Three-Stage Stochastic Market-Clearing Model for the Swiss Reserve Market

Transcription

1 power systems eehlaboratory 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

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3 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). iii

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5 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 v

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7 Contents List of Figures List of Tables List of Acronyms List of Symbols x xi xiii xv 1 Introduction Balancing Reserves Reserve Market Stochastic Programming Structure of the Thesis Reserve Market in Switzerland Self-scheduling Market Overview of Ancillary Services Structure of Reserve Market Primary Control Reserves Secondary and Tertiary Control Reserves Bid Structure Indivisible Bids Conditional Bids Dimensioning Criteria Probabilistic Approach Deterministic Approach Remuneration Scheme Remuneration of Capacity Remuneration of Energy Two-Stage Market-Clearing Model Background Stochastic Market-Clearing Model Decision Variables vii

8 viii CONTENTS Objective Function Constraints Formulation Improvements of Two-Stage Model Linearization of Bid Curves Selection of Scenarios Three-Stage Market-Clearing Model Introduction Stochastic Market-Clearing Model Decision Variables Objective Function Non-anticipativity Matrix Constraints Formulation Scenarios for Hourly Market Modelling of Hourly Bid Curves Scenario Construction Simulation Results Case Study: Impact of Hourly Market Complete Scenario Simulations Conclusions and Outlook Summary Future Work A Hourly Discretizing Factors 69 Bibliography 71

9 List of Figures 1.1 Real-time electricity consumption and demand forecast of November 16, 2015 [1] Example of a scenario tree for three-stage problems 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] 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 Overview of scenario selection methods Cost difference w.r.t. perfect information scenario Cost difference w.r.t. perfect information scenario (Methods 2, 5 and 6) 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 Amount of reserves procured for Week Estimated cost savings w.r.t two-stage model ix

10 x LIST OF FIGURES 4.10 Scenario tree of complete model Reserve amount of complete model Substitution between TCR+ and TCR products Substitution between daily and hourly products

11 List of Tables 2.1 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] 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) 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 A.1 Hourly discretizing factors xi

12 xii LIST OF TABLES

13 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

14 xiv LIST OF ACRONYMS

15 List of Symbols x w x w S x w T + x w T x d x d T + x d T x h x h T + x h T ω d ω h Ω d Ω h π d (ω d ) π h (ω h ) 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 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 c w Cost vector of bids in weekly market λ T + (ω d ) Cost vector of TCR+ in daily market for scenario ω d λ T (ω d ) Cost vector of TCR in daily market for scenario ω d ζ T + (ω d, ω h ) Cost vector of TCR+ in hourly market for scenario combination (ω d, ω h ) ζ T (ω d, ω h ) Cost vector of TCR in hourly market for scenario combination (ω d, ω h ) α + i α i β i + 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

16 xvi LIST OF SYMBOLS ρ + i ρ i ϕ + i ϕ i 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 a (i) s+ Slope of ith piece of linearized deficit curve for secondary positive reserves a (i) s Slope of ith piece of linearized deficit curve for secondary negative reserves a (i) o+ Slope of ith piece of linearized deficit curve for overall positive reserves a (i) o Slope of ith piece of linearized deficit curve for overall negative reserves b (i) s+ Intercept of ith piece of linearized deficit curve for secondary positive reserves b (i) s Intercept of ith piece of linearized deficit curve for secondary negative reserves b (i) o+ Intercept of ith piece of linearized deficit curve for overall positive reserves b (i) o ε s+ ε s ε o+ ε o 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 x min d,t + (ωd ) x max d,t + (ωd ) x min d,t (ωd ) x max d,t (ωd ) x min h,t + (ωh ) x max h,t + (ωh ) x min h,t (ωh ) x max h,t (ωh ) 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

17 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 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

18 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

19 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 The economic saving of this new approach is estimated to be around 12 million Swiss francs (CHF) in 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

20 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: uncertain information. decisions made without any realization of 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 N Ω scenarios λ(1),, λ(ω),, λ(n Ω ), each with a length of 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

21 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.

22 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.

23 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 unfavor- 7

24 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]

25 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

26 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

27 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 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 PCR Demand Max. Export Switzerland 71 MW 90 MW Austria 67 MW 90 MW Germany 578 MW 173 MW The Netherlands 67 MW 90 MW

28 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 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.

29 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 Bid Structure In weekly and daily market, bidding rules are listed in Table 2.2. explanations of Table 2.2 are also listed below. Some Table 2.2: Bid structure of the Swiss reserve market [5] Type of Control SCR TCR Symmetry Symmetrical Asymmetrical Offer Size 50 MW/Bid 100 MW/Bid Min. Output Window ±5 MW +5 MW or -5 MW Conditional Bids Allowed, min. 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 P max = 100 MW and technical minimum P min = 10 MW. The planned production is P plan = 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.

30 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 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: ξ(i) = { 1, bid i is accepted 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: = 670 CHF. Instead, if bids #1 and #3 are accepted, the total cost will be: = 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.

31 2.5. DIMENSIONING CRITERIA 15 Table 2.3: Example bids for demonstration of indivisibility Bid # Volume Price [MW] [CHF/MW] 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 Bid # Provider Bid ID Volume Price [MW] [CHF/MW] 1 A A A A B B B 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

32 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] 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 P s, 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 P o. 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]

33 2.6. REMUNERATION SCHEME 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 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] 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.

34 18 CHAPTER 2. RESERVE MARKET IN SWITZERLAND Table 2.5: Remuneration of activated SCR [5] Direction Upward Downward Price SwissIX a +20% b SwissIX 20% c Cash Flow Swissgrid Bidder Bidder Swissgrid Energy Flow Bidder Swissgrid Swissgrid Bidder a Hourly price index for Swiss day-ahead auction in EPEX Spot Market b at least weekly base c at most weekly base 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.

35 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

36 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 s.t. expected total procurement cost 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, and The complete model will be presented again as a whole in Section

37 3.2. STOCHASTIC MARKET-CLEARING MODEL 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: where x w = [x w S, x w T +, x w T ], (3.2) x w Υ N w 1 is the decision variable vector of bids in weekly market x w S ΥN S w 1 is the decision variable vector of SCR bids in weekly market x w T + Υ N w T + 1 is the decision variable vector of TCR+ bids in weekly market x w T Υ N w T 1 is the decision variable vector of TCR bids in weekly market Υ = {0, 1} is the set of binary variables indicating acceptance and rejection of a bid N w S is the number of SCR weekly bids N w T + N w T is the number of TCR+ weekly bids is the number of TCR weekly bids N w = N w S + N w T + + N w T 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 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

38 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 x d (ω d ) is assigned: where x d (ω d ) = [x d T + (ω d ), x d T (ω d )], (3.3) ω d Ω d is the index of scenarios for daily market Ω d contains N d Ω scenarios for daily market x d T + (ω d ) R 42 1 is the amount of TCR+ to be procured in daily market in each 4-hour time block for scenario ω d x d T (ω d ) R 42 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 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 Λ w + E[Λ d (ω d )], ω d Ω d, (3.4) where Λ w is the total procurement cost in the weekly market and Λ d (ω d ) is the total procurement cost in the daily market for the given week with input data of scenario ω d. E[Λ d (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: where Λ w = c wx w = (κ p) x w (3.5)

39 3.2. STOCHASTIC MARKET-CLEARING MODEL 23 c w R N w 1 is the cost vector of all weekly bids κ R N w 1 is a vector containing unit prices (CHF/MW) of each bid p R N w 1 = [p S, p T+, p T ] is a vector consisting of volume (MW) of each bid p S is a vector consisting of volume (MW) of each SCR bid p T+ p T is a vector consisting of volume (MW) of each TCR+ bid 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.

40 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: { α 1 x + β 1, x min x x B λ =, (3.6) α 2 x + β 2, x B < x x max where λ is the procurement cost of this time block x denotes the amount of reserve to be procured in the daily market (α 1, β 1 ) and (α 2, β 2 ) are fitting parameters of the first and second part of the linearized curve respectively x min and x max are lower and upper bound of the bid curve x B is the breaking point of the linearized bid curve Coefficients α 1 and α 2 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 α 2 > α 1. Therefore, the piecewise linear bid curve can be considered as a convex function. Equation (3.6) can be reformulated as: λ = max{α 1 x + β 1, α 2 x + β 2 }, x [x min, x max ]. (3.7)

41 3.2. STOCHASTIC MARKET-CLEARING MODEL 25 Equation (3.7) can be written as an optimization problem: min λ x s.t. α 1 x + β 1 λ, α 2 x + β 2 λ, x min x x max. (3.8) In reality, daily procurement costs are calculated for each 4-hour time block and summed up. Fitting parameters α 1, α 2, β 1 and β 2 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: where min x d T + (ω d ), x d T (ω d ) λ T+ (ω d ) + λ T (ω d ) s.t. α + 1 (ωd )x d T + (ω d ) + β + 1 (ωd ) λ T+ (ω d ), α + 2 (ωd )x d T + (ω d ) + β + 2 (ωd ) λ T+ (ω d ), α 1 (ωd )x d T (ω d ) + β 1 (ωd ) λ T (ω d ), α 2 (ωd )x d T (ω d ) + β 2 (ωd ) λ T (ω d ), x d T + (ω d ) [x min d,t + (ω d ), x max d,t + (ω d )], x d T (ω d ) [x min d,t (ω d ), x max d,t (ω d )], ω d Ω d, λ T+ (ω d ) R 42 1 is the cost vector of TCR+ λ T (ω d ) R 42 1 is the cost vector of TCR (3.9) α + 1 (ωd ), β + 1 (ωd ), α + 2 (ωd ), β + 2 (ωd ) R 42 1 are fitting parameters of TCR+ bid curves α 1 (ωd ), β 1 (ωd ), α 2 (ωd ), β 2 (ωd ) R 42 1 are fitting parameters of TCR bid curves x min d,t + (ω d ), x max d,t + (ω d ) are lower and upper bound of TCR+ bid curve x min d,t (ω d ), x max d,t (ω d ) are lower and upper bound of TCR bid curve Therefore, the term E[Λ d (ω d )] in the Equation (3.4) can be reformulated as: E[Λ d (ω d )] = ( ω d) [ ] λ T+ (ω d ) λ T (ω d ), (3.10) ω d Ω d π d ω d Ω d, where π d ( ω d ) is the probability of scenario ω d and is the summation vector whose elements are all 1.

42 26 CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL 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 To guarantee that this rule will not be violated in the optimization problem, it is translated into the following constraint: where A c x w b c, (3.11) A c R N c w N w 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 b c R N w c 1 is a vector whose elements are all 1 Nc w 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 = [x 1, x 2,, x 7 ], where x i {0, 1} is the binary decision variable for Bid #i. According to the definition of conditional bids, at most one of x 1, x 2 and x 3 can have a value of 1. The rest should be 0. This is also applicable for x 6 and x 7. The rule of conditional bids can thus be written as: { x 1 + x 2 + x 3 1 x 6 + x 7 1. In this case, A c = [ ] [ 1, and b c = 1 ].

43 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 ( P s R s ) + P ( P o R o ) 0.2%, (3.12) where P s and P o are spontaneous and overall power imbalances respectively, R s and R o 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 ( P s+ R s+ ) + P ( Ps R s ) + P ( Po+ R o+ ) where +P ( P o R o ) 0.2%, (3.13) P s+ > 0 and P s < 0 are positive and negative spontaneous power imbalances respectively P o+ > 0 and P o < 0 are positive and negative overall power imbalances respectively R s+ and R s refer to positive and negative SCR (currently in Switzerland R s+ = R s ) R o+ and R o refer to overall positive and negative control reserves If we define cumulative distribution functions of power imbalances as such: F s+ (p) = P ( P s+ p ) F s (p) = P ( P s p ) F o+ (p) = P ( P o+ p ), (3.14) F o (p) = P ( P o p ) then Equation (3.15) can be reformulated as: F s+ (R s+ ) + F s (R s ) + F o+ (R o+ ) + F o (R o ) (3.15)

44 28 CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL The amount of procured reserve R s+, R s, R o+ and R o can be replaced by decision variables in the following terms: R s+ = R s = p S xw S, R o+ = p S xw S + p T + x w T + + x d T + (ω d ), R o = p S xw S + p T x w T + x d T (ω d ), ω d Ω d. Inserting Equation (3.16) into (3.15), Equation (3.15) becomes: ) F s+ (p S xw S ) + F s (p S xw S ) + F o+ (p S xw S + p T + x w T + + x d T + (ω d ) ) +F o (p S xw S + p T x w T + x d T (ω d ) 0.002, ω d Ω d. (3.16) (3.17) Deficit curves are derived from statistical behavior of power imbalance measurements. Consequently, the exact formulation of F s+ ( ), F s ( ), F o+ ( ) and F o ( ) is not known. To incorporate Equation (3.17) into the MILP problem, deficit curves need to be approximated and piecewise linearized. 10 x Before Linearization After Linearization 8 Cumulative Probability Volume (MW) 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

45 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: where a (1) s + R s+ + b (1) s +, a (2) s + R s+ + b (2) s +, ε s+ =.. [ R s+ + b (m) s +, R s+ a (m) s + a (1) s R s + b (1) s, a (2) s R s + b (2) ε s =. s, a (n) s R s + b (n) s, a (1) o + R o+ + b (1) o +, a (2) o + R o+ + b (2) ε o+ =. o +, a (j) o + R o+ + b (j) o +, a (1) o R o + b (1) o, a (2) o R o + b (2) ε o =. a (k) o, o R o + b (k) o, [ R s+ r s (0) +, r (1) [ ] R s+ r s (1) +, r s (2) + r (m 1) s + s + ] ], r s (m) + [ R s r s (0), r (1) [ ] R s r s (1), r s (2) s ]. [ ] R s r s (n 1), r s (n) [ ] R o+ r o (0) +, r o (1) + [ ] R o+ r o (1) +, r o (2) +. [ ] R o+ r o (j 1) +, r o (j) + [ ] R o r o (0), r o (1) [ ] R o r o (1), r o (2). [ R o r (k 1) o ], r o (k) (3.18) (3.19) (3.20) (3.21) 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 a (i) and b(i) deficit curve are linearization parameters of piece i of the corresponding

46 30 CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL m, n, j, k are the total number of pieces for each linearized deficit curve [ r (i 1), r ] (i) valid is the range of reserve volume in which linear piece i is 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 Thus, constraints with respect to the probabilistic dimensioning criterion can be formulated as: a (i) ( ) (i) s + Rs+ + b s + ε s+, i = 1,, m, a (i) ( ) (i) s Rs + b s ε s, i = 1,, n, a (i) ( ) (i) o + Ro+ + b o + ε o+, i = 1,, j, a (i) ( ) (i) o Ro + b o ε o, i = 1,, k, R s+ R max R min s + R min s Ro min + R min s +, R s R max s, R o+ R max o +, o R o Ro max, ε s+ + ε s + ε o+ + ε o (3.22) In this set of constraints, ε s+, ε s, ε o+ and ε o are regarded as decision variables and appended to those described in Section They do not appear in objective function, though. R s+, R s, R o+ and R o 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: 400 p T + x w T + + x d T + (ω d ), ω d Ω d. (3.23)

47 3.2. STOCHASTIC MARKET-CLEARING MODEL Formulation The complete mathematical formulation of the current two-stage stochastic market-clearing model is presented as follows: min c wx w + ( π d ω d) [ ] λ T+ (ω d ) λ T (ω d ) x w, λ T+ (ω d ), λ T (ω d ) ω d Ω d (3.24) s.t. A c x w b c, (3.25) α + 1 (ωd )x d T + (ω d ) + β 1 + (ωd ) λ T+ (ω d ), ω d Ω d, (3.26) α + 2 (ωd )x d T + (ω d ) + β 2 + (ωd ) λ T+ (ω d ), ω d Ω d, (3.27) α 1 (ωd )x d T (ω d ) + β1 (ωd ) λ T (ω d ), ω d Ω d, (3.28) α 2 (ωd )x d T (ω d ) + β2 (ωd ) λ T (ω d ), ω d Ω d, (3.29) a (i) ( s + p ) S xw S + b (i) s + ε s+, i = 1,, m, (3.30) a (i) ( s p ) S xw S + b (i) s ε s, i = 1,, n, (3.31) ) a (i) o + (p S xw S + p T + x w T + + x d T + (ω d ) + b (i) o + ε o+, ω d Ω d, i = 1,, j, (3.32) ) a (i) o (p S xw S + p T x w T + x d T (ω d ) + b (i) o ε o, ω d Ω d, i = 1,, k, (3.33) ε s+ + ε s + ε o+ + ε o 0.002, (3.34) 400 p T + x w T + + x d T + (ω d ), ω d Ω d, (3.35) x d T + (ω d ) [x min d,t + (ω d ), x max d,t + (ω d )], ω d Ω d, (3.36) x d T (ω d ) [x min d,t (ω d ), x max d,t (ω d )], ω d Ω d. (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.

48 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 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. ŷ i y i γ, α 1 x i + β 1, x i [x (0) B, x(1) B ]. ŷ i = α j x i + β j, x i [x (j 1) B, x (j) B ]. α M x i + β M, x i [x (M 1) B i = 1,, N,, x (M) B ], (3.38) where (x i, y i ) represents an individual bid on bid curve, and x i, y i denote cumulative volume and cost at this point respectively ŷ i is the estimated cost after linearization at x i 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 α j and β j are linearization parameters of piece j x (j 1) B and x (j) B indicate the lower and upper bound of piece j M is the total number of pieces of the linearized curve

49 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 (ŷ i y i ) 2 i=1 α 1 x i + β 1, x i [x (0) B, x(1) B ]. s.t. ŷ i = α j x i + β j, x i [x (j 1) B, x (j) B ]. α M x i + β M, x i [x (M 1) B i = 1,, N,, x (M) B ], (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

50 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. Example Bid Curve Cost (CHF) 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).

51 3.3. IMPROVEMENTS OF TWO-STAGE MODEL 35 Residual = Data Fit (CHF) Method 1 Method 2 Method 3 Method 4 Residual of Fitted Bid Curve Bid Number (X) Figure 3.6: Residual of fitted bid curve Table 3.1: Performance of linearization methods Computation Time [s] Average Residual [CHF] Method Method Method Method 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.

52 36 CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL Amount of Procured Reserves Method 4 Method 3 Method 2 Method Volume (MW) SCR TCR Weekly TCR Daily Figure 3.7: Amount of procured reserves using four fitting methods Total Cost of Reserves Weekly Daily Method 4 Method 3 Method 2 Method Cost (MCHF) Figure 3.8: Total procurement cost of reserves using four fitting methods 0.5 Cost of Reserves in Daily Market Cost (MCHF) 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

53 3.3. IMPROVEMENTS OF TWO-STAGE MODEL 37 By applying the improved linearization method on 34 weeks in 2015 (Week 02 35), a total cost saving of 455,372 CHF can be gained, as is shown in Table 3.2. This corresponds to 0.65% and an average saving of 13,393 CHF per week. Table 3.2: Estimation of cost savings by improved linearization method (Week 02 35, 2015) Total Savings in CHF 455,372 Total Savings in % 0.65 Average Savings per Week in CHF 13, 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: α 1, α 2 and α 3 (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

54 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 α 1. 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 α 1 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: where ρ(x, y) = 1 N 1 N ( ) ( ) xi µ x yi µ y, (3.40) i=1 N is the size of vector x and y σ x µ x and µ y are mean values of x and y σ x and σ y 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 W 1 Y 1 W 2 W 3 W 4 Slope of TCR+ α Slope of TCR α 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 σ y

55 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] W 1 Y 1 W 2 W 3 W 4 Slope of TCR+ α Slope of TCR α 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.

56 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.

57 3.3. IMPROVEMENTS OF TWO-STAGE MODEL 41 Table 3.6: Information of selected weeks [6] Week Number Date Avg. Temperature [ C] Week Jan /-0.7 Week Feb /-2.9 Week Mar /1.4 Week Apr /3.1 Week May /7.7 Week Jun /10.3 Week Jul /15.7 Week Aug /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

58 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 Method 6 (Current Practice) Method 2 (W 1, Y 1) Method 5 (W 1, Y 1, W 2, W 3 with different probability) Avg. [CHF] Std. [CHF] Total [MCHF] 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 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.

59 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 757,795 Total Savings in % 1.09 Average Savings per Week in CHF 22,288 Table 3.9: Estimation of savings by implementing both improvements (Week 02 35, 2015) Total Savings in CHF 1,213,167 Total Savings in % 1.74 Average Savings per Week in CHF 35,681

60 44 CHAPTER 3. TWO-STAGE MARKET-CLEARING MODEL

61 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 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

62 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

63 4.2. STOCHASTIC MARKET-CLEARING MODEL 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 and Section is dedicated to the non-anticipativity considered in this three-stage problem. Section adapts dimensioning criteria to the three-stage model and updates the constraints. The complete formulation of the model is presented in Section 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 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: x h (ω d, ω h ): where x h (ω d, ω h ) = [x h T + (ω d, ω h ), x h T (ω d, ω h )], (4.1) ω h Ω h is the index of scenarios for hourly market Ω h contains N h Ω scenarios for hourly market x h T + ( ω d, ω h) R is the amount of TCR+ to be procured in the hourly market in each hour for scenario combination ( ω d, ω h) x h T ( ω d, ω h) R is the amount of TCR to be procured in the hourly market in each hour for scenario combination ( ω d, ω h) 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 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

64 48 CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL The framework of the objective function for the three-stage model can be formulated as follows: min Λ w + E[Λ d (ω d )] + E[Λ h (ω d, ω h )], ω d Ω d, ω h Ω h, (4.2) where Λ h (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[Λ h (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 as: Similar to Equation (3.6), procurement cost in one hour can be written ζ = { ρ 1 x + ϕ 1, x min x x B, (4.3) ρ 2 x + ϕ 2, x B < x x max where ζ is the procurement cost of this hour x is the amount of reserve to be procured in hourly market

65 4.2. STOCHASTIC MARKET-CLEARING MODEL 49 (ρ 1, ϕ 1 ) and (ρ 2, ϕ 2 ) are fitting parameters of the first and second part of the hourly bid curve respectively x min and x max are lower and upper bound of the bid curve x B is the breaking point of the linearized bid curve Provided that ρ 2 > ρ 1, Equation (4.3) can be reformulated as a optimization problem: min ζ x s.t. ρ 1 x + ϕ 1 ζ, (4.4) ρ 2 x + ϕ 2 ζ, x min x x max. 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 x h T + (ω d,ω h ), x h T (ω d,ω h ) where ζ T+ (ω d, ω h ) + ζ T (ω d, ω h ), s.t. ρ + 1 (ωh )x h T + (ω d, ω h ) + ϕ + 1 (ωh ) ζ T+ (ω d, ω h ), ρ + 2 (ωh )x h T + (ω d, ω h ) + ϕ + 2 (ωh ) ζ T+ (ω d, ω h ), ρ 1 (ωh )x h T (ω d, ω h ) + ϕ 1 (ωh ) ζ T (ω d, ω h ), ρ 2 (ωh )x h T (ω d, ω h ) + ϕ 2 (ωh ) ζ T (ω d, ω h ), x h T + (ω d, ω h ) [x min h,t + (ω h ), x max h,t + (ω h )], x h T (ω d, ω h ) [x min h,t (ω h ), x max h,t (ω h )], ω d Ω d, ω h Ω h, ζ T+ (ω d, ω h ) R is the hourly cost vector of TCR+ ζ T (ω d, ω h ) R is the hourly cost vector of TCR (4.5) ρ + 1 (ωh ), ϕ + 1 (ωh ), ρ + 2 (ωh ), ϕ + 2 (ωh ) R are linear coefficients of TCR+ hourly bid curves and are only dependent on hourly scenario ω h ρ 1 (ωh ), ϕ 1 (ωh ), ρ 2 (ωh ), ϕ 2 (ωh ) R are linear coefficients of TCR hourly bid curves and are only dependent on hourly scenario ω h x min h,t + (ω h ), x max h,t + (ω h ) are lower and upper bound of TCR+ bid curve x min h,t (ω h ), x max h,t (ω h ) are lower and upper bound of TCR bid curve

66 50 CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL Therefore, the term E[Λ h (ω d, ω h )] in Equation (4.2) can be reformulated as: E[Λ h (ω d, ω h )] = ( π d ω d) π h (ω h) [ ] ζ T+ (ω d, ω h ) ζ T (ω d, ω h ), ω d Ω d, ω h Ω h where π h ( ω h ) is the probability of hourly scenario ω h Non-anticipativity Matrix (4.6) 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 A n. The number of rows corresponds to the product of 168 (hours in a week), the number of daily scenarios NΩ d, and the number of hourly scenarios N Ω h. The number of columns is the product of 42 (time blocks in a week) and the number of daily scenarios NΩ d. The value of a single element in the non-anticipativity maxtrix A n (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: ω1 d, ωd 2, and 2 scenarios for hourly market: ωh 1, ωh 2. Hence, there are in total 4 scenario combinations: (ω d 1, ω h 1 ), (ω d 1, ω h 2 ), (ω d 2, ω h 1 ), (ω d 2, ω h 2 ) (4.7) Consider 2 time blocks with 4 hours each: p 1 = (t 1, t 2, t 3, t 4 ), p 2 = (t 5, t 6, t 7, t 8 ) (4.8)

67 4.2. STOCHASTIC MARKET-CLEARING MODEL 51 Decision variable vector for second stage is: [ x d = = x (1) d, x(2) d, x(3) d ], x(4) d [ x d p 1 (ω d 1), x d p 2 (ω d 1), x d p 1 (ω d 2), x d p 2 (ω d 2) ]. (4.9) Decision variable vector for third stage is: x h = [ x h t 1 (ω d 1, ω h 1 ), x h t 2 (ω d 1, ω h 1 ), x h t 3 (ω d 1, ω h 1 ), x h t 4 (ω d 1, ω h 1 ), x h t 5 (ω1, d ω1 h ), x h t 6 (ω1, d ω1 h ), x h t 7 (ω1, d ω1 h ), x h t 8 (ω1, d ω1 h ) x h t 1 (ω1, d ω2 h ), x h t 2 (ω1, d ω2 h ), x h t 3 (ω1, d ω2 h ), x h t 4 (ω1, d ω2 h ), x h t 5 (ω1, d ω2 h ), x h t 6 (ω1, d ω2 h ), x h t 7 (ω1, d ω2 h ), x h t 8 (ω1, d ω2 h ), x h t 1 (ω2, d ω1 h ), x h t 2 (ω2, d ω1 h ), x h t 3 (ω2, d ω1 h ), x h t 4 (ω2, d ω1 h ), x h t 5 (ω2, d ω1 h ), x h t 6 (ω2, d ω1 h ), x h t 7 (ω2, d ω1 h ), x h t 8 (ω2, d ω1 h ), x h t 1 (ω2, d ω2 h ), x h t 2 (ω2, d ω2 h ), x h t 3 (ω2, d ω2 h ), x h t 4 (ω2, d ω2 h ), ] x h t 5 (ω2, d ω2 h ), x h t 6 (ω2, d ω2 h ), x h t 7 (ω2, d ω2 h ), x h t 8 (ω2, d ω2 h ). (4.10) The first four variables in Equation (4.10) correspond to second-stage scenario ω1 d and time block 1, and are thus linked to second-stage decision variable x (1) d. The next four variables in Equation (4.10) are related to second-stage scenario ω1 d and time block 2, and are thus linked to secondstage decision variable x (2) d, and so on.

68 52 CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL Therefore, the non-anticipativity matrix in this case is: Constraints The binding factors in the three-stage market-clearing model are identical with those in two-stage model, as explained in Section The constraint for conditional bids in the first stage has exactly the same formulation as Equation (3.11).

69 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: R s+ = R s = p S xw S, R o+ = p S xw S + p T + x w T + + A n x d T + (ω d ) + x h T + (ω d, ω h ), R o = p S xw S + p T x w T + A n x d T (ω d ) + x h T (ω d, ω h ), ω d Ω d, ω h Ω h. (4.11) Inserting Equation (4.11) into Equation (3.15), we get: F s+ (p S xw S ) + F s (p S xw S ) +F o+ (p S xw S + p T + x w T + + A n x d T + (ω d ) + x h T + (ω d, ω h ) ) +F o (p S xw S + p T x w T + A n x d T (ω d ) + x h T (ω d, ω h ) 0.002, (4.12) ω d Ω d, ω h Ω h. Linearization of deficit curves remains the same as in Equation (3.22). R s+, R s, R o+ and R o 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: 400 p T + x w T + + A n x d T + (ω d ) + x h T + (ω d, ω h ), ω d Ω d, ω h Ω h. (4.13) Formulation The complete mathematical formulation of the three-stage stochastic marketclearing model is presented as follows:

70 54 CHAPTER 4. THREE-STAGE MARKET-CLEARING MODEL min x w, λ T+ (ω d ), λ T (ω d ), ζ T+ (ω d,ω h ), ζ T (ω d,ω h ) c wx w + ( π d ω d ) [ λ T+ (ω d ) λ T (ω d ) ] ω d Ω d + ( ω d ) ( π h ω h ) [ ζ T+ (ω d, ω h ) ζ T (ω d, ω h ) ] π d ω d Ω d, ω h Ω h (4.14) s.t. A c x w b c, (4.15) α + 1 (ωd )x d T + (ω d ) + β + 1 (ωd ) λ T+ (ω d ), ω d Ω d, (4.16) α + 2 (ωd )x d T + (ω d ) + β + 2 (ωd ) λ T+ (ω d ), ω d Ω d, (4.17) α + 3 (ωd )x d T + (ω d ) + β + 3 (ωd ) λ T+ (ω d ), ω d Ω d, (4.18) α 1 (ωd )x d T (ω d ) + β 1 (ωd ) λ T (ω d ), ω d Ω d, (4.19) α 2 (ωd )x d T (ω d ) + β 2 (ωd ) λ T (ω d ), ω d Ω d, (4.20) α 3 (ωd )x d T (ω d ) + β 3 (ωd ) λ T (ω d ), ω d Ω d, (4.21) ρ + 1 (ωh )x h T + (ω d, ω h ) + ϕ + 1 (ωh ) ζ T+ (ω d, ω h ), ω d Ω d, ω h Ω h, (4.22) ρ + 2 (ωh )x h T + (ω d, ω h ) + ϕ + 2 (ωh ) ζ T+ (ω d, ω h ), ω d Ω d, ω h Ω h, (4.23) ρ 1 (ωh )x h T (ω d, ω h ) + ϕ 1 (ωh ) ζ T (ω d, ω h ), ω d Ω d, ω h Ω h, (4.24) ρ 2 (ωh )x h T (ω d, ω h ) + ϕ 2 (ωh ) ζ T (ω d, ω h ), ω d Ω d, ω h Ω h, (4.25) a (i) ( s + p ) S xw S + b (i) s + ε s+, i = 1,, m, (4.26) a (i) ( s p ) S xw S + b (i) s ε s, i = 1,, n, (4.27) ) a (i) o + (p S xw S + p T + x w T + + A n x d T + (ω d ) + x h T + (ω d, ω h ) + b (i) o + ε o+, ω d Ω d, i = 1,, j, (4.28) ) a (i) o (p S xw S + p T x w T + A n x d T (ω d ) + x h T (ω d, ω h ) + b (i) o ε o, ω d Ω d, i = 1,, k, (4.29) ε s+ + ε s + ε o+ + ε o 0.002, (4.30) 400 p T + x w T + + A n x d T + (ω d ) + x h T + (ω d, ω h ), ω d Ω d, (4.31) x d T + (ω d ) [x min d,t + (ω d ), x max d,t + (ω d )], ω d Ω d, (4.32) x d T (ω d ) [x min d,t (ω d ), x max d,t (ω d )], ω d Ω d, (4.33) x h T + (ω d, ω h ) [x min h,t + (ω h ), x max h,t + (ω h )], ω d Ω d, ω h Ω h, (4.34) x h T (ω d, ω h ) [x min h,t (ω h ), x max h,t (ω h )], ω d Ω d, ω h Ω h. (4.35)

71 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 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.

72 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 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 MW Week Number Mon Tue Wed Thu Fri Sat Sun Figure 4.3: Free TCE+ volume in 2015 (Week 01 35) Free TCE Capacity MW Week Number 10 0 Sun Sat Fri Thu Wed Tue Mon 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

73 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 p 1, p 2, p 3, p 4 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: η + i = η i = 120 p i max (120 p j), 1 j 4 p i max p, j 1 j 4 (4.36) where η + i and η i are hourly discretizing factors for tertiary positive and negative reserves in hour i respectively, and p i is the hourly peak load level in hour i.

74 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 Probability Minimum 5% Median 45% Mean 45% Maximum 5% 4.4 Simulation Results This section presents simulation results of the proposed three-stage stochastic market-clearing model. Section 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 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 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.

75 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).