Optimal Bidding Strategy of a Decentralized Storage Device in both Energy and Reserve Markets

Transcription

1 Power Systems P L Laboratory Ludovico Nati Optimal Bidding Strategy of a Decentralized Storage Device in both Energy and Reserve Markets Master Thesis PSL1703 EEH Power Systems Laboratory ETH Zurich Examiner: Prof. Dr. Gabriela Hug Supervisor: Xuejiao Han, Dr. Evangelos Kardakos Zurich, September 19, 2017

2 Abstract To face the challenge of higher uncertainty and variability in the power system as a consequence of increasing penetration of intermittent energy resources, rapid expansion of distributed energy resources (DER) like storage devices can be forecasted due to their potential ability to contribute to flexibility provision. To realize a promising revenue stream for the storage providers, design of optimal bidding strategies considering different market frameworks (e.g. day-ahead energy market and reserve market) are required. This thesis analyzes the bidding strategy of a private-owned and storage device which participates as a price taker in the wholesale electricity market in the same way as all the other market players. Two different electricity market frameworks will be analyzed and two optimal bidding strategies suited for these framework will be derived. The first electricity market framework embeds the feature of a joint clearing of the energy and reserve products. The optimization problem for this kind of electricity market derives the optimal strategy for energy and reserve at the same time. This optimization problem is a four-stage stochastic model formulated as a Mixed-Integer-Linear problem and incorporates the uncertainties of eventual reserve energy deployment. The second electricity market framework embeds the feature that the energy and reserve products are cleared in sequence. The first optimization problem yields the optimal strategy for the reserve capacity market. This formulation is a two-stage stochastic problem incorporating the uncertainty of reserve capacity price realization. The second optimization problem determines the optimal strategy for the day-ahead market once that the commitment of reserve capacity is known. This formulation is a fourstage stochastic problem incorporating the uncertainties of eventual reserve energy deployment. Both problems have a Mixed-Integer linear formulation. Case of studies and simulations are eventually carried out. It can be observed that up to 90% of the total revenues derive from the reserve market. It is thus justified a strategy taking into account both the involvement in the energy and reserve markets. i

3 Contents 1 Introduction Overview Structure of the Thesis Literature Review Swiss Electricity Market Short-Term Energy-only Markets Spot Markets Balancing Market Reserve Market Stand-By-Power Market Regulating Energy Compensation Joint Dispatch Framework Introduction Market and Model Framework Model Formulation Case Study Sequential Dispatch Framework Introduction Reserve Capacity Offering Strategy Primary Reserve Product Primary and Secondary Reserve Products Day-ahead Offering Strategy Case Study Conclusions Outlook A Scenario Generation Methodologies 73 A.1 Scenarios of Reserve Price Realization A.2 Scenarios of Reserve Energy Deployment ii

4 List of Figures 2.1 Example of a pay-as-bid reimbursement mechanism in one of the reserve market capacity Example of a bidding curve and an offering quantity. We assume that the specification price of an offering quantity is 0 CHF/MW h Decision framework of the model. Note the decisions stages of the variables involved Logic of the state of charge management for the storage device Results from the optimization problem. The 24 Day-ahead bidding quantities and the Reserve bidding quantity are the first-stage decisions variables Reserve realization in the first, second and 503rd scenarios Day-ahead prices, day-ahead bidding quantities and state of charge when the day-ahead market is the only available energy market Scheme of the strategy An example of the clearing criterion assumed in this paper. If the threshold price is 6.5 CHF/MWh, the bid that it is accepted is (6 CHF/MWh, 6 MW) If the threshold price is 7.5 CHF/MWh, the bid that it is accepted is (7 CHF/MWh, 7 MW). This the same logic of figure Bidding curve defined with the parameters shown in Example Two stage decision framework of the producers Example of the power constraint of the storage device Offering curve of primary reserve Illustration of the total revenues for each primary capacity price realization, divided in components from energy and reserve market Offering Curve Secondary Reserve Capacity iii

5 LIST OF FIGURES iv 4.10 Revenues in the scenarios of secondary prices realization Framework of the four stages decision model with the correspondent variables assigned to each decision stage Logic of the state of charge management for the storage device Parameters from the optimization problem Bidding quantities for the first 24 hours Secondary reserve realization in the first, second and 503rd scenarios for the first 24 hours Values of interest when the day ahead market is the only available energy market A.1 Relative frequency histogram of primary capacity price realization with 3 bins. The data underlying this histogram are the historical price data of primary reserve capacity in A.2 Example of eight scenarios generated with the methodology. In this case n = 2. Note the points of discontinuity at hour 8 and A.3 Schematic view of the scenarios defined in Figure A.2. Note the indistinguishable parts between the scenarios defined... 79

6 List of Tables 2.1 Products on reserve market. Brief overview. Adapted from [3] and [18] Key technological properties of the battery Key technological properties of the battery Bid-blocks of the bidding curve shown in Figure Key technological properties of the battery Values of the bid-curve shown in Figure Key technological properties of the battery A.1 Edges of the bins of the relative frequency histogram shown in Figure A A.2 Scenarios generated from the histogram defined in Figure A v

7 Chapter 1 Introduction 1.1 Overview The variability and the inter-temporal variations of the renewable energy sources make their integration into the electricity grid particularly defying. In particular, the capability of the system to ensure a systematic match between generation and consumption of electricity is seriously jeopardized. To address this challenge, electric energy storage is considered to be one of the greatest solution to provide more flexibility to the system and is expected to expand significantly in the future [1]. It has to be taken into account that in competitive electrical systems such as the ones developed over the last decades all over the world, the bulk of the energy storage facilities will be operated by merchant and for-profit entities. In such frameworks, the diffusion of electric energy storage derives directly from the profitability of the investments for storage providers. In addition, we must consider the possibility that in competitive environments subsidies to electric energy storage could not be available and storage providers are treated in same way as all the other market players. Optimal bidding strategies in liberalized electricity markets without subsidies are crucial in order to ensure to the storage providers an attractive revenue stream which ultimately will foster the diffusion of storage facilities. The design of optimal bidding strategies will be the main topic of the thesis and we will restrict our analysis to Battery Energy Storage Systems (BESS) and to the short-term scheduling of such devices in the day-ahead market of energy and in the fast reserve market 1. BESS, in particular the ones based on Li-Ions cells, are indeed well suited for fast regulation and this is a market which can ensure an high profitability to storage devices [21]. On the other hand, the main challenge for energy constrained storage devices when ensuring control reserve is the State of 1 Primary and Secondary reserve market, if we consider the European definition of reserve products. 1

8 CHAPTER 1. INTRODUCTION 2 Charge (SOC) management [26] and this problem will be addressed in this thesis as well. We will focus our analysis to price takers BESS which participate in two different electricity market frameworks. The fist electricity market framework embeds the feature of a joint clearing of the energy and reserve products and both products are remunerated according to an uniform pricing mechanism. This first market framework can be interpreted as a simplified version of the day-ahead scheduling reserve market in PJM as the one described [4]. The second electricity market framework embeds the feature that the energy and reserve products are cleared in sequence. At first the reserve capacity market is cleared and remunerated according to a pay-as-bid mechanism while only later the energy day-ahead market is cleared and remunerated according to an uniform pricing mechanism. This second market framework can be interpreted as a simplified version of the Swiss electricity market. The bidding strategy suited for the first market framework derives the optimal quantities for energy and reserve at the same time, as a result of an optimization problem. This optimization problem is a four-stage stochastic model formulated as a Mixed-Integer-Linear problem and incorporates the uncertainties of eventual reserve energy deployment. By including these four decision stages the model ensures that the excess of energy due to the different reserve deployment scenarios will be traded in the market floors following the day-ahead market and at the same time a proper SOC management is ensured. The strategy suited for the second market framework is structured with two different optimization problems. The first problem aims to derive the optimal bidding curves 2 for reserve capacity. This is a two-stage stochastic problem incorporating the uncertainties of reserve capacity price realization and includes as well the feature that the higher is the reserve capacity cleared, the lower are the opportunities of profit in the energy market. The second problem instead seeks to determine the optimal quantity of energy to sell in the day-ahead market once that the commitment of reserve capacity is known. This is a four-stage stochastic problem incorporating the uncertainties of eventual reserve energy deployment. By representing these four decision stages this model as well ensures that the excess of energy due to the different reserve deployment scenarios will be traded in the market floors following the day-ahead market and at the same time a proper SOC management is ensured. Both problems have a Mixed-Integer linear formulation. 2 The difference between bidding quantities and bidding curves will be discussed in details in Chapter 3.

9 CHAPTER 1. INTRODUCTION Structure of the Thesis This thesis will have the following structure: Chapter 3 shows the strategy for the market framework which is a simplified version of the day-ahead scheduling reserve market in PJM and presents as well a case study. Chapter 4 shows instead the strategy for the market framework which is a simplified version of the Swiss electricity market and presents as well another case study. In case of need, Chapter 2 quickly reviews the Swiss electricity market framework and its reading is suggested to who does not have enough experience with the Swiss electricity market. Chapter 3 and Chapter 4 can be read independently from one-another and the interested reader can move directly to each of them without consulting the other one first. 1.3 Literature Review The optimal offering strategy of a storage facility has already been widely discussed in literature. First of all, [2] presents the overview of how currently energy and reserve markets are modeled. Then [4] addresses the optimal operation of an independent storage unit in both energy and reserve markets. A non-convex stochastic optimization problem is developed and a certain amount of practical assumptions are considered in order to make the problem computationally tractable. Reference [5] proposes bidding strategies for an energy storage facility in both perfectly and imperfectly competitive markets. Nevertheless, only the energy markets are taken into account in [5] and we consider this assumption as a limiting factor. Reference [6] proposes as well a bidding and operation strategy only for multi-temporal energy markets but it analyzes more in detail the case of a battery storage systems. Reference [7] provides instead a bidding strategy for a risk adverse virtual power plant offering in both energy and reserve markets. It considers a market framework of joint clearing between energy and reserve and it sets up a two stage stochastic problem considering the balancing market as the only market floor available after the day-ahead market. Reference [26] illustrates the main issues to take into account when an energy constrained storage system is responsible for control reserve. Moreover, [8] and [21] describe the main problems of an energy constrained storage system when providing control reserve and give as well some feedback strategies in order to tackle these challenges. References [9], [10] and [11] address the issue of the self scheduling problem of a pumped-storage hydro plant in both energy and reserve market. The main characteristic of the pumped storage plant are the technical operating constraints, such as the three-dimensional non linear relationship

10 CHAPTER 1. INTRODUCTION 4 between the power produced, the energy stored, and the head of the associated reservoir. Such issues can be overcome by piecewise linear models of such a relation, as described in [12], but anyway in this thesis we will take into account only BESS with far less computational requirements for what concerns the feasible operating region. Finally, [13] illustrates an optimal scheduling of Vehicle-to-Grid energy and ancillary services. It is discussed an efficient way of managing a fleet of plugged in electric vehicles (EV) from a utility or an aggregator perspective. It is observed that the aggregator can profit significantly by using the fleet of EV for ancillary services purposes or for peak load shaving.

11 Chapter 2 Swiss Electricity Market This chapter provides a short description of the Swiss short-term wholesale electricity market. As such, no long-term contracts such as futures or forward agreements are inquired in this chapter. The over-the-counter agreements are not analyzed as well. Thus, we will focus on the day-ahead, intra-day, and balancing energy markets and on the operating reserves markets. A deeper review can be found in [3], [14], [15], [16], [32] and [17] and these are the main sources of information of this chapter. The structure of this chapter will follow widely [18, chapter 2 ]. As any other electricity market-place, the Swiss market faces the challenge of balancing demand and supply of energy at any time in order to avoid the failure of the electricity system. The main idea of the wholesale electricity market, in order to avoid the failure of the system, is to trade not only the main commodity, which is electricity, but to trade as well other kind of products which guarantee the real-time balance of the system. Within this family of products needed to ensure the secure operation of the system we will focus only on the operating reserves, which are the products of interest in the offering strategy described in Chapter 3 and Chapter 4. A more detailed analysis of further markets (e.g. black start capability, congestion management via re-dispatching, capacity market...) is out of the scope of this thesis. It is necessary now to introduce the concept of the balancing group (BG), which is a key component in the Swiss electricity market. Basically, a BG is a virtual entity that combines generators and consumers for billing and measurement purposes. Each BG is represented by a single balancing responsible party (BRP) and the production as well as the consumption from each generator and load in Switzerland is assigned to a specific balance group. Measurements of overall flow of energy takes place at the BG level, as the sum of the net positions of generators and consumers belonging to a specific BG, and the BRP is responsible for communications between the TSO and the BG. There are more than 100 BGs in Switzerland. 5

12 CHAPTER 2. SWISS ELECTRICITY MARKET 6 The TSO receives a daily schedule from each balance group until 14:30 the day before delivery, although adjustments are possible with 15 (45) lead time for national (international) exchanges, and also ex-post. This schedule considers all over-the-counter and spot markets agreements that the balance group might have. Thus, each balance group has an overall scheduled net position (positive, negative or neutral) that must be followed throughout the energy delivery time. Eventually, Swissgrid verifies that an overall equilibrium exists between the schedules of the different balance groups, and controls that the overall surplus energy and deficit energy matches. The rest of Chapter 2 will be structured as follows: Section 2.1 describes the main features of the energy-only markets, whose goal is to trade the electric energy commodity. Section 2.2 describes instead the main feature of the operating reserve markets, whose goal is stand-by power in order to guarantee at any time the real time balance between the supply and demand of electricity. 2.1 Short-Term Energy-only Markets The main short-term energy-only markets in Switzerland are: Day-ahead Spot Market Intra-day Spot Market Balancing (real-time) Market Spot Markets In the energy spot market players can trade their energy position. These markets represent a volume equal to 40% of the Swiss national consumption. The bulk of the energy transaction is operated in the day-ahead spot market. It operates as an uniform auction with hourly contracts. It opens 45 days before the delivery time and it is cleared the day before at 11:00 am. Besides hourly products, it is possible to trade block contracts for base and peak load as well as block orders linking several hours. The mechanism of remuneration of this market relies on a uniform market clearing price, the same for parties in long and short positions 1. Such a price index is known as SwissIX. The intra-day spot market allows a continuous trading mechanism of energy. It opens the day before the physical energy delivery at 15:00 and closes one hour before the delivery time. This market is mainly used to adjust the position from the day-ahead market due to outages or forecasts 1 In other spot-markets, such as the Italian one, this statement does not necessarily true. See the mechanism of the Prezzo Unico Nazionale for more information [20]

13 CHAPTER 2. SWISS ELECTRICITY MARKET 7 errors. The traded volume is about 10% of the one traded in the day-shead spot market and prices are very volatile Balancing Market In an ideal scenario where each BG follows its scheduled net position for all the time period involving energy delivery, production and consumption of electricity matches perfectly and the system is in equilibrium without the need to activate expensive reserves. Due to the uncertainties involved in the production/consumption of electricity, it should be clear that such a situation does not regularly occur. Thus, a mechanism to compensate the imbalances of the BG is needed, while economic efficiency needs to be guaranteed as well. This is the role of the balancing market. The balancing market provides energy to cover both generation excess and deficit, and constitutes a real-time market to balance power production and consumption. After the ex-post adjustments where the BGs can further trade their imbalances within one-another, Swissgrid calculates the deviations of each BG from its scheduled net position. This process is known as imbalance settlement. Furthermore, the deviations of each BG are evaluated from the economical point of view according to the imbalance pricing mechanism. The imbalance pricing aims to solve an adverse selection/hidden-information problem: BRP has private information about their demand and supply possibilities, which can be improved with costly forecast technology. Thus, the imbalance pricing have to ensure the right economic signals to motivate as much as possible the correct self-forecasts of the BRP. The goal is achieved by a so-called dual pricing system, where the positive and negative imbalances are priced. In particular, the imbalance spread, which is the deviation between prices in the spot markets and the balancing market, determines the true ability of the BRPs to make good self-forecasts. The dual pricing system currently set up in the Swiss framework is the following: Additional consumption or reduced production are charged: 1.1 [max{swissix, λ SCRE+, λ T CRE+ } + 10 CHF/MW h] Where SwissIX is the hourly spot price, λ SCRE+ is the price of positive secondary energy, and λ T CRE+ is the price of positive tertiary energy, if positive secondary or tertiary energy was activated. Additional production or reduced consumption are remunerated at the price of:

14 CHAPTER 2. SWISS ELECTRICITY MARKET [min{swissix, λ SCRE, λ T CRE }+5 CHF/MW h] Where SwissIX is the hourly spot price, λ SCRE is the price of negative secondary energy, and λ T CRE+ is the price of negative tertiary energy, if negative secondary or tertiary energy was activated. 2.2 Reserve Market The availability of stand-by power (operating reserve) is crucial to compensate the deviations from expected generation and consumption and ultimately to guarantee the overall balance of inflows and withdrawals of energy in the grid. Three different kinds of reserves are defined in Switzerland according to the time leaps of activations and according to their overall roles in maintaining the system balance: Primary control reserve: Also referred to as frequency containment reserve. It is a symmetrical upwards and downwards power product. The role of such a reserve is to contain the frequency of the synchronized grid within a certain band thus avoiding the frequency collapse or soar. Activation takes place directly at in the power stations by means of turbine regulators and the total contracted capacity must be provided if the grid frequency deviation is ±200mHz or larger. Response is required within 15 seconds. Primary reserve is activated all throughout the international synchronized grid, according to a solidarity principle. Secondary control reserve: Also referred to as frequency restoration reserve. It is contracted as a symmetrical upwards and downwards power product, and it is automatically activated by Swissgrid through a control signal 2 in online power plants. One goal of such a reserve is to restore the frequency close to its nominal value until additional control actions are taken. Another goal is to maintain the desired level of energy exchange of a control area with the rest of the grid. It is activated some seconds after the disrupting event and only in the Control Area where such event has occurred. A full response within 5 minutes is required. Tertiary control reserve: Also referred to as replacement reserve. The goal of this reserve is to restore the margins of the secondary reserve energy. Swissgrid dispatches tertiary energy by or phone call, in merit order of the cheapest energy and a complete response is required within 15 minutes. This time leap of activation allows some generators to provide tertiary reserve capacity in an offline status. 2 In the cases where the secondary reserve is automatically activated, it is referred as Automatic Generation Control (AGC).

15 CHAPTER 2. SWISS ELECTRICITY MARKET 9 Figure 2.1: Example of a pay-as-bid reimbursement mechanism in one of the reserve market capacity Stand-By-Power Market The supply of primary, secondary and tertiary control reserve capacity is organized via tenders carried out by Swissgrid. Table 2.1 illustrates the main features of these tenders. A bidding curve must be submitted, which is a set of non-decreasing combination of power and price per MW, to participate in the reserve capacity market. Swissgrid can choose at most one combination from each bid curve as shown in Fig In Fig. 2.1 Swissgrid accepts the bid of (6 CHF/MWh, 6MW) from the bidding curve depicted Regulating Energy Compensation The three reserve products just described have a different compensation mechanism of their respective reserve energy and Swissgrid takes care of such remuneration mechanism. An overview can be tracked in Table 2.1. In brief the main concepts are the following: Primary reserve energy is not reimbursed. Secondary energy is charged and averaged over a 15-minutes time span, and it is compensated as follows: Positive secondary energy is reimbursed with a 20% premium above the hourly SwissIX spot price. Anyway, at least the weekly base price is awarded. Negative secondary energy is charged with a 20% discount above the hourly SwissIX spot price. Anyway, at least the weekly base price is awarded For tertiary reserve energy, bidders who were awarded for providing stand-by power, must submit energy offers for the complete period in four-hours block. Dispatch of tertiary energy is thus determined in merit order, under a pay-as-bid mechanism of the energy offers.

16 CHAPTER 2. SWISS ELECTRICITY MARKET 10 Table 2.1: Products on reserve market. Brief overview. Adapted from [3] and [18] Primary Reserve Secondary Reserve Tertiary Reserve Product Type Symmetric Symmetric positive/negative Gate Closure 1 week ahead 1 week ahead 1 Week or 2 days ahead Contract Length 1 Week 1 Week 1 Week or 4 hours Capacity Payment pay as-bid pay as-bid pay as-bid Minimum bid for capacity +-1MW +-5MW +-5MW Maximum bid for capacity +-25MW +-50MW +-100MW Energy Payment not reimbursed SwissIX +- 20% pay as-bid

17 Chapter 3 Joint Dispatch Framework 3.1 Introduction The purpose of Chapter 3 is to formulate an optimal bidding strategy of a storage device in both energy and reserve market, with the specification that the reserve capacity market as well as the day-ahead energy market are cleared jointly and with the specification that an uniform pricing mechanism of reimbursement is in place for the energy ad reserve markets. Our goal will be to derive the optimal offering quantities for the 24 hours concerning the day-ahead market and the offering quantity of the reserve capacity. We assume that the reserve market where we are interested in participating is the so-called regulation market. The market framework considered in this problem can be interpreted as a simplified version of the day-ahead scheduling reserve market where the exact utilization of the reserve bids is decided by the market. This is a market framework similar to the one described in [4], but it will be further discussed in section 3.2. The only stochastic process that is considered in this formulation is the aleatory reserve deployment. It is important now to underline the meaning offering quantities or bidding quantities and the meaning of bidding curve. A bidding curve is a set of non decreasing combinations of prices and quantities of a given commodity or a non-decreasing combination of prices and quantities of a given product. In the case of the electricity markets the underlined commodity or product can be the amount of the electricity or the amount of reserve. An offering quantity instead is just the quantity of a commodity or a product. Figure 3.1 illustrates the concept. Producers must submit in general bidding curves to participate in the electricity market. Thus, if their offering strategy determines only an offering quantity, as it is the case in our discussion in Chapter 3, producers have to specify a price to attach to this quantity. We suggest that the price to attach to this quantity is 0 CHF/MW h, in such a way that the quantity 11

18 CHAPTER 3. JOINT DISPATCH FRAMEWORK 12 Figure 3.1: Example of a bidding curve and an offering quantity. We assume that the specification price of an offering quantity is 0 CHF/MW h.

19 CHAPTER 3. JOINT DISPATCH FRAMEWORK 13 is always accepted by the market 1. At this point, it should be clear why an offering strategy determining offering quantities is only suited if the remuneration mechanism of the accepted bids is not a pay-as-bid one. A pay-asbid remuneration mechanism, as in Chapter 4, yields that the combinations offered in the market are always remunerated according to their own price specifications so that an offering quantity specified with 0 CHF/M W h is not profitable at all. Instead, a uniform pricing as the one assumed in Chapter 3, ensures that the accepted bids are remunerated according to the market clearing price, and not according to their own specification price. Thus, a strategy yielding offering quantities can be suited for an uniform pricing reimbursement mechanism. 3.2 Market and Model Framework As already introduced, the energy and the reserve markets are cleared at the same time in the market framework of this chapter. The joint dispatch of energy and reserve is indeed one of the most used approach in electricity markets all over the world and seems to be economically more efficient compared to a sequential dispatch of energy and reserve [2]. In addition, an uniform pricing mechanism of reimbursement is supposed to hold in both the markets of reserve and energy. The market structure underlying the formulation of the optimal bidding strategy comprehends 24 time periods, and each time period corresponds to 1 hour. As such, there is full correspondence between the value of power and energy (MW and MW h). The available products to be traded are: One symmetric reserve capacity product, equal for all the considered time periods. Basically, there is no difference between capacities for up and down regulation. We assume that this reserve product is the one traded in the so-called regulation market. It covers the continuous fast and frequent changes in load and generation that create energy imbalances and frequency fluctuations [2]. Day-ahead market, intra-day market and balancing market for what concerns the energy markets. It is assumed that the closure of trading floors is structured as follows: The day-ahead market and the regulation market are cleared at the same time before the delivery day. For example, the day-ahead and the regulation market can be cleared at 10 a.m. the day before the energy delivery. This is in practice the logic of a joint dispatch. 1 For simplicity, we do not assume the possibility of having negative prices, so that the quantity offered at 0 CHF/MW h is always accepted. This assumption generally holds true, although negative prices have been observed in rare circumstances in electricity markets.

20 CHAPTER 3. JOINT DISPATCH FRAMEWORK 14 The intra-day markets are closed one hour before the delivery time. It means, for example, that the intra-day market of energy delivery between a.m. is cleared at 9 a.m. As such, there are 24 gate closures, one for each intra-day market. Balancing market are always available. The reimbursement of the reserve product consists of two parts: 1. Reimbursement of the capacity only, just for the fact that it is left available for an aleatory future deployment. The price of such a reimbursement is assumed to be a parameter resulting from a price forecast and an uniform pricing mechanism holds in this market. 2. Reimbursement of the energy eventually deployed. The amount of energy eventually deployed is the only uncertain parameter in this formulation which is modeled via stochastic scenarios. The mechanisms for pricing the reserve energy are actually different from market to market (i.e. the Swiss market mechanism prices it based on the dayahead market price plus a 20% of margin in addition/subtraction). In this model, the Swiss mechanism for reserve energy reimbursement is implemented. The payment mechanism of the energy markets instead is as follows: Day-ahead markets and intra-day markets have a single pricing mechanism. In our perspective perfect forecasts are assumed for prices of both markets. Since the prices are known in advance, there is a possibility to profit from arbitrage between the two markets (Buy energy in the market at lower price and sell energy in the market at higher price). The balancing (real-time) market has a dual pricing mechanism depending whether the imbalances are positive or negative. The mechanism is structured in such a way that arbitrage between the day-ahead and the balancing market is not profitable. More in details, the imbalance mechanism is implemented as follows: The positive imbalance is priced 50% less than the day-ahead price in the given time period. The amount of negative imbalance is evaluated 50% more than the day-ahead price in the given time period. Please note that the remuneration mechanism illustrated in the two bullet points is slightly different from the one currently in place in the Swiss market.

21 CHAPTER 3. JOINT DISPATCH FRAMEWORK 15 A reasonable assumption is that such a strict dual pricing mechanism of the balancing market prevents arbitrage between the intra-day market and the balancing market as well. In fact, a profitable arbitrage between the intra-day market and the balancing market occurs when the intra-day prices are higher than the prices of negative imbalances or lower than the price of positive imbalances. Since the intra-day prices are supposed to lie within the range of ±50% with respect to the day-ahead prices for a given time period, the possibility of a profitable arbitrage is not verified. Moreover, we need to discuss about the timing of the strategic decisions throughout the day and how we can incorporate these features into the optimization problem. In particular, keeping always in mind that the only stochastic process being involved is the eventual reserve deployment, we set up a four-stages decision model with the characteristics depicted in Fig These characteristics can be listed as follows: The day-ahead market quantities and the reserve capacity quantity are the 1 st stage variables. These decisions are made at the beginning of the stochastic process. The intra-day market decisions from hour 9 to hour 16 correspond to the 2 nd stage variables. These are made 8 hours after the beginning of the stochastic process. The intra-day market decisions from hour 17 to hour 24 correspond to the 3 rd stage variables. These are made 16 hours after the beginning of the stochastic process. The Balancing market decisions correspond to the 4 th and last stage variables. These are made at the end of the stochastic process. Such a multi-stage formulation ensures that the excess of energy due to the different reserve energy deployment scenarios is traded in the market floors following the day-ahead one. Decisions variables referred to a later stage, and thus to a later market floor, are made with less uncertainties about the reserve energy deployment compared to the decision variables referred to an earlier stage. In other words, our storage device is able to recognize that its state of charge, in a certain scenario and at a certain time, has been influenced by the reserve energy deployment up to that time and thus it decides its position in later market floors having at disposal these information. This is the logic of how the state of charge management is carried out in this optimization problem. Figure 3.3 clarifies further this idea. Furthermore, it should be noted that the intra-day markets in the first 8 hours are neglected and that the four-stage decision framework does not

22 CHAPTER 3. JOINT DISPATCH FRAMEWORK 16 Figure 3.2: Decision framework of the model. Note the decisions stages of the variables involved. represent exactly the actual gate closure of the intra-day markets 2. Such simplifications are needed to render the problem computationally tractable. Moreover, non-arbitrage constraints have been set up in the formulation and consist of a limitation of the day-ahead and intra-day energy sales in a given hour. Lastly, we manage to keep the model mixed integer linear. 2 It was specified previously that the gate closure of the intra-day markets is one hour before the energy delivery of the interested hour. Thus, there are actually 24 intra-day markets gate closures in one day and, a model representing each closure would be a 24- stages decision model. Needless to say, such a model would be computationally intractable.

23 CHAPTER 3. JOINT DISPATCH FRAMEWORK 17 Figure 3.3: Logic of the state of charge management for the storage device.

24 CHAPTER 3. JOINT DISPATCH FRAMEWORK Model Formulation Sets Ω T Set of scenarios of reserve energy deployment. Set of time periods 3. Each time period corresponds to one hour. Thus the power and energy measures correspond in this formulation. Continuous Variables P ch t,ω P dis t,ω P da t,ω R P id t,ω P ba t,ω P baplus t,ω P baminus t,ω Es t,ω Revenues t,ω Actual energy withdrawn by the device in time t and scenario ω, MWh. Actual energy delivered by the device in time t and scenario ω, MWh. Energy sold/bought in day-ahead energy market in time t and scenario ω, MWh. Reserve capacity sold for the day. MW. Energy sold/bought in intra-day energy market in time t and scenario ω, MWh. Energy sold/bought in the balancing (real-time) energy market in time t and scenario ωpc, MWh. Energy sold in intra-day energy market in time t and scenario ωpc, MWh. This variable is needed to ensure a dual pricing system in the balancing market expressed in a linear form. Energy bought in intra-day energy market in time t and scenario ωpc, MWh. This variable is needed to ensure a dual pricing system in the balancing market expressed in a linear form. Energy stored at the end of t in scenario ωpc, MWh. Revenues in time t and scenario ωpc, MWh. Binary Variables Sch t,ω Charging state of the device in time t and scenario ω, (1 for charging state, 0 otherwise). Sdis t,ω Discharging state of the device in time t and scenario ω, (1 for discharging state, 0 otherwise). 3 The time periods here are 24 and they represent one day.

25 CHAPTER 3. JOINT DISPATCH FRAMEWORK 19 Parameters π ω Probability of occurrence of scenario ω. ef f Es ini P max P min Emax Emin P Rda t P Rid t k t,ω P Rrc P Rre t,ω Charging (round-trip) efficiency of the device. Energy stored at the beginning of the period. Maximum amount of power that can be delivered by the device (>0), MW. Minimum amount of power that can be withdrawn by the device (<0), MW. Maximum amount of energy that can be stored by the device, MWh. Minimum amount of energy that can be stored by the device, MWh. Day-ahead prices in time t, CHF/MWh. Day-ahead prices in time t, CHF/MWh. Percentage of reserve capacity eventually deployed in time t and scenario ω. This is the parameter representing the stochastic process of reserve energy eventual deployment. k t,ω ( 1, 1). Hourly price of reserve capacity reimbursement. Remuneration/Payment price of reserve energy in time t and scenario ω. It depends whether the reserve energy deployment k t,ω was positive or negative in time t and scenario ω. If positive, the reimbursement price is 20% higher than the day-ahead price in the same time t. If negative, the payment price is 20% lower than the day-ahead price in the same time t. It is almost the same remuneration mechanism implemented in the Swiss market.

26 CHAPTER 3. JOINT DISPATCH FRAMEWORK 20 max t,ω π ω Revenues t,ω (3.1) s.t Revenues t,ω = P Rda t P da t,ω + P Rid t P id t,ω P Rda t P baplus t,ω 1.5 P Rda t P baminus t,ω + P Rrc R + P Rre t,ω k t,ω R t, ω (3.2) P ba t,ω = P baplus t,ω P baminus t,ω t, ω (3.3) P dis t,ωpc P ch t,ωpc = P da t,ωpc + P id t,ω + P ba t,ω + k t,ω R t, ω (3.4) 0 P ch t,ω P min Sch t,ω t, ω (3.5) 0 P dis t,ω P max Sdis t,ω t, ω (3.6) Sdis t,ω + Sch t,ω 1 t, ω (3.7) Es 1,ω = Es ini + eff P ch 1,ω P dis 1,ω ω (3.8) Es t,ω = Es t 1,ω + eff P ch t,ω P dis t,ω t = 2...T, ω (3.9) Es T,ω = Es ini ω (3.10) Emin Es t,ω Emax t, ω (3.11) P da t,ω + P id t,ω + P ba t,ω + R P max t, ω (3.12) P da t,ω + P id t,ω + P ba t,ω R P min t, ω (3.13) P min P da t,ω P max t, ω (3.14) 0.15 P min P id t,ω 0.15 P max t, ω (3.15) P id t,ω = 0 t = 1...8, ω (3.16) P da t,ω = P da t,ω t = 1...T, ω, ω Ω (3.17) P id t,ω = P id t,ω t = , ω, ω Ω : ksc 1:8,ω = ksc 1:8,ω (3.18) P id t,ω = P id t,ω t = 17...T, ω, ω Ω : ksc 1:16,ω = ksc 1:16,ω (3.19) P ch t,ω, P dis t,ω, P baplus t,ω, P baminus t,ω 0 (3.20) The stochastic problem ( ) seeks to maximize the expected value of the revenues over time periods t and the scenarios ω. The revenues consist of the the day-ahead market sales (1 st term of (3.2)), the intra-day market sales (2 nd term of (3.2)), the balancing market sales (3 rd and 4 th terms of (3.2)), and the remuneration for the reserve capacity and energy (5 th and 6 th terms of (3.2)). In particular, the 5 th term of (3.2) shows the revenues concerning the reimbursement of reserve capacity while the 6 th term of (3.2) shows the term concerning the reimbursement of reserve energy.

27 CHAPTER 3. JOINT DISPATCH FRAMEWORK 21 Furthermore, the balancing market sales (3 rd and 4 th terms of (3.2)) in combination with (3.3) ensure a dual pricing system in the balancing market while keeping the model linear. Equations ( ) define the constraints concerning the charging and discharging energy/states of the storage device and such a formulation ensures that problem remains as mixed integer linear. Note that the eventual reserve energy deployment k t,ω R appears in Equation (3.4). The constraints describing the available energy in the storage device at the end of each time period are ( ). Note that the amount of energy at the end of period is bound to be at the same level as the one at the beginning of it by (3.10). Equation (3.11) instead limits the amount of stored energy within the technical feasibility. Finally, Equations ( ) define the available power constraints. These equations state that the maximum available power should be always at least as the sum of the power committed in the energy markets, and the sum of reserve that have been committed. That is, at any moment the device is able to provide the maximum amount of reserve if it is called to do so by the TSO. Equation (3.14) limits the day-ahead market commitment as the maximum technical power of the device. Equations (3.15) limits instead the intra-day market commitment as a percentage of the maximum technical power of the device. With equations ( ) it is thus possible to limit the possibility of arbitrage between the day-ahead and intra-day markets. Equation (3.16) reduce the number of the intra-day markets available. In particular, there are no intra-day markets available for the first eigth hours, as previously explained. Equation (3.17) defines the non-anticipativity constraints for what concern the first-stage decisions. In particular the day-ahead market variables are the first-stage and they assume the same values for all the set of scenarios involved 4. Equation (3.18) defines the non-anticipativity constraints for what concern the second-stage decisions. These are the intra-day market variables from hour 9 to hour 16. Equation (3.19) defines the non-anticipativity constraints for what concern the third-stage decisions. These are the intra-day market variables from hour 17 to hour 24, that is to the end of the time periods. Equation (3.20) finally defines the non-negativity of the interested variables. 4 Actually the reserve variable R is a first stage decision variable, because it varies nor through the time periods nor through the scenario set. For the sake of simplicity and clearness we have defined the variable R as an R ω, so that we do not need the nonanticipativity constraints for variable R.

28 CHAPTER 3. JOINT DISPATCH FRAMEWORK 22 Table 3.1: Key technological properties of the battery Property 3.4 Case Study Value Power (Charging and Discharging) 8 MW Maximum State of Charge 20 MWh Minimum State of Charge 2 MWh Technology Li-Ion Efficiency (Round Trip) 95% In this section, we consider a case study of model ( ). Model ( ) is implemented in Matlab R2016a - academic use using an optimization interface called YALMIP [31]. We take the day-ahead and intra-day prices of the Swiss market in an off-working day in summer, that is Saturday, July 16 th This day will be our 24 hours framework of the model. We suppose to have a battery with the characteristics specified in Table 3.1. Such battery is a realistic one in line with the current available storage technologies for power systems illustrated in [30]. We generate 1331 scenarios of reserve realization, structured in such a way that the reserve realization is coherent with a four-stage decision model. For detailed information about the scenario generation methodology please refer to Appendix A. To summarize, scenarios are derived from a random distribution B (α = 5, β = 5) adapted for a four-stages decision model, as it is the case for ( ). We formulate our optimization problem setting the variable R as an integer variable, and not as continuous variable like it was specified in Section 3.3, because in several markets (e.g. the Swiss market) the offer for reserve capacity has to be specified with integer values. Moreover, we define the price of reserve capacity as 22 CHF/MW h as it is a realistic price, at least for the Swiss market of primary and secondary reserve capacity. The time required for solving this mixed integer linear optimization problem on a 3.40 GHz Intel(R) processor with 16 GB of RAM is 147 seconds 5. The solver chosen for solving this problem is CPLEX-IBM. Figure 3.4 plots the day-ahead prices, the day-ahead bidding quantities and the reserve capacity bidding quantity for the 24 hours. Note that the reserve bidding quantity and the day ahead bidding quantities are the firststage decision variables and thus these values are equal for all the scenarios involved. Figure 3.5 compares the reserve realization in the first, second and 503 rd scenarios. Note that the reserve realization in the first and second scenario 5 About two minutes and an half

29 MW -- CHF CHAPTER 3. JOINT DISPATCH FRAMEWORK Day Ahead Prices Day Ahead Bidding Quantities Reserve Bidding Quantity Time Figure 3.4: Results from the optimization problem. The 24 Day-ahead bidding quantities and the Reserve bidding quantity are the first-stage decisions variables. is the same until hour 16 and then it diverges 6. It means that in the first and second scenario, the second-stage and third-stage variables are the same, as a consequence of the non-anticipativity constraints ( ). This is not the case for the first or second and the 503 rd scenario where the reserve deployment process is not the same in any hour, and thus there are no variables in common between the two scenarios, apart from the first-stage ones. Moreover, the expected values of the overall revenues of the storage device is 3444 CHF for the day considered. It is interesting that 3168 CHF of such revenues comes directly from the reserve capacity market 7. It means basically that about 90% of the revenues of a storage device derive from the remuneration of stand-by capacity only, at least in this formulation and with these price parameters. It is worth then to analyze a situation where the storage device can participate only in the energy market, in order to have a valuation of the rev- 6 For more information about the scenarios generation methodology refer to Appendix A. 7 Such a number comes basically multiplying the price of the reserve capacity (22 CHF/MW h) times the number of hours of the period considered (24h) times the amount of cleared capacity (6MW ). 22 CHF/(MW h) 24h 6MW = 3168 CHF

30 Percentage of Reserve Deployed CHAPTER 3. JOINT DISPATCH FRAMEWORK st scenario reserve realization 2nd scenario reserve realization 503rd scenario of reserve realization Time Figure 3.5: Reserve realization in the first, second and 503rd scenarios. enues without the involvement of the storage device into the reserve market. We propose to use the optimization problem ( ) in order to carry out such an appraisal, with slight modifications. In particular, we set the variables concerning the reserve, the intra-day market and the balancing market bidding quantities to 0. In other words, we should add to the optimization problem ( ) the following constraint: R, P id t,ω, P ba t,ω = 0 t, ω. The rest of the parameters/variables are not at all involved in this modification. Figure 3.6 shows the day-ahead prices, the day-ahead bidding quantities and the state of charge when the storage device can participate only in the day-ahead energy market. The strategy followed by the storage device is to sell energy (discharge) in the hours where the price is high and to buy energy (charge) when the price is low. In such a way, it is possible to realize an inter-temporal energy arbitrage exploiting the price difference between the time periods as Fig. 3.6 shows. Moreover, note that the storage device can make profit in the energy market only by exploiting the differences between prices in different hours. It means that the profits of the device are not affected by the magnitude of the prices, but only by their relative difference. A counter-intuitive consequence of this concept is that, if the prices are very high and constant throughout the period (e.g. 100 CHF/M W h constant during period), the storage device makes no profit at all. This idea is actually confirmed by simulations

31 CHAPTER 3. JOINT DISPATCH FRAMEWORK 25 carried out always with model ( ) constrained to the day-ahead energy market, which confirm that there is no possibility of profit in the energy markets with constant high prices during the time span of a day. With the price parameters set as the ones on Saturday, July 16 th 2016, the expected profit of the storage device, when it is allowed to participate only in the day-ahead market, is 240 CHF. Compared to profit of 3444 CHF when the storage device is allowed to participate in the reserve market as well, it means that the reserve market is the most important stream of revenues for the storage device. It is thus justified from the economical point of view the involvement of the storage device in the reserve market, as well as in the energy one.

32 MW -- CHF -- MWh CHAPTER 3. JOINT DISPATCH FRAMEWORK Day Ahead Prices. Only Day-ahead market available Day Ahead Bidding Quantities. Only Day-ahead market available State of charge Time Figure 3.6: Day-ahead prices, day-ahead bidding quantities and state of charge when the day-ahead market is the only available energy market.

33 Chapter 4 Sequential Dispatch Framework 4.1 Introduction The purpose of Chapter 4 is to formulate an optimal bidding strategy of a storage device in both energy and reserve markets with the specification that the reserve capacity market as well as the day-ahead energy market are cleared in sequence and with the specification that a pay-as-bid reimbursement mechanism is in place for the reserve market. This is a market mechanism similar to the Swiss electricity market described in Chapter 2. The offering strategy takes into account the primary and secondary reserve capacity markets, neglecting on purpose the tertiary reserve capacity market. The primary and secondary reserve capacity markets ensure indeed higher possibility of profits for the storage device as [21] states, due to the higher remuneration of the stand-by power. The optimal offering strategy described in Chapter 4 is derived via the solution of two different stochastic optimization problems in sequence. At first the bidding curve for the reserve capacity market is determined and only later the bidding quantities for the day-ahead energy market are defined 1. The choice of splitting the decision making process in two different steps is a consequence of the Swiss market sequential structure. As Chapter 2 points, the primary and secondary reserve capacity markets are cleared on Tuesday the week-ahead while the day-ahead market is cleared one day before the energy delivery. Thus, it is justified a strategy which takes into account the fact that further information is available for the day-ahead market compared to the reserve market. In particular, the amount of accepted reserve capacity is already known when the decisions of the day-ahead market are made. Figure 4.1 illustrates the structure of strategy implemented in this chapter. 1 For the readers not familiar with the concepts of bidding curve and bidding quantities a short recall can be found in Section

34 CHAPTER 4. SEQUENTIAL DISPATCH FRAMEWORK 28 Figure 4.1: Scheme of the strategy. In particular, the model described in Section 4.2, which is the one that has to be solved first, aims to derive the optimal bidding curves of the primary and the secondary reserve capacity, with the specification that the reimbursement mechanism is in a pay-as-bid manner. Since a uniform clearing price does not exist in pay-as-bid markets, it is necessary to build up a strategy which determines bidding curves taking into account the fact that each price-quantity combination is always remunerated according to its own price specification. Scenarios of reserve capacity prices are thus needed to define a bidding curve that mitigates the risk of high-price combinations can not be accepted and low-price price combinations result in low profits. The second model described in Section 4.3 seeks to determine the optimal bidding quantity to be sold in the day-ahead market once that the accepted reserve capacities are known. Scenarios of eventual reserve deployment are used in order to set up a four-stage decision model. By including these four decision stages the model ensures that the excess of energy due to the different reserve deployment scenarios will be traded in the market floors following the day-ahead market. It has to be further specified that a pay-as-bid market mechanism does not rely on a uniform clearing price, which is equal for all the market participants. Usually in the pay-as-bid mechanism the bids of the producers are accepted only if they are offered at a price equal or lower than the resulted market price. Actually in the case of the Swiss market, it is not described in any document whether the pay-as-bid mechanism is implemented in such a way or some other mechanisms determine the bid-blocks accepted. In general primary and secondary frequency control should be distributed

35 CHAPTER 4. SEQUENTIAL DISPATCH FRAMEWORK 29 Figure 4.2: An example of the clearing criterion assumed in this paper. If the threshold price is 6.5 CHF/MWh, the bid that it is accepted is (6 CHF/MWh, 6 MW). Figure 4.3: If the threshold price is 7.5 CHF/MWh, the bid that it is accepted is (7 CHF/MWh, 7 MW). This the same logic of figure 4.2.

36 CHAPTER 4. SEQUENTIAL DISPATCH FRAMEWORK 30 as uniformly as possible throughout the grid in such a way that its action is independent from the origin of the imbalances and from the contingent distribution of loads and generations [22]. Thus, it is reasonable that Swissgrid takes into account, apart from pure economic criteria, some other locational effects when clearing the market for primary and secondary control. Thus, in this thesis it is assumed that does not exist a market price for primary and secondary reserve capacity. Instead, we assume that Swissgrid decides the bids to be accepted based on locational effects, apart from pure economic order. Nevertheless, from the point of view of a producer located in a precise geographical location, it can be assumed that there exist a threshold price in such a way that the first bid in the bidding curve with the price lower than a threshold price is accepted. Figure 4.2 shows an example of such a threshold price. In this case, the threshold price is supposed to be 6.5 CHF/MWh. Thus, the combination that is accepted is the first one with the price specification lower than 6.5 CHF/MWh. The bidding curve depicted in Fig. 4.2 yields that the bidblock accepted is the one defined for (6 CHF/MWh, 6 MW). On the other hand Fig. 4.3 describes a scenario where the threshold price is supposed to be 7.5 CHF/MWh. In this case, the bid that is accepted from Swissgrid is the one specified for (7 CHF/MWh, 7 MW). As it is already possible to spot, such threshold price will be a source of uncertainty because it defines which bid-blocks from a bidding curve are accepted. From now on however, the terms threshold price and reserve price realization will be used with the same meaning, just to indicate this source of uncertainty in the reserve market.

37 CHAPTER 4. SEQUENTIAL DISPATCH FRAMEWORK Reserve Capacity Offering Strategy The formulation of an offering strategy which takes into account a pay-asbid market remuneration mechanism needs to be formulated in a specific way, in order to catch the main features of such a reality, while guaranteeing the computational tractability. As it will be shown later in this section guaranteeing computational tractability is particularly challenging, and simplifications need to be carried out. Let s define the threshold price λ as a random variable following the density function f λ : R + R +. Supposing that a producer submits a single price-quantity offer (p, q) 2, the acceptance probability of the offer is [23]: P[λ P ] = p f λ (l)dl (4.1) where l is an auxiliary integration variable. Needless to say, under a pay as bid pricing mechanism, the expected remuneration price with which the bid is accepted is p. The expected return ρ of the producer is thus: Replacing (4.1) into (4.2) we obtain: E[ρ] = P[λ p] p q (4.2) E[ρ] = p q p f λ (l)dl (4.3) Now, the idea is to extend the formulation of the return written in (4.3) valid only for a single price-quantity offer, to a complete step-wise non-decreasing bid-curve. Eventually, the ultimate goal is to find the generic optimization model describing the pay-as-bid remuneration mechanism [23]. In general, the offers from the generation-side can be modeled through a set of B non-decreasing price-quantity offers {(p b, q b ), b = 1...B}. Equation (4.3) is thus extended to the general formulation of a pay-as-bid model which is based on the one described in [23]: max q b,p b,b B E[ρ b ] E[c b ] (4.4) b=1 s.t E[ρ b ] = p b q b E[c b ] = h(q b ) pb+1 p b f λ (l)dl b (4.5) pb+1 p b f λ (l)dl b (4.6) 2 The quantity refers to the amount of capacity offered in the reserve market and the price to the unitary value of such amount. It is the same concept as the one described in Fig. 4.2 and Fig. 4.3.

38 CHAPTER 4. SEQUENTIAL DISPATCH FRAMEWORK 32 q b Ψ b (4.7) q b > q b 1 b (4.8) p b > p b 1 b (4.9) where p B+1 = and Ψ is the feasible operating region of the device. The parameter c b is the opportunity cost of providing a certain amount of reserve quantity q b, whose value is computed through the function h( ). The opportunity cost here refers to the revenues that the storage device loses in the energy market, if it commits the power capacity q b for ancillary services provision [24] [25]. In fact, the reserve capacity committed is not available for the energy only market and thus the profit in the energy only market is lower. Equation (4.8) and (4.9) specify the non-decreasing constraints of the offering curve. Note that p b and q b represent the price and quantity specification of each bid-block of the bidding curve, while B represents the amount of bid-blocks of the bidding curve. In other words, B defines the amount of stairs of the bidding curve. Note that the formulation ( ) is non-linear and our ultimate goal is to derive a computationally tractable mixed integer linear problem out of ( ). To do this, we take wide inspiration from a methodology illustrated in [18]. The first step is to represent the threshold price λ using a set of possible realization scenarios {λ ω, ω = 1...Ω}. Each threshold price scenario λ ω is associated with a probability of occurrence π ω an it holds that ω π ω = 1. The basic idea is to consider this set of threshold price realization scenarios as the potential offer prices that the price-taker producer can submit into the market. Thus, the parameter {λ ω, ω = 1...Ω} not only represent the scenario realizations of the threshold price parameter, but also a basket of prices that can be chosen for each bid-block. It means that, with this simplification the decision variables q b, p b and B, which were defined in the optimization problem are simplified and reduced as follows: Variables p b can be only assume values out of the set {λ ω, ω = 1...Ω}. It means that the bid-blocks will be in a number lower or equal than Ω. Variable B is constrained to be lower or equal than Ω. As explained in the previous bullet point, it means that the maximum amount of bid-blocks or stairs are lower or equal than Ω. Variables q b are not affected by this simplification and they are free to assume any value. The only limitation is that the minimum increment of power between two consecutive bid-blocks is 1 MW, as it is the actual case for the Swiss market. Now, in order to generate the linearized model, we have to define the following variables and parameters which will be eventually used in the mixed

39 CHAPTER 4. SEQUENTIAL DISPATCH FRAMEWORK 33 Figure 4.4: Bidding curve defined with the parameters shown in Example 1. integer linear formulation ( ). Such a formulation takes inspiration from the one described in [18]: Sets Ω Variables Scenarios of reserve capacity prices realization I Set of bid blocks (Same cardinality as Ω) q ω Reserve capacity cleared in scenario ω. MW. q block ω,i Reserve capacity from bid block i, cleared in scenario ω. MW. Bin ω Bin block ω,i Binary variable which specifies whether there is an increment of power in scenario ω compared to scenario ω 1. If this is the case, a bid block is specified as (λ ω, q ω ) Binary variable which specifies whether bid block i is cleared in scenario ω c ω ρ ω Opportunity cost of reserve capacity q ω Revenues from the reserve market in scenario ω Parameters λ ω M Clearing prices of reserve capacity in scenario ω. Sorted in ascending order Large enough constant The meaning of the variables and parameter just defined and used in the optimization problem ( ) will be further clarified in Example 1.