Calibration of Operating Reserve Demand Curves Using a System Operation Simulator

The objective of this paper is to analyse the trade-off between the cost of operation and system reliability resulting from different shapes of Operating Reserve Demand Curves under scarcity pricing. We implement a model of the short-term operation of Belgium and we validate it against historical realisations of operation. The model of short-term operation is implemented using 4 embedded optimization problems. The model simulates the trade-off between the lag and cost of mobilizing flexible resources versus the increased reliability that these resources ensure for system operation and allows us to quantity the resulting effect on the level of scarcity pricing adders. We compare eight variants of operating reserve demand curves, and use them as the basis for supporting a recommendation to the Belgian regulatory authority for the implementation of scarcity pricing in Belgium.


I. INTRODUCTION
T HE increased integration of renewable resources in power systems has led to a tension in electricity markets. On the one hand, it has increased the need for flexible assets by increasing the variability and unpredictability of supply in the system. On the other hand, it has decreased the remuneration of flexible assets by pushing them further down the merit order curve and by consequently depressing the real-time energy price. This paradox raises concerns about the adequate remuneration of flexible resources, where flexible resources should be understood as resources that can provide automatic and manual frequency restoration reserves with a full activation times of five to fifteen minutes. 1 Scarcity pricing is a mechanism that has been considered as a possible remedy to the aforementioned lack of adequate incentives for investing in flexible assets. Manuscript  Its introduction contributes towards a market-based shift in the capacity mix that is required by the energy transition. It also reduces the need and scope for administrative out-ofmarket interventions, such as strategic reserve.

A. Literature Review and Context
Scarcity pricing is a mechanism for better remunerating energy and reserve in periods of scarcity. It relies on an operating reserve demand curve (ORDC) that introduces price elasticity in the procurement of reserve in the real-time market. Scarcity pricing induced by ORDCs exhibits less pronounced and more frequent price spikes than scarcity pricing induced by the value of loss load alone (VOLL pricing). The resulting "well-behaved" energy price creates a more stable environment for investment. Scarcity pricing also mitigates the missing money problem by uplifting the real-time price for energy during periods of scarcity. Early references to this mechanism are found in Stoft in [1] and the theory was formally anchored to the loss of load probability (LOLP) and the value of loss load by Hogan in [2] and [3]. Although scarcity pricing is a real-time mechanism, the uplifted real-time energy and reserve prices are expected to back-propagate to forward (e.g. day-ahead) energy and reserve markets, thereby generating robust investment signals in the market [4].
Scarcity pricing based on operating reserve demand curves has commonly been associated with US markets, and the adaptation to European market design requires the introduction of EUspecific terminology. Real-time markets and real-time energy in Europe are referred to as balancing markets and balancing energy respectively. Note that real-time reserve is not traded in balancing markets, which is a serious oversight of European electricity market design [4]. The European term for reserve is balancing capacity. For the remainder of the paper, balancing capacity and balancing energy are used as synonyms for reserve and energy. We consider two types of balancing capacity; automatic and manual frequency restoration reserve (aFRR and mFRR respectively). aFRR is driven by an automatic controller and can react within a few seconds in order to restore frequency deviations. It can thus be considered as synchronised reserve. The full activation time of aFRR is typically five minutes. mFRR is manually activated in order to cope with more significant system disturbances, including contingencies. These resources need to be fully available within 15  Scarcity pricing has been increasingly considered for implementation by a number of US independent system operators, such as ERCOT and PJM. In these markets, there has been a transition from an ORDC with a fixed reserve requirement to a downward sloping ORDC based on the VOLL and LOLP [5]. ISO-NE and MISO are also considering this transition, following the recommendation of their respective market monitors.
TSOs and regulators in Europe have also considered variations of scarcity pricing mechanisms. A Reserve Scarcity Pricing function based on the VOLL and the amount of leftover reserve based on hour-ahead measurements is implemented in Great Britain [6]. Ireland has also implemented a scarcity pricing mechanism, nevertheless it is not based on ORDC and LOLP but rather triggered by stress conditions in the system [7], [8].
Poland is planning to implement a scarcity pricing scheme by the first half of 2023. The Polish scheme would be based on an ORDC based on VOLL and LOLP, and would remunerate non-contracted but available balancing capacity [9].
In Belgium, a number of studies have been performed on behalf of the Belgian regulatory authority for energy, in order to assess the potential of scarcity pricing based on ORDC [10], [11]. In more recent work, [4] investigates the adaptation of the mechanism, which has been inspired by US-style two-settlement systems, to European market design. A central measure proposed in this work is the implementation of a real-time market for reserve capacity [12], [13]. The current work advances this research effort for analyzing the implementation of the mechanism in EU markets by focusing on the calibration of the operating reserve demand curves. This is a key design parameter of the mechanism.
The calibration of ORDCs has largely been restricted to open-loop analyses in the existing literature [11], [14], [15]. An open-loop analysis is defined by its lack of feedback from the introduction of elastic ORDCs. It calculates adders given a dispatch as opposed to a closed-loop analysis whose dispatch would be affected by the ORDCs. The Belgian transmission system operator, ELIA, studies scarcity pricing in Belgium in [15]. Papavasiliou et al. [11] provide a sensitivity analysis of the induced scarcity prices based on the parametrization of the operating reserve demand curve. The scarcity prices are derived from the remaining reserve capacity of a unit commitment model that uses fixed reserve requirements. Zarnikau et al. [14] analyses the impact of scaling ORDCs horizontally, as well as the effect of the ORDC on the real-time market price and investment incentives for natural-gas-fired generation in the Texas electricity market.
References [11], [14] highlight the influence of the shape of ORDC on the price and how the parametrization of the ORDC can affect the remuneration of different technologies through its effect on prices. Nevertheless, the open-loop approach proposed in those papers is not able to capture the dispatch and commitment incentives created by different calibrations of the ORDC. This shortcoming of open-loop analyses motivates our proposal for a nested modelling approach for simulating the short-term operation of the system and for modelling the interplay between reserve prices and operational efficiency.
Our simulator models the operation of a perfectly coordinated system, where a centralized optimization model commits and dispatches resources in a coordinated fashion. Uncertainty is assumed to stem from the actual load that needs to be served by the system. The sequential optimization of system scheduling aims at replicating the real-time controllability of the different assets present in the system, with a specific focus on quantifying the interplay between lags in decision making and the revelation of uncertain information in the system. The constraints on the scheduling of assets are inspired by the detailed commitment model of [16]. This allows us to quantify the fundamental tradeoff that ORDCs aim at balancing: incurring non-negligible fixed costs for committing flexible resources that can allow the system to operate reliably in real time, versus running the risk of not covering imbalances fully.
The proposed quantitative methodology could be used in the future to adapt the parametrization of ORDCs to the state of the system. An increase in renewable generation may introduce significant additional uncertainty to the system, resulting in ORDCs that are broader, and which can provide stronger signals for investment. Such a shift has already been discussed in ERCOT [14] and our methodology can be used to assess such policies.
The nested modelling of system operation also appears in [17] and [18], albeit less detailed. Nested models also appear in other contexts in [19], [20] and [21]. We proceed to discuss the relation of these publications to our work.
Zhou and Botterud [17] develop their model in order to analyze an ORDC which is based on the loss of load probability, and which accounts for the uncertainty caused by wind and load forecast errors, as well as generation contingencies. Lavin et al. [18] introduce an LOLP which is a function of the ambient temperature, in order to represent the higher probability of forced generator outages under extreme temperature conditions. Those models use short-term system operation simulators in order to compare different ORDC schemes. Nevertheless, these publications mostly leave aside the aspect of calibrating the ORDC. More specifically, the results of Zhou and Botterud [17] and Lavin et al. [18] are limited by the hourly temporal granularity of their model, as they cannot quantify the tradeoff between short-term dispatch adjustments and the lagged activation of reserves. The increased precision of our model remedies this shortcoming, and advances the state of the art by assessing more faithfully the interplay between costs and lags versus reliability of operation for different calibrations of ORDC.
In this sense, the precision of our model is closer to the Smart-ISO model of Simao and Powell [19]. The model of [19] is inspired by system operator practices and it is used in order to assess the reliability of the PJM system under different levels of wind power integration. We can also draw similarities with the model of Bakirtzis et al. in [20] where the authors propose a number of short-term operating models in order to cope with the increased uncertainties of power system operations. More recently, Atakan et al. [21] use their model to compare the cost performance of different levels of reserve requirement. Their calibration of the level of reserve requirement based on simulations in a stochastic hierarchical planning framework is simular in spirit to our approach. Note, however, that none of the aforementioned papers is focused on the question of the calibration of ORDCs.
Discussions on the shape of demand curves also appear in the capacity market literature. Cramton summarizes lessons learned from the early capacity market designs, and provides recommendations for the design of capacity demand curves in [22]. References [23] and [24] focus on the parametrization of capacity demand curves. Brown [23] derives curve parameters that aim at maximizing welfare in a 3-stage model where two firms can (i) choose their production capacity, (ii) compete in a capacity auction and (iii) compete in an energy market. Zhao, Zheng and Litvinov [24] propose a framework for rigorously determining capacity curves based on the value of reliability obtained by simulating different levels of capacity in the market.
Our paper can also be linked to Mays [25]. The author advocates for a more direct connection between the construction of ORDCs and the expected outcome of a stochastic market.

B. Contributions and Structure of the Paper
The main contributions of this paper are thus threefold. First, we provide of a quantitatively sound methodology for calibrating ORDCs. Previous works in the scarcity pricing literature were either not accounting for the feed-back effect of introducing ORDCs due to being open-loop [11], [14], or they are focusing on the comparison between fixed reserve requirements and price elastic ORDCs and not on the parametrization of such ORDCs [17], [18]. Our method anchors the calibration to a closed-loop simulation model and examines three specific design criteria for ORDCs that emerge in a realistic implementation of the mechanism.
Second, we implement detailed system operation models for quantifying the tradeoff between incurring fixed costs for committing flexible resources and allowing the system to operate reliably instead of running the risk of shedding load.
Third, the current modeling effort is contributing directly to the implementation of a scarcity pricing mechanism in the Belgian electricity market. The results of the analysis constitute the basis for our recommendation to the Belgian regulatory authority for the possible implementation of scarcity pricing in Belgium in the short to medium term. This recommendation supports their latest study [26].
The paper is structured as follows. Section II summarizes the principles of scarcity pricing, with a specific focus on certain dilemmas pertaining to the calibration of ORDCs. Section III presents the short-term operating model that we have developed in order to support the calibration of an ORDC. Our case study of the Belgian market is presented in Section IV. The results of the case study are presented and analysed in Section V. Finally, we conclude and discuss future research perspectives in Section VI.

II. SCARCITY PRICING AND ORDC
Scarcity pricing is introduced and justified in Section II-A. SubSection II-B considers the case of multiple reserve products and Section II-C introduces variants of ORDC.

A. Rationale of Scarcity Pricing
The argument that rationalizes the implementation of scarcity pricing with ORDCs based on LOLP and VOLL relies on the intrinsic stochasticity of an economic dispatch and in its deterministic approximation [3]. In this setting, the value of an additional MW of balancing capacity is linked to the value of the improved reliability that it provides to the system by reducing the likelihood of load shedding. This marginal value is characterised by Hogan in [3] as a function of the value of lost load (V OLL), the loss of load probability (LOLP (·)) given the level of reserve in the system (r) and the marginal cost of the marginal unit in the system ( MC): The most complete integration of scarcity pricing based on ORDC to electricity market operations would correspond to the co-optimization of reserve and energy in real time. An ORDC based on (1) would then be the explicit demand curve for reserve and would be inserted in the multi-product auction that trades energy and reserve simultaneously. Co-optimized markets would then produce one price for each product: (i) a reserve price for the available reserve and (ii) an energy price for the energy traded. Note that, in the absence of binding ramp constraints, the price of energy will be coupled to the price of reserve in order to ensure an equivalence between the marginal profit on the energy and reserve market for a marginal generator that supplies both reserve and energy.
In the absence of a co-optimization of energy and reserves, we could use the expression of (1) to compute adders based on the amount of reserve that is available in real time, as measured by system telemetry. This adder reflects the level of stress in the system, and would correspond to the price of reserve. The coupling between reserve and energy prices would then be implemented by adding this adder as a price component to the real-time energy price in the absence of co-optimization (the balancing energy price in EU nomenclature).

B. Multiple Reserve Products
Formula (1) has been generalized by Hogan in [27] to the case of multiple reserve products of different quality. The quality of reserve refers to the delivery time that is required for this specific reserve product to be fully available, which in EU jargon is referred to as full activation time. This generalization is based on the split of a real-time dispatch interval in two parts. In the first part of the interval, it is assumed that only high-quality resources can respond, whereas in the second part of the interval, all reserve types are assumed to be able to respond. We consider the full interval as an imbalance interval.
In the EU, the reference duration for an imbalance interval is 15 minutes and [28] suggests the following split of the real-time dispatch based on the products that have been historically available in the EU balancing market. The first part of the interval would last 7.5 minutes and imbalances would be resolved by balancing capacity that can be fully activated in no longer than 7.5 minutes (which corresponds in our analysis to aFRR capacity 2 ). The second part of the interval would also last for 7.5 minutes. In this time interval, imbalances would be resolved by balancing capacity that can be fully activated in 15 minutes (which correspond in our analysis to mFRR capacity).
Based on this split of an imbalance interval, the authors in [28] suggest the introduction of two ORDCs: (i) a 7.5-minute ORDC (2) for the first part of the interval and (ii) the 15-minute ORDC (3) for the second part of the interval: Here, LOLP x (·) corresponds to the loss of load probability after x minutes, and r x is the amount of reserve that can be activated within x minutes. The loss of load probability after x minutes is described in (4) and represents the probability of the imbalance after x minutes exceeding the balancing capacity that can be made available in x minutes: The imbalance is assumed to be drawn from a normal distribution with mean μ x and standard deviation σ x . These parameters can be estimated from the historical system imbalance. They are computed per 4-hour block and per season, in order to account for seasonality.
The settlement in a co-optimized market can be understood by analysing the following model in a convex setting. The model presented below is solely for the sake of explaining the settlement in system with multiple products of reserve. Assuming a benefit function for demand (B(·)) and a constant production cost for generator g ∈ G (C g ), where G is the set of generators in the system, our goal is to maximize the welfare of the system as a function of the demand (d), reserve available after 7.5 and 15 minutes (r 7.5 and r 15 ) 3 and the production (p g ) and supply of fast and slow reserve (r F g and r S g ) for every generator g: The co-optimization must obey the market clearing constraint for energy and fast and slow reserve (the associated dual variables are provided in brackets): (λ 15 ) : The operating constraints of generator g are characterised by the set X g : It is worth noting that the fast reserve supplied by the generators is eligible not only for the pool of reserve available after 7.5 minutes (7) but also for the pool of reserve available after 15 minutes (8).
The profit maximization problem faced by a generator g can be obtained by first relaxing the market clearing constraints of the co-optimization model (6)-(8): and then decomposing the relaxed problem by g ∈ G: From the generator point of view, λ, λ 7.5 and λ 15 are exogenous parameters representing respectively the price of energy, fast reserve, and slow reserve for the first and second interval of an imbalance period. From the system operator point of view, these prices are can be obtained by solving the initial co-optimization problem.
The prices of reserve for the first and second interval, λ 7.5 and λ 15 , are set by the ORDCs. This is demonstrated by the KKT conditions of the initial co-optimization problem relative to the complementarity constraints implicating variables r 7.5 and r 15 in (14) and (15): Generators are then remunerated according to (12) under cooptimization.
r Balancing capacity that can be made available in 7.5 minutes is remunerated with the fast reserve price: r Balancing capacity that can be made available in 15 minutes is remunerated with the slow reserve price: r Energy is remunerated in real time, with the energy price, λ.
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A notable challenge of integrating scarcity pricing in the EU and certain past US markets is the lack of co-optimization in the European market. Without co-optimization, a system operator cannot rely anymore on the dual variables of the co-optimization problem in order to characterize the prices and needs to resort to approximations. In the past, ERCOT has used adders as proxies to couple the reserve and energy prices, although the evolution of the ERCOT design is towards a co-optimization model [30].
In the spirit of the original ERCOT design, [28] suggests to introduce 3 adders in order to approximate pricing under cooptimization. The fast adder and slow adder would be equal to the fast and slow reserve price and would remunerate balancing capacity that can be made available in 7.5 and 15 minutes. The energy adder would be equal to the fast adder and would be added to the real-time balancing price for remunerating energy.
Assuming that the energy adder is equal to the fast adder is an approximation that presumes non-binding ramp constraints. The reader is referred to the supplement of [4] for a more detailed discussion about this approximation. The explanation can be summarized as follows. The price of energy is fixed by the KKT conditions of the initial co-optimization problem relative to the generation variables p g , r F g and r S g . Depending on which constraints of the model are binding, the energy price may be offset by a constant with respect to the fast adder, the slow adder, or neither.

C. Variants of ORDC
(2) and (3) depend on a number of design parameters. Different variants of ORDCs can be produced depending on these assumptions. In this paper, we consider the following design parameters, which have been discussed in the context of the implementation of scarcity pricing in Belgium: (i) different values for VOLL, (ii) whether the argument of the LOLP operator is the reserve capacity remaining before or after the activation of reserve and (iii) whether imbalance increments within an imbalance interval are assumed to be correlated or not.
1) VOLL At 8300 € /MWh Versus 13500 € /MWh: The Belgian federal planning bureau has estimated the Belgian VOLL at 8300 € /MWh in [31]. This value has been used as the reference value of the VOLL in [28]. The value of 13500 € /MWh has also been suggested because it represents the current bidding limit of the imbalance price [32] and as such the market players' assumed highest VOLL.
2) Pre-Versus Post-Activation: Reference [4] points out that the pre-and post-activation variants correspond to different interpretations of what making a certain quantity of reserve available in real time would mean in terms of system operator expectations. The pre-activation interpretation means that 1 MW of reserve implies that a resource has been afforded time to recover from its balancing dispatch during the previous imbalance interval. The post-activation interpretation means that the resource is prepared to offer 1 MW even if it has not been afforded time to return to its originally scheduled setpoint. The effect of the assumption is found to be significant in the context of the stochastic equilibrium formulation presented in [4]. As the time step of the real-time / balancing market becomes shorter If the post-activation reserve capacity margin is denoted as r, then the pre-activation margin is r − imb, with imb being the difference between the scheduled and actual demand. This allows us to value balancing capacity at the beginning of an interval before absorbing the imbalance.
3) Independent Versus Correlated Imbalance Increments: When splitting an imbalance periods into two intervals, the full imbalance that needs to be covered also has to be split into two imbalance increments. Each interval is then responsible for causing one of the two imbalance increments. The assumption then is on how imbalance increments correlate in these two separate time steps. On the one extreme, we can assume correlated increments which implies that the total imbalance over both stages evolves linearly from the beginning to the end of the interval. On the other extreme, the assumption of independent imbalances implies that the total imbalance over both stages is the sum of two independently distributed imbalance increments occurring at stages 1 and 2 respectively.
The distinction affects the implied standard deviation of the imbalance that is used in the 7.5-minute version of (4). Given σ and μ, the standard deviation and mean of the 15-minute imbalance, the standard deviation of the 7.5-minute increments is either σ/ √ 2 if the increments are independent, or σ/2 if the increments are perfectly correlated. The mean of both the independent and correlated 7.5-minute increments is μ/2. Fig. 1 presents different variants of ORDC. Each variant can be interpreted as a geometric transformation. The variation of VOLL, of the imbalance distribution, and of the activation time are respectively equivalent to vertical scaling, horizontal scaling and horizontal translation.

III. SIMULATOR FOR SHORT-TERM OPERATION
The structure of the simulator of short-term operation is described in Section III-A. We then characterize the generic unit commitment problem that is the building block of the different components of the simulator in Section III-B. The modifications to this generic unit commitment problem in order to account for the specificities of the intraday and real-time models are discussed in Section III-C.

A. Structure of the Simulators
The short-term operating model that we develop for the purpose of our analysis is composed of 4 embedded optimization problems that are solved in sequence throughout the day. Simulations begin in the day ahead by scheduling inelastic production, and unfold in intraday and real time by solving a sequence of unit commitment and economic dispatch problems with different scheduling windows. Each problem is employed for the commitment and / or dispatch of specific types of plants, depending on their response speed.
Particular care is given to (i) the operational constraints of the individually modelled generation plants, (ii) the revelation of real-time uncertainty and the scheduling of the system based on forecast information, and (iii) the effect of each decision-making stage on subsequent optimization problems.
Depending on the characteristics of an asset, its commitment plan and dispatch decisions will be obtained by different optimization problems. Assets can be partitioned into 3 broad categories, based on their real-time controllability.
1) DA scheduled generators cannot modify their planned day-ahead dispatch. This might be due to the inflexibility of the technology, or a link between electricity production and other processes, such as heating. The electricity production of these generators is typically determined in forward processes, and these units are not participating in a balancing market. 2) Fast balancing capacity generators require a nonnegligible lag to start up (between 1 and 3 hours) but are very reactive once committed. CCGT generators constitute the bulk of this category. 3) Slow balancing capacity generators include all emergency generators and demand response resources. These generators are typically costly to start up, but can be activated in a very short time, in order to free up some of the fast balancing capacity. 4 The 4 dispatch and commitment models that we develop optimize different subsets of the aforementioned assets. The dependencies are described hereunder. The sequencing of the models in the simulator is indicated in Fig. 2. r The day-ahead unit commitment (DA-UC) model is used for scheduling the inelastic production that will not vary in real time relative to its day-ahead set-point. The model is launched once, before the beginning of the day, with a scheduling horizon of 72 hours. The model assumes a fixed initial dispatch of units for the day, which will be identical for every variant of ORDC that is tested in our analysis. The parameters of the simulation include the day-ahead load forecast, as well as settings that determine the reactivity and availability of the generation pool. This problem also determines the hydro storage target for the real-time models. The system is allowed to deviate from this target in order to address balancing issues, but such deviations are penalized.  r The real-time economic dispatch (RT-ED) dispatches the generators that are committed in the previous optimization problems. All of these models are necessary for simulating efficient system operation. The DA-UC allows us to account for uncertainty in day-ahead scheduling. The Inter-RUC provides real-time flexibility on the commitment of CCGT units. The PRT-RUC alone would be insufficient for those units because they require a long scheduling window to be worth committing. The PRT-RUC allows the system operator to react to the very short-term revelation of uncertainty by activating emergency measures. The RT-ED is necessary for determining a final position for the system, for quantifying scarcity adders, and for quantifying the cost of operating the system.

B. Generic Unit Commitment Problem
All the optimization problems are based on a standard unit commitment problem that aims at minimizing the total cost of the system under a series of constraints for both classical and pumped hydro generation.

1) Sets, Variables and Parameters:
We define T = {t 1 , t 2 , . . . t T } as the scheduling window for a problem, D as the demand, G = {1, 2, . . . N} as the set of generators and S = {7.5, 15} as the type of ORDC considered in this analysis. Let us also denote V R it as the marginal benefit function of reserve type i at period t. The segments of the ORDC are obtained by approximating (2) and (3) with a stepwise constant function 5 .
The set of decisions concerning a generator g at period t is characterized by the point x g,t = (p g,t , r F g,t , r S g,t , r NS,F g,t , r NS,S g,t , u g,t , v g,t , w g,t , s g,t ). This vector is the concatenation of the production, fast reserve, slow reserve, fast non-spinning reserve, slow non-spinning reserve and binary variables for the commitment, activation, shut-down and start-up of generator g at time t. The vector x g,t belongs to the set X = R 5 + × B 4 . Each generator g is characterized by its technical parameters P + g , P − g , R g , R NS,F g , R NS,S g , UT g , DT g and SU g which are respectively the maximum and minimum production limit, the ramp rate, the limit for fast and slow non-spinning reserve, the minimum up time and down time, the start-up time of the unit, and its cost function C g : X → R. The cost function includes the minimum load cost of keeping a generator online, the start-up cost of starting it up, and the production cost with variable heat-rate. The cost function is described in the appendix.
We represent demand for energy and reserve using the vector m t = (z t , r 7.5 t , r 15 t ). This tuple consists of the shortage in energy and the system supply for reserve for both the first and second half of an imbalance interval interval for period t. , u H t ) represents the production, consumption, energy stored, fast and slow reserve supplied by pumped hydro, and the pumping mode of a pumped hydro unit for period t. Note that m t ∈ R 3 + and h t ∈ R 5 + × B 1 . We further introduce the notation t − to characterize the period preceding period t.
2) Objective Function: The system operator aims at minimizing the sum of the production cost and shortage cost minus the benefit from reserve: (18) The cost of shedding load is valued at the VOLL.

3) Market Clearing Constraints:
5 A method to approximate (1) by a stepwise constant function is proposed in [17].
Constraint (19) ensures that the market is balanced at all times, and constraints (20) and (21) define the market clearing conditions for fast and slow reserve. Reserve can be sourced from online generators (spinning reserve), from offline generators (non-spinning reserve) or from hydro (hydro reserve).

4) Generation Constraints:
Generation constraints represent the ramp constraints of fast and slow reserve and the maximum and minimum technical production limits of a unit.

5) Non-Spinning Reserve Constraints:
(26) and (27)  6) Transition Constraints: The transition constraints (28) and (29) represent the ramp constraint for production and the commitment transition constraint. The production ramp constraint has two possible modes: one for normal operation and one for activation. The parameter T is the number of minutes of one period in a specific problem. For example, T = 60 for the DA-UC problem. 7) Operating Constraints: The operating constraints (30), (31) and (32)

8) Hydro Generation Constraints:
The pumped hydro generation constraints restrict the maximum hydro consumption, energy stored and hydro production in constraints (33) . Note that a unit is either pumping or producing, as a function of the pumping mode u H t . Eq (36) restricts the hydro reserve to the total stored energy. Constraint (37) describes the evolution of energy stored in the reservoir as a function of pumping and production decisions in the previous period, as well as the efficiency of the plant.

C. Modifications for the Intraday and Real-Time Problems
The intraday problem (Inter-RUC) and real-time problems (PRT-RUC and RT-ED) extend the standard unit commitment problem described previously to account for the day-ahead schedule and the particularities of the balancing market.
1) Day-Ahead Constraints: Given the set G I representing the DA-scheduled generators, constraints (38) and (39) restrict their ability to supply reserve. Constraint (40) characterizes their real-time inflexibility by equalizing the real-time production to the scheduled day-ahead production p DA, * g,t .

2) Status Constraints:
The commitment decisions of previous optimization models are enforced by this set of constraints. We introduce the notation S U,t , S S,t , S A,t and S F,t to represent the set of generators that are unavailable, in the start-up process, activated or free in period t. 6 u g,t = 0 ∀t ∈ T , ∀g ∈ S U,t ∪ S S,t (41) Constraints (41) and (42) imply that a generator is either off (u g,t = 0) or on (u g,t = 1) because of the minimum down time and minimum up time constraints of previous optimization problems. Similarly, (43) enforces the start-up variables dictated by a start-up decision in a previous problem and its start-up time. 6 No generator can be in two sets at the same time and all generators must be in a set at every period t.

3) Hydro-Deviation Constraints:
The opportunity cost of hydro in real time is modelled by a hydro storage target e DA, * t and the variable q t representing the deviation from that target: The term t∈T q t 0 HD(q)dq is subtracted from the objective function of the real-time problem in order to penalize deviations.

4) Start-Up Constraints:
The scheduling window for the prereal-time problem is defined as T = {t 0,0 , t 0,1 , t 1 , . . . , t w−1 }. It represents two 7.5-minute periods (t 0,0 and t 0,1 ) and w − 1 15-minute periods. This window plans over w · · · 15 minutes. The first period is split, in order to account for the start-up profile of emergency generators and how much of their generation is available after 7.5 minutes. Given the initial position of a generator p 0 g , the initial transition constraint of (28) needs to be reformulated: The start-up constraints (47) and (48) ensure that generators comply with their start-up profile. This start-up profile is characterized by the maximum production 7.5 minutes and 15 minutes after activation (R SU,0 g and R SU,1 g ). The start-up ramp profile is similar to the limit on non-spinning reserve for emergency generators and demand response.
Note that the two 7.5-minute dispatch periods only account for one 15-minute commitment period (49)-(50).

5) Economic Dispatch Constraints:
The real-time economic dispatch is similar to the first and second period of the pre-realtime unit commitment except that all the generators are either activated, unavailable or in start-up. No commitment decision is taken in this problem.

IV. MODEL VALIDATION AGAINST HISTORICAL DATA
SubSection IV-A describes the case study of the simulation and Section IV-B the validation of the day-ahead module.

A. Case Study
The investigation is performed on the Belgian power system with the historical load of 2018.
1) Generation Pool: The generation pool modelled in the simulator includes all the controllable assets of Belgium and the load flexibility considered by the Belgian transmission system operator, Elia, and is presented in Table I.
The controllable assets are mainly based on the database of installed capacity by unit, which is publicly available on the Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  [33]. In addition to these resources, this pool also includes demand response, pumped-hydro and some foreign balancing capacity 7 . Demand response varies from month to month and is extracted from historical data. Foreign balancing capacity is estimated at 50 MW.
The technical parameters relative to the operating constraints of the flexible generators are largely aligned with [34], except for the minimum production of emergency generators. We consider emergency generators and demand response as "all-or-nothing" generators.
2) Net Load: Net load is modelled as the power that must be served by flexible and controllable assets. It corresponds to the difference between grid load and renewable energy and imports / exports.
The data that we use for net load is obtained from ELIA [35] and the ENTSO-E transparency platform [36]. The data resolution of the ELIA website and ENTSO-E platform is respectively 15-minute and hourly.
3) Imbalance: The mean and standard deviation of the distribution of the imbalances used for the LOLP in (4) is obtained from the historical system imbalance recorded in [35]. For simulating 2018, we use the historical imbalance of the 3 preceding years. The full characterization of the mean and standard deviation of the system imbalance can be found in the appendix.

B. Validation
We validate our model by assessing the quality of the forward position computed by the day-ahead unit commitment compared to the historical records of day-ahead positions. The validation is restricted to the day-ahead unit commitment because the cooptimization of reserve and energy in real time in our simulator is a closed-loop investigation that is expected to produce a different dispatch depending on the ORDC that we analyse.
The comparison is performed over the aggregated forecast production per type of fuel. There are five types of fuel, namely 7 Demand response and foreign balancing capacity are referred to as Non-CIPU generation and Inter-TSO in the Belgian framework. (i) nuclear, (ii) gas, (iii) hydro, (iv) liquid fuel and (v) other. We will mostly focus on gas and hydro production. Nuclear and other technologies are mainly driven by the maximum available output and liquid fuel is used as an emergency measure and is rarely scheduled in the day ahead. The performance of the simulator is compared to that of [10]. Whereas [10] aims at computing scarcity prices and does not perform a sensitivity analysis on the shape of the ORDC, both the present model and that of [10] are calibrated against the Belgian system. The current work improves [10] by (i) refining and extending the generation pool, (ii) reducing the granularity of the dispatch, and (iii) proposing a more realistic modeling of the dispatch and commitment decisions. These enhancements allow us to analyse the tradeoff between the commitment of fast balancing capacity and the cost of operating the system with more realism. Table II details this comparison and demonstrates that the increased modeling detail does not come at the cost of accuracy in replicating past observations of the Belgian electricity system. Discrepancies between our simulations and the historical dispatch may originate from: (i) The difficulty of assessing the opportunity cost of pumped-hydro and thermal assets that belong to a portfolio, (ii) the impact of climatic conditions on the availability of certain assets and (iii) difference in the commitment of particular CCGTs between the simulated and historical dispatch.

V. RESULTS AND ANALYSIS
The results presented in this section are obtained by simulating the historical demand of Belgium for 2018. The analysis is based on a reference scenario and on a sensitivity analysis on the availability of the slow balancing capacity for contributing towards satisfying the demand of the 7.5-minute ORDC.
Our analysis focuses on comparing the total operating cost of the different variants of ORDCs (Section V-A), and on analyzing the impact of these variants on the level of the scarcity adder (Section V-B). The comparison focuses largely on the influence of the level of conservatism of the variants. More conservative variants (value of lost load at 13500 € /MWh and / or independent 7.5-minute imbalance increments) are compared against less conservative variants (value of lost load at 8300 € /MWh and / or correlated 7.5-minute imbalance increments). Some of the slow balancing capacity is assumed to partly cover the demand of the 7.5 minute ORDC and [15] uncovers the importance of this assumption. We investigate further in this direction with the sensitivity analysis that is performed in Section V-C. Note that the reference scenario assumes an availability of 28%.

A. Cost Analysis of the Reference Scenario
The total cost of the variants is reported in Table III. The values reported here are obtained by adding the production cost, minimum load cost, activation cost and shortage cost of the system, and do not include the cost of the price-inelastic generators, since the latter is identical across different scenarios. The last columns report the cost of emergency measures. The total cost varies from 1.697 M € per day to 1.683 M € per day. Thus, we find a difference of up to 14 k € per day between the different variants. This corresponds to a variation of up to 0.8% of the mean total flexible cost, which can be considered quite stable.
Despite the stability of the total cost, we can analyse the differences between the variants in order to better understand the impact of the ORDC on the commitment and dispatch decisions. Based on Table III, we can observe that more conservative variants are typically less costly. This trend is more accentuated for the variation of the VOLL, where the 13500 variants are consistently lower in cost than their 8300 counterpart. More conservative ORDCs tend to result in higher minimum load costs, and this is balanced out by their lower production cost. This is particularly the case when comparing the variants with different value of lost load. For instance, the 13500/Postactivation/Independent variant incurs 352,000 € of minimum load cost and 1,304,000 € of production cost, compared to the 8300/Post-activation/Independent that incurs 343,000 € of minimum load cost cost and 1,316,000 € of production cost.

B. Price Analysis of the Reference Scenario
The values of the adders that result from the different variants under the reference scenario are presented in Table IV. The fast adder varies from 2.7 € /MWh to 6.5 € /MWh and the slow adder Firstly, conservative ORDCs (13500 variants and Independent variants) produce higher adders than their counterparts. Note that the most significant difference is caused by the distribution of the 7.5-minute imbalance increments, with the independent variants producing fast reserve adders that are approximately twice the value of their counterparts.
Secondly, correlated variants result in a higher slow reserve adder. This is driven by the fact that CCGTs have lower incentives for commitment, which decreases the committed balancing capacity and increases the value of the slow adder.
This last point highlights a fundamental difference between the variations in terms of distribution of imbalance increments versus the variations of the VOLL. The independent and correlated variants only impact the 7.5-minute ORDC and increase or decrease the incentives for committing CCGTs, while keeping the slow reserve demand constant. In comparison, variations of the VOLL impact both the 7.5-minute and 15-minute ORDCs. Fig. 3 provides an indication about the persistence of the price signal generated by the ORDC in terms of profitability for owners of flexible assets. The figure compares the price  signal obtained by 4 variants, beginning with the most conservative variant that produces the highest adder (13500/Postactivation/Independent) and modifying each of the design parameters in turn. The y-axis displays the measure of the adder under the metric of conditional value at risk as a function of the risk aversion of the agents on the x-axis. Depending on the risk aversion α of an agent, the agent will only consider the 100 − α worst adders for computing its expected payoffs from the adder. The risk aversion can range from 0% to 100%, where 0% is a completely risk-neutral agent and 100% is a completely risk-averse one. We observe a notable drop in the value of the payoff curve for low values of the x axis, which corresponds to the impact of a very high adder resulting from very stressed conditions in the system. These highly stressed conditions constitute less than 1% of the total possible outcomes in the system. It is possible to assess the quality of the signal produced by a variant by analysing the persistence of the adder when the risk aversion increases.
In Fig. 3 we observe that the correlated variant is the least persistent by a wide margin. The variants related to the value of lost load and the pre/post-activation capacity produce similar levels of persistence. Note that, for these variants, the decrease can be considered constant until a risk aversion level of 7.5%, which indicates a mean adder that is generated by the repetition of a large number of occurrences of small adders in the market, which is desirable from the perspective of mitigating investment risk. Fig. 4 presents the mean fast adder per day for 2018. It shows that the mean price per day is between 0 and 10 € /MWh for the majority of the time, while during approximately 50 days the average adder is higher than 10 € /MWh. Scarcity adders turn out to be zero between 75 and 86% of the time in the simulations depending on the variant.

C. Sensitivity Analysis for the Variation of the Availability of Slow Balancing Capacity for the 7.5-Minute ORDC
The eligibility of slow balancing capacity for the 7.5-minute ORDC is an important determinant for the remuneration of flexible assets. In principle, the eligibility of these resources should be plant-dependent and should reflect as accurately as possible their reactivity. For reasons of simplicity, the current Belgian scarcity pricing proposal only assumes a generic value for the availability. ELIA [15] assumes an availability of 50%, and our analysis considers availabilities ranging from 0% to 28%.
Given an availability ρ, the parameters R SU,0 g in (47) and R NS,F g in (26) for the emergency generators and demand response are modified as follows: The effects of modifying the availability on the adder are twofold, and can be observed in Table V. Increasing ρ (i) reduces notably the level of the fast adder by increasing the fast balancing capacity pool, and (ii) increases marginally the level of the slow reserve adder. Increasing the availability of mFRR for covering the demand of the 7.5-minute ORDC reduces the need for aFRR from CCGTs, and has a direct effect on their commitment. This compresses the committed balancing capacity, which in turn increases the level of the slow adder.

VI. CONCLUSION
We develop a detailed unit commitment and economic dispatch simulation model of the Belgian power system in order to analyze the effect of different design choices for Operating Reserve Demand Curves on the cost of system operation and the price of aFRR and mFRR capacity. Our simulator attempts to emulate a best-case, fully coordinated operation of the system from the day ahead to real time. We propose four modules that are interleaved and implemented as a rolling horizon optimization.
The precision of our model allows us to account for the tension between incurring fixed costs for committing flexible resources that can allow the system to operate reliably in real time, versus running the risk of shedding load. We validate our model against historically observed data of the Belgian market for 2018. We then perform a case study on the impact of ORDCs on scarcity prices and system costs for 2018. We also perform a sensitivity analysis on the extent to which mFRR reserves are assumed to contribute towards satisfying the demand for 7.5-minute reserves.
The main findings of our analysis can be summarized as follows: 1) The total flexible operating cost for a day is stable, regardless of the chosen ORDC variant. It is also stable for the specific generation pool of Belgium that is investigated in our work.
2) The fast adder varies from 2.8 € /MWh to 6.5 € /MWh in the reference scenario. The main driver of the price is the assumption related to the distribution of the 7.5-minute imbalance increments, followed by the value of lost load.

3) The level of the fast adder is sensitive to assumptions
about what resources can contribute towards covering the demand of the 7.5-minute ORDC. Note that this sensitivity was already reported in [15]. Note that, even though the methodology is generic, the conclusions regarding shape of the ORDC are system-dependent and cannot be generalized to other systems.
In future work, we are interested in developing a Monte Carlo simulation model for the Belgian system which draws samples of system uncertainty, instead of relying on historical data. This would allow us to enhance the statistical reliability of our results, by exposing the system to multiple years of hypothetical operation.

APPENDIX SEASONAL IMBALANCE
The parameters of the ORDCs, which are extracted from the historical seasonal imbalances, can be found in Table VI. Block 1 is 10 pm to 2 am, block 2 is 2 am to 6 am, block 3 is 6 am to 10 am, block 4 is 10 am to 2 pm, block 5 is 2 pm to 6 pm and block 6 is 6 pm to 10 pm.

Cost Function
The cost function of generators with variable outputs and variable heat rates can be expressed as in (52): C g (p, u, v) = C P (p) + C ML (u) + C SU (v).
The composition of the cost function is similar to the threepart bids of ERCOT with a production cost, a minimum load cost and a start-up cost.
1) The production cost is a function of production. It includes the emission cost, which depends on the emission price of a ton of CO 2 (C C0 2 ) and the emission rate of the plant (ER g ). It also depends on the production cost in function of the price of the fuel (C F g ) and the variable heat rate as a function of the power output (HR g (p)).
C P (p) = p 0 C CO 2 · ER g + C F g · HR g (x)dx (53) 2) The minimum load cost is a function of the commitment variable u and the minimum load of the plant (Q ML g ) and the price of the fuel.
3) The start-up cost is a function of the activation variable v and the start-up load of the plant (Q SU g ), as well as the price of the fuel.
The plant parameters are extrapolated from the real parameters of a subset of all the plants from a private database. This extrapolation is performed in order to adapt our realistic data to capacity levels that are not explicitly represented in our database. The fuel cost is based on the monthly historical spot TTF prices [37]. The heat rate is piecewise constant.