Economic Model of Ancillary Services in Real Time for Frequency Control

Abstract

Modern power systems integrate ancillary services (ASs) to provide security and quality of service in real-time operation (RTO) due to the intense frequency variations caused by the uncertainty of solar–wind generation. To this end, the ancillary services market focuses on power reserves for secondary and tertiary frequency control. The adjustment and dispatch of reserves from plants are manual instructions executed by the system operator in order to maintain the frequency within the normal operating range (49.80 ≤ f ≤ 50.20 Hz). This work proposes a methodology for the economic modeling of the ancillary services market in real time using a dynamic hourly mathematical model that integrates the variability of solar–wind generation, a demand monitoring curve, and the trajectory of marginal cost (MgC). This is a segmented methodology in which plants with costs close to the marginal cost are identified in real time using the Supramarginal (SMg) and Inframarginal (IMg) methods. Finally, this economic model for real-time power reserve reallocation represents an innovation in the ancillary services market because its results allow for the operation costs (OC) of the reserves to be reduced by up to 60% and for the displacement of marginal costs to be reduced by 10 to 40% with respect to traditional methodologies such as the economic merit list and the technical minimum methodology, which cause plants to operate without economic justification.

1. Introduction and Literature Review to the Ancillary Services Market

The energy transition and the introduction of environmental policies in response to climate change are aimed towards achieving carbon neutrality, which presumes the definitive retirement of coal-fired thermal power plants in favor of the development of an emissions-free energy matrix in the short and medium term [1]. The energy transition challenge is focused on the development of modern technologies, such as solar and wind power, to replace fossil fuel thermal generation. Chile has been developing its solar and wind generation for over a decade, and the penetration of renewable energies reached more than 40% of its total generation in 2022, making it a world leader as the country with the most significant potential for and the most extensive development of solar generation [2].

However, the full implementation of energy transition is linked to the energy market, which aims to supply demand using new economical and emissions-free generation technologies. The problem lies in the decoupling of thermal generation from and the integration of renewable generation into this new ancillary services (ASs) market. This ancillary services market was designed to offer security and quality of service to electric power systems for the same reasons that have afforded conventional hydrothermal generation—a monopolistic participation in the energy market: the great inertial capacity and the immediate stability that it has in comparison to solar–wind generation in the event of failures in the electric power system.

The ramifications of renewable energies present an opportunity for ancillary services that offer the same conditions in terms of technical and economic competency delivered by conventional generation. However, there are high levels of uncertainty with respect to renewable generation, demand, and system failures, causing severe frequency deviations that can force system operators to apply power reserves arbitrarily to mitigate frequency instability. Therefore, this inefficient mechanism of arbitrary use of power reserves with unjustified dispatches from power plants translates to additional costs to the market, making the overall operation of the system more expensive. Frequency control is the most critical variable in the ancillary services market. It is responsible for ensuring system stability through the action of three categories of operation, in the following order: primary frequency control (PFC); secondary frequency control (SFC); and tertiary frequency control (TFC). Therefore, frequency control depends on those power plants that are able to deliver power reserves in the shortest response time ramping their generation up and down [3]. However, the market for ancillary services must be able to incentivize and integrate conventional and renewable generation through an economic model that is able to control the frequency in real time while monetizing investment costs, implementing control design, and avoiding arbitrary and unjustified dispatches from plants [4,5].

In both Latin America and Chile, there are unlimited resources for the development of renewable energies from solar and wind sources, allowing for the delivery of large capacities of power reserves for secondary and tertiary frequency control, as shown in Figure 1, where the annual evolution of the total capacity of the power reserves is depicted for renewable resources in Chile that have not yet been fully allocated to frequency control.

The challenge of this market is the development of technical and economic mechanisms for the distribution of these power reserves to realize frequency control in real-time operation, enabling the balancing of the energy market as well as the market for ancillary services. Therefore, the objective of this work is to develop and implement a real-time economic model for the ancillary services market that allows the system operator to optimally reallocate power reserves for secondary and tertiary frequency control, the results of which will reduce the operation costs and the marginal costs of the system.

Table 1. Nomenclature and definition of variables.

Abbreviation	Definition
AGC	Automatic generation control
AS	Ancillary services
Dx	Demand curve
EML	Economic merit list
IMg	Inframarginal
MgC	Marginal cost
OC	Operation cost
PFC	Primary frequency control
RTO	Real-time operation
SFC	Secondary frequency control
SMg	Supramarginal
TFC	Tertiary frequency control
TM	Technical minimum
TSO	Transmission system operators
VC	Variable cost of generation
Variables	Definition
$CP$	Start-up cost
$HOpe$	Operating hours
$Res$	Power reserve
${TC}_t^{SFC}$	Total cost of the power reserve for secondary frequency control
${TC}_{n,t}^{SFC\text{-}SMg}$	Costs of power reserves for secondary frequency control of Supramarginal
${TC}_{n,t}^{SFC\text{-}IMg}$	Costs of power reserves for secondary frequency control of Inframarginal
${TC}_t^{TFC,up}$	Total cost of the power reserve reallocations for up generation of the tertiary frequency control
${TC}_{n,t}^{TFC,up\text{-}SMg}$	Cost of power reserves for up generation Supramarginal of tertiary frequency control
${TC}_{n,t}^{TFC,up\text{-}IMg}$	Cost of power reserves for up generation Inframarginal of tertiary frequency control
${TC}_t^{TFC,down}$	Total cost of the power reserve reallocations for down generation of the tertiary frequency control
${TC}_{n,t}^{TFC,down\text{-}SMg}$	Cost of power reserves for down generation Supramarginal of tertiary frequency control
${TC}_{n,t}^{TFC,down\text{-}IMg}$	Cost of power reserves for down generation Inframarginal of tertiary frequency control

describes the nomenclature and definition of variables used in this article.

1.1. State of the Art Economic Models for Real-Time Frequency Control

The beginnings of the ancillary services markets were always linked to conventional hydrothermal generation without market rules for its use. The massive entry of solar photovoltaic and wind generation requires participation in this ancillary services market to displace thermal generation completely, which is highly polluting. This is the case in China, which is developing an environmental regulation and corporate green technology innovation policy that helps identify new drivers of corporate green technology innovation and confirms the combined effects of the new media environment and environmental regulation in stimulating corporate green behavior, which can facilitate the construction of an ecological civilization [6]. Otherwise, protocols that include financial sanctions should be applied, as is the case with Russia, but also on the global climate schedule, which is known to reflect the global sustainable development strategy [7]. Chile, for example, is one of the world leaders in developing solar generation and has reduced its emissions exponentially (Figure 2). However, in countries with higher energy development, the ancillary services market should allow liberalized business models for renewable generation with options to choose different operators for coordination and management [8]. In addition, a liberalized energy market model allows for the integration of mechanisms for remuneration, investment, energy sales, operating costs, and even economic incentives for power reserves for frequency control [8]. Next, this topic’s most relevant bibliographic selection for this proposal is presented, showing the state-of-the-art works associated with the ancillary services market that use power reserves for real-time frequency control.

1.1.1. Bidding and Auction Market for Frequency Control Power Reserves

The availability of renewable generation implies modifications to the energy market rules to integrate its participation in ancillary services [9]. The ancillary services market should be designed with economically attractive bidding and an auction model to decouple renewable generation that can deliver power reserves for frequency control from the energy market.

In Europe, transmission system operators (TSOs) reduce costs and optimize reserves for the SFC in real-time operation, with bidding models and hourly auctions, as is the case of the Iberian Electricity System of Portugal [10]. Also, the TSOs define the market contracts in three operational aspects for power reserves (capacity, allocation, and activation) to avoid deserted auctions when no generation is available [11]. In Portugal, Spain, and France, the bids and auctions of power reserves for frequency control are traded in EUR/MW, involving three transmission system operators and the English TSO of the National Grid [12,13]. Although the bidding and auction model is efficient in a programmed systemic scenario, in real-time, it lacks flexibility because the bid values are static for each plant and do not change over time, causing an operating cost overrun, thereby always leaving the option to those plants that monopolize the bids and auctions in the programmed operation.

These ancillary service models operate before dispatch. The prices in the submitted bids are fixed, and participants can change the energy volumes in their bid between 1 and 5 min before the scheduled dispatch interval begins [14]. Like the pre-dispatch problem, the real-time ancillary service market is complex to implement due to its size and human and technological efforts [15] because variable bids are received every couple of minutes, and the market is settled accordingly to obtain the price and quantity values every 5 min before the dispatch and retirement of the plants [16]. This bidding and auction mechanism for real-time operation demonstrates that power reserve adjustments can be made because of system deviations that limit the resources to perform frequency control. The limitations of this methodology are associated with the fact that the system operator’s technological and human efforts are subjected to high stress levels when testing real-time scenarios with reallocations of less than 5 min. This extreme methodology of the real-time availability of bids and auctions is practical if an oversizing of ancillary services is applied to instantly adjust the frequency control reserves and avoid the stress of real-time operation. This would immediately cause an operation cost overrun and erroneous decisions by the system operators.

1.1.2. Mechanisms and Economic Models to Reallocate Power Reserves in Real Time

The complex search for economic mechanisms to theoretically support the power reserve reallocations for real-time frequency control, especially with specific cases in secondary and tertiary frequency control, allows for the relating of the problem to other models with some similarities in their solution. For example, game theory algorithms are based on the opportunity cost arising between energy payments and bid manipulation [17]. This game theory methodology applies to those plants dedicated to the sale of energy in a spot market when they are not governed by the marginalist theory, i.e., in a power market based on auctions or power exchanges. However, this method is slightly similar to the ancillary services market in definitions of opportunity cost based on the fixed offer of power reserves. Its primary deficit is the absence of an economic mechanism in real time to encourage those plants that are generating close to the marginal cost and dedicate part of their power in reserves for frequency control.

Other similar market techniques perform an optimized allocation to distribute power reserves with a commercial categorization obtained through long-term contracts and regular reserves purchased with daily or hourly auctions with frequency regulation market prices [18]. Also, ERCOT, California ISO, and Midcontinent ISO systems present an optimal model not in real-time but executing a co-optimization of reserves at different frequency controls with varied time scales (0.5 s, 10 s, 10 min, 15 min, and 30 min) [19]. The optimization of reserves in scheduled operations performed by system operators is efficient and optimizes the generation of resources dedicated to providing power reserves. However, adapting it to a real-time model is complex due to the size and operational effort involved. Even this mechanism challenges the flexibility of the generation, and the technical restrictions of switching on and off increase, causing an increase in the significant maintenance of the power plants and an extra cost for purchasing unscheduled energy in the spot market.

Other authors optimize generation resources by combining secondary and tertiary frequency control reserve reallocations for the same plant, using an economic dispatch solved by the lambda method that satisfies the supply demand balance [20]. This method also solves the real-time power reserve reallocation by running a distributed economic dispatch by linear programming with Wasserstein’s metric-based robust distribution optimization technique [21]. This methodology is excellent for reallocating power reserves in real time since it uses the theoretical foundations of economic dispatch to find the lowest cost of reserve reallocation. The problem with this model is its static structure that does not perceive the changes of the new power plants that can acquire the reserve reallocations through an opportunity cost that can follow the marginal cost. This instantaneous method partially solves the reserve changes for frequency control in real time. However, it still lacks improvements that guarantee its optimization in time.

1.1.3. Model of Monopolistic Structure of Power Reserves for Frequency Control

Several transmission system operators apply the use of reserves for frequency control, being influenced by generator eligibility rules. This methodology causes energy prices to increase and each power plant’s revenues to be affected, even for units not actively providing reserves [22]. Currently, there are models for ancillary services markets that avoid generator eligibility rules, such as the agent-based modeling framework, which includes design parameters and strategies to the market operator in real time, such as clearing algorithms for the day-ahead market, intraday dispatch, and redispatch [23]. That is, when the conditions of competition in a market equilibrium are not met, generation companies tend to have a monopolistic structure in providing their services. The problem in this methodology is the reallocation of power reserves for frequency control in those plants of the same company that modify the scenarios in real time in the energy market to be awarded the ancillary services and receive the maximum profits to the detriment of other plants without the possibility of awarding. Thus, monitoring competition in real time is essential to stop this type of action and has the mission to regulate the energy market and the ancillary services.

Linear optimization can mitigate this problem by calculating the optimal daily and intraday market bids through a real-time predictive control model for uncertain reserve auctions due to the energy market’s priority [24]. In ref. [25], a real-time integral post-operation method is proposed that minimizes the gap between the net payments of power plants and a penalty factor at different time scales (5 s, 15 s, up to 60 s) when they do not meet the market bids at the time of allocating frequency regulation. Both methods are effective when applied to supply the demand, and the minimum hourly reserves are not met in the scheduled dispatch. In the case of ancillary services in real-time, their implementation becomes more complex because the specific calculation of costs or economic gaps between the reallocation of reserves in less than one minute becomes impracticable due to the system’s dynamics.

Classical pre-dispatch models with security constraints are also used with power reserve for frequency control and then the co-optimization of energy with power reserve for ancillary services in the 24 h day-ahead market schedule, and then migrate to a 15 min or hourly real-time redispatch model using marginal busbar costs [26]. This real-time model applies a mechanism for reallocating reserves through direct instruction by the system operator based solely on availability, capacity, and the time to provide reserves with plants out of economic order.

The costs involved in this direct instruction mechanism are reflected in the global operation of the system due to the incense burner dephasing of the marginal cost and the excess of plants operating at the technical minimum with ancillary services without justification, increasing the operating cost of supplying the demand.

1.1.4. Efficient Real-Time Power Reserve Reallocation Model

Implementing economic mechanisms dedicated to using the power reserves in frequency control is complex. For generation plants, remuneration and profitability are essential to enter this ancillary services market. Currently, there are models based on offers and auctions to establish the power reserves for frequency control. However, most reserve reallocation models have a static profile in the prices set by bids and auctions. This mechanism satisfactorily fulfills the reallocation of power reserves but does not follow a dynamic price metric that evidences a minimum total cost of the reserves in the system.

Unlike Chile, Latin American power systems such as Peru, Argentina, Colombia, and others still need to develop a fully developed ancillary services market to unify power reserves for frequency control and integrate renewable generation. Although a broad scientific field exists in ancillary services markets, the problem still needs to be fully solved. Therefore, this work proposes a model adapted to the ancillary services market for frequency control that overcomes the traditional auction and hourly bidding methods with modeling in non-real scheduling scenarios. This model is a dynamic method that allows for the minimization of the total hourly costs of power reserve reallocations of the candidate plants called Supramarginal (SMg) and Inframarginal (IMg) for secondary and tertiary frequency control, which depend exclusively on the marginal cost path (MgC), demand trend and the opportunity cost presented to the awarded plants guaranteeing the minimum cost of system operation.

2. Theoretical Framework and Methodology of a Real-Time Reserve Allocation Model for Frequency Control

The theoretical framework defines the structure and technical conditions for applying the power reserves of power plants intended to raise or lower generation in the ancillary services market for secondary and tertiary frequency control. This theoretical scope governs the proposed economic methodology and the validation of the model using real case studies, intending to respect the margins of symmetrical, asymmetrical power reserves, activation times, delivery times, and power ramps.

2.1. Secondary Frequency Control Symmetrical Reserves

The ancillary service for secondary frequency control, in the future SFC(±), corresponds to the capacity of power reserves to raise or lower generation to maintain the frequency in a normal operating state from 49.80 to 50.20 Hz. This frequency control operates in manual mode or automatic generation control (AGC). These SFC(±) reserves are symmetrical, with a value of ±120 MW during their use and a power ramp ranging between ±24 MW/min, as shown in Figure 3. In addition, this ancillary service is governed by a total activation time of 5 min and a minimum delivery time of 15 min, as shown in Figure 4.

The economic mechanism of this ancillary service is subdivided into two categories of operation, the first case being the secondary frequency control to increase generation, hereinafter SFC(+). According to

Table 2. List of economic merit and priority of TFC/down.

Gx-n	SFC(±) (MW)	TFC(±) (MW)	Type	CMg (USD/MWh)	TFC(+) Ranking	TFC(−) (USD/MW)
G-1	100	140	Solar	0	-	43.4
G-2	90	90	Solar	0	-	43.4
G-3	60	80	Wind	0	-	43.4
G-4	28	45	Gas	28.6	-	49.0
G-5	14	20	Coal	28.7	-	-
G-6	14	20	Coal	28.7	-	-
G-7	35	15	Coal	29.2	-	-
G-8	8	15	Coal	29.2	-	-
G-9	8	0	Coal	29.8	-	-
G-10	30	0	Coal	30.2	-	-
G-11	8	0	Coal	31.4	-	-
G-12	8	0	Coal	31.7	-	-
G-13	35	60	Gas	32.4	-	-
G-14	30	40	Gas	35.2	-	41.4
G-15	30	40	Gas	38.7	-	41.6
G-16	100	100	Hydro	40.9	1°	-
G-17	5	20	Coal	44.5	-	-
G-18	100	100	Hydro	47.2	2°	-
G-19	40	40	Gas	77.0	3°	-

, the increase in the generation of this ancillary service SFC(+) is in increasing order of the variable costs of the plants that were awarded from the economic merit list. The second case is the secondary frequency control to lower generation, hereinafter SFC(−). The decrease in generation of this ancillary service SFC(−) is in increasing order of the values offered by the plants that were awarded from the priority list of activation of lowering of generation, as shown in Table 2.

2.2. Asymmetrical Tertiary Frequency Control Reserves

The ancillary service for tertiary frequency control, called TFC(±), corresponds to the spinning or cold power reserves for raising or lowering generation. This ancillary service acts in an emergency state when the system frequency oscillates between ±0.7 Hz concerning its nominal value of 50.00 Hz. The purpose of the TFC(±) is to restore the frequency when the power reserves of the primary and secondary frequency control are depleted. The operation of the tertiary frequency control is manual. The TFC(±) reserves are asymmetrical, with values from ±150 to ±320 MW in different operating time blocks, as shown in Figure 3. In addition, this ancillary service is governed by a maximum synchronization time between 5 and 15 min and a total delivery time of one hour, as shown in Figure 4.

The economic mechanism of this ancillary service is subdivided into two categories of operation, the first case being the tertiary frequency control to increase generation, hereinafter TFC(+). The increase in the generation of this ancillary service TFC(+) is in increasing order of the variable costs of the plants that were awarded from the economic merit list, according to Table 2. The second case is the tertiary frequency control to lower generation, hereinafter TFC(−). The decrease in the generation of this ancillary service TFC(−) is in increasing order of the values offered by the plants that were awarded from the priority list of the activation of the lowering of generation, as shown in Table 2.

2.3. Methodology of a Real-Time Reserve Allocation Model for Frequency Control

The methodology develops an hourly dynamic mathematical model that minimizes the total cost of power reserves for the ancillary services market, using reallocations of power reserves of those plants enabled for real-time secondary and tertiary frequency control. The model uses the available plants for real-time operation ordered from the lowest to the highest variable generation cost. The selection of each plant is categorized as Supramarginal when the variable cost of generation is greater than the marginal cost of the system, and Inframarginal if the variable cost of generation is less than the marginal cost. Therefore, the marginal cost of the system is a consequence of the dispatch of plants due to the variation in demand.

This methodology allows for the hourly optimization of reserve costs in real time based on marginal cost (MgC) and demand curve (Dx), as opposed to the current static and inefficient methods of power reserve reallocations, which do not indicate actual reserve costs. Firstly, it is necessary to detect whether a failure in the generation system causes power reserve deficits in the ancillary service market in real time. Secondly, it is required to identify whether the missing power reserves deficit corresponds to the SCF(±) and the TFC(±), or both cases, respectively. Thirdly, the candidate plants SMg and IMg close to the MgC in real time must be selected. Fourth, the reserve capacity of each candidate plant SMg and IMg is accounted for separately until the power reserve deficit is covered. Fifth, the total cost of reserves for the SCF(±) and TCF(±) of the SMg and IMg candidate plants must be minimized to execute the reallocation of missing reserves in real-time operation. Finally, if the demand curve and marginal cost trajectory shift, a new reallocation of power reserves must be performed, as shown in Figure 5 by a flow chart and in Figure 6 by a time plot.

2.4. Mathematical Method for Reallocation Secondary Frequency Control Reserves

Mathematically, the methodology is a real-time economic model that minimizes hourly the total cost of power reserve reallocations for the new candidate plants selected as Supramarginal and Inframarginal for secondary frequency control to step-up and step-down generation as indicated in Equations (1)–(3).

$${\mathrm{TC}}_{\mathrm{t}}^{\mathrm{SFC}}=\min {\displaystyle \sum _{\mathrm{n}\in \mathrm{N}}}\left({\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{SFC}\text{-}\mathrm{SMg}}+{\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{SFC}\text{-}\mathrm{IMg}}\right)\forall \mathrm{t}\in \mathrm{T}$$

(1)

$${\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{SFC}\text{-}\mathrm{SMg}}=\left({\mathrm{VC}}_{\mathrm{n}}-\mathrm{M}{\mathrm{gC}}_{\mathrm{t}}\right)·{\mathrm{Res}}^{\left(-\right)}·\mathrm{HOpe}$$

(2)

$${\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{SFC}\text{-}\mathrm{IMg}}=\left(\mathrm{M}{\mathrm{gC}}_{\mathrm{t}}-{\mathrm{VC}}_{\mathrm{n}}\right)·{\mathrm{Res}}^{(+)}·\mathrm{HOpe}$$

(3)

The objective Equation (1) represents the minimization of the total cost ${\mathrm{TC}}_{\mathrm{t}}^{\mathrm{SFC}}$ of the power reserve for secondary frequency control for the $\mathrm{N}$ power plants named SMg and IMg in an hourly period $\mathrm{t}$. Equation (2) corresponds to the costs of power reserves ${\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{SFC}\text{-}\mathrm{SMg}}$ for secondary frequency control of Supramarginal power plants. Equation (3) represents the cost of power reserves ${\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{CSF}\text{-}\mathrm{IMg}}$ of secondary frequency control of Inframarginal power plants. Finally, Equations (2) and (3) depend on the hourly marginal cost $\mathrm{M}{\mathrm{gC}}_{\mathrm{t}}$, variable cost ${\mathrm{VC}}_{\mathrm{n}}$, power reserve amount ${\mathrm{Res}}^{(-)}$ or ${\mathrm{Res}}^{(+)}$, and operating hours $\mathrm{HOpe}$.

2.5. Mathematical Method for Reallocation Tertiary Frequency Control Reserves

Mathematically, the power reserves for tertiary frequency control for up and down generation are calculated using a real-time economic model that minimizes hourly the total cost of power reserve reallocations to the new candidate plants selected as Supramarginal and Inframarginal, as shown in Equations (4)–(9).

$${\mathrm{TC}}_{\mathrm{t}}^{\mathrm{TFC},\mathrm{up}}=\min {\displaystyle \sum _{\mathrm{n}\in \mathrm{N}}}\left({\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{TFC},\mathrm{up}\text{-}\mathrm{SMg}}+{\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{TFC},\mathrm{up}\text{-}\mathrm{IMg}}\right)\forall \mathrm{t}\in \mathrm{T}$$

(4)

$${\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{TFC},\mathrm{up}\text{-}\mathrm{SMg}}=\left({\mathrm{VC}}_{\mathrm{n}}-\mathrm{M}{\mathrm{gC}}_{\mathrm{t}}\right)·{\mathrm{Res}}^{(+)}·\mathrm{HOpe}+\mathrm{CP}$$

(5)

$${\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{TFC},\mathrm{up}\text{-}\mathrm{IMg}}=\left(\mathrm{M}{\mathrm{gC}}_{\mathrm{t}}-{\mathrm{VC}}_{\mathrm{n}}\right)·{\mathrm{Res}}^{(+)}·\mathrm{HOpe}$$

(6)

$${\mathrm{TC}}_{\mathrm{t}}^{\mathrm{TFC},\mathrm{down}}=\min {\displaystyle \sum _{\mathrm{n}\in \mathrm{N}}}\left({\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{TFC},\mathrm{down}\text{-}\mathrm{SMg}}+{\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{TFC},\mathrm{down}\text{-}\mathrm{IMg}}\right)\forall \mathrm{t}\in \mathrm{T}$$

(7)

$${\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{TFC},\mathrm{down}\text{-}\mathrm{SMg}}=\left({\mathrm{VC}}_{\mathrm{n}}-\mathrm{M}{\mathrm{gC}}_{\mathrm{t}}\right)·{\mathrm{Res}}^{(-)}·\mathrm{HOpe}$$

(8)

$${\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{TFC},\mathrm{down}\text{-}\mathrm{IMg}}=\left(\mathrm{M}{\mathrm{gC}}_{\mathrm{t}}-{\mathrm{V}}_{\mathrm{n}}\mathrm{C}\right)·{\mathrm{Res}}^{(-)}·\mathrm{HOpe}$$

(9)

Equations (4) and (7) represent the minimization of the total cost ${\mathrm{TC}}_{\mathrm{t}}^{\mathrm{CTF},\mathrm{up}}$ and ${\mathrm{TC}}_{\mathrm{t}}^{\mathrm{CTF},\mathrm{down}}$ of the power reserve reallocations for up/down generation of the tertiary frequency control of the $\mathrm{N}$ power plants named SMg and IMg for a period $\mathrm{t}$. Equations (5) and (6) correspond to the costs ${\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{CTF},\mathrm{up}\text{-}\mathrm{SMg}}$ and ${\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{CTF},\mathrm{up}\text{-}\mathrm{IMg}}$ of the candidate plants SMg and IMg intended as power reserves for raising generation. Finally, Equations (8) and (9) represent the costs ${\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{CTF},\mathrm{down}\text{-}\mathrm{SMg}}$ and ${\mathrm{TC}}_{\mathrm{n},\mathrm{t}}^{\mathrm{CTF},\mathrm{down}\text{-}\mathrm{IMg}}$ of the candidate SMg and IMg plants destined as power reserves to lower generation.

3. Simulation of Real-Time Scenarios

The methodology is validated using three scenarios of real-time operation to apply the reallocation and activation of power reserves for frequency control with authentic data [27], as shown in Table 2. The methodology and validation of the model can be tested in any real system or experimental system using the initial conditions of variable generation costs, marginal cost, auctions for ancillary services, and the power reserves of the secondary control SFC(±) and tertiary control TFC(±) of frequency of each system.

The first real scenario is reallocating power reserves for generation raising using the secondary frequency control for generation raising SFC(+).The second real scenario is the activation of the power reserves for generation raising/lowering, known as the spinning reserves, and obeying the tertiary frequency control TFC(±). Finally, the third real scenario is reallocating power reserves for raising generation in the tertiary frequency control TFC(+).

Currently, real-time power reserve resignations are solved with inefficient methods that increase the overall system operation cost. The most usual method is using the economic merit list (EML), which uses the most economical plants in the energy market, directly harming the plants in a deficit state to fulfill energy sale contracts with plants in the spot market. This EML method is used in Latin America, such as Peru, Argentina, Colombia, and other Central American countries. The second method corresponds to using plants at minimum technical cost (TM) or higher variable generation cost, causing an unnecessary displacement of the marginal cost and the exponential increase in the operating cost due to the excess of plants operating at a minimum technical cost. This TM method is applied in countries with higher energy development, such as Portugal, Spain, and France, in power exchanges between interconnected countries using the most expensive generation of each electrical system in each country. However, this work proposes a novel and efficient solution using the SMg and IMg candidate plants that minimizes the total cost of reserves for reallocation. It is important to note that the plants selected as candidates for reserve reallocation acquire an opportunity cost as the variable cost of generation (VC) is close to the system’s actual marginal cost. The validation of the model with the inefficient reserve reallocation methods and the proposed methodology using IMg and SMg plants is presented below, as shown in

Table 3. Methods for resolving power reserve reallocations.

Method	Symb.	State	Marginal Cost	Operation Cost
Economic merit list	EML	Inefficient	High	High
Technical minimum	TM	Inefficient	Down	High
Inframarginal methodology	IMg	Proposal	Optimum	Optimum
Supramarginal methodology	SMg	Proposal	Optimum	Optimum

.

3.1. Gas Depletion in Combined Cycle Plants Implies Reallocation Reserves to Increase Generation in Secondary Frequency Control

A reallocation for SFC(+) is performed with real-time operation data [27], using the actual hourly marginal cost of 29.2 USD/MWh corresponding to the G-8 plant, according to Table 2. Therefore, in the event of a failure in the generation system, power reserve reallocations must be performed for frequency control. Thus, if the G-4 plant is without power reserve for the SFC(+) in 28 MW for one hour or several hours, then the 28 MW of SFC(+) of the G-4 plant must be replaced.

3.1.1. Economic Merit List Method for Calculating the Cost of Raising the Generation in SFC

This technique involves using the most economical plants in the merit list to reallocate reserves according to the amount of reserves available for each plant in order of lowest to highest variable generation cost. The plants with the lowest marginal cost, i.e., G-1, G-2, G-3, and G-4, are used by calculating the reserve cost of Equation (3).

• Plant G-1 (PV):

${TC}_{SFC\text{-}EML}=\left(29.2-0\right)·28·1=\mathrm{USD}818$. It has a maximum margin of $SFC\left(+\right)=100\mathrm{M}\mathrm{W}$.

• Plant G-2 (PV):

${TC}_{SFC\text{-}EML}=\left(29.2-0\right)·28·1=\mathrm{USD}818$. It has a maximum margin of $SFC\left(+\right)=90\mathrm{M}\mathrm{W}$.

• Plant G-3 (Wind):

${TC}_{SFC\text{-}EML}=\left(29.2-0\right)·28·1=\mathrm{USD}818$. It has a maximum margin of $SFC\left(+\right)=60\mathrm{M}\mathrm{W}$.

• Plant G-4 (Gas):

${TC}_{SFC\text{-}EML}=\left(29.2-28.6\right)·28·1=\mathrm{USD}17$ It has a maximum margin of $SFC\left(+\right)=28\mathrm{M}\mathrm{W}$ and runs out of SFC(+) reserve.

3.1.2. Technical Minimum Method to Calculate the Cost of Raising Generation in SFC

This technique involves using the plants operating at a technical minimum or out of service. It corresponds to the plants with the highest variable generation cost in the energy market since their provision is the ancillary services market. In order to reallocate the power reserves, plants G-16, G-17, G-18, and G-19 are used by calculating the reserve cost of Equation (2).

• Plant G-16 (Hydro):

${TC}_{SFC\text{-}TM}=\left(40.9-29.2\right)·28·1=\mathrm{USD}328$. It has a maximum margin of $SFC\left(+\right)=100\mathrm{M}\mathrm{W}.$

• Plant G-17 (Coal):

${TC}_{SFC\text{-}TM}=\left(44.5-29.2\right)·28·1=\mathrm{USD}428$. It has a maximum margin of $SFC\left(+\right)=5\mathrm{M}\mathrm{W}$.

• Plant G-18 (Hydro):

${TC}_{SFC\text{-}TM}=\left(47.2-29.2\right)·28·1=\mathrm{USD}504$. It has a maximum margin of $SFC\left(+\right)=100\mathrm{M}\mathrm{W}$.

• Plant G-19 (Gas):

${TC}_{SFC\text{-}TM}=\left(77.0-29.2\right)·28·1=\mathrm{USD}1338$. It has a maximum margin of $SFC\left(+\right)=40\mathrm{M}\mathrm{W}$.

3.1.3. Methodology with Inframarginal Plants to Calculate the Cost of Increasing Generation in SFC

The IMg plants with $MgC>VC$ that are close to the hourly $M{gC}_t$ (G-8 = 29.2 USD/MWh) are used to award the SFC(+) reserve reallocation for one hour, according to Table 2. The selected plants are G-2, G-3, G-5, G-6, and G-7 by calculating the reserve cost of Equation (3) and as shown in Figure 7.

• Plant G-2 (PV):

${TC}_{\mathrm{2,1}}^{SFC\text{-}IMg}=\left(29.2-0\right)·28·1=\mathrm{USD}818$. It has a maximum margin of $SFC\left(+\right)=90\mathrm{M}\mathrm{W}$.

• Plant G-3 (Wind):

${TC}_{\mathrm{3,1}}^{SFC\text{-}IMg}=\left(29.2-0\right)·28·1=\mathrm{USD}818$. It has a maximum margin of $SFC\left(+\right)=60\mathrm{M}\mathrm{W}$.

• Plant G-5 (Coal):

${TC}_{\mathrm{5,1}}^{SFC\text{-}IMg}=\left(29.2-28.7\right)·28·1=\mathrm{USD}14$. It has a maximum margin of $SFC\left(+\right)=14\mathrm{M}\mathrm{W}$.

• Plant G-6 (Coal):

${TC}_{\mathrm{6,1}}^{SFC\text{-}IMg}=\left(29.2-28.7\right)·28·1=\mathrm{USD}14$. It has a maximum margin of $SFC\left(+\right)=14\mathrm{M}\mathrm{W}$.

• Plant G-7 (Coal):

${TC}_{\mathrm{7,1}}^{SFC\text{-}IMg}=\left(29.2-29.2\right)·28·1=\mathrm{USD}0$. It has a maximum margin of $SFC\left(+\right)=35\mathrm{M}\mathrm{W}$.

The results indicate that the Inframarginal G-2 and G-3 solar/wind renewable power plants have sufficient capacity to replace the SFC(+) reserve reallocations. However, they are not selected as candidates as their reserve cost (USD 818) exceeds the G-5 and G-6 plants. Although the G-5 and G-6 plants are candidate plants for reserve reallocation, and together they can achieve the SFC(+) of 28 MW, contributing 14 MW each, they still do not satisfy the minimum value of the reserve power reallocation cost (USD 28 in total for G-5 and G-6) compared to the G-7 plant. Finally, the optimum corresponds to the G-7 power plant that has a SFC(+) capacity of 35 MW in total to replace the 28 MW SFC(+) of the G-4 power plant with a cost ${TC}_{\mathrm{7,1}}^{SFC\text{-}IMg}$ of USD 0 due to the proximity (opportunity cost) it has with the real-time marginal cost MgC (USD/MWh 29.2). Then, the system operator considers the calculation obtained for the Inframarginal plants as a reallocation alternative before being executed in real time. That is to say:

• The G-7 power plant can perform SFC(+) of 28 MW in replacement of the G-4 power plant;
• Total cost of the reserve to be replaced: ${TC}_{\mathrm{7,1}}^{SFC\text{-}IMg}=$ USD 0.

3.1.4. Methodology with Supramarginal Plants to Calculate the Cost of Increasing Generation in SFC

The candidate SMg plants with $MgC<VC$, which are close to the hourly $M{gC}_t$ (G-8 = 29.2 USD/MWh), as shown in Table 2, are used to reallocate 28 MW of SFC(+) reserve power from the G-4 plant. The selected plants are G-9, G-10, G-11, G-12, and G-13 by calculating the reserve cost of Equation (2) and as shown in Figure 7.

• Plant G-9 (Coal):

${TC}_{\mathrm{9,1}}^{SFC\text{-}SMg}=\left(29.8-29.2\right)·28·1=$ USD 17. It has a maximum margin of $SFC\left(+\right)=8\mathrm{M}\mathrm{W}.$

• Plant G-10 (Coal):

${TC}_{\mathrm{10,1}}^{SFC\text{-}SMg}=\left(30.2-29.2\right)·28·1=$ USD 28. It has a maximum margin of $SFC\left(+\right)=30\mathrm{M}\mathrm{W}$.

• Plant G-11 (Coal):

${TC}_{\mathrm{11,1}}^{SFC\text{-}SMg}=\left(31.4-29.2\right)·28·1=$ USD 62. It has a maximum margin of $SFC\left(+\right)=8\mathrm{M}\mathrm{W}$.

• Plant G-12 (Coal):

${TC}_{\mathrm{12,1}}^{SFC\text{-}SMg}=\left(31.7-29.2\right)·28·1=\mathrm{U}\mathrm{S}\mathrm{D}70$. It has a maximum margin of $SFC\left(+\right)=8\mathrm{M}\mathrm{W}$.

• Plant G-13 (Gas):

${TC}_{\mathrm{13,1}}^{SFC\text{-}SMg}=\left(32.4-29.2\right)·28·1=\mathrm{U}\mathrm{S}\mathrm{D}90$. It has a maximum margin of $SFC\left(+\right)=35\mathrm{M}\mathrm{W}$.

The results indicate that the Supramarginal G-11, G-12, and G-13 thermal plants have sufficient capacity to replace the 28 MW SFC(+) reserve reallocations of the G-4 plant. However, they are not selected as candidate plants as their standby cost ranges from USD 62 to 90, exceeding the reallocation costs offered by G-9 and G-10 plants. Finally, the optimum power reserve reallocation corresponds to the G-9 and G-10 plants, which have a maximum SFC(+) capacity of 8 and 30 MW, respectively, with a reserve cost per plant of ${TC}_{\mathrm{9,1}}^{CSF\text{-}SMg}$ = 17 USD and ${TC}_{\mathrm{10,1}}^{CSF\text{-}SMg}$ = USD 28. Then, the system operator considers the calculation obtained for the Supramarginal plants as a reallocation alternative before being executed in real-time. That is:

• The G-9 plant can perform eight MW SFC(+) in replacement of the G-4 plant;
• The G-10 power plant can perform 20 MW SFC(+) in replacement of the G-4 power plant;
• Total cost of reserve to be replaced: ${TC}_{\mathrm{9,1}}^{CSF\text{-}SMg}+{TC}_{\mathrm{10,1}}^{CSF\text{-}SMg}=\mathrm{USD}45$.

3.2. Severe Frequency Variations Involve Activating the Power Reserves of the TFC

For this type of factual scenario, an empirical methodology is used that is supported by the economic mechanisms of the merit list to increase generation and the TFC(−) auction list for plants that have to decrease generation, as shown in Table 2.

3.2.1. Method for Activating Power Reserves to Raise TFC Generation

If the actual MgC of the system is 44.5 USD/MWh corresponding to the G-17 plant and the frequency variations are lower than 49.80 Hz with a duration time of more than 5 min, then the methodology of activating the TFC(+) reserves is applied using the ranking of the plants by the economic merit list, according to Table 2. Therefore, the candidate plant to increase generation is the G-16 plant (40.9 USD/MWh). If the reserves are insufficient, the next plant is ranked according to the same economic price criteria. That is, the G-18 plant (47.2 USD/MWh) and then the G19 plant (77.0 USD/MWh), as shown in Figure 8. If the real MgC of the system is 44.5 USD/MWh corresponding to the G-17 plant and the frequency variations are lower than 49.80 Hz with a duration time of more than 5 min. Then, according to Table 2, the methodology of activating the TFC(+) reserves is applied using the ranking of the plants through the economic merit list. Therefore, the candidate plant to increase generation is the G-16 plant (40.9 USD/MWh). If the reserves are insufficient, the next plant in the ranking is followed by the next plant in the same economic price criterion. That is, the G-18 plant (47.2 USD/MWh) and the G19 plant (77.0 USD/MWh), as shown in Figure 8.

3.2.2. Method to Activate Power Reserves to Decrease Generation in TFC

If the actual MgC of the system is 77.0 USD/MWh corresponding to the G-19 plant and the frequency variations exceed 50.20 Hz for more than 5 min, then the methodology of activating the TFC(−) reserves is applied using the TFC(−) activation priority list, which goes from the lowest to the highest price offered to provide the TFC(−) service, as shown to Table 2. The candidate plant for reducing generation is the G-14 plant (41.4 USD/MWh). If the reserves are not restored, the next plant is followed by the next plant with the same economic criteria of auctioned prices. That is, G-15 (41.6 USD/MWh), G-1/G-2/G-3 (43.4 USD/MWh), and G-4 (49.0 USD/MWh), as shown in Figure 8.

3.3. Bidding and Auction Market for Frequency Control Power Reserves

A reallocation of reserves for TFC(+) is performed with real-time operation data [27], using a real hourly MgC of 29.2 USD/MWh corresponding to the G-8 plant, according to Table 2. Therefore, in the event of a failure in the generation system, power reserve reallocations must be performed for frequency control. If the G-18 plant lacks power reserve for the TFC(+) in 100 MW for one hour, then the 100 MW of TFC(+) of the G-18 plant must be replaced.

Similarly to the problem of power reserve reallocation for secondary frequency control, this problem of power reserve reallocation for tertiary frequency control is solved using the economic merit list method and the technical minimum power plant method. The proposed methodology is also applied using the Supramarginal and Inframarginal candidate plants, which allows for the minimization of the total cost of reserves for tertiary reserve reallocation. It is essential to highlight that the plants selected as Supramarginal and Inframarginal candidates for reserve reallocation acquire an opportunity cost as the variable cost of generation is close to the actual marginal cost of the system, as shown in Table 2. Next, the validation of the model with the inefficient methods of reserve reallocation and the proposed methodology using Inframarginal and Supramarginal plants is presented.

3.3.1. Economic Merit List Method for Calculating the Cost of Raising Generation in TFC

This technique involves using the most economical plants in the merit list to reallocate reserves according to the amount of reserves available for each plant in order of lowest to highest variable generation cost. The plants with the lowest marginal cost, i.e., G-1, G-2, G-3, and G-4, are used by calculating the reserve cost of Equation (6).

• Plant G-1 (PV):

${TC}_{TFC,up\text{-}EML}=\left(29.2-0\right)·100·1=\mathrm{USD}2920$. It has a maximum margin of $TFC\left(+\right)=140\mathrm{M}\mathrm{W}$.

• Plant G-2 (PV):

${TC}_{TFC,up\text{-}EML}=\left(29.2-0\right)·100·1=\mathrm{USD}2920$. It has a maximum margin of $TFC\left(+\right)=90\mathrm{M}\mathrm{W}$.

• Plant G-3 (Wind):

${TC}_{TFC,up\text{-}EML}=\left(29.2-0\right)·100·1=\mathrm{USD}2920$. It has a maximum margin of $TFC\left(+\right)=80\mathrm{M}\mathrm{W}$.

• Plant G-4 (Gas):

${TC}_{TFC,up\text{-}EML}=\left(29.2-28.6\right)·100·1=\mathrm{USD}60$. It has a maximum margin of $TFC\left(+\right)=45\mathrm{M}\mathrm{W}$.

3.3.2. Technical Minimum Method to Calculate the Cost of Raising the Generation in TFC

This technique involves using the plants operating at a technical minimum or out of service and corresponding to the plants with the highest variable generation cost in the energy market. To reallocate the power reserves, the G-16, G-17, G-18, and G-19 plants are used to calculate the reserve cost of Equation (5).

• Plant G-16 (Hydro):

${TC}_{TFC,up\text{-}TM}=\left(40.9-29.2\right)·100·1=\mathrm{USD}1170$. It has a maximum margin of $CSF\left(+\right)=100\mathrm{M}\mathrm{W}$.

• Plant G-17 (Coal):

${TC}_{TFC,up\text{-}TM}=\left(44.5-29.2\right)·100·1=\mathrm{USD}1530$. It has a maximum margin of $CSF\left(+\right)=20\mathrm{M}\mathrm{W}$.

• Plant G-18 (Hydro):

${TC}_{TFC,up\text{-}TM}=\left(47.2-29.2\right)·100·1=\mathrm{USD}1800$. It has a maximum margin of $CSF\left(+\right)=100\mathrm{M}\mathrm{W}$ and runs out of TFC(+) reserve.

• Plant G-19 (Gas):

${TC}_{TFC,up\text{-}TM}=\left(77.0-29.2\right)·100·1=\mathrm{USD}4780$. It has a maximum margin of $CSF\left(+\right)=40\mathrm{M}\mathrm{W}$.

3.3.3. Methodology with Inframarginal Plants to Calculate the Cost of Increasing Generation in TFC

The IMg candidate plants to deliver the missing 100 MW reserve of the G-18 plant must lower their generation and evaluate the total cost of the reserve to reallocate the TFC(+) service. Then, from Table 2, the candidate plants G-7, G-6, G-5, and G-4 are selected due to the closer proximity concerning the actual MgC of 29.2 USD/MWh, corresponding to the G-8 plant. Their calculation methodology is obtained according to Equation (6). The IMg candidate plants, in order to deliver the missing 100 MW reserve of the G-18 plant, must lower their generation and evaluate the total cost of the reserve to reallocate the TFC(+) service. Then, from Table 2, the candidate plants G-7, G-6, G-5, and G-4 are selected due to the closer proximity concerning the actual MgC of 29.2 USD/MWh, corresponding to the G-8 plant. Their calculation methodology is obtained according to Equation (6) and as shown in Figure 9.

• Plant G-7 (Coal):

${TC}_{\mathrm{7,1}}^{TFC,up\text{-}IMg}=\left(29.2-29.2\right)·100·1=\mathrm{USD}0$. It has a maximum margin of $TFC\left(+\right)=15\mathrm{M}\mathrm{W}$.

• Plant G-6 (Coal):

${TC}_{\mathrm{6,1}}^{TFC,up\text{-}IMg}=\left(29.2-28.7\right)·100·1=\mathrm{USD}50$. It has a maximum margin of $TFC\left(+\right)=20\mathrm{M}\mathrm{W}$.

• Plant G-5 (Coal):

${TC}_{\mathrm{5,1}}^{TFC,up\text{-}IMg}=\left(29.2-28.7\right)·100·1=\mathrm{USD}50$. It has a maximum margin of $TFC\left(+\right)=20\mathrm{M}\mathrm{W}$.

• Plant G-4 (Gas):

${TC}_{\mathrm{4,1}}^{TFC,up\text{-}IMg}=\left(29.2-28.6\right)·100·1=\mathrm{USD}60$. It has a maximum margin of $TFC\left(+\right)=45\mathrm{M}\mathrm{W}$.

• Plant G-3 (Wind), G-2 (Solar) and G-1 (Solar):

${TC}_{(\mathrm{3,2},1),1}^{TFC,up\text{-}IMg}=\left(29.2-0\right)·100·1=\mathrm{USD}2920$ Each renewable plant (G-3, G-2, and G-1) has a maximum margin of TFC(+) = 80, 90, and 100 MW, respectively. However, these plants are left out of the selection due to their high cost.

To supply the missing 100 MW reserve of the G-18 plant, the candidate IMg plants to replace the TFC(+) are the G-7, G-6, G-5, and G-4 plants with a total cost of ${TC}_{(\mathrm{7,6},\mathrm{5,4}),1}^{CSF,up\text{-}IMg}$ = USD 160. However, the renewable power plants G-3, G-2, and G-1 have variable costs of 0 USD/MWh and 100% TFC(+) reserve capacity. However, unfortunately, the cost of power reserve reallocations is higher than the rest of the conventional power plants, reaching a value of ${TC}_{(\mathrm{3,2},1),1}^{CSF,up\text{-}IMg}$ = USD 2920 for each renewable power plant. Then, the system operator considers the calculation obtained for the Inframarginal power plants as a reallocation alternative before being executed in real time. Thus:

• G-7, G-6, G-5, and G-4 power plants can perform a TFC(+) of 100 MW to replace the G-18 power plant;
• Total cost of the reserve to be replaced: ${TC}_{(\mathrm{7,6},\mathrm{5,4}),1}^{CSF,up\text{-}IMg}$ = USD 160.

3.3.4. Methodology with Supramarginal Power Plants to Calculate the Cost of Increasing the Generation in TFC

SMg candidate plants from Table 2 are used to deliver the missing 100 MW reserve of the G-18 plant. The selected plants must evaluate the total cost of raising generation to deliver the TFC(+) reserve capacity. Then, the candidate plants, according to their available reserve capacity, are G-13, G-14, G-15, and G-16 because they have the closest proximity to the actual MgC of 29.2 USD/MWh corresponding to the G-8 plant. Therefore, their calculation methodology is obtained according to Equation (5) and as shown in Figure 9.

• Plant G-13 (Gas):

${TC}_{\mathrm{13,1}}^{TFC,up\text{-}SMg}=\left(32.4-29.2\right)·100·1=\mathrm{USD}320$. It has a maximum margin of $CTF\left(+\right)=60\mathrm{M}\mathrm{W}$.

• Plant G-14 (Gas):

${TC}_{\mathrm{14,1}}^{TFC,up\text{-}SMg}=\left(35.2-29.2\right)·100·1=\mathrm{USD}600$. It has a maximum margin of $CTF\left(+\right)=40\mathrm{M}\mathrm{W}$.

• Plant G-15 (Gas):

${TC}_{\mathrm{15,1}}^{TFC,up\text{-}SMg}=\left(38.7-29.2\right)·100·1=\mathrm{USD}950$. It has a maximum margin of $CTF\left(+\right)=40\mathrm{M}\mathrm{W}$.

• Plant G-16 (Hydro):

${TC}_{\mathrm{16,1}}^{TFC,up\text{-}SMg}=\left(40.9-29.2\right)·100·1=\mathrm{USD}1170$. It has a maximum margin of $CTF\left(+\right)=100\mathrm{M}\mathrm{W}$.

The results indicate that the optimum for power reserve reallocation corresponds to plants G-13 and G-14, which have a maximum TFC(+) capacity of 60 and 40 MW, respectively, with a reserve cost per plant of $C{T}_{\mathrm{13,1}}^{CTF,up\text{-}SMg}$ = USD 320 and $C{T}_{\mathrm{14,1}}^{CTF,up\text{-}SMg}$ = USD 600. Then, the system operator considers the calculation obtained for the Supramarginal plants as a reallocation alternative before being executed in real time. That is:

• The G-13 plant can perform 60 MW TFC(+) in replacement of the G-18 plant;
• The G-14 power plant can perform a TFC(+) of 40 MW to replace the G-18 power plant;
• Total cost of the reserve to be replaced ${TC}_{\mathrm{13,1}}^{CTF,up\text{-}SMg}+{TC}_{\mathrm{14,1}}^{CTF,up\text{-}SMg}=\mathrm{USD}920$.

4. Discussion and Analysis of the Results Obtained

The following is the discussion and analysis of the results obtained for the real case studies of Section 3.1, Section 3.2 and Section 3.3. For a better interpretation, the results are supported using tables and graphs.

4.1. Analysis of the Calculation Methods and the Results of the Reallocation of Reserves for SFC

The Economic Merit List and Technical Minimum methods can satisfy the missing reserves for reallocating the 28 MW secondary frequency control of the G-4 plant. However, the selected plants of both methods fail to compete with the costs of the proposed methodology because the results of the reserve costs range from 17 to 818 USD/MW for the merit list method. The reserve cost ranges from 328 to 1338 USD/MW for the technical minimum method.

The results proved more economical in the proposed model tested than the other inefficient methods. The proposed methodology results show that the total costs of power reserve reallocations for real-time secondary frequency control using candidate Inframarginal and Supramarginal plants range from 0 to 45 USD/MW. As the selection of candidate plants moves away from the marginal cost in the direction above or below the reference value, the opportunity cost increases, and therefore, the total costs of reserves for real-time reallocation are not optimal because they range from 90 to 817 USD/MW, which corresponds to the high opportunity costs of renewable energies.

Mathematically, the results respond to the methodology of the real-time economic model that minimizes the total cost ${TC}_t^{CSF}$ of power reserve reallocations of the set of the selected candidate power plants of type SMg and IMg, as described in Equation (1). As the real-time marginal cost MgC changes, the cost minimization calculation of the power reserve reallocations cost should be modified by selecting of new candidate power plants SMg and IMg, respectively. Finally, Figure 7 and

Table 4. Results of the reallocation of reserves for SFC(+).

Plant	Type	Method	${\mathbf{TC}}_{\mathbf{t}}^{\mathbf{CSF}}$(USD)	Reserve SFC(+) (MW)	Reallocation SFC(+)
G-1	Solar	EML	817	28	Non-optimal
G-2	Solar	EML	817	28	Non-optimal
G-3	Wind	EML	817	28	Non-optimal
G-4	Gas	EML	817	28	No reserve
G-5	Coal	IMg	14	14	Candidate 1
G-6	Coal	IMg	14	14	Candidate 2
G-7	Coal	IMg	0	28	Optimal
G-9	Coal	SMg	17	8	Candidate 3
G-10	Coal	SMg	28	20	Candidate 4
G-11	Coal	SMg	62	8	Non-optimal
G-12	Coal	SMg	70	8	Non-optimal
G-13	Gas	SMg	90	35	Non-optimal
G-16	Hydro	TM	338	28	Non-optimal
G-17	Coal	TM	428	5	Non-optimal
G-18	Hydro	TM	504	28	Non-optimal
G-19	Gas	TM	1.338	28	Non-optimal

detail all the results of the reserve calculations for reallocating the 28 MW of the G-4 plant.

4.2. Analysis of the Results of the Activation of Reserves for TFC

The numerical results of this methodology of activating power reserves for tertiary frequency control for raising and reducing generation satisfy the equilibrium conditions between the energy market and ancillary services. For the activation of tertiary reserves to raise generation, the costs range from 40.9 to 77.0 USD/MWh near the marginal cost with the economic merit list mechanism. For the activation of tertiary reserves to lower generation, the costs range from 41 to 43.4 USD/MWh, depending on the bidding auction list, as shown in Figure 8.

4.3. Analysis of the Calculation Methods and the Result of the Reallocation of Reserve for TFC

The Economic Merit List and Technical Minimum methods can satisfy the missing reserves for reallocating the 100 MW tertiary frequency control of the G-18 plant. However, the selected plants from both inefficient methods fail to compete with the proposed methodology because the results of the reserve costs range from 60 to 2920 USD/MW for the Merit List Method (LME), and the Technical Minimum Method (TM) the reserve cost ranges from 1170 to 4780 USD/MW.

Therefore, the proposed economic optimization model suits real-time power reserve reallocations for tertiary frequency control. From the results obtained, the candidate plants with spinning and cold reserves are identified. The main difference of this ancillary service is the duration time, which must sustain the required or reallocated power reserve of less than or equal to one hour. This implies that the reserve reallocation is with the minimum adjustment error by the system operator as it is a manual action mechanism. The execution of an undesired manual control action of tertiary reserves immediately causes an increase in the cost of operation in unnecessary time due to the inefficient dispatch of plants that operate at the technical minimum and out of price.

Mathematically, the real-time economic model methodology results allow for the minimization of the total cost ${TC}_t^{CTF,up}$ of power reserve reallocations of the set of candidate power plants of type SMg and IMg, as described in Equation (4). As the real-time marginal cost MgC shifts, the cost minimization calculation of power reserve reallocation cost should be modified by selecting new candidate power plants SMg and IMg, respectively.

Table 5. Results of the reallocation of reserves for TFC(+).

Plant	Type	Method	${\mathbf{TC}}_{\mathbf{t}}^{\mathbf{TFC},\mathbf{up}}$/${\mathbf{TC}}_{\mathbf{t}}^{\mathbf{TFC},\mathbf{down}}$ (USD)	Reserve TFC(+) (MW)	Reallocation TFC(+)
G-1	Solar	EML	2920	14	Non-optimal
G-2	Solar	EML	2920	14	Non-optimal
G-3	Wind	EML	2920	14	Non-optimal
G-4	Gas	IMg	60	45	Optimum
G-5	Coal	IMg	50	20	Optimum
G-6	Coal	IMg	50	20	Optimum
G-7	Coal	IMg	0	15	Optimum
G-13	Gas	SMg	320	60	Candidate 1
G-14	Gas	SMg	600	40	Candidate 2
G-15	Gas	SMg	950	40	Non-optimal
G-16	Hydro	TM	1170	100	Non-optimal
G-17	Coal	TM	1530	20	Non-optimal
G-18	Hydro	TM	1800	100	No reserve
G-19	Gas	TM	4780	40	Non-optimal

and Figure 9 detail all the results of the reserve calculations for reallocating the 100 MW of the G-18 power plant.

5. Conclusions and Future Work

The real-time power reserve reallocation model is optimal for system operator decisions, overcoming traditional static methods, such as the economic merit list, technical mini-mum, and auction/bidding methodologies, that lead to high operating and marginal costs. This real-time reallocation model is a novelty for the ancillary services market. It reduces the operating costs of reserves by more than 60% concerning non-candidate plants. It minimizes marginal cost shifts of 10 to 40% concerning the demand trend and randomly unjustified dispatches from the economic merit list.

Regarding the case studies, the model validation can optimally respond to power reserve reallocations for secondary frequency control with the use of inframarginal and supramarginal candidate plants with reserve costs ranging from 0 to 45 USD/MW as opposed to renewable plant reserve costs ranging from 90 to 817 USD/MW. In contrast, power reserve reallocations for tertiary frequency control avoid cost overruns ranging from 60 to 1170 USD/MW. However, this work can identify the available solar and wind generation resources to actively participate in this market, with reserve capacities ranging from 70 to 100 MW for secondary frequency control and reserves from 1000 to 1900 MW for tertiary frequency control.

Finally, this proposed model solves a large part of this problem of reallocating power reserves at minimum cost. However, the opportunity costs of the plants for the reallocation of reserves are subject to conventional generation and the negligent use of the system’s inertia that conditions the plants out-of-economic-order operating at the technical minimum. The model proposed in this situation does not allow for the selection of these plants since they operate far from marginal cost and are not cataloged as SMg and IMg candidate plants. Therefore, in a future work, it is important to define a model that can monitor and avoid monopolistic actions with the use of unjustified plants in both markets and a real-time economic model for plants with reactive power reserve capacities to optimize the costs of modifying the P-Q curves in a dynamic state of the dispatches of plants that maintain voltage control in real time.