
investigated and improved, mainly from the aspects of scenario modelling and scenario selection method. The improvements are simulated
with real market data and can be readily implemented in current Swiss
reserve market.
2. A three-stage stochastic market-clearing model for Swiss reserve market is proposed, which is so far novel in applications of stochastic
programming. Based on analysis of current market reference data,
scenarios for third stage are created in order to simulate possibilities
in future Swiss reserve market.
Therefore, the thesis is organized as follows:
Chapter 2 opens with an overview of ancillary services in Switzerland
and the organization of their procurement. The focus is then shifted to the
market design of secondary and tertiary control reserves. Explanations on
bid structure, dimensioning criteria and remuneration scheme follow.
Chapter 3 presents the current two-stage stochastic market-clearing model.
Formulation of the model is depicted in Section 3.2, while Section 3.3 exhibits improvements on scenarios used in the two-stage model.
6
CHAPTER 1. INTRODUCTION
In Chapter 4, formulation of the three-stage stochastic market-clearing
model is illustrated, followed by explanations on modelling and selection
method of third-stage scenarios in Section 4.3. Then, simulations regarding
the three-stage model are run and results are presented.
Chapter 5 covers the conclusion and the outlook of this project.
Chapter 2
Reserve Market in
Switzerland
In this chapter, the Swiss ancillary services market is presented. The focus
will be specially on the procurement of frequency control reserves. Market
structure, bid structure, dimensioning criteria and remuneration scheme of
reserve market will be explained in Sections 2.3, 2.4, 2.5 and 2.6 respectively.
2.1
Self-scheduling Market
In most European countries, wholesale trading of electricity is separated
from dispatch by TSOs. This type of market design is named self-scheduling
market, or decentralized market, as opposed to centralized market in North
America. Switzerland belongs to one of these countries.
In a self-scheduling market, generation schedule of each power plant is
determined on their own based on their economic interest. In most cases,
these schedules are based on results from bilateral, day-ahead and intraday
power trading. Schedules are submitted to TSOs after trading closes. TSOs
will only intervene when there is a need for balancing or when security
criteria are violated. In such cases, TSO will deploy Ancillary Services (AS)
to ensure system security and reliability.
In Switzerland, connection between energy markets and TSO (Swissgrid)
is established through Balance Group (BG) model [18]. A balance group is
a virtual aggregation of several feed-in and feed-out points. Energy transactions are carried out by each BG as an entity. After transaction closes, BGs
are obliged to deliver their schedules to Swissgrid. Meanwhile, Swissgrid
balances the production and consumption within the entire Swiss network.
However, due to forecast errors and inexacts, actual delivery of energy is
likely to deviate from schedules, creating imbalances in the system (a.k.a.
balance energy). The settlement of balance energy is a two-price system
depending on the direction of discrepancy. The prices are usually unfavor7
8
CHAPTER 2. RESERVE MARKET IN SWITZERLAND
able to BGs as a measure to discourage such discrepancies. Formula for
calculating imbalance settlement price can be found in [18]. The prices are
calculated by Swissgrid and posted on Swissgrid’s website every month.
2.2
Overview of Ancillary Services
As national TSO, Swissgrid guarantees the secure and reliable operation of
power system with the help of ancillary services from providers. Ancillary
services organized by Swissgrid primarily include frequency control, voltage support, compensation of active power losses and black start [3], which
are explained below. The main focus of this thesis is on frequency control
reserves.
Frequency Control
Frequency control can also be referred to as active power control, where
imbalances between electricity production and consumption are balanced
by deploying control reserves. According to European Network of Transmission System Operators for Electricity (ENTSO-E), system frequency control
can be divided into three levels: primary control, secondary control and tertiary control [19]. Correspondingly, three types of frequency control reserves
are defined: Primary Control Reserves (PCR), Secondary Control Reserves
(SCR) and Tertiary Control Reserves (TCR). In [20], the conventional terms
are replaced by modern terms of control reserves. In this thesis, conventional
terms of reserves will be kept since they are still more familiar to audience
in the industry nowadays. Figure 2.1 presents the activation sequence of
the three types of frequency control. Technical features of the three-level
frequency control are summarized as follows.
Figure 2.1: Temporal structure of frequency control after a disturbance [2]
2.2. OVERVIEW OF ANCILLARY SERVICES
9
1. Primary Frequency Control
The purpose of primary frequency control is mainly to stabilize system
frequency after a disturbance at steady state. Full activation time is
usually 30 seconds after disturbance in Continental Europe [21]. Primary frequency control is activated through automatically adjusting
setpoints for frequency and power at a local generator and is therefore
decentralized control. Since this type of control is purely proportional,
it merely prevents system frequency from further deviating, but cannot restore frequency to normal value. All online generators should
be technically available for the provision of primary frequency control
through installation of speed governors [2]. Some frequency sensitive
loads such as induction motors also participate in this control by counteracting frequency deviations [22, 23].
Primary control reserves are also known as Frequency Containment
Reserves (FCR). In the synchronous area of Continental Europe, the
overall amount of primary control reserve is 3000 MW [19]. This
amount is shared between member states and the demand in each
country is designated by European Network of Transmission System
Operators for Electricity (ENTSO-E) every year.
2. Secondary Frequency Control
Secondary frequency control is also referred to as Automatic Generation Control (AGC) or Load Frequency Control (LFC). Secondary
control is activated to restore system frequency and power exchanges
between areas in case of frequency noises under normal operation or after a large incident. The activation of secondary control usually starts
30 seconds after the disturbance and is completed within 15 minutes
at the latest [19]. In contrast with primary control, secondary control
can be regarded as a type of centralized control. However, only generators in the control area where frequency disturbance occurred will
participate in this control.
In modern terms of ENTSO-E, secondary control reserves are categorized into Frequency Restoration Reserves (FRR), or more specifically, automatic Frequency Replacement Reserves (aFRR), as some
documents indicate [24]. Secondary control reserves can also release
primary control reserves as they can sustain for longer periods.
3. Tertiary Control Reserves
The purpose of tertiary control is usually twofold: to assist secondary
control reserve to recover system frequency and to replace primary
and secondary control reserves. The activation of tertiary frequency
control is manual, in the forms of calls from local TSO. After receiving
10
CHAPTER 2. RESERVE MARKET IN SWITZERLAND
the activation signal, power plant operators will adjust the setpoint
values of power output manually. In some countries, tertiary control
is also activated for the purpose of managing congested lines, which is
often referred to as re-dispatch.
According to modern ENTSO-E definitions, fast tertiary control reserves (those can be fully activated within 15 minutes) are grouped
into Frequency Restoration Reserves (FRR), or more specifically manual Frequency Restoration Reserves (mFRR), while slower units participate as Replacement Reserves (RR) (activation time can be from
15 minutes up to 1 hour).
Voltage Support
In power systems analysis, voltage at each node is usually coupled with the
exchange of reactive power. Maintaining certain voltage level is also crucial
in system operation, since large deviations will cause damages to electrical equipment and further jeopardize system security. Therefore, as TSO,
Swissgrid should ensure that voltage at each node remains in an acceptable
range.
Voltage support is mainly realized by reactive power control. Unlike
active power, reactive power cannot be transmitted. Thus, reactive power
control is rather local [2].
So far, there is no tendering process for reactive power in Switzerland.
All power stations online must provide a certain volume of reactive power
in order to keep the voltage within the range indicated by Swissgrid. The
exchanged reactive energy is remunerated at a fixed rate (CHF/Mvarh) [25].
Compensation of Active Power Losses
Resistance in power transmission lines inevitably leads to losses of active
power. These energy losses must be compensated in the network in order to
deliver the desired amount of energy to end-consumers. In Switzerland, the
tendering process for active power losses and inadvertent deviations takes
place once per month. Any balance group in the Swiss control area can participate and compensated energy will be remunerated at its bid price based
on exchange schedules [5].
Black Start and Island Operation
Black start refers to the ability of power generators to start operating without the need for power injection from the grid. Island operation is the
capability of a power station to operate continuously without requiring any
connection to the synchronous grid. Both services are guarantees for the
2.3. STRUCTURE OF RESERVE MARKET
11
restoration of power grid after large incidents. Currently, the provision of
black start and island operation services is secured via bilateral agreement
between the provider and Swissgrid [25].
Figure 2.2 depicts the relationship between Swissgrid, power plants and
end consumers in the ancillary services market.
Figure 2.2: Simplified diagram of Swiss ancillary services market [3]
2.3
2.3.1
Structure of Reserve Market
Primary Control Reserves
The procurement of primary control reserves in Switzerland is now in cooperation with Germany, Austria and the Netherlands. Since April 2015, a
total PCR of 783 MW is procured on this common market platform, making
it the largest reserve market in Europe [26]. The demand for PCR and the
maximum export quantity of each individual country is shown in Table 2.1.
Table 2.1: Volume of PCR Cooperation in 2015 [5]
Country
Switzerland
Austria
Germany
The Netherlands
PCR Demand
71 MW
67 MW
578 MW
67 MW
Max. Export
90 MW
90 MW
173 MW
90 MW
12
CHAPTER 2. RESERVE MARKET IN SWITZERLAND
In this common market, power plants from all four countries can submit
their bids into the pool. The tender call takes place every Tuesday afternoon
[26]. After gate closure, the market will be cleared and those bids with lowest
price will be accepted, regardless of their geographical location (provided
that the maximum export quantity of each country is not exceeded). The
procured PCR will be utilized across the four participating countries.
2.3.2
Secondary and Tertiary Control Reserves
SCR and TCR are procured together in a national reserve market in Switzerland. This reserve market is the main focus of this thesis.
Currently, SCR is procured on a weekly basis, whereas the procurement
of TCR is split between a weekly auction and daily auction. Figure 2.3
illustrates the scheme of the two-stage Swiss reserve market.
Figure 2.3: Scheme of a two-stage reserve market in Switzerland [4]
Weekly Market
The weekly auction for SCR and TCR is closed every Tuesday afternoon
at 13:00 [27], before the gate closure of PCR market. At this stage, AS
providers who are willing to participate in this market will submit their bids
for SCR and/or TCR into system. The bids in the weekly market must be
valid for a horizon of the whole week, i.e. 168 hours.
2.4. BID STRUCTURE
13
Daily Market
TCR can also be procured from a daily market, which provides power plants
with more flexibility. In a daily market, there are six auctions, each of them
comprising a 4-hour block. Power plants can bid for any of these blocks.
Accepted bids must be available for a duration of 4 hours.
Traditionally, the share of reserves in the weekly and daily market is determined prior to the tender call. However, this approach does not allow any
flexibility in the amount procured in weekly and daily market, hence leading
to higher procurement costs. The stochastic approach, on the other hand,
takes into account options from both markets and finds the optimal solution,
i.e. the procurement combination with the lowest cost. This approach will
be elaborated in Section 3.2.
2.4
Bid Structure
In weekly and daily market, bidding rules are listed in Table 2.2. Some
explanations of Table 2.2 are also listed below.
Table 2.2: Bid structure of the Swiss reserve market [5]
Type of Control
Symmetry
Offer Size
Min. Output Window
Conditional Bids
SCR
Symmetrical
50 MW/Bid
±5 MW
Allowed, min.
TCR
Asymmetrical
100 MW/Bid
+5 MW or -5 MW
volume increment is ±1 MW
• Symmetry refers to the symmetry of provided control power bands.
For the time being, SCR is a symmetrical product, which requires ancillary services provider to hold the reserved capacity available for both
upward and downward regulation. TCR, however, is non-symmetrical.
Ancillary services providers will have to specify whether they bid for
upward or downward regulating capacity. TCR+ refers to upward
tertiary control reserve, whereas TCR− refers to downward tertiary
control reserve thereafter.
Illustrative Example: Figure 2.4 shows two options of a generator with
maximum generation capacity Pmax = 100 MW and technical minimum Pmin = 10 MW. The planned production is Pplan = 60 MW. If
the generator participates in SCR provision (left bar), it has to provide a power band of ±40 MW, which means it is capable of adjusting
its power output by max. 40 MW both upwards and downwards upon
call. If it plans to offer all its remaining capacity to TCR market (right
bar), it will be able to bid for 40 MW TCR+ and an additional 60
MW of TCR−.
14
CHAPTER 2. RESERVE MARKET IN SWITZERLAND
Figure 2.4: Example of SCR and TCR provision
• Offer Size means the maximum volume of a bid.
• Minimum Output Window is namely the minimum volume of each bid.
• Conditional Bids are bids which allow different price/volume combinations. Minimum volume increment is the resolution of these combinations. More details will be explained in Section 2.4.2.
2.4.1
Indivisible Bids
In current design of Swiss reserve market, bids are not divisible. This means
that a bid can either be rejected or accepted. There is no such result that
a bid is partially accepted or split. As a result, decision variables for each
individual bid are binary variables as such:
(
1, bid i is accepted
ξ(i) =
0, bid i is rejected
Illustrative Example: Assume that there are four bids in the pool and total
demand for reserve capacity is 100 MW.
In this illustrative example, if we select bids purely based on the merit
order of bid prices, Bids #1 and #2 will be selected. Since Bid #2 cannot
be split to match the total demand, the total cost in this case will be:
50 × 5 + 70 × 6 = 670 CHF. Instead, if bids #1 and #3 are accepted, the
total cost will be: 50 × 5 + 50 × 7 = 600 CHF.
Therefore, from the perspective of minimizing total cost, the second
combination of Bids (#1 and #3) is more favorable than the first (#1 and
#2), although the unit price of Bid #2 is lower than Bid #3.
2.5. DIMENSIONING CRITERIA
15
Table 2.3: Example bids for demonstration of indivisibility
Bid #
1
2
3
4
2.4.2
Volume
[MW]
50
70
50
80
Price
[CHF/MW]
5
6
7
8
Conditional Bids
Conditional bids are a set of mutually exclusive bids of which only one can
be accepted by TSO. This type of bid offers providers the opportunity of
bidding various price/volume combinations.
Illustrative Example: In Table 2.4, bids with identical bid ID and from the
same provider are recognized as a set of conditional bids. In this case, bids
{#1, #2, #3} are a set of conditional bids submitted by Provider A, while
bids {#6, #7} are another set of conditional bids submitted by Provider B.
Bid #4 is another non-conditional bid from Provider A. Bid #5 is a nonconditional bid from Provider B. Bids {#1, #2, #3} as well as {#6, #7}
are mutually exclusive, meaning that at most one of the bids within the set
can be accepted. The acceptance of Bids {#1, #2, #3} and {#6, #7},
however, does not have any influence on any other bids such as Bids #4 and
#5.
Table 2.4: Example of conditional bids
2.5
Bid #
Provider
Bid ID
1
2
3
4
5
6
7
A
A
A
A
B
B
B
1
1
1
2
1
2
2
Volume
[MW]
10
30
50
50
20
10
30
Price
[CHF/MW]
9
7.5
6
5.5
8
7
6.5
Dimensioning Criteria
Dimensioning criteria are applied to determine the adequate amount of reserve in a system. According to Continental Europe Operation Handbook
by ENTSO-E [19], there are mainly two methods of dimensioning secondary
16
CHAPTER 2. RESERVE MARKET IN SWITZERLAND
and tertiary control reserves: probabilistic approach and deterministic approach. In Switzerland, criteria for dimensioning reserves are a hybrid of
probabilistic and deterministic approach [4].
2.5.1
Probabilistic Approach
The probabilistic approach of dimensioning reserves is based on the recommendation by ENTSO-E that the Area Control Error (ACE) has to be
regulated to zero in a certain amount of hours within a year [19]. The percentage of hours is not strictly specified by ENTSO-E and can be determined
by each TSO individually. Switzerland, for example, requires that ACE shall
be smoothened to zero in 99.8% of all hours during a year. In other words,
the deficit of reserves should not occur with a probability of more than
0.2% [4], which can also be interpreted as the deficit rate of reserves.
To determine the deficit rate of reserves, cumulative probability distribution curves of power imbalances (deficit curve) are used. The deficit curve
normally takes into account all possible causes of failures, forecast errors
and fluctuations of Renewable Energy Sources (RES).
In Switzerland, two sets of deficit curves are built with respect to the
dimensioning of secondary reserves solely and total amount of secondary
and tertiary reserves respectively. For the former curve, AGC signals and
remaining ACE are aggregated to form spontaneous power imbalance ∆Ps ,
which is expected to be compensated by secondary reserves. The latter curve
considers AGC signals, remaining ACE as well as activated tertiary reserves,
which yields to the so-called open-loop ACE, or overall power imbalance ∆Po .
This open-loop ACE will be covered by the sum of secondary and tertiary
control reserves. Figure 2.5 demonstrates the deficit curves for dimensioning
secondary and overall reserve amount in Switzerland.
Figure 2.5: Deficit curves for dimensioning reserves in Switzerland [4]
2.6. REMUNERATION SCHEME
2.5.2
17
Deterministic Approach
The deterministic approach of sizing reserves recommended by ENTSO-E
is based on the largest possible generation incident, which includes power
plant outage, tripping of power lines, and so on [19]. In Switzerland, the
largest power generation incident is the outage of the nuclear power plant
Leibstadt, whose installed capacity is 1250 MW [4]. Hence, the total amount
of reserves to prevent further incident under this circumstance should be at
least 1250 MW.
Meanwhile, Switzerland also has contractual Mutual Emergency Assistance Service (MEAS) with neighboring countries. This type of contract
guarantees the availability of maximum 400 MW reserve given that Switzerland holds the same amount of reserve ready at the same time, which adds up
to 800 MW positive reserves. Since the amount of SCR procured is empirically around 400 MW, the deterministic criteria of securing supply in case of
largest power plant outage and MEAS contract can be altogether converted
to procuring 400 MW of TCR+ for MEAS contract at the moment [4].
2.6
Remuneration Scheme
2.6.1
Remuneration of Capacity
After bids are accepted in the reserve market, corresponding ancillary service providers will be remunerated for holding the accepted reserve capacity.
This remuneration is based on the bid price of the capacity (pay-as-bid ). Essentially, bid price should reflect the opportunity cost of holding the reserves
instead of selling the produced electricity on the spot market. Since the majority of ancillary services providers in Switzerland is hydro power plant
owners, this payment also implies the value of water in the reservoir [28].
2.6.2
Remuneration of Energy
The remuneration scheme of activated secondary and tertiary energy can be
different, due to the way they are deployed and some historical reasons.
Secondary Control
The activation of SCR is triggered automatically and centrally after ACE
exceeds a certain limit. The amount of activated SCR in each balance group
is calculated ex post and is usually proportional to the amount of reserve
capacity it has been accepted in the reserve market. Since activation of
SCR does not require any manual dispatch from TSO, it is remunerated at
a flat rate coupled with spot market price. Detailed remuneration rate can
be found in Table 2.5.
18
CHAPTER 2. RESERVE MARKET IN SWITZERLAND
Table 2.5: Remuneration of activated SCR [5]
Direction
Price
Cash Flow
Energy Flow
Upward
SwissIXa +20%b
Swissgrid → Bidder
Bidder → Swissgrid
Downward
SwissIX −20%c
Bidder → Swissgrid
Swissgrid → Bidder
a
Hourly price index for Swiss day-ahead auction in EPEX Spot Market
at least weekly base
c
at most weekly base
b
Tertiary Control
In contrast with SCR, TCR is deployed manually. When a dispatcher in shift
observes continuous power imbalance that needs to be balanced, he/she will
manually call ancillary services providers to activate their tertiary control.
The lead time is currently 15 minutes in Switzerland.
For this reason, an separate market for Tertiary Control Energy (TCE)
arises to allow ancillary services providers to bid for the regulating energy
they provide. Providers with TCR bids accepted in the reserve market must
submit energy bids to TCE market. Apart from compulsory bids, additional
energy can also be offered voluntarily without previously accepted reserve
bids [5]. Dispatchers will activate TCE based on demand and Merit Order
List (MOL). Bids with lowest price will be activated first. Currently, payment of activated tertiary energy is calculated based on activated amount
and bid price.
Chapter 3
Two-Stage Market-Clearing
Model
This chapter focuses on the current two-stage stochastic market-clearing
model and the formulation of the model. Contributions of this thesis are
investigation and improvement of second-stage scenarios used in this twostage model. These improvements result in higher accuracy of predicting
bidding behavior in daily market using historical data and thus decrease total procurement cost.
3.1
Background
As is briefly stated in Sections 1.2 and 2.3, traditional reserve procurement process decouples dimensioning from procurement and neglects the
link between weekly and daily market. By introducing a two-stage stochastic market-clearing model, dimensioning and procurement are coupled, and
substitution between weekly and daily products is also enabled. The chief
objective is to minimize total expected procurement cost while respecting
all constraints.
Figure 3.1 illustrates the decision-making model of this two-stage market. Weekly market is cleared once per week. At this point in time, a
decision on the procurement in weekly market has to be made with only
weekly bids available. It is still uncertain how market participants will bid
in daily market. Unknown daily bids are therefore modelled by a finite
number of scenarios, each representing a possible set of inputs with a given
probability. Based on these daily scenarios and known bids from the weekly
auction, weekly procurement decision is made with regard to weekly bids to
be accepted and suggested amount of reserve to be procured in daily market. After the two-stage clearing of weekly market, results are then passed
onto the daily market, where daily bids are received. At this stage, a deterministic market-clearing process is carried out with actual daily bids and
19
20
CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL
Figure 3.1: Two-stage stochastic market-clearing scheme
procurement results from previous stage.
In the context of this thesis, first stage refers to weekly market, and
second stage refers to daily market. In this chapter, only the two-stage
stochastic model for weekly market clearing will be elaborated. The deterministic clearing model for daily market is a simple optimization problem
based on given bids.
3.2
Stochastic Market-Clearing Model
This section will introduce the formulation of the two-stage stochastic programming model for current reserve market. The objective of this optimization problem is to minimize expected total cost of procurement of all
scenarios. Constraints include bidding behavior of known market stages,
probabilistic and deterministic dimensioning criteria, etc. An overview of
the optimization model is presented in Equation (3.1).
min
expected total procurement cost
s.t.
conditional bids in weekly market
probabilistic dimensioning criteria
deterministic dimensioning criteria
(3.1)
Please note that bids are usually submitted by BGs (portfolio-based )
instead of a specific power plant (unit-based ). Thus, this model does not
include any power flow and power balance equations. The selection of bids
is purely based on bid price and bid volume.
Components of this optimization problem will be built step by step in
Sections 3.2.1, 3.2.2 and 3.2.3. The complete model will be presented again
as a whole in Section 3.2.4.
3.2. STOCHASTIC MARKET-CLEARING MODEL
3.2.1
21
Decision Variables
In this two-stage stochastic programming problem, there are two different
decision variable vectors, each representing a stage.
First-Stage Decision Variables
Considering different products in the weekly reserve market, the decision
variable vector of first stage consists of three components:
w
w
xw = [xw
S , xT+ , xT− ],
(3.2)
where
• xw ∈ ΥN
w ×1
is the decision variable vector of bids in weekly market
w
NS ×1 is the decision variable vector of SCR bids in weekly
• xw
S ∈ Υ
market
NTw ×1
• xw
T+ ∈ Υ
market
+
is the decision variable vector of TCR+ bids in weekly
w
NT − ×1
• xw
is the decision variable vector of TCR− bids in weekly
T− ∈ Υ
market
• Υ = {0, 1} is the set of binary variables indicating acceptance and
rejection of a bid
• NSw is the number of SCR weekly bids
• NTw+ is the number of TCR+ weekly bids
• NTw− is the number of TCR− weekly bids
• N w = NSw + NTw+ + NTw− is the total number of bids in the weekly
market (including SCR, TCR+ and TCR−)
As real bids from the weekly market are considered in this model, the
indivisibility of bids should preserved in the decision-making process. This
feature can be interpreted as binary decision variables for weekly bids, as is
already discussed in Section 2.4.1.
In case of conditional bids, each single bid within the set of conditional
bids will still be counted as one bid and is assigned with a binary variable.
An additional constraint is added to ensure the property of conditional bids,
which will be explained in Section 3.2.3.
22
CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL
Second-Stage Decision Variables
In the second stage, a finite number of scenarios are constructed in order
to model the uncertainty. For each daily scenario ω d , an individual secondstage decision variable vector xd (ω d ) is assigned:
xd (ω d ) = [xdT+ (ω d ), xdT− (ω d )],
(3.3)
where
• ω d ∈ Ωd is the index of scenarios for daily market
• Ωd contains NΩd scenarios for daily market
• xdT+ (ω d ) ∈ R42×1 is the amount of TCR+ to be procured in daily
market in each 4-hour time block for scenario ω d
• xdT− (ω d ) ∈ R42×1 is the amount of TCR− to be procured in the daily
market in each 4-hour time block for scenario ω d
In the second stage, TCR market is cleared for each 4-hour time block.
Since the results from two-stage market clearing are valid for one week (24 ×
7 = 168 hours), length of second-stage parameters and variables is often
168/4 = 42, which corresponds to 42 time blocks in a week.
3.2.2
Objective Function
The objective of this optimization model is to minimize expected procurement cost. The total cost can be calculated as the sum of first-stage cost
(deterministic cost of weekly market) and second-stage cost (expected value
of daily costs with respect to different scenarios):
min Λw + E[Λd (ω d )],
∀ω d ∈ Ωd ,
(3.4)
where Λw is the total procurement cost in the weekly market and Λd (ω d )
is the total procurement cost in the daily market for the given week with
input data of scenario ω d . E[Λd (ω d )] is the expectation of daily procurement
costs in all scenarios.
First-Stage Cost Function
Since the first stage of this problem is deterministic, cost incurred in the
weekly market is calculated as the cumulative cost of accepted bids:
Λw = c|w xw = (κ · p)| xw
where
(3.5)
3.2. STOCHASTIC MARKET-CLEARING MODEL
• cw ∈ R N
• κ ∈ RN
w ×1
w ×1
23
is the cost vector of all weekly bids
is a vector containing unit prices (CHF/MW) of each bid
w
• p ∈ RN ×1 = [pS , pT+ , pT− ] is a vector consisting of volume (MW) of
each bid
• pS is a vector consisting of volume (MW) of each SCR bid
• pT+ is a vector consisting of volume (MW) of each TCR+ bid
• pT− is a vector consisting of volume (MW) of each TCR− bid
Second-Stage Cost Function
To render the computation time affordable for real market clearing, this
stochastic programming problem is limited to a Mixed Integer Linear Programming (MILP) problem. Therefore, all components are to be linearized.
To model daily costs, bid curves are derived from actual bids and then linearized via the following steps:
1. For each 4-hour time block, bids are obtained for TCR+ and TCR−
respectively.
2. As for conditional bids, the bid with minimum price within the set is
selected to represent this set. If there are multiple bids with the same
minimum price, the one with the largest quantity is chosen.
3. Bids are sorted in ascending order according to their bid prices.
4. Cumulative volume and cost are calculated respectively.
5. A bid curve is drawn based on cumulative volume and cost. The
left curve in Figure 3.2 shows an example of bid curve. Each blue
circle represents a bid. The x-coordinate corresponds to the cumulative
volume up to this bid, whereas the y-coordinate is the cumulative cost
for such volume.
6. To incorporate daily bids into the linear model, bid curves are linearized. The curve on the right in Figure 3.2 depicts how a bid curve
(blue line) can be approximated by two straight lines (green dashed
lines).
Two-piece linearization is selected by the current market-clearing algorithm, since it represents bidding behavior in a most efficient manner.
24
CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL
Figure 3.2: Example of a bid curve (before and after linearization)
After linearization, procurement cost derived from this linearized bid
curve (associated with a 4-hour time block for a specific scenario) can be
written as:
(
α1 x + β1 , xmin ≤ x ≤ xB
λ=
,
(3.6)
α2 x + β2 , xB < x ≤ xmax
where
• λ is the procurement cost of this time block
• x denotes the amount of reserve to be procured in the daily market
• (α1 , β1 ) and (α2 , β2 ) are fitting parameters of the first and second
part of the linearized curve respectively
• xmin and xmax are lower and upper bound of the bid curve
• xB is the breaking point of the linearized bid curve
Coefficients α1 and α2 represent the slopes of the piecewise linear bid
curve, which can also be regarded as the “unit price” for reserves within a
certain range. Since bids are sorted from the cheapest to the most expensive
ones, it is obvious that α2 > α1 . Therefore, the piecewise linear bid curve
can be considered as a convex function. Equation (3.6) can be reformulated
as:
λ = max{α1 x + β1 , α2 x + β2 },
∀x ∈ [xmin , xmax ].
(3.7)
3.2. STOCHASTIC MARKET-CLEARING MODEL
25
Equation (3.7) can be written as an optimization problem:
min
λ
s.t.
α1 x + β1 ≤ λ,
α2 x + β2 ≤ λ,
xmin ≤ x ≤ xmax .
x
(3.8)
In reality, daily procurement costs are calculated for each 4-hour time
block and summed up. Fitting parameters α1 , α2 , β1 and β2 vary according to each piecewise linear bid curve derived for each time block and are
constructed for each scenario. Therefore, Equation (3.8) can be extended to
the full week horizon and all scenarios in the daily market:
min
λT+ (ω d ) + λT− (ω d )
s.t.
+ d
d d
d
d
α+
1 (ω )xT+ (ω ) + β1 (ω ) ≤ λT+ (ω ),
xdT (ω d ),
+
xdT (ω d )
−
+ d
d
d d
d
α+
2 (ω )xT+ (ω ) + β2 (ω ) ≤ λT+ (ω ),
− d
d
d d
d
α−
1 (ω )xT− (ω ) + β1 (ω ) ≤ λT− (ω ),
(3.9)
− d
d
d d
d
α−
2 (ω )xT− (ω ) + β2 (ω ) ≤ λT− (ω ),
d
max
d
xdT+ (ω d ) ∈ [xmin
d,T+ (ω ), xd,T+ (ω )],
d
max
d
xdT− (ω d ) ∈ [xmin
d,T− (ω ), xd,T− (ω )],
∀ω d ∈ Ωd ,
where
• λT+ (ω d ) ∈ R42×1 is the cost vector of TCR+
• λT− (ω d ) ∈ R42×1 is the cost vector of TCR−
+ d
+ d
+ d
42×1 are fitting parameters of
d
• α+
1 (ω ), β1 (ω ), α2 (ω ), β2 (ω ) ∈ R
TCR+ bid curves
− d
− d
− d
42×1 are fitting parameters of
d
• α−
1 (ω ), β1 (ω ), α2 (ω ), β2 (ω ) ∈ R
TCR− bid curves
d
max
d
• xmin
d,T+ (ω ), xd,T+ (ω ) are lower and upper bound of TCR+ bid curve
d
max
d
• xmin
d,T− (ω ), xd,T− (ω ) are lower and upper bound of TCR− bid curve
Therefore, the term E[Λd (ω d )] in the Equation (3.4) can be reformulated
as:
X
E[Λd (ω d )] =
h
i
πd ω d 11×42 λT+ (ω d ) + 11×42 λT− (ω d ) ,
(3.10)
ω d ∈Ωd
d
d
∀ω ∈ Ω ,
d
where πd ω is the probability of scenario ω d and 11×42 is the summation vector whose elements are all 1.
26
CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL
3.2.3
Constraints
The constraints considered in the stochastic market-clearing model primarily
encompass three aspects:
1. Conditional bids in weekly market
2. Probabilistic dimensioning criterion
3. Deterministic dimensioning criterion
The mathematical formulation of three types of constraints will be explained individually in this section.
Conditional Bids in Weekly Market
According to current bidding rules in the Swiss reserve market, providers
are allowed to submit a set of conditional bids. Definition and example of
conditional bids are given in Section 2.4.2.
To guarantee that this rule will not be violated in the optimization problem, it is translated into the following constraint:
Ac xw ≤ bc ,
(3.11)
where
w
w
• Ac ∈ RNc ×N is a matrix whose element is either 1 or 0 depending
on whether the corresponding bid is within the set of conditional bids
or not
w ×1
• bc ∈ RNc
is a vector whose elements are all 1
• Ncw is the number of conditional bid sets amongst all bids in the weekly
market (including SCR, TCR+ and TCR−)
Illustrative Example: Consider the bids in Table 2.4. Bids {#1, #2, #3}
and {#6, #7} are two sets of conditional bids. The decision vector for these
seven bids is:
x = [x1 , x2 , · · · , x7 ]| ,
where xi ∈ {0, 1} is the binary decision variable for Bid #i.
According to the definition of conditional bids, at most one of x1 , x2 and
x3 can have a value of 1. The rest should be 0. This is also applicable for
x6 and x7 . The rule of conditional bids can thus be written as:
(
x1 + x2 + x3 ≤ 1
.
x6 + x7
≤1
1 1 1 0 0 0 0
1
In this case, Ac =
, and bc =
.
0 0 0 0 0 1 1
1
3.2. STOCHASTIC MARKET-CLEARING MODEL
27
Probabilistic Dimensioning Criterion
The probabilistic criterion for dimensioning reserves states that the portion
of time within a year when secondary control reserve alone is not able to
cover spontaneous power imbalances and when secondary and tertiary control reserve together are not able to cover overall power imbalances should
not exceed 0.2% [4]. This percentage can be translated as the probability of
power imbalances being greater than the amount of reserves procured:
P (∆Ps ≥ Rs ) + P (∆Po ≥ Ro ) ≤ 0.2%,
(3.12)
where ∆Ps and ∆Po are spontaneous and overall power imbalances respectively, Rs and Ro are the amount of secondary and overall reserves
(including SCR and TCR).
Considering both positive and negative power imbalances, Equation (3.12)
can be rewritten as:
P ∆Ps+ ≥ Rs+ + P ∆Ps− ≤ −Rs− + P ∆Po+ ≥ Ro+
(3.13)
+P ∆Po− ≤ −Ro− ≤ 0.2%,
where
• ∆Ps+ > 0 and ∆Ps− < 0 are positive and negative spontaneous power
imbalances respectively
• ∆Po+ > 0 and ∆Po− < 0 are positive and negative overall power
imbalances respectively
• Rs+ and Rs− refer to positive and negative SCR (currently in Switzerland Rs+ = Rs− )
• Ro+ and Ro− refer to overall positive and negative control reserves
If we define cumulative distribution functions of power imbalances as
such:


F
(p)
=
P
∆P
≥
p

s
s
+
+


F (p) = P ∆P ≤ −p
s−
s−
,
(3.14)

Fo+ (p) = P ∆Po+ ≥ p



F (p) = P ∆P ≤ −p
o−
o−
then Equation (3.15) can be reformulated as:
Fs+ (Rs+ ) + Fs− (Rs− ) + Fo+ (Ro+ ) + Fo− (Ro− ) ≤ 0.002.
(3.15)
28
CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL
The amount of procured reserve Rs+ , Rs− , Ro+ and Ro− can be replaced
by decision variables in the following terms:
Rs+ = Rs− = p|S xw
S,
|
w
d
d
Ro+ = p|S xw
S + pT+ xT+ + xT+ (ω ),
(3.16)
|
w
d
d
Ro− = p|S xw
S + pT− xT− + xT− (ω ),
∀ω d ∈ Ωd .
Inserting Equation (3.16) into (3.15), Equation (3.15) becomes:
| w
| w
|
w
d
d
x
+
x
(ω
)
(p
x
)
+
F
p
x
+
p
)
+
F
Fs+ (p|S xw
o
s
T+
S
+
−
S S
S S
T+ T+
| w
|
d
d
d
d
+Fo− pS xS + pT− xw
T− + xT− (ω ) ≤ 0.002, ∀ω ∈ Ω .
(3.17)
Deficit curves are derived from statistical behavior of power imbalance
measurements. Consequently, the exact formulation of Fs+ ( ), Fs− ( ),
Fo+ ( ) and Fo− ( ) is not known. To incorporate Equation (3.17) into the
MILP problem, deficit curves need to be approximated and piecewise linearized.
·
·
·
·
−4
10
x 10
Before Linearization
After Linearization
9
Cumulative Probability
8
7
6
5
4
3
2
300
350
400
450
500
Volume (MW)
550
600
Figure 3.3: Example of piecewise linearized deficit curve
Figure 3.3 shows an example of a piecewise linearized deficit curve, where
the blue line is the original curve, and the red dashed line is the linearized
curve. As can be seen from the figure, linearization can be a relatively
good approximation of the original curve, while releasing computational
3.2. STOCHASTIC MARKET-CLEARING MODEL
29
burden of the problem. The linearization process takes place once per half
a year along with the update of original deficit curve based on the most
recent measurement data. After linearization, parameters are obtained and
inserted into constraints:

i
h
(1)
(0)
(1)
(1)

,
r
∈
r
,
R
+
b
R
a

s
s
s
s
s
s
+
+
+
+
+
+



i
h


(2)
(1)
(2)
(2)

Rs+ ∈ rs+ , rs+
as+ Rs+ + bs+ ,
(3.18)
εs + =
..
..


.
.




i
h

(m)
(m−1)
(m)

a(m)
, rs+
s+ Rs+ + bs+ , Rs+ ∈ rs+
εs− =
εo + =
εo − =

(1)
(1)

as− Rs− + bs− ,




 (2)
(2)

as− Rs− + bs− ,
i
h
(1)
(0)
∈ rs− , rs−
i
h
(2)
(1)
∈ rs− , rs−
Rs−
Rs−
..
..


.
.




h
i

(n)
(n−1)
(n)

a(n)
R
+
b
,
R
∈
r
,
r
s− s−
s−
s−
s−
s−

h
i
(1)
(1)
(0)
(1)

a
R
+
b
,
R
∈
r
,
r

o+ o+
o+
o+
o+
o+



h
i
 (2)

(2)
(1)
(2)

ao+ Ro+ + bo+ , Ro+ ∈ ro+ , ro+
..


.





(j)

a(j)
o+ Ro+ + bo+ ,

(1)
(1)

ao− Ro− + bo− ,






(2)
(2)

ao− Ro− + bo− ,
..


.





(k)

a(k)
o− Ro− + bo− ,
..
.
(3.19)
(3.20)
h
i
(j−1)
(j)
Ro+ ∈ ro+ , ro+
h
i
(0)
(1)
Ro− ∈ ro− , ro−
h
i
(1)
(2)
Ro− ∈ ro− , ro−
..
.
(3.21)
h
i
(k−1)
(k)
Ro− ∈ ro− , ro−
where
• subscripts s+ , s− , o+ , o− denote variables/parameters of secondary
positive, secondary negative, overall positive and overall negative reserves
·
• ε (R) is the probability of deficit with a reserve amount of R according
to the corresponding deficit curve
(i)
·
(i)
·
• a and b are linearization parameters of piece i of the corresponding
deficit curve
30
CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL
• m, n, j, k are the total number of pieces for each linearized deficit
curve
h
i
(i−1)
(i)
• r
, r
is the range of reserve volume in which linear piece i is
valid
·
·
Theoretically, deficit curves should be close to convex functions. Under
this assumption, Equations (3.18)−(3.21) can be treated similarly as the
linearized bid curves in Section 3.2.2. Thus, constraints with respect to the
probabilistic dimensioning criterion can be formulated as:
 (i)
(i)

as+ Rs+ + bs+ ≤ εs+ , i = 1, · · · , m,



(i)
(i)


as− Rs− + bs− ≤ εs− , i = 1, · · · , n,



(i)
(i)


ao+ Ro+ + bo+ ≤ εo+ , i = 1, · · · , j,



(i)
a(i) R

 o− o− + bo− ≤ εo− , i = 1, · · · , k,
,
≤ Rs+ ≤ Rsmax
Rsmin
+
+


max
min

Rs− ≤ Rs− ≤ Rs− ,



max
min


Ro+ ≤ Ro+ ≤ Ro+ ,


max
min


Ro− ≤ Ro− ≤ Ro− ,


εs+ + εs− + εo+ + εo− ≤ 0.002.
(3.22)
In this set of constraints, εs+ , εs− , εo+ and εo− are regarded as “decision variables” and appended to those described in Section 3.2.1. They do
not appear in objective function, though. Rs+ , Rs− , Ro+ and Ro− can be
expressed by decision variables according to Equation (3.16).
Deterministic Dimensioning Criterion
According to Section 2.5.2, the deterministic criterion of sizing reserves in
Switzerland can be described as a minimum TCR+ of 400 MW according to
MEAS contract. This criterion can thus be written with weekly and daily
decision variables as follows:
d
d
400 ≤ p|T+ xw
T+ + xT+ (ω ),
∀ω d ∈ Ωd .
(3.23)
3.2. STOCHASTIC MARKET-CLEARING MODEL
3.2.4
31
Formulation
The complete mathematical formulation of the current two-stage stochastic
market-clearing model is presented as follows:
min
c|w xw +
xw , λT+ (ω d ), λT− (ω d )
X
i
h
πd ω d 11×42 λT+ (ω d ) + 11×42 λT− (ω d )
ω d ∈Ωd
(3.24)
s.t.
w
Ac x ≤ bc ,
(3.25)
+ d
d
d
d
d d
d
α+
1 (ω )xT+ (ω ) + β1 (ω ) ≤ λT+ (ω ), ∀ω ∈ Ω ,
(3.26)
d d
d
α+
2 (ω )xT+ (ω )
+
β2+ (ω d )
+
β1− (ω d )
d
d
d
d
d
d
≤ λT+ (ω ), ∀ω ∈ Ω ,
(3.27)
d d
d
α−
1 (ω )xT− (ω )
≤ λT− (ω ), ∀ω ∈ Ω ,
(3.28)
− d
d d
d
d
d
d
α−
2 (ω )xT− (ω ) + β2 (ω ) ≤ λT− (ω ), ∀ω ∈ Ω ,
(3.29)
| w
(i)
a(i)
s+ pS xS + bs+ ≤ εs+ , i = 1, · · · , m,
| w
(i)
a(i)
s− pS xS + bs− ≤ εs− , i = 1, · · · , n,
| w
|
w
d
d
p
x
+
p
a(i)
x
+
x
(ω
)
+ b(i)
o+
T+
o + ≤ εo + ,
S S
T+ T+
(3.30)
(3.31)
∀ω d ∈ Ωd , i = 1, · · · , j,
|
| w
w
d
d
(i)
a(i)
o− pS xS + pT− xT− + xT− (ω ) + bo− ≤ εo− ,
(3.32)
∀ω d ∈ Ωd , i = 1, · · · , k,
(3.33)
εs+ + εs− + εo+ + εo− ≤ 0.002,
(3.34)
d
d
d
d
400 ≤ p|T+ xw
T+ + xT+ (ω ), ∀ω ∈ Ω ,
d
d
d
max
xdT+ (ω d ) ∈ [xmin
d,T+ (ω ), xd,T+ (ω )], ∀ω
(3.35)
∈ Ωd ,
(3.36)
d
max
d
d
d
xdT− (ω d ) ∈ [xmin
d,T− (ω ), xd,T− (ω )], ∀ω ∈ Ω .
(3.37)
Constraint (3.25) corresponds to the conditional bids received in weekly
market. Constraints (3.26)−(3.29) are related to piecewise linearized daily
bid curves. (3.30)−(3.34) are associated with probabilistic dimensioning criterion. Constraint (3.35) is for deterministic dimensioning criterion (MEAS
constraint). Last but not least, constraints (3.36) and (3.37) define feasible
ranges for daily procurement amounts.
32
CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL
3.3
Improvements of Two-Stage Model
This section presents two major improvements made to the current twostage market-clearing model: linearization of bid curves and selection of
daily scenarios. Different methods are investigated and compared. Results
show that cost savings can be achieved with these improvements, which can
be readily implemented in Swissgrid.
3.3.1
Linearization of Bid Curves
The linearization method of bid curves refers to how the piecewise linearized
bid curve in Figure 3.2 is obtained. It encompasses two aspects: the number
of curve pieces and the method of obtaining linearization parameters.
In [4], a two-piece linearization method is applied and this method has
been implemented since the launch of two-stage stochastic market-clearing
model in Switzerland. This linearization method is based on minimizing the
maximum fitting error on a curve, which is referred to as maximum error
estimation hereafter.
The process of searching for optimal fitting parameters using maximum
error estimation can be described as:
min γ
s.t. |ŷi − yi | ≤ γ,

(0)
(1)
α1 xi + β1 , xi ∈ [xB , xB ]




..


.


(j)
ŷi = αj xi + βj , xi ∈ [x(j−1)
, xB ]
B


..



.



(M −1)
(M )
αM xi + βM , xi ∈ [xB
, xB ]
∀i = 1, · · · , N ,
,
(3.38)
where
• (xi , yi ) represents an individual bid on bid curve, and xi , yi denote
cumulative volume and cost at this point respectively
• ŷi is the estimated cost after linearization at xi
• N is the total number of data points to be fitted on the bid curve
• γ denotes the maximum error between fitted curve and original curve
• αj and βj are linearization parameters of piece j
(j−1)
• xB
(j)
and xB indicate the lower and upper bound of piece j
• M is the total number of pieces of the linearized curve
3.3. IMPROVEMENTS OF TWO-STAGE MODEL
33
Another commonly used method in curve fitting is least squares estimation. It can be described via the following formulation:
min
N
X
(ŷi − yi )2
i=1
s.t.

(0)
(1)
α1 xi + β1 , xi ∈ [xB , xB ]




..


.


(j)
ŷi = αj xi + βj , xi ∈ [x(j−1)
, xB ]
B


..



.



(M −1)
(M )
αM xi + βM , xi ∈ [xB
, xB ]
∀i = 1, · · · , N ,
(3.39)
,
Notations in Equation (3.39) follow those in Equation (3.38).
In the linearization process, optimizations based on Equation (3.38) or
(3.39) are repeated for each combination of bids taken as breaking point. The
optimal solution is then the minimum value of objective functions among all
iterations.
Sometimes a two-piece linearization of the bid curve can result in inaccurate estimation of daily procurement cost. This usually occurs when
the shape of bid curve is close to quadratic, for example the one in Figure
3.2. In such cases, more pieces should be considered while linearizing the
bid curve. However, too many pieces will significantly increase the number of iterations due to more combinations of breaking points. Considering
the computational burden, the maximum number of curve pieces considered
here is three.
Therefore, four linearization methods (Figure 3.4) concerning two dimensions (number of pieces and estimation method) are analyzed and compared.
The number of pieces refer to the value of M in Equations (3.38) and (3.39).
Figure 3.4: Overview of bid curve linearization methods
Figure 3.5 shows an example of bid curve linearization using the abovementioned four different methods. As can be seen from the figure, three-piece
34
CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL
fitting is generally closer to real bids. As for estimation method, using least
squares estimation can improve fitting performance to a certain extent.
Cost (CHF)
Example Bid Curve
Bids
Method 1
Method 2
Method 3
Method 4
Volume (MW)
Figure 3.5: Example of bid curve linearization by four methods
Figure 3.6 and Table 3.1 illustrate the residuals and computation time
of the example bids in Figure 3.5. In this context, residual is defined as
the cost derived from original bid curve subtracted by the fitted cost from
the linearized curve. A negative average residual suggests that generally the
fitted curve is overestimating procurement cost.
Looking from residual’s perpective, Method 4 yields the most accurate
approximation of the bid curve, with an overestimated cost of only 59 CHF
on average. The current practice, Method 1, can result in an overestimation
of over 800 CHF on average. By switching the estimation method from maximum error estimation to least squares estimation, the average residual can
be significantly reduced to approximately 184 CHF. Increasing the number
of curve pieces can further improve linearization accuracy, as can be deduced
from the comparisons between Methods 1 and 3, as well as Methods 2 and
4.
Computationally, two-piece fitting is generally quicker than three-piece
fitting. This is due to the iterative search for an optimal combination of
breaking points. Nevertheless, the computation time of 4 seconds using
Method 4 is still affordable as a trade-off with the accuracy it yields.
To investigate the impact of linearization method on the optimization
results, these four methods are applied in a two-stage market-clearing model.
The scenario input data for daily market is the data of the delivery week
(Perfect Information Scenario).
3.3. IMPROVEMENTS OF TWO-STAGE MODEL
35
Residual = Data − Fit (CHF)
Residual of Fitted Bid Curve
3000
2000
1000
Method 1
Method 2
Method 3
Method 4
0
−1000
−2000
−3000
0
5
10
15
Bid Number (X)
20
25
Figure 3.6: Residual of fitted bid curve
Table 3.1: Performance of linearization methods
Method
Method
Method
Method
1
2
3
4
Computation Time [s]
0.1622
0.3000
1.4339
4.7684
Average Residual [CHF]
-827.2980
-183.9706
-81.6516
-59.1752
Figures 3.7 and 3.8 are market-clearing results of the four linearization
methods. In this case, weekly decisions are made based on estimations
of daily market. As can be seen from the figures, the more accurate the
linearization method is (from Method 1 to Method 4), the larger is the share
of daily procurement. This can be explained by the fact that linearization
methods with less accuracy usually tend to overestimate daily cost, which
gives way to more procurement in the weekly market. Moreover, Figure 3.8
indicates that linearization methods with more accuracy also result in less
total procurement cost.
If we set the weekly procurement to the same level for all four methods,
the cost of reserves in the daily market is shown in Figure 3.9. The case
“Real Bids” provides a benchmark as the true cost in daily market, which
is calculated as the summed cost of all accepted bids. Since the amount
of reserves needed for the second stage to satisfy the probabilistic dimensioning criteria is almost the same in four methods (minor difference might
exist due to substitution between TCR+ and TCR− products), it is quite
straightforward from the results that current practice Method 1 incurs the
most cost, whereas the cost of Method 4 is the closest to that using real
bids.
Based on the investigations above, Method 4 (three-piece least squares
estimation) is selected as the linearization method for the improved twostage market-clearing model and the three-stage model in the coming chapters.
36
CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL
Amount of Procured Reserves
Method 4
Method 3
Method 2
Method 1
−600
−400
−200
0
200
Volume (MW)
SCR
TCR Weekly
400
600
800
TCR Daily
Figure 3.7: Amount of procured reserves using four fitting methods
Weekly
Total Cost of Reserves
Daily
Method 4
4.197
Method 3
4.216
Method 2
4.231
Method 1
4.237
0
0.5
1
1.5
2
2.5
Cost (MCHF)
3
3.5
4
4.5
Figure 3.8: Total procurement cost of reserves using four fitting methods
Cost of Reserves in Daily Market
0.5
Cost (MCHF)
0.4
0.403
0.348
0.287
0.3
0.257
0.233
0.2
0.1
0
Method 1
Method 2
Method 3
Method 4
Real Bids
Figure 3.9: Procurement cost in daily market using four fitting methods
after fixing weekly decision
3.3. IMPROVEMENTS OF TWO-STAGE MODEL
By
(Week
shown
13,393
37
applying the improved linearization method on 34 weeks in 2015
02−35), a total cost saving of 455,372 CHF can be gained, as is
in Table 3.2. This corresponds to 0.65% and an average saving of
CHF per week.
Table 3.2: Estimation of cost savings by improved linearization method
(Week 02−35, 2015)
Total Savings in CHF
Total Savings in %
Average Savings per Week in CHF
3.3.2
455,372
0.65
13,393
Selection of Scenarios
In stochastic programming, scenarios are crucial in obtaining an optimal
solution to the problem. Therefore, various data analyses and simulations
have been carried out within the framework of this thesis in order to improve
the performance of scenarios in the current model.
Although bids in reserve market do reflect price level of reserves to some
extent, they are essentially different from other price indices, especially when
the remuneration is pay-as-bid. Linearized bid curves can be affected by
various factors with large randomness. Thus, it is practically infeasible to
derive a forecast model for linearization parameters of bid curves. As a
result, only historical data can be counted upon while generating scenarios
to simulate possibilities in daily market.
In [4], three scenarios are taken into account in this two-stage stochastic market-clearing model. Indeed, three equi-probable scenarios are defined based on the bids in the week prior to the delivery week. This selection method is based on previous experience before this stochastic marketclearing model was introduced. However, it is necessary to investigate potential correlations between delivery week and other historical weeks and
explore the possibility of reducing procurement cost by introducing more
effective scenario selection methods.
Relationship between Weeks
In this section, the relationship between delivery week W and previous weeks
W−1, W−2, W−3, W−4 and Y−1 are investigated. Explanations of week
names can be found in Table 3.3.
As is discussed previously, bids in each time block can be represented
by a set of linearization parameters of the bid curve, or more specifically,
the slopes of different linear pieces: α1 , α2 and α3 (since three-piece fitting
method is chosen). Empirically, the first piece of the linearized bid curves is
most relevant to the procurement cost, as it usually covers a large portion of
38
CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL
Table 3.3: Explanation of week names
Notation
W
W−1
W−2
W−3
W−4
Y−1
Explanation
the week in which procured reserves are used (delivery week)
one week before the delivery week
two weeks before the delivery week
three weeks before the delivery week
four weeks before the delivery week
the same week in last year
reserve needed in the daily market. Given this implication, we will mainly
focus on the slope of the first piece, namely α1 .
As one week contains 42 time blocks, the profile of that week can usually
be represented by a curve or a vector with 42 data points. The input data
here is the values of α1 in each block. To investigate whether a particular
week can satisfyingly resemble the profile of an unknown week, two dimensions need to be considered: the correlation between those curves, and the
error between the corresponding data points.
The correlation coefficient of two vectors x and y is defined as:
N
1 X
ρ(x, y) =
N −1
i=1
x i − µx
σx
yi − µ y
σy
,
(3.40)
where
• N is the size of vector x and y
• µx and µy are mean values of x and y
• σx and σy are standard deviations of x and y
The correlation coefficient indicates the degree of similarity between
curve shapes. A time horizon of 34 weeks (Week 02−35, 2015) is selected
based on available data. For each week in the time horizon, correlation coefficients between that week and historical weeks (W−1, W−2, W−3, W−4,
Y−1) are calculated respectively. In the end, results of all 34 weeks are
synthesized by taking the median value of the correlation coefficients.
Table 3.4: Correlation coefficients between weeks
Slope of TCR+ α1+
Slope of TCR− α1−
W−1
0.780
0.797
Y−1
0.789
0.717
W−2
0.713
0.711
W−3
0.726
0.782
W−4
0.737
0.773
Table 3.4 shows the results of the correlation analysis. From this table
we can conclude that slopes of first piece are linearly correlated between
3.3. IMPROVEMENTS OF TWO-STAGE MODEL
39
weeks. These results provide the indication that directly using historical
data as input or reproducing series with similar pattern can be effective
ways of scenario generation method for this problem.
Despite that correlation coefficients imply the linear relationship between
vectors, they do not provide any link between the magnitude of the values.
Especially in our case, it is important to know how close the price level
of delivery week is to historical weeks. Therefore, the Root Mean Square
Errors (RMSEs) between two weeks are calculated.
Table 3.5: RMSE between weeks [CHF/MW]
Slope of TCR+ α1+
Slope of TCR− α1−
W−1
6.765
7.360
Y−1
9.375
9.351
W−2
7.756
11.560
W−3
9.115
10.330
W−4
12.815
12.611
From Table 3.5 we can see that RMSEs between W and W−1, Y−1,
W−2, W−3 remain within a certain range, whereas W−4 already deviates
more from W. Hence, conclusion can be drawn that W−1, Y−1, W−2 and
W−3 are most representative historical data for delivery week W. In the
next part of analysis, only these four historical weeks will be taken into
account as input data.
Comparison of Different Selection Methods
Based on the four selected historical weeks, a variety of scenario selection
methods are compared in this section. Figure 3.10 gives an overview of all
experimented methods.
In total 10 selection methods are compared and analyzed. They can be
primarily categorized into three groups:
1. Direct Input of Historical Data
This type of methods directly uses historical data. Parameters of bid
curves in the selected weeks are directly fed into the model as scenarios. The number of scenarios varies according to the number of
selected weeks. The probability of each scenario is usually equal (except Method 5).
• Method 0: Direct input of the delivery week W, also considered
as Perfect Information Scenario. This method is in reality not
possible, and only acts as an benchmark.
• Method 1: Direct input of W−1.
• Method 2: Direct input of W−1 and Y−1.
• Method 3: Direct input of W−1, W−2 and W−3.
• Method 4: Direct input of W−1, Y−1, W−2 and W-3.
40
CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL
Figure 3.10: Overview of scenario selection methods
• Method 5: Direct input of W−1, Y−1, W−2 and W−3, each
scenario assigned with a different probability factor.
2. Historical Data with Given Range
The idea of these methods originates from [4]. An upper and lower
bound is generated based on historical data. Here, the percentage is
assumed to be 20%, which means that three levels are generated: 80%,
100% and 120% of historical data.
• Method 6: 80%, 100% and 120% of W−1. This method is
currently used in the two-stage stochastic market-clearing model.
• Method 7: 80%, 100% and 120% of weeks W−1, Y−1, W−2,
W−3 respectively.
3. Average of Historical Data
Here, historical weeks are averaged. Based on the average, a random
term may be added to generate multiple scenarios.
• Method 8: Mean value of weeks W−1, Y−1, W−2, W−3.
• Method 9: Mean value of weeks W−1, Y−1, W−2, W−3, added
with a random variable term in proportion to standard deviation
of data.
To test these 10 methods, the last full week of each month from January
to August in 2015 is selected to form a sample set of 8 weeks. Information
regarding these 8 weeks are listed in Table 3.6.
3.3. IMPROVEMENTS OF TWO-STAGE MODEL
41
Table 3.6: Information of selected weeks [6]
Week Number
Week 04
Week 08
Week 13
Week 17
Week 22
Week 26
Week 30
Week 35
Date
19-25 Jan 2015
16-22 Feb 2015
23-29 Mar 2015
20-26 Apr 2015
25-31 May 2015
22-28 Jun 2015
20-26 Jul 2015
24-30 Aug 2015
Avg. Temperature [◦ C]
1.3/-0.7
4.4/-2.9
12.4/1.4
19.9/3.1
19.7/7.7
23.4/10.3
29/15.7
28/13.4
Actual procurement costs are calculated as the sum of the weekly procurement cost and the actual cost in the daily market after the realization
of daily bids. For each method, cost difference is calculated as the actual
cost of the method subtracted by actual cost of Method 0.
Figure 3.11: Cost difference w.r.t. perfect information scenario
Figure 3.11 presents the difference in actual costs using different methods
compared to perfect information scenario (Method 0). The 8 bars represent
results of 8 different weeks. The violet line shows the average cost difference in percentage. Since Method 0 is obtained using the actual data, it
is intuitive that actual cost of Method 0 should be the lowest amongst all
10 methods. Methods that have the lowest cost difference with regard to
Method 0 should be considered as the best methods.
From the results in Figure 3.11, we can conclude that the best two methods are Method 2 and Method 5, with the average cost difference below 5%.
Although Method 2 seems more accurate than Method 5 in terms of average
cost difference, the standard deviation of cost differences by Method 2 is
larger than Method 5. Therefore, we will have a closer look at these two
methods.
Figure 3.12 illustrates simulation results of Methods 2, 5 and 6 on 34
weeks in 2015 (Week 02−35). Method 6 corresponds to current practice
42
CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL
Figure 3.12: Cost difference w.r.t. perfect information scenario (Methods 2,
5 and 6)
and provides a benchmark. The height of the bars is namely difference in
actual procurement cost between selected method and perfect information
scenario.
According to Figure 3.12, current practice has led to significant cost
increase in three weeks (Week 16, 20 and 24). Although statistically these
data can be considered as outliers, this must be prevented and considered
as risks from the real-world implementation point of view. Based on results
in Table 3.7 Method 5 can be viewed as the most “stable” method with the
lowest average cost difference and standard deviation.
Table 3.7: Comparison of Methods 2, 5 and 6 w.r.t. perfect information
Avg. [CHF]
Std. [CHF]
Total [MCHF]
Method 6
(Current Practice)
8915.1
141861.4
3.031
Method 2
(W−1, Y−1)
7412.4
76943.9
2.520
Method 5
(W−1, Y−1, W−2, W−3
with different probability)
6686.3
54382.7
2.273
Based on the analysis above, Method 5 is selected as the improved scenario selection method. Table 3.8 lists the savings of implementing Method
6 in comparison with current practice (after improving linearization model).
For Week 02−35 in 2015, a total saving of 757,795 CHF (1.09%) can be
achieved. By combining results in Tables 3.2 and 3.8, the overall saving
potential is calculated, as is shown in Table 3.9. If both improvements on
linearization method and scenario selection method can be implemented, a
total saving of over 1.2 million CHF (1.74%) can be gained.
3.3. IMPROVEMENTS OF TWO-STAGE MODEL
43
Table 3.8: Estimation of savings by improved scenario selection method
(Week 02−35, 2015)
Total Savings in CHF
Total Savings in %
Average Savings per Week in CHF
757,795
1.09
22,288
Table 3.9: Estimation of savings by implementing both improvements (Week
02−35, 2015)
Total Savings in CHF
Total Savings in %
Average Savings per Week in CHF
1,213,167
1.74
35,681
44
CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL
Chapter 4
Three-Stage Market-Clearing
Model
In this chapter, a three-stage stochastic market-clearing market is proposed.
Firstly, motivation and decision making process are introduced in Section
4.1. Section 4.2 mainly focuses on the formulation of the three-stage model.
In Section 4.3, how third-stage scenarios are generated and selected is illustrated. Simulation results are demonstrated in Section 4.4.
4.1
Introduction
In Switzerland, a majority of ancillary services providers are hydro units.
The available generation power of hydro power plants are very much dependent on seasonal pattern and weather conditions, which can be rather
unpredictable. Based on market observations, it is probable that during water peak seasons, hydro units still have unsold power in real-time operation
and have to either get rid of the additional power at a very low price or to
pay a penalty for power imbalances it incurs. Furthermore, there has been
a growing trend of RES units participating in ancillary services provision.
These units are mainly wind or photovoltaic generators, whose power output
is highly unforeseeable and intermittent.
The current market model which clears at most once per day is not
adapted for such changes. Therefore, an additional market stage where redundant power can be sold or bought as reserves is highly desirable, from
both TSO’s and provider’s perspective. Providers gain more flexibility in
bidding and will be exempted from power imbalance payments if trading
succeeds. For Swissgrid, having the possibility of procuring cheaper reserves
closer to real-time operation can potentially reduce the amount of more
expensive reserves bought in weekly and daily market, thus achieving financial gains in reserve procurement cost. Based on these motivations, such a
market is very likely to appear within the framework of the Swiss ancillary
45
46
CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL
services market.
In this thesis, we assume that there will be a new market stage: hourly
market. This hourly market will be cleared every hour after the trading in
spot market closes. With this assumption, a three-stage stochastic marketclearing model needs to be developed, which is exactly the main purpose of
this thesis.
The decision-making process of the three-stage market can be illustrated
by Figure 4.1. At the first stage when the weekly market is cleared, only bids
in weekly market are known. The rest of the bids (daily market and hourly
market) are yet uncertain and are therefore represented by scenarios. After
market-clearing, results pertaining to the acceptance of weekly bids as well as
suggested procurement volume in the daily and hourly market are obtained.
Weekly decisions are then given to the clearing of daily market as input. At
this second stage, the daily market is to be cleared with available daily bids
and still unknown hourly bids, which are again modelled as scenarios (can
be different from those used in the first stage as more information could
be collected). At the final stage, the hourly market will be cleared with
decisions from the previous two stages as well as actual bids in the hourly
market.
Since this three-stage reserve market is still conceptual in Switzerland,
no concrete market data are available to help construct hourly scenarios, let
alone the update and realization of hourly bids. Therefore, this thesis will
only focus on the weekly clearing model, which is the three-stage stochastic
market-clearing model.
Figure 4.1: Three-stage stochastic market-clearing scheme
4.2. STOCHASTIC MARKET-CLEARING MODEL
4.2
47
Stochastic Market-Clearing Model
This section will step by step unveil the three-stage stochastic marketclearing model. Decision variables and objective function are explained in
Sections 4.2.1 and 4.2.2. Section 4.2.3 is dedicated to the non-anticipativity
considered in this three-stage problem. Section 4.2.4 adapts dimensioning
criteria to the three-stage model and updates the constraints. The complete
formulation of the model is presented in Section 4.2.5.
4.2.1
Decision Variables
Intuitively, decision variables in the three-stage model can be grouped into
first-stage decision variables, second-stage decision variables and third-stage
decision variables. The definition and notation of first- and second-stage
decision variables are exactly identical with those in the two-stage model,
as introduced in Section 3.2.1.
For the third stage, a set of scenarios are created to represent uncertainty in the hourly market. Here, we assume that there does not exist any
correlation between daily and hourly scenarios. Therefore, scenarios used in
third stage can be formulated
as a combination of scenarios for daily and
hourly market: ω d , ω h . Each scenario combination is associated with a set
of third-stage decision variables: xh (ω d , ω h ):
xh (ω d , ω h ) = [xhT+ (ω d , ω h ), xhT− (ω d , ω h )],
(4.1)
where
• ω h ∈ Ωh is the index of scenarios for hourly market
• Ωh contains NΩh scenarios for hourly market
• xhT+ ω d , ω h ∈ R168×1 is the amount of TCR+ to be
hourly market in each hour for scenario combination
• xhT− ω d , ω h ∈ R168×1 is the amount of TCR− to be
hourly market in each hour for scenario combination
procured in the
ωd, ωh
procured in the
ωd, ωh
In the third stage, TCR market is cleared for each individual hour. Since
the three-stage market-clearing model is valid for one week (24 × 7 = 168
hours), length of third-stage parameters and variables is often 168.
4.2.2
Objective Function
Similar to the two-stage model, the objective function in the three-stage
market-clearing model can be split into three terms, each representing one
stage. The terms of previous two stages are exactly the same as in Section
48
CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL
3.2.2. The framework of the objective function for the three-stage model
can be formulated as follows:
min Λw + E[Λd (ω d )] + E[Λh (ω d , ω h )],
∀ω d ∈ Ωd , ω h ∈ Ωh ,
(4.2)
where Λh (ω d , ω h ) is the procurement cost in the hourly market for the
given week with input data of scenarios (ω d , ω h ). E[Λh (ω d , ω h )] is the expectation of hourly procurement costs for all daily-hourly scenario combinations.
To model third-stage costs, we use a similar method as second-stage
costs. Though behavior of market participants in hourly market is still
unpredictable, it can be deduced from the motivation of this market stage
that a certain amount of reserve power will be available either for free or
at a very low price. For any amount exceeding this limit, bid prices may
increase drastically, which represents the emergency assistance reserves when
procured reserves are not sufficient for the need of system security. This
market behavior can be const ructed as a two-piece linear bid curve, as is
shown in Figure 4.2.
Figure 4.2: Hypothetical hourly bid curve
Similar to Equation (3.6), procurement cost in one hour can be written
as:
(
ρ1 x + ϕ1 ,
ζ=
ρ2 x + ϕ2 ,
xmin ≤ x ≤ xB
,
xB < x ≤ xmax
where
• ζ is the procurement cost of this hour
• x is the amount of reserve to be procured in hourly market
(4.3)
4.2. STOCHASTIC MARKET-CLEARING MODEL
49
• (ρ1 , ϕ1 ) and (ρ2 , ϕ2 ) are fitting parameters of the first and second
part of the hourly bid curve respectively
• xmin and xmax are lower and upper bound of the bid curve
• xB is the breaking point of the linearized bid curve
Provided that ρ2 > ρ1 , Equation (4.3) can be reformulated as a optimization problem:
min
ζ
x
s.t.
ρ1 x + ϕ1 ≤ ζ,
ρ2 x + ϕ2 ≤ ζ,
xmin ≤ x ≤ xmax .
(4.4)
Extending the single-hour optimization to 168 hours within a week and
considering all scenario combinations, the full version of Equation (4.4) is
presented in Equation (4.5).
min
h
d h
d h
xh
T (ω ,ω ), xT (ω ,ω )
ζT+ (ω d , ω h ) + ζT− (ω d , ω h ),
−
+
s.t.
+ h
d
h
h h
d
h
ρ+
1 (ω )xT+ (ω , ω ) + ϕ1 (ω ) ≤ ζT+ (ω , ω ),
+ h
d
h
h h
d
h
ρ+
2 (ω )xT+ (ω , ω ) + ϕ2 (ω ) ≤ ζT+ (ω , ω ),
− h
d
h
h h
d
h
ρ−
1 (ω )xT− (ω , ω ) + ϕ1 (ω ) ≤ ζT− (ω , ω ),
− h
d
h
h h
d
h
ρ−
2 (ω )xT− (ω , ω ) + ϕ2 (ω ) ≤ ζT− (ω , ω ),
h
max
h
xhT+ (ω d , ω h ) ∈ [xmin
h,T+ (ω ), xh,T+ (ω )],
h
max
h
xhT− (ω d , ω h ) ∈ [xmin
h,T− (ω ), xh,T− (ω )],
∀ω d ∈ Ωd , ∀ω h ∈ Ωh ,
(4.5)
where
• ζT+ (ω d , ω h ) ∈ R168×1 is the hourly cost vector of TCR+
• ζT− (ω d , ω h ) ∈ R168×1 is the hourly cost vector of TCR−
+ h
+ h
+ h
h
168×1 are linear coefficients of
• ρ+
1 (ω ), ϕ1 (ω ), ρ2 (ω ), ϕ2 (ω ) ∈ R
TCR+ hourly bid curves and are only dependent on hourly scenario
ωh
− h
− h
− h
h
168×1 are linear coefficients of
• ρ−
1 (ω ), ϕ1 (ω ), ρ2 (ω ), ϕ2 (ω ) ∈ R
TCR− hourly bid curves and are only dependent on hourly scenario
ωh
h
max
h
• xmin
h,T+ (ω ), xh,T+ (ω ) are lower and upper bound of TCR+ bid curve
h
max
h
• xmin
h,T− (ω ), xh,T− (ω ) are lower and upper bound of TCR− bid curve
50
CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL
Therefore, the term E[Λh (ω d , ω h )] in Equation (4.2) can be reformulated
as:
E[Λh (ω d , ω h )] =
X
h
i
πd ω d πh ω h 11×168 ζT+ (ω d , ω h ) + 11×168 ζT− (ω d , ω h ) ,
ω d ∈Ωd ,
ω h ∈Ωh
(4.6)
where πh ω h is the probability of hourly scenario ω h .
4.2.3
Non-anticipativity Matrix
Non-anticipativity can be explained as the link between different stages.
This is particularly important in multi-stage problems. The principle of
non-anticipativity is: “if realizations of stochastic processes are identical up
to stage k, the values of decision variables must be identical up to stage
k” [11].
In our problem, although daily scenarios and hourly scenarios are considered as “decoupled”, it is still important to link third-stage decision variables
with the corresponding second-stage decision variables. This link is established through a non-anticipativity matrix An . The number of rows corresponds to the product of 168 (hours in a week), the number of daily scenarios
NΩd , and the number of hourly scenarios NΩh . The number of columns is the
product of 42 (time blocks in a week) and the number of daily scenarios NΩd .
The value of a single element in the non-anticipativity maxtrix An (ij, mn)
is 1, if:
1. hour i is included in time block m, and
2. scenario combination j is identical with scenario n at second stage
where
• i is the index of the corresponding hour
• j is the index of the corresponding scenario combination
• m is the index of the corresponding time block
• n is the index of the corresponding scenario in second stage
Illustrative Example: Assume that there are 2 scenarios for daily market:
ω1d , ω2d , and 2 scenarios for hourly market: ω1h , ω2h . Hence, there are in total
4 scenario combinations:
(ω1d , ω1h ), (ω1d , ω2h ), (ω2d , ω1h ), (ω2d , ω2h )
(4.7)
Consider 2 time blocks with 4 hours each:
p1 = (t1 , t2 , t3 , t4 ), p2 = (t5 , t6 , t7 , t8 )
(4.8)
4.2. STOCHASTIC MARKET-CLEARING MODEL
51
Decision variable vector for second stage is:
h
i
(1) (2) (3) (4)
xd = xd , xd , xd , xd
h
i
= xdp1 (ω1d ), xdp2 (ω1d ), xdp1 (ω2d ), xdp2 (ω2d ) .
(4.9)
Decision variable vector for third stage is:
h
xh = xht1 (ω1d , ω1h ), xht2 (ω1d , ω1h ), xht3 (ω1d , ω1h ), xht4 (ω1d , ω1h ),
xht5 (ω1d , ω1h ), xht6 (ω1d , ω1h ), xht7 (ω1d , ω1h ), xht8 (ω1d , ω1h )
xht1 (ω1d , ω2h ), xht2 (ω1d , ω2h ), xht3 (ω1d , ω2h ), xht4 (ω1d , ω2h ),
xht5 (ω1d , ω2h ), xht6 (ω1d , ω2h ), xht7 (ω1d , ω2h ), xht8 (ω1d , ω2h ),
xht1 (ω2d , ω1h ), xht2 (ω2d , ω1h ), xht3 (ω2d , ω1h ), xht4 (ω2d , ω1h ),
(4.10)
xht5 (ω2d , ω1h ), xht6 (ω2d , ω1h ), xht7 (ω2d , ω1h ), xht8 (ω2d , ω1h ),
xht1 (ω2d , ω2h ), xht2 (ω2d , ω2h ), xht3 (ω2d , ω2h ), xht4 (ω2d , ω2h ),
i
xht5 (ω2d , ω2h ), xht6 (ω2d , ω2h ), xht7 (ω2d , ω2h ), xht8 (ω2d , ω2h ) .
The first four variables in Equation (4.10) correspond to second-stage
scenario ω1d and time block 1, and are thus linked to second-stage decision
(1)
variable xd . The next four variables in Equation (4.10) are related to
second-stage scenario ω1d and time block 2, and are thus linked to second(2)
stage decision variable xd , and so on.
52
CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL
Therefore, the non-anticipativity matrix in this case is:

1
1

1

1

0

0

0

0


1

1

1

1

0

0

0

0

0

0

0

0

0

0


0

0

0

0

0

0

0

0

0
0
4.2.4
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0

0
0

0

0

0

0

0

0


0

0

0

0

0

0

0

0

0

0

0

0

1

1


1

1

0

0

0

0

1

1

1
1
Constraints
The binding factors in the three-stage market-clearing model are identical
with those in two-stage model, as explained in Section 3.2.3. The constraint
for conditional bids in the first stage has exactly the same formulation as
Equation (3.11).
4.2. STOCHASTIC MARKET-CLEARING MODEL
53
Probabilistic Dimensioning Criterion
The probabilistic criterion remains the same as Equation (3.15). However,
the amount of overall reserve differs, as the additional amount of reserve
procured in hourly market should be taken into account:
Rs+ = Rs− = p|S xw
S,
|
w
d
d
h
d
h
Ro+ = p|S xw
S + pT+ xT+ + An xT+ (ω ) + xT+ (ω , ω ),
|
w
d
d
h
d
h
Ro− = p|S xw
S + pT− xT− + An xT− (ω ) + xT− (ω , ω ),
(4.11)
∀ω d ∈ Ωd , ∀ω h ∈ Ωh .
Inserting Equation (4.11) into Equation (3.15), we get:
| w
Fs+ (p|S xw
S ) + Fs− (pS xS )
|
w
d
d
h
d
h
+Fo+ p|S xw
+
p
x
+
A
x
(ω
)
+
x
(ω
,
ω
)
n T+
S
T+
T+ T+
|
w
d
d
h
d
h
+
p
x
+
A
x
(ω
)
+
x
(ω
,
ω
)
≤ 0.002,
+Fo− p|S xw
n T−
S
T−
T− T−
(4.12)
∀ω d ∈ Ωd , ∀ω h ∈ Ωh .
Linearization of deficit curves remains the same as in Equation (3.22).
Rs+ , Rs− , Ro+ and Ro− will be replaced by Equation (4.11).
Deterministic Dimensioning Criterion
Considering reserves procured in the hourly market, Equation (3.23) can be
modified as follows:
d
d
h
d
h
400 ≤ p|T+ xw
T+ + An xT+ (ω ) + xT+ (ω , ω ),
(4.13)
∀ω d ∈ Ωd , ∀ω h ∈ Ωh .
4.2.5
Formulation
The complete mathematical formulation of the three-stage stochastic marketclearing model is presented as follows:
54
CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL
c|w xw +
min
xw , λT+ (ω d ),
λT− (ω d ),
ζT+ (ω d ,ω h ),
ζT−
(ω d ,ω h )
P
πd ω d
11×42 λT+ (ω d ) + 11×42 λT− (ω d )
ω d ∈Ωd
+
P
πd ω d πh ω h 11×168 ζT+ (ω d , ω h ) + 11×168 ζT− (ω d , ω h )
ω d ∈Ωd ,
ω h ∈Ωh
(4.14)
s.t.
Ac xw ≤ bc ,
(4.15)
+ d
d d
d
d
d
α+
1 (ω )xT+ (ω ) + β1 (ω ) ≤ λT+ (ω ), ∀ω ∈
+ d
d d
d
d
d
α+
2 (ω )xT+ (ω ) + β2 (ω ) ≤ λT+ (ω ), ∀ω ∈
+ d
d d
d
d
d
α+
3 (ω )xT+ (ω ) + β3 (ω ) ≤ λT+ (ω ), ∀ω ∈
− d
d
d
d d
d
α−
1 (ω )xT− (ω ) + β1 (ω ) ≤ λT− (ω ), ∀ω ∈
− d
d
d
d d
d
α−
2 (ω )xT− (ω ) + β2 (ω ) ≤ λT− (ω ), ∀ω ∈
− d
d
d
d d
d
α−
3 (ω )xT− (ω ) + β3 (ω ) ≤ λT− (ω ), ∀ω ∈
+ h
d
h
h h
d
h
ρ+
1 (ω )xT+ (ω , ω ) + ϕ1 (ω ) ≤ ζT+ (ω , ω ),
d
d
h
h
d
Ω ,
(4.16)
Ωd ,
(4.17)
d
(4.18)
d
Ω ,
(4.19)
Ωd ,
(4.20)
d
(4.21)
Ω ,
Ω ,
∀ω ∈ Ω , ∀ω ∈ Ω ,
h h
d
h
ρ+
2 (ω )xT+ (ω , ω ) +
d
d
h
h
h
ϕ+
2 (ω )
h h
d
h
ρ−
1 (ω )xT− (ω , ω ) +
d
d
h
h
h
ϕ−
1 (ω )
d
h
d
h
(4.22)
≤ ζT+ (ω , ω ),
∀ω ∈ Ω , ∀ω ∈ Ω ,
(4.23)
≤ ζT− (ω , ω ),
∀ω ∈ Ω , ∀ω ∈ Ω ,
(4.24)
− h
h h
d
h
d
h
ρ−
2 (ω )xT− (ω , ω ) + ϕ2 (ω ) ≤ ζT− (ω , ω ),
∀ω d ∈ Ωd , ∀ω h ∈ Ωh ,
(4.25)
| w
(i)
a(i)
(4.26)
s+ pS xS + bs+ ≤ εs+ , i = 1, · · · , m,
|
w
(i)
a(i)
(4.27)
s− pS xS + bs− ≤ εs− , i = 1, · · · , n,
| w
|
w
d
d
h
d
h
(i)
a(i)
o+ pS xS + pT+ xT+ + An xT+ (ω ) + xT+ (ω , ω ) + bo+ ≤ εo+ ,
∀ω d ∈ Ωd , i = 1, · · · , j,
(4.28)
| w
|
w
d
d
h
d
h
(i)
a(i)
o− pS xS + pT− xT− + An xT− (ω ) + xT− (ω , ω ) + bo− ≤ εo− ,
∀ω d ∈ Ωd , i = 1, · · · , k,
(4.29)
εs+ + εs− + εo+ + εo− ≤ 0.002,
d
d
h
d
h
d
400 ≤ p|T+ xw
T+ + An xT+ (ω ) + xT+ (ω , ω ), ∀ω
d
max
d
d
d
xdT+ (ω d ) ∈ [xmin
d,T+ (ω ), xd,T+ (ω )], ∀ω ∈ Ω ,
d
max
d
d
d
xdT− (ω d ) ∈ [xmin
d,T− (ω ), xd,T− (ω )], ∀ω ∈ Ω ,
h
max
h
d
d
xhT+ (ω d , ω h ) ∈ [xmin
h,T+ (ω ), xh,T+ (ω )], ∀ω ∈ Ω ,
(4.30)
d
∈ Ω , (4.31)
(4.32)
(4.33)
∀ω h ∈ Ωh ,
(4.34)
h
max
h
d
d
h
h
xhT− (ω d , ω h ) ∈ [xmin
h,T− (ω ), xh,T− (ω )], ∀ω ∈ Ω , ∀ω ∈ Ω .
(4.35)
4.3. SCENARIOS FOR HOURLY MARKET
55
Constraint (4.15) corresponds to the conditional bids received in the
weekly market. Constraints (4.16)−(4.21) are related to three-piece daily
cost curve. Constraints (4.22)−(4.25) are related to hourly cost curve.
(4.26)−(4.30) are associated with probabilistic dimensioning criterion. Constraint (4.31) is for deterministic dimensioning criterion (MEAS constraint).
Constraints (4.32) and (4.33) define feasible ranges for daily procurement
amounts, whereas constraints (4.34) and (4.34) define feasible ranges for
hourly procurement amounts.
4.3
Scenarios for Hourly Market
4.3.1
Modelling of Hourly Bid Curves
As is mentioned previously, the hourly market has not yet been established
in Switzerland and no market data are available, which is the biggest challenge in building scenarios for hourly market. Figure 4.2 presented how a
hypothetical hourly bid curve is constructed. Key parameters are slopes of
both pieces and the breaking point.
Breaking Point
In the current market setting, providers with additional available power may
submit bids into TCE market, which will be referred to as free TCE bids
hereafter. These bids are not bound with a previously accepted reserve bid
and can represent real-time available capacity of power plants to a certain
extent. Nevertheless, since not many BGs are submitting such bids, estimations based on these data may turn out to be conservative.
Making use of these free TCE bids does not require any procurement of
the corresponding reserve volume, which can save reserve costs. Therefore,
the total volume of free TCE bids can be in a way regarded as the breaking
point of the bid curve presented in Figure 4.2. Prior to this breaking point,
the curve has a very small slope, which corresponds to the cheap or even
free reserves that can be procured in the hourly market.
Slope
Although breaking point is the key binding parameter in the hourly bid
curve, it is still important to select adequate slopes for the first and second
linear pieces.
The slope of the first piece has to be very low in order to reflect the
nature of these capacity. However, it cannot be set to 0, since it would
result in multiple solutions of the optimization problem.
The slope of the second piece needs to be relatively large, as it represents
the cost of emergency reserves in case the procured reserves are not sufficient.
56
CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL
This price has to be realistic as well. Otherwise, the solution will always
be driven to the first piece, which undermines the motivation of having the
second piece.
Based on the criteria above and sensitivity analysis, the slope of the
first piece is defined as 0.1 CHF/MW, while the slope of the second piece is
chosen to be 200 CHF/MW.
4.3.2
Scenario Construction
Since slopes of hourly bid curve are assumed to be constants, key parameter
in hourly scenarios is the breaking point, or the volume of free TCE bids.
Free TCE+ Capacity
250
MW
300
200
200
150
100
100
0
0
10
20
Week Number
30
40
Mon
Tue
Wed
Thu
Fri
Sat
50
Sun
0
Figure 4.3: Free TCE+ volume in 2015 (Week 01−35)
Free TCE− Capacity
200
300
MW
200
150
100
100
0
40
30
20
Week Number
10
0
Sun
Sat
Fri
Thu
Wed
Tue
Mon
50
0
Figure 4.4: Free TCE− volume in 2015 (Week 01−35)
Figures 4.3 and 4.4 illustrate the free TCE volume in the current market.
It is noticeable that both positive and negative products are subject to a
4.3. SCENARIOS FOR HOURLY MARKET
57
seasonal and temporal pattern. Free TCE+ volume increases during offpeak hours and summer time, when water reservoirs are relatively full. In
contrast, free TCE− volume drops during off-peak hours. This is obvious
due to the operating point of power generators during peak and off-peak
periods.
To preserve such seasonal pattern of free TCE volume, data from a time
horizon of 4 weeks are taken into consideration to simulate possibilities in
the delivery week. W−1, W−2, W−3 and W−4 are provided as inputs
into the model, and the minimum, mean, median and maximum values of
each hour will be calculated respectively. In this way, we obtain 4 scenarios,
namely minimum scenario, mean scenario, median scenario and maximum
scenario. The construction process is illustrated in Figure 4.5.
Figure 4.5: Scenario construction process
Currently, the resolution of TCE market bids is also 4 hour. In order to
accommodate it with the hourly clearing design, a set of hourly discretizing
factors are derived using hourly peak load data from [29]. Assume that
peak generation in the system equals to 1.2 times peak load. For each time
block, assume that the peak load levels for each hour within the block are
p1 , p2 , p3 , p4 respectively. The available amount in third stage for each
hour is the product of corresponding discretizing factor η and the free TCE
volume for that time block. Discretizing factors for hourly tertiary positive
and negative reserves can be calculated as:
ηi+ =
120 − pi
,
max (120 − pj )
1≤j≤4
ηi−
pi
=
,
max pj
(4.36)
1≤j≤4
where ηi+ and ηi− are hourly discretizing factors for tertiary positive and
negative reserves in hour i respectively, and pi is the hourly peak load level
in hour i.
58
CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL
The calculated hourly discretizing factors are shown in Appendix A.
As hourly market has not yet been established, scenarios selected above
represent different possible levels of available bid volume in this new market.
In a full set of scenarios for hourly market, all four scenarios are incorporated
and assigned with different probability factors, which are shown in Table 4.1.
These probabilities are selected based on sensitivity analysis.
Table 4.1: Probability factors of hourly scenarios
Scenario
Minimum
Median
Mean
Maximum
4.4
Probability
5%
45%
45%
5%
Simulation Results
This section presents simulation results of the proposed three-stage stochastic market-clearing model. Section 4.4.1 studies the impact of adding a
third stage by investigating different hourly scenarios . In Section 4.4.2, a
full set of daily and hourly scenarios will be considered and results will be
demonstrated.
4.4.1
Case Study: Impact of Hourly Market
The objective of this case study is to investigate the influence of hourly
market scenarios on optimization results. Therefore, six cases are simulated
and compared. For convenience, perfect information scenario is selected
for second stage. These six cases only differ in the selection of third-stage
scenarios and are defined in Figure 4.6.
• Case “Two-Stage” refers to the two-stage model with perfect information as second-stage scenario, which is a deterministic market-clearing.
• Cases “Min”, “Median”, “Mean” and “Max” are deterministic threestage market-clearing models where minimum, median, mean and maximum scenarios are inputs for third-stage scenario respectively.
• Case “Full” refers to the case where a full set of hourly scenarios (minimum, median, mean, maximum) are imported and assigned with different probability factors.
The 6 cases are simulated on 8 selected weeks in 2015. Information
regarding these 8 weeks can be found in Table 3.6. Results in terms of
reserve procurement and total cost of procurement are presented as follows.
4.4. SIMULATION RESULTS
59
Figure 4.6: Definition of cases
Reserve Amount
For each selected week, six optimization models are run and decisions are
obtained regarding accepted bids in weekly market and suggested procurement volume in daily and hourly market. Since decisions regarding daily and
hourly market are associated with each hour/time block and each scenario,
an average procurement volume is calculated. Two typical weeks (Week
04 and Week 35) are selected here to illustrate the impact of third stage
scenarios on procurement decisions.
Figure 4.7 shows the amount of SCR and TCR procured in each market.
As can be seen from the figure, as the available volume in hourly market
increases (from “Two-Stage” to “Max”), the volume of reserves procured in
hourly market increases correspondingly. This is due to the fact that hourly
reserves are generally more attrative in terms of pricing. As a consequence,
the amount of reserves procured in daily market reduces, which can be regarded as the substitution effect between daily and hourly products. The
amount of SCR and overall reserves, however, remains constant, as it is the
optimum to satisfy dimensioning criteria.
In Figure 4.8, we can observe that the total amount of reserves changes
according to different cases. A slight difference in the procured amount
of SCR explains this phenomenon. As SCR is generally more expensive
(3-5 times) than TCR, it is possible that more TCR is procured in order
to compensate for SCR, which leads to lower procurement cost. This can
be considered as an outcome of dimensioning reserves using probabilistic
criterion in Equation (3.15).
60
CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL
Reserve Amount for Week 35
Full
Max
Mean
Median
Min
Two Stage
−800
−600
−400
−200
0
200
Volume (MW)
SCR
TCR Weekly
TCR Daily
400
600
800
1000
TCR Hourly
Figure 4.7: Amount of reserves procured for Week 35 2015
Reserve Amount for Week 04
Full
Max
Mean
Median
Min
Two Stage
−800
−600
−400
SCR
−200
0
200
Volume (MW)
TCR Weekly
TCR Daily
400
600
800
1000
TCR Hourly
Figure 4.8: Amount of reserves procured for Week 04 2015
4.4. SIMULATION RESULTS
61
Total Cost of Procurement
Total procurement cost is calculated as the sum of weekly, daily and hourly
reserve costs. For full scenario case, the hourly cost is the expected cost of
all hourly scenarios. The total procurement cost of each three-stage case
is benchmarked with the two-stage case. Cost savings in percentage are
obtained and presented in Table 4.2 and Figure 4.9.
From the results we can see that incorporating an hourly market can
achieve an average saving of up to 3.11% in the maximum case. This can
be an indicator for future market design, as it reveals the potential of having an hourly market. If the liquidity of third-stage market is similar to
the free TCE volume nowadays, “mean” and “median” cases can represent
future market situation, in which around 1.5% can be saved. To be more
conservative, the minimum case is considered, which still provides 0.35%
savings. The “full” scenarios case is somewhat close to minimum case, but
with slightly higher savings potential. Therefore, we can conclude that the
“full scenarios” case is a rather conservative estimation of the third-stage
market.
Table 4.2: Potential cost savings w.r.t. two-stage model
Week 04
Week 08
Week 13
Week 17
Week 22
Week 26
Week 30
Week 35
Average
4.4.2
Min
0.09%
0.31%
0.31%
0.03%
0.11%
0.48%
0.38%
1.10%
0.35%
Median
0.86%
1.31%
1.01%
0.92%
0.61%
1.33%
1.00%
2.26%
1.16%
Mean
1.06%
1.63%
1.49%
1.36%
1.14%
1.80%
1.20%
2.72%
1.55%
Max
2.33%
3.25%
3.35%
3.04%
1.89%
4.09%
2.24%
4.66%
3.11%
Full
0.15%
0.38%
0.36%
0.19%
0.27%
0.49%
0.41%
1.16%
0.43%
Complete Scenario Simulations
In this section, simulation results with a complete set of second-stage and
third-stage scenarios are presented. Method 5 in Figure 3.10 is selected as
second-stage scenarios. All four hourly scenarios (min, median, mean, max)
are incorporated into third stage with probabilities given in Table 4.1. In
total, 16 scenario combinations are considered. The whole scenario tree is
presented in Figure 4.10. Notations of scenarios are defined in Table 4.3.
The size of the optimization problem is shown in Table 4.4. For Week 04
2015, there are in total 17,090 decision variables and 108,905 constraints (the
exact number may vary according to number of weekly bids). The problem
is solved with CPLEX using a commercial package TOMLAB in Matlab.
62
CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL
Cost Saving w.r.t. Two−Stage Model
5%
4%
Cost Saving
3.11%
3%
2%
Week 04
Week 08
Week 13
Week 17
Week 22
Week 26
Week 30
Week 35
1.55%
1.16%
1%
0.43%
0.35%
0
Min
Median
Mean
Max
Full
Figure 4.9: Estimated cost savings w.r.t two-stage model
Figure 4.10: Scenario tree of complete model
Table 4.3: Scenario notations
Daily Scenarios
Number Scenario
I
W−1
II
Y−1
III
W−2
IV
W−3
Hourly Scenarios
Number
Scenario
1
Minimum
2
Median
3
Mean
4
Maximum
4.4. SIMULATION RESULTS
63
The average computation time is around 20-25 minutes on ordinary personal computers, which is satisfying for three-stage stochastic programming
problems and affordable for real market clearing.
Table 4.4: Problem size
First-Stage Variables
Second-Stage Variables
Third-Stage Variables
Variables for Deficit Rate
Constraints for Conditional Bids
Constraints for MEAS
Constraints for Deficit Curve
Constraints for Daily Cost
Constraints for Hourly Cost
Constraints for Reserve Amount
288
(Depending on number of bids)
42 × 4 × 2 + 42 × 4 × 2 = 672
168 × 16 × 2 + 168 × 16 × 2 = 10, 752
2 + 168 × 16 × 2 = 5, 378
22
(Depending on number of bids)
168 × 16 = 2, 688
88, 722
(Depending on number of pieces)
1, 344
10, 752
5, 377
Reserve Amount
Figure 4.11 and Table 4.5 show simulated reserve procurement amount for
the 8 selected weeks. Daily and hourly reserve amount is calculated as the
mean value of different scenarios and hour/time blocks. We can get the
following takeaways from these results:
1. During summer (July and August), more reserve is available in the
hourly market. This leads to less procurement in the daily market,
and even reduces the amount of SCR procured in the weekly market.
However, a growing share of hourly TCR+ is not witnessed in Week
30 and 35. This is partially due to the unusual hot summer in Europe
in 2015. Most Swiss producers exported their produced electricity to
Italy, where demand is extraordinarily high due to air conditioning.
2. During winter and spring seasons, a scarcity of water is expected.
Thus, not much available power in the hourly market is anticipated.
As a result, more weekly and daily products are procured in order to
guarantee system security.
3. Comparing Week 22 and 30, we can roughly estimate that 1 MW of
TCR− can compensate for 2 MW of TCR+. The comparison between Week 26 and 30 shows that 1 MW SCR can be replaced by
approximately 11 MW TCR−. These relationships provide basis for
64
CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL
substitutions between different products while obeying system security
criteria.
Reserve Amount with Complete Scenarios
Week 35
Week 30
Week 26
SCR
Week 22
TCR Weekly
TCR Daily
Week 17
TCR Hourly
Week 13
Week 08
Week 04
−800
−600
−400
−200
0
200
Volume (MW)
400
600
800
1000
Figure 4.11: Reserve amount of complete model
Table 4.5: Amount of reserves procured in complete model
Week
Week
Week
Week
Week
Week
Week
Week
04
08
13
17
22
26
30
35
SCR [MW]
385
375
371
380
370
375
370
372
TCR+ [MW]
415
428
451
408
413
403
405
424
TCR− [MW]
159
219
248
190
271
221
275
241
Total Cost of Procurement
The total expected cost of procurement is calculated for each week and then
compared with that of the two-stage model with identical daily scenarios.
Results are presented in Table 4.6.
Table 4.6 conveys the message that on average 10,767 CHF can be saved
by using a three-stage market-clearing. This result is based on the hypothesis
that all improvements suggested in Section 3.3 are implemented on the twostage model. If we count in savings brought by these improvements, more
financial benefits could be gained.
Moreover, these results are based on the conservative third-stage selection method (which considers minimum scenario). In the future when the
third-stage market is established, it is highly probable that Swissgrid will
4.4. SIMULATION RESULTS
65
Table 4.6: Total cost of procurement of complete model
Week
Week
Week
Week
Week
Week
Week
Week
04
08
13
17
22
26
30
35
Two-Stage [MCHF] Three-Stage [MCHF]
1.816
1.810
1.638
1.629
3.247
3.235
2.692
2.681
1.717
1.708
1.556
1.545
1.381
1.375
1.854
1.832
Average
Saving [CHF]
6,879
8,440
11,468
10,794
8,453
11,702
6,508
21,887
10,767
receive more bids than what is available in the minimum case. Taking this
possibility into account, potential savings will look more optimistic.
Substitution Effect
One particular advantage of using this three-stage stochastic market-clearing
model versus traditional approach is the substitution effect. It can be observed from two aspects:
1. Substitution between SCR, TCR+ and TCR− products, which is enabled through probabilistic dimensioning criteria.
2. Substitution between weekly, daily and hourly products, which is enabled through three-stage stochastic model.
Figure 4.12 shows an example of substitution between TCR+ and TCR−
procured in the daily and hourly market. Scenario (I, 1) for Week 30 is
selected here. As weekly reserves remain constant throughout the entire
week, we only focus on the total amount of reserves procured in daily and
hourly market. From this figure, it is obvious that TCR+ is procured less
during peak hours when TCR− is more attractive in cost. During off-peak
hours, TCR− is replaced by TCR+. The outcome of this substitution is the
minimum total procurement cost.
Figure 4.13 illustrates the substitution between daily and hourly products, most prominently in TCR−. Daily scenario I and its four sub-scenarios
are demonstrated here. First of all, suggested procurement amount for each
hourly scenario differs slightly. The decision is accommodated to each scenario input and the overall objective is to optimize the expected outcome.
Another observation is that daily procurement amount drops significantly
during hours when much reserve is available in the hourly market, especially in TCR− procurement. This verifies the link between different stages
established by this three-stage stochastic market-clearing model.
66
CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL
Amount of TCR+ and TCR− Procured in Daily and Hourly Market
260
TCR+
TCR−
Volume (MW)
240
220
200
180
Mon
Tue
Wed
Thu
Fri
Sat
Sun
Figure 4.12: Substitution between TCR+ and TCR− products
TCR+ Amount
300
200
Volume (MW)
100
Mon
Tue
Wed
Thu
Fri
TCR− Amount
Sat
Sun
Tue
Wed
Thu
Sat
Sun
300
Daily: Scen. I
Hourly: Scen. (I, 1)
Hourly: Scen. (I, 2)
Hourly: Scen. (I, 3)
Hourly: Scen. (I, 4)
200
100
Mon
Fri
Figure 4.13: Substitution between daily and hourly products
Chapter 5
Conclusions and Outlook
5.1
Summary
This thesis focuses on the market-clearing model of the Swiss reserve market.
The current design of the Swiss reserve market is composed of a weekly
auction and a daily auction. Secondary reserves (fast reserves that are high
quality products) are procured in the weekly market, while the amount of
tertiary reserves (slow reserves) is split between the weekly and daily market.
On the basis of a two-stage stochastic market-clearing model which has been
implemented since January 2014, this thesis proposes some improvements
of the current two-stage model and develops a novel three-stage stochastic
market-clearing model to accommodate with future market design. More
specifically, major contributions of this thesis can be summarized as follows:
1. Scenarios for daily market are investigated. Firstly, different linearization methods of daily bid curve are compared. A three-piece linearization method based on least squares estimation is chosen, which is more
accurate than current practice and is able to reduce total cost. Then,
9 scenario selection methods are experimented. Based on simulation
results, the method containing W−1, Y−1, W−2, W−3 with different
probability factors assigned to each scenario is selected. Based on the
estimation of 34 weeks in 2015, these improvements can achieve an
overall cost saving of 1.74%. The improvements can be readily implemented on the current two-stage market-clearing model in Switzerland.
2. The framework and the formulation of a three-stage stochastic marketclearing model are proposed. In addition to the current market stages,
an hourly market is considered where reserves are available at a very
low cost or even for free. Scenarios for hourly market are constructed
and selected based on analysis of free TCE bids in the current market.
These data provide a reference for future market situation.
3. The proposed three-stage model is simulated with different scenario
67
68
CHAPTER 5. CONCLUSIONS AND OUTLOOK
inputs. Firstly, single scenario in hourly market is considered in order
to gain insight into the impact of third stage. Results show that in a
most optimistic case, over 3% saving can be achieved by having a third
stage. Secondly, a complete set of scenarios (including 4 daily scenarios
and 4 hourly scenarios) are implemented. Reserve amount and total
procurement cost are presented. On average the three-stage model can
save 10,767 CHF per week in comparison with the improved two-stage
model. Lastly, substitution effects in the stochastic market-clearing
models are illustrated with examples.
So far, the application of a three-stage stochastic market-clearing model
in the reserve market, especially from TSO perspective, is novel. The whole
work of this thesis is based on real-world market data. The improvements of
two-stage model can be directly implemented in the current market-clearing
system. The three-stage model can be utilized as soon as the design of the
third-stage market goes live.
5.2
Future Work
The proposed framework and improvements are closely related to solving
real-world problem faced by TSOs. However, a more realistic consideration
will be to include certain risk factors in the objective function. Risk-averse
models can mitigate market risks caused by abnormal network conditions
and trading behavior. To develop this model, more information of market behavior needs to be collected and analyzed, which can be rather non-trivial.
Appendix A
Hourly Discretizing Factors
According to Equation (4.36), hourly discretizing factors are obtained as
follows:
69
70
APPENDIX A. HOURLY DISCRETIZING FACTORS
Table A.1: Hourly discretizing factors
Hour
00-01h
01-02h
02-03h
03-04h
04-05h
05-06h
06-07h
07-08h
08-09h
09-10h
10-11h
11-12h
12-13h
13-14h
14-15h
15-16h
16-17h
17-18h
18-19h
19-20h
20-21h
21-22h
22-23h
23-24h
Winter Weeks
1-8 & 44-52
Weekday Weekend
0.8689
0.7778
0.9344
0.8889
0.9836
0.9630
1.0000
1.0000
1.0000
1.0000
0.9836
0.9821
0.7541
0.9643
0.5574
0.8929
1.0000
1.0000
0.9600
0.8000
0.9600
0.7500
1.0000
0.7250
0.9259
0.9091
0.9259
0.9697
1.0000
1.0000
0.9630
1.0000
0.8750
1.0000
0.8333
0.6897
0.8333
0.7241
1.0000
0.7931
0.5088
0.6667
0.6491
0.7179
0.8246
0.8462
1.0000
1.0000
Summer Weeks
18-30
Weekday Weekend
0.8750
0.8364
0.9375
0.9091
0.9688
0.9818
1.0000
1.0000
1.0000
0.9655
0.9688
1.0000
0.8750
1.0000
0.6875
0.9310
1.0000
1.0000
0.7576
0.8718
0.6364
0.7436
0.6061
0.6923
0.9130
0.9310
0.8696
0.9655
0.8696
1.0000
1.0000
1.0000
0.8571
1.0000
0.8571
0.9286
0.9643
0.8929
1.0000
0.8929
0.5833
0.5000
0.5625
0.6750
0.6875
0.8000
1.0000
1.0000
Spring/fall Weeks
9-17 & 31-43
Weekday Weekend
0.9194
0.8333
0.9355
0.8704
0.9677
0.9444
1.0000
1.0000
1.0000
1.0000
0.9016
1.0000
0.7869
0.9455
0.5738
0.8364
1.0000
1.0000
0.8400
0.8378
0.8000
0.7568
0.8400
0.7027
0.8438
0.8529
0.8750
0.8824
0.9375
0.8824
1.0000
1.0000
1.0000
1.0000
0.9333
0.9143
0.8000
0.8000
0.7333
0.5714
0.4800
0.6571
0.6000
0.7143
0.8000
0.8571
1.0000
1.0000
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