Technical-Economic Model in the Real-Time Ancillary Services Market for the Reallocation of Power Reserves in Primary Frequency Control

Abstract

Chile’s National Electric System is one of the countries in South America with the greatest potential for the development of solar–wind generation, allowing for the acceleration of the energy transition with the definitive withdrawal of conventional fossil fuel thermal generation. However, the integration of the market of ancillary services requires security, stability, and quality of service to the electricity system. In this context, the primary frequency control (PFC) is considered as the first line of defense of an electric power system, due to its immediate action in severe frequency variations when they exceed ±0.7 Hz with respect to the nominal operating frequency of 50.00 Hz, allowing the safe and efficient integration of large blocks of solar–wind renewable generation in spite of the uncertainty or forecast errors that could cause its massive dispatch. The principal contribution of this work is the implementation of a technical-economic mathematical model that minimizes the total costs of real-time power reserve reallocations for primary frequency control, using the dynamic factors of stationarity in those conventional and renewable solar–wind generation plants. The validation of the model is consolidated through real scenarios, specifically the deficit of power reserves, which necessitates a dynamic response in primary frequency control over 10 s and 5 min. In terms of expected results, the proposed model contributes to the Supra-/Infra-Marginal methodology, reducing the total costs of power reallocation reserves for primary frequency control, compared to other inefficient methods, such as the Maximum Power Method, the Minimum Technical Method, and the Random Direct Instruction Method.

1. Introduction to Power Reserve Reallocation Models for Primary Frequency Control

Power systems that use the availability of conventional electricity generation and solar–wind renewable generation to integrate primary frequency control in the ancillary services market are synonymous with high investment and costs for the companies that own the plants [1]. The primary frequency control, also known as primary regulation, is activated quickly when the power system frequency deviates from its nominal value. In this type of control, the generators, through their speed regulators, adjust their active power to restore the balance between generation and demand. In South America, investments in technologies applied to primary frequency control mechanisms are limited compared to those in other more developed countries, such as those in Europe and North America [2]. However, Chile at the South American level is a pioneer in the regulations of the ancillary services (AS) markets, demanding in the short term the participation of renewable generation companies in the primary frequency control, through the investment of technology and adequacy of control of their generation plants to minimize the dependence on the rotational inertia of conventional thermal generation [3].

The system operator is responsible for managing all available generation resources in real-time to supply demand in a safe and economical manner. However, the potential of solar–wind generation and the low demands of the system make it difficult to operate the market for ancillary services, which involves the reallocation of power reserves. This forces the dispatch of conventional thermal generation plants at a minimum generation power (technical minimum) that remains constant over time due to inertia requirements [4]. This constant action of thermal power plants generating at a technical minimum power maximizes the probability of renewable resource curtailment (reduction in economic generation) in the form of generalized pro-rating at the system level or with economic decoupling of marginal costs in different electrical zones that increase the total operating costs of the system [5].

Therefore, the economic vulnerability of the electricity system increases, due to the technological deficits of the economic generation resource to reallocate the primary frequency control reserves to solar–wind generation, due to its inefficient technological development for this type of ancillary service, causing zero values of the marginal cost in the bars of the system with higher renewable generation and high marginal costs in the rest of the bars of the system [6]. While in Chile, the news is the installation of large blocks of free solar energy that has traveled the world, undermining the benefits and sustainability of this technology that it could offer to the market of ancillary services [7]. In addition, Figure 1 shows the “duck curve” of the Chilean electricity system, which shows a short-term future projection of solar generation expansion, with an installed capacity close to 60% of the system’s total demand. This condition triggers the system operator’s alarms due to the weak growth in demand (free and regulated customers), high penetration of solar generation, and the increase in conventional generation that ramps up to replace solar generation. Critical situation of the system that evidences the vulnerability of security and the reason why power reserves should be allocated for primary frequency control in renewable generation.

The first objective of this work is to implement a dynamic response mathematical model for the droop adjustments of conventional power plants that are delivering power reserves for primary frequency control, while the second objective is the implementation of a mathematical model for economic purposes in the market of ancillary services, which allows the integration of solar–wind generation, to minimize the total costs of power reserves reallocations in those plants assigned as supramarginal (MgS: those plants with marginal cost lower than the real-time marginal cost) and infra-marginal (MgI: those plants with marginal cost higher than the real-time marginal cost). From the proposed mathematical models, a methodology is developed that consists of randomly selecting generation plants classified as supramarginal and inframarginal, and grouping them into two segments. The first segment is conventional generation (coal, gas, diesel, and hydro) to balance the response capacities for power reserves through a mathematical model that determines the percentage adjustments of the steady state by generation technology. While the second segment involves solar–wind generation integrated into a dynamic economic model, minimizing power reserve costs over time through an opportunity cost that depends on the real-time marginal cost, demand curve tracking, and renewable resource uncertainty.

Regarding the expected results of the methodology, by reallocating power reserves for the PFC to the solar–wind generation resource [7], it would avoid dispatching conventional generation plants that are not established by economic order in the preparation of the unit commitment model the day ahead. In other words, it reduces the dependence on hydroelectric reservoir generation, which are usually declared in a state of exhaustion (drought) and the use of coal-fired power plants (polluting emissions) that are in a definitive retirement stage due to the decarbonization plan [8], where it is also possible to withdraw all gas-fired thermal generation that is in a state of operation at technical minimum unnecessarily, taking advantage of the technical flexibility they offer in the minimum operation and stop times. Additionally, solar–wind generation can open another line of research for future researchers in the ancillary services market, integrating reactive power reserves to exert voltage control at the system busbars in the absence of dynamic elements provided by conventional generation [9]. For example, in Chile, the maximum amounts of solar–wind generation in reactive power that are recorded in real-time operation to control voltage at 220–500 kV busbars vary from ±70 MVAr of injection/absorption with voltage magnitudes varying in the order of ±7 kV [10].

Finally, the proposed methodology must be validated with other traditional methods of power reserve reallocations, such as: (a) Maximum Power Method; (b) Technical Minimum Method; (c) Random Direct Instruction Method; and (d) Supra-/Infra-Marginal Method (proposed in this work). For each of these validation conditions, the proposed model must demonstrate an effectiveness in minimizing its costs that guarantees system stability, with the optimal use of power reserves for primary frequency control, being able to respond in an activation time of ten seconds (PFC-10s) and a maximum delivery time up to five minutes (PFC-5min), when the frequency variations exceed ±0.7 Hz, respectively. For a better mathematical and methodological follow-up of the proposal inthis work,

Table 1. Nomenclature and definition of variables.

Abbreviation	Definition
PFC	Primary frequency control
MgC	Marginal costs
VC	Variable cost of generation
AS	Ancillary services
MgS	Supramarginal
MgI	Inframarginal
MP	Maximum power
TM	Technical minimum
RDI	Random direct instruction
BESS	Battery energy storage system
β	Bias
EDAC-BF	Automatic load disconnection system for low frequency
PFC-10s	Response of the primary frequency control at 10 s
PFC-5min	Response of the primary frequency control at 5 min
Variables	Definition
$\mathrm{T}\mathrm{N}\mathrm{R}{(\pm )}_{\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}$	Represents the total net reserve of active power to up/down generation of each plant in the reallocation of reserve for primary frequency control at 10 s of activation
$\mathrm{T}\mathrm{N}\mathrm{R}{(\pm )}_{\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$	Represents the total net reserve of active power to up/down generation of each plant in the reallocation of reserve for primary frequency control, with a maximum delivery time of 5 min
$\mathrm{G}\mathrm{A}\mathrm{P}\mathrm{R}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}$	Represents the gross active power reserve for raising/lowering generation of each plant for primary frequency control at 10 s of activation without considering the dynamic power response factor
$\mathrm{G}\mathrm{A}\mathrm{P}\mathrm{R}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$	Represents the gross active power reserve for raising/lowering the generation of each plant for primary frequency control, with a maximum delivery time of 5 min, without considering the dynamic power response factor
$\mathrm{D}\mathrm{F}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}$	Represents a percentage or dynamic power response factor as a function of the droop settings for each generation technology at 10 s of activation
$\mathrm{D}\mathrm{F}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$	Represents a percentage or dynamic power response factor as a function of the droop settings for each generation technology, with a maximum delivery time of 5 min
$\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )}$	Represents the minimum total cost of net power reserve reallocation, considering the minimization of supra/infra-marginal power plant reserve costs, for primary frequency control
$\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}\mathrm{M}\mathrm{g}{\mathrm{S}}_{\mathrm{n},\mathrm{t}}}$	It represents the cost of the net power reserves of the n supramarginal power plants (MgS) in hour t
$\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}\mathrm{M}\mathrm{g}{\mathrm{I}}_{\mathrm{n},\mathrm{t}}}$	It represents the cost of the net power reserves of the n infra-marginal power plants (MgI) in hour t
$\mathrm{M}\mathrm{g}{\mathrm{C}}_{\mathrm{t}}$	It represents the marginal cost of the system in hour t
$\mathrm{V}{\mathrm{C}}_{\mathrm{n}}$	Represents the variable cost of generation of each power plant n by technology
${\mathrm{R}}_{\mathrm{t}}$	Represents the amount of net power reserve to be reallocated in hour t for each plant n by technology

shows the abbreviations, variables, and their definitions.

2. State of the Art in the Market of Ancillary Services for Real-Time Operation

The energy transition is a worldwide commitment to transform the energy matrix of each electrical system into a carbon-neutral one. In South America, Chile is a leading reference in the development of renewable energies, with a significant solar generation capacity that challenges traditional market remuneration mechanisms, starting from investment, energy sales, operating costs, and even economic incentives, as well as new models of ancillary services dedicated to frequency control [11]. It is important to highlight that the energy transition must involve the transmission system with optimal power flow exchanges with solar–wind renewable generation, as in other countries with higher development such as the British electric system, where the challenges are focused on the dispatch of power plants to optimize power flow exchanges with renewable energies, such as energy storage, green hydrogen and biomass to accelerate decarbonization [12]. This optimal redistribution of power flows, in the face of the massive entry of renewable generation, enables the minimization of high marginal cost decoupling in the different areas of the power system, ensuring the n-1 security criterion in transmission lines and meeting the minimum rotational inertia requirements of the system [13].

In power systems with massive solar–wind generation, they immediately alter the safety conditions of the system operator, causing ramps up/down generation from 25 kW/s as in the case of the Spanish power system [14] and the case of the Chilean power system in the order of 50 MW/min for solar generation over 6000 MW in its maximum peak [15]. This operational situation causes high frequency change rate slopes, compromising system security and evidencing the insufficiency of the reserve for primary frequency control, as is the case in Nordic countries such as Sweden, Finland, and Norway, which is dominated by hydroelectric generation, where a novel solution is proposed to address this problem of frequency change rates, which is to include wide area measurements for monitoring (WAM) and rate of change in frequency sharing (RoCoF) in remote areas with lower inertia to improve their primary frequency control [16]. However, in countries with higher wind generation penetration, the system operator faces the challenge of quickly restoring frequency due to the uncertainty of renewable generation, which unfortunately lacks a control mechanism in the speed regulator to contribute to the PFC [17]. Therefore, the solution to this challenge is to analyze the technical value of the generating units that participate in primary frequency control. Subsequently, a formulation for calculating the active power of the generating units in response to frequency variations is developed. In addition, the concept of weighted average frequency active power change (WAFAPC) is introduced, by means of continuous and discrete formulations with different types of weighting functions [17].

While the Irish electricity system mitigates variations in power ramps through energy storage elements (BESS), a mathematical formulation based on the Fourier transform is used to synthesize a real frequency signal from variations in electricity consumption and generation [18]. Also, the impact of BESS is evaluated in relation to the installed storage capacity and the adopted regulation strategy, allowing for a comparison of the regulation provided by conventional hydro generation [18]. It is important to note that energy storage systems can be used to support reserves for primary frequency control and can be integrated into the market to monetize their benefits. The fluctuations infrequency change ramps by wind generation can be damped with hydro generation in collaboration with the BESS. For such a proposal, a ramp control strategy is created that utilizes the charging and discharging power of the BESS in real-time operation during primary frequency regulation [19]. To evaluate the performance of BESSs in supporting hydro generation in frequency regulation, a comprehensive analysis combining physical characteristics and economic indicators related to ancillary services markets is performed [19].

It is evident that inertia in an electrical system is fundamental for the correct use of power reserves for primary frequency control. The market for ancillary services must be sufficiently effective for power generation plants to be able to provide them profitably. Non-synchronous wind generation in the British power system poses a challenge for system operators in terms of inertia control [20]. Therefore, the solution is based on formulating a non-convex two-level optimization problem, where the upper-level problem represents the operation of a unit commitment model with frequency constraints. The lower-level problem incorporates the generation dispatches and frequency responses of a group of virtual power generation plants, guided by the dual price signals of the dispatched generation and the prices of the ancillary services, using the Karush–Kuhn–Tucker optimality conditions [20]. In contrast, the Italian power system in Sardinia Island, the solution to the inertia deficits is through a methodology that quantifies the required inertia capacity for each ancillary service, in order to ensure the stability and security of the system, when the regulation margins of conventional generation for primary frequency control have already been exploited [21]. In terms of control theory, the required inertia quantities are identified, and the values of the gains associated with the related controllers of the generation plants are calculated [21]. The methodology for this problem is solved with nonlinear optimization; the objective function to be minimized is related to the maximum value of the exchanged power for each service after a frequency imbalance and to the relative costs of such maximum power exchanges, while constraints are defined to limit the amplitude and the gradient of the power system frequency response [21]. Conventional gas-fired thermal generation, on the other hand, can stabilize the uncontrolled second frequency droop and provide inertia quickly through its good response to droop settings, exploring in depth the frequency response potential of inertial resources in electricity-heat-gas (IES) coupling power systems, depending on the slow dynamics characteristics of gas-fired thermal power plants [22]. Therefore, this work utilizes gas inertia as a form of energy storage to mitigate the second frequency drop, as natural gas can be temporarily stored in the pipeline and released in large quantities in the event of emerging power generation shortages [22]. However, stabilizing inertia with thermoelectric power plants generates the problem of high emissions, which is why the author proposes a model and simulation of a numerical thermoelectric with BiTe-based materials to reduce emissions [23], applying a robust design for thermoelectric power plants, through structural and service behavior of composite slabs with high content of recycled aggregates [24]. Anyway, the system operator must have the necessary monitoring, control, and supervision resources to quantify in real-time operation the importance of inertia in an electrical system and, at the same time, ensure competitive conditions in the market for ancillary services to justify the costs incurred in security [25]. This research, the author aims to establish a complete system inertia monitoring system, which consists of five modules, being the first module in an inertia estimation system, the second module is an adaptive system, the third module corresponds to an event-based algorithm, fourth module is the inertia prognosis and the fifth module is the human–machine interface (HMI) [25].

It is important to note that the first solar–wind generation plants were not part of, nor were they intended to encourage or integrate into the ancillary services market. The technical-economic capabilities were very limited for providing inertia, primary frequency control, and voltage control. However, conventional thermal generation has the flexibility to adapt to the ancillary services market. Such is the case with gas-fired thermal generation, which uses a universal evaluation mechanism to assess the capability of primary frequency control for conventional thermal generation under various operating conditions by decomposing and quantifying the power state, especially under flexible operating processes [26]. A transient heat current model describing the dynamic process is implemented by constructing the relationship of different parameters in superheated components in various energy states. Also, the authors have developed dynamic simulation models and mathematical models of the boiler to improve and study the performance of primary frequency control. For example, a computational system identification method is used to describe the dynamic characteristics of the rapidly changing turbine regulation system and the main steam pressure during frequency fluctuations [26]. While the coal-fired thermal generation, with a design featuring high–low position shafts, participates in the ancillary services market through a primary frequency regulation control strategy, by decoupling the power characteristics, and the dynamic behavior of the high–low position shafts arranged in the coal-fired unit. In addition, the dynamic coupling characteristics between the high- and low-level units are quantitatively analyzed, evidencing a time delay of 14 s between the two power outputs. A revised regulation strategy employing a load distribution factor and steam flow feedforward compensation is proposed [27]. However, it is essential in each power system to establish the regulatory requirements for conventional generating plants to contribute to primary frequency control. As renewable generation advances, a market rule design for real-time operation becomes increasingly necessary to incentivize electricity generation owners. As is the case with the Electric System of Texas (ERCOT), which addresses this problem by implementing a series of market rule changes, including the introduction of a new type of fast frequency response (FFR) reserve in the electricity market [28]. This type of FFR reserve is designed to complement the traditional primary frequency response (PFR) reserve, helping to prevent frequency droop in the event of a major generator outage. The authors of this paper implement reserve requirements to ensure a sufficient reserve to stop the frequency droop before reaching the critical frequency threshold while coupling PFR reserve, FFR reserve, and system inertia [28].

Finally, the literature evidences the need to increase the technological mechanisms of generation plants to provide primary frequency control, as a result of the massive entry of solar–wind renewable generation, and to cover the deficits of inertia of the system. However, there is a limited field of research in the ancillary services market, which allows to demonstrate research findings in economic models for real-time operation, using renewable power plants in the reallocation of power reserves for primary frequency control, specifically in those solar–wind generation plants and the weak development of market balance mechanisms for conventional thermal and hydroelectric plants that can make this service profitable in the security and stability in the system with the appropriate distribution of statism. It is essential to note that the markets for ancillary services have been consolidating in several countries across North America, Europe, and South America over the past few years. Chile has been a pioneer in advancing this market, with exponential growth in solar–wind generation, as illustrated in Figure 2. In addition, the results of each research study in the reviewed literature provide opening lines of research for other authors, albeit with different structures and methodologies. The economic sustainability of ancillary services provided by various power generation technologies is very important in the system and has been addressed through various control models and market rules. The alternatives for organizing the sector that can be applied are multiple and vary from economic models based on tenders [29], compensatory mechanisms for services [30], previous offers or final offers using the concept of “gate closure” [31], or a regression model of uniform prices between the energy market and the ancillary services market [32]. Also, the economic compliance of hourly auctions must be ensured through incentives and/or penalties that evaluate the performance factor of renewable and conventional generation for frequency control [33]. However, the lack of a well-established operational framework and a cost-sharing model between conventional and renewable power plants has hindered their widespread implementation and large-scale development. Cost sharing can be solved using three different methods: the uniform allocation method, the predictive weighted allocation method, and the dynamic weighted allocation method [34]. In these cases, the economic modeling of energy costs, whether using sequential, simultaneous, or co-optimization methodologies, plays a fundamental role in achieving convergence of the global operational cost. It is key to establish a point of equilibrium between the energy market and the ancillary services market, mainly with zero marginal cost renewable generation [35], to make costs and contracts transparent to future investors [36], in view of the market rules for ancillary services in the reliability, quality, and safety of the electricity system.

3. Evolutionary Market of the Use of Real-Time Power Reserves

Chile’s national electricity system has a long history of using primary frequency control, initially with the allocation of power reserves arbitrarily distributed between conventional thermal and hydroelectric generation, where the lack of resources to invest in and adapt control mechanisms in generation plants predominated. The vulnerability of primary frequency control was evident due to the fact that there was no market regulation to justify its investment and profitability. System security was considered the only background to support the use of primary frequency control, to the detriment of the additional costs involved by the companies owning the generation plants, in order to meet this system security requirement. It is critical to mention that in the northern zone of Chile’s national electricity system (Figure 2), the predominance of coal-gas thermal generation with high seasonality adjustments (3 to 12%), cause a constant inflexibility of generation in real time to the system operator, insufficient to cushion the uncertainty of solar–wind renewable generation and the large variations in demand derived from mining processes concentrated in the same northern zone. However, the massive penetration of solar–wind generation, the integration of the ancillary services market, and an accelerated energy transition are expected to occur. They have encouraged market agents to develop regulations that enable generation plant owners to make conventional thermal-hydraulic generation more flexible and even integrate renewable generation, through active participation in power reserves destined for primary frequency control, thereby guaranteeing system security in the real-time ancillary services market.

3.1. Inefficient Use of Primary Frequency Control in Real-Time Operation

Without any market regulation of ancillary services prevailing to establish a balance of competences between generation owning companies to guarantee the security of the electric system, statistically, an equitable distribution of power reserves for primary frequency control was used in those gas and coal-fired thermal generation plants, established by the independent system operator, with a factor of 7% of their maximum generation power. That is, if a thermal power plant has a maximum power of 250 MW, the adjustment factor established by the independent system operator is 7%; therefore, the maximum generation power for economic dispatch must be adjusted to approximately 233 MW, leaving a total contribution of ±17 MW available for primary frequency control. Therefore, the amount of total power reserve destined to the primary frequency control that allows attenuating the frequency variations that oscillate between ±0.7 Hz, must consider the value of the “Bias (β)”, which depends on the historical records of events or failures that have occurred in the system where the value of demand imbalance (MW) in the system due to loss of load or generation and the behavior of the frequency (Hz) is taken into account. The Bias (β) of the system changes as the electrical system evolves; therefore, it is recommended to calculate it periodically to verify the validity of the value used or if its update is required. For the case of the Chilean electrical system, the Bias (β) ranges between 60 and 70 MW/0.1 Hz.

For a more detailed understanding of the importance of Bias (β) and its relationship with the primary frequency control, a real case that occurred on 1 June 2016 at 04:46 h is analyzed (Figure 3), describing the frequency behavior for a system with high presence of coal–gas thermal generation, when the failure of a coal-fired thermal power plant with 273 MW occurs. The frequency drops to 48.94 Hz, immediately triggering the automatic under-frequency load shedding operation (EDAC-BF) set in the first step. Subsequently, the frequency is restored to 49.40 Hz. In this generation failure, the primary frequency control reserve contributes 90 MW to the system. This power reserve response value is obtained empirically for the Chilean national electric system, as defined by the system operator, based on experience with numerous failures that occurred in the system, where the frequency incurs variations of 20 MW for each 0.1 Hz, respectively. Finally, 140 MW are destined to a theoretical reserve for primary frequency control, and the reality shows that it provides a maximum of 90 MW in reserve for primary frequency control due to the high stationarity of the generation plants; otherwise, the operation of the EDAC-BF in the first step would have been avoided.

3.2. Integration of Frequency Control in the Market of Ancillary Services

The global energy matrix, including Chile, requires actions and restructuring of energy policies in the fight against climate change, which is one of the most significant challenges that humanity must face in the 21st century. The energy transition refers to a shift in the models of energy production, distribution, and consumption aimed at achieving greater sustainability through the achievement of zero carbon emissions. From an energy perspective, the challenge focuses on transforming the electricity system with renewable generation technologies to displace the use of fossil fuel-based electricity generation and reduce emissions

It is important to recognize that the ancillary services market plays a crucial role in the energy transition, as it ensures the provision of quality and safety in electric power systems, thereby crafting market opportunities for companies to advance in technological innovations for solar–wind renewable generation. Frequency control is one of the most important ancillary services that ensures system stability. However, conventional thermal generation plays an important role in primary frequency control, due to its dynamic response design, damping, inertia, and stability, limiting the participation of solar–wind renewable generation due to its limited technological development. At present, solar generation already has grid-forming and grid-following technologies that allow it to actively participate in primary frequency control, while wind generation, driven by the kinetic energy of the wind, can adapt its torque and angle control designs for primary frequency control.

It is essential to define the role of power generation in the ancillary services market in terms of the magnitude of power reserves required to counteract frequency deviations. The dynamic behavior of the frequency in the face of deviations is categorized as underfrequency or over frequency, which evolves from a stationary state of the system to a fault state. In this case, the power reserves of the primary frequency control must act to return the system to its stationary state. In addition, the actuation times of the power reserves show the performance of each generation technology to be awarded in an economic model that guarantees its participation in the market; that is why the response times are categorized in activation times and delivery times in a range that goes from 10 s to 5 min for frequency deviations exceeding ±0.7 Hz (Figure 4).

4. Technical-Economic Methodology of Real-Time Ancillary Services

The challenges in regulatory policies, market modifications, the implementation of new power generation technologies, and the participation of demand are the main axes of the energy transition with the objective of reducing emissions and dependence on fossil fuels. Modern electric power systems require a high level of system quality and security, supported by the ancillary services market. Therefore, the need to involve solar–wind renewable generation is fundamental in minimizing the use of conventional thermal generation, such as coal, gas, and oil.

It is important to clarify that different generation technologies with a rotational axis and that operate in a steady state exhibit dynamic behavior over time due to frequency variations. This means that the proposed methodology should be segmented into two parts, the first one being a technical methodology for those plants that use rotor and that can respond to frequency changes in different times at 10 s and later at 5 min depending on the level of stationarity for each generation technology and with technological complements such as grid forming/grid following inverters for solar generation. While the second methodology is of an economic nature, it aims to minimize the total costs of the reserves prior to selecting candidate plants for the reallocation of power reserves. Finally, to merge both technical and economic methodologies, it is necessary to categorize the generation plants according to the positioning of their variable cost of generation (VC) with respect to the marginal cost (MgC) of the system in real time (Figure 5). That is, the selection of these plants is categorized as Supramarginal (MgS), when the variable cost of generation of each plant is higher than the marginal cost of the system and Inframarginal (MgI), if the variable cost of generation is lower than the marginal cost, understanding that the trajectory of the marginal cost of the system is a consequence of the dispatch of plants due to the variation in the demand.

4.1. Technical Methodology Applied to the Dynamic Model of Power Reserve Response Time

This technical methodology considers the dynamic power response of each power plant due to its generation technology. Therefore, the variable of interest is the response time that allows the plant to deliver a given power, through a subdivision of primary reserves that act at different times from 10 s (PFC-10s) to 5 min (PFC-5min), in proportion to the percentages or dynamic response factors of each plant’s statistics (Figure 6). The dynamic reserve response at 10 s (PFC-10s) is intended to dampen frequency variations ranging from ±0.2 Hz to those exceeding ±0.7 Hz, thereby avoiding the activation of automatic load shedding schemes due to low frequency and generation shedding due to over frequency. In addition, the total activation time should be less than 10 s after the contingency has occurred. Therefore, the proposed technical methodology is defined by a total net reserve delivered by each generation plant by technology, based on its gross active power reserve and the dynamic response factor by droop, according to the generation technology, being mathematically expressed as follows:

$$\mathrm{T}\mathrm{N}\mathrm{R}{(\pm ) }_{\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}={\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{n}}\left[\mathrm{G}\mathrm{A}\mathrm{P}\mathrm{R}{(\pm ) }_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}\times \mathrm{D}\mathrm{F}{(\pm ) }_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}\right]}$$

(1)

where

• $\mathrm{T}\mathrm{N}\mathrm{R}{(\pm )}_{\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}$ represents the total net reserve of active power to up/down generation of each plant in the reallocation of reserve for primary frequency control at 10 s of activation.
• $\mathrm{G}\mathrm{A}\mathrm{P}\mathrm{R}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}$ represents the gross active power reserve for raising/lowering generation of each plant for primary frequency control at 10 s of activation without considering the dynamic power response factor.
• $\mathrm{D}\mathrm{F}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}$ represents a percentage or dynamic power response factor as a function of the droop settings for each generation technology at 10 s of activation.

While the dynamic reserve response at 5 min (PFC-5min), its objective is to dampen frequency variations exceeding ±0.7 Hz and recover demand by activation of automatic load shedding schemes for under frequency and reduce generation shedding for over frequency. In addition, the total delivery time should be less than 5 min after the contingency has occurred (Figure 6). This delivery time corresponds to the period in which the plants must be able to maintain the total reserve committed to complementary services, counted from the moment in which the total activation time (PFC-10s) elapsed, mathematically expressed as follows:

$$\mathrm{T}\mathrm{N}\mathrm{R}{(\pm )}_{\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}={\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{n}}\left[\mathrm{G}\mathrm{A}\mathrm{P}\mathrm{R}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}\times \mathrm{D}\mathrm{F}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}\right]}$$

(2)

where

• $\mathrm{T}\mathrm{N}\mathrm{R}{(\pm )}_{\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ represents the total net reserve of active power to up/down generation of each plant in the reallocation of reserve for primary frequency control with a maximum delivery time of 5 min.
• $\mathrm{G}\mathrm{A}\mathrm{P}\mathrm{R}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ represents the gross active power reserve for raising/lowering generation of each plant for primary frequency control with a maximum delivery time of 5 min, without considering the dynamic power response factor.
• $\mathrm{D}\mathrm{F}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ represents a percentage or dynamic power response factor as a function of the droop settings for each generation technology with a maximum delivery time of 5 min.

Figure 6. Structure of the technical methodology that establishes the ranges of the power reserves in the activation times of 10 s and the maximum delivery time of 5 min.

Figure 6. Structure of the technical methodology that establishes the ranges of the power reserves in the activation times of 10 s and the maximum delivery time of 5 min.

4.2. Economic Methodology That Minimizes the Total Costs of Power Reserve Reallocations

The economic methodology uses a dynamic hourly mathematical model that minimizes the total cost of power reserve reallocations for the ancillary services market in real-time primary frequency control. This methodology uses the renewable and conventional generation plants that are available to supply the demand in real time operation. The economic dispatch of all generation plants follows a merit list of variable generation costs. The candidate generation plants for the reallocation of power reserves for primary frequency control are categorized as supramarginal (MgS), when the variable generation cost of each plant is greater than the marginal cost of the system, while if the variable generation cost is less than the marginal cost, they are categorized as infra-marginal (MgI). Therefore, the variation in demand causes a displacement of the marginal cost, which immediately generates an opportunity cost for the economic merit list participants in the power reserve market of primary frequency control in the ancillary services market.

In other words, the methodology allows minimizing the total hourly cost of those plants selected as candidates to replace the missing power reserves in real-time primary frequency control, previously considering the dynamic response percentages according to the droop of each generation technology. A detailed description of the economic methodology specializes in the use of supramarginal and inframarginal candidate plants positioned in the vicinity of the marginal cost of the system, then the minimization of the costs of the reserves of each plant is calculated, prior technical methodology, and finally the power value of the reserve to be replaced is reallocated. This methodology must consider the time horizon on an hourly basis, since the demand curve and marginal costs shift hourly, and a new reallocation of power reserves is performed to validate the new candidate plants in the supramarginal and inframarginal conditions (Figure 7).

Mathematically, the economic methodology is a real-time economic model that minimizes the total cost of power reserve reallocations hourly for the new candidate plants selected as supramarginal and inframarginal for primary frequency control to ramp up and/or ramp down generation as indicated by the following equations:

$$\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )}=\mathrm{M}\mathrm{i}\mathrm{n}\left(\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}\mathrm{M}\mathrm{g}{\mathrm{S}}_{\mathrm{n},\mathrm{t}}},\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}\mathrm{M}\mathrm{g}{\mathrm{I}}_{\mathrm{n},\mathrm{t}}}\right)$$

(3)

$$\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}\mathrm{M}\mathrm{g}{\mathrm{S}}_{\mathrm{n},\mathrm{t}}}=\left(\mathrm{V}{\mathrm{C}}_{\mathrm{n}}\text{-}\mathrm{M}\mathrm{g}{\mathrm{C}}_{\mathrm{t}}\right)\times {\mathrm{R}}_{\mathrm{t}}\times \mathrm{T}$$

(4)

$$\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}\mathrm{M}\mathrm{g}{\mathrm{I}}_{\mathrm{n},\mathrm{t}}}=\left(\mathrm{M}\mathrm{g}{\mathrm{C}}_{\mathrm{t}}\text{-}\mathrm{V}{\mathrm{C}}_{\mathrm{n}}\right)\times {\mathrm{R}}_{\mathrm{t}}\times \mathrm{T}$$

(5)

where

• $\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )}$ represents the minimum total cost of net power reserve reallocation, considering the minimization of supra-/infra-marginal power plant reserve costs, for primary frequency control.
• $\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}\mathrm{M}\mathrm{g}{\mathrm{S}}_{\mathrm{n},\mathrm{t}}}$ represents the cost of the net power reserves of the n supramarginal power plants (MgS) in hour t.
• $\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}\mathrm{M}\mathrm{g}{\mathrm{I}}_{\mathrm{n},\mathrm{t}}}$ represents the cost of the net power reserves of the n infra-marginal power plants (MgI) in hour t.
• $\mathrm{M}\mathrm{g}{\mathrm{C}}_{\mathrm{t}}$ represents the marginal cost of the system in hour t.
• $\mathrm{V}{\mathrm{C}}_{\mathrm{n}}$ represents the variable cost of generating power at each power plant n by technology.
• ${\mathrm{R}}_{\mathrm{t}}$ represents the amount of net power reserve to be reallocated in hour t for each plant n by technology.
• $\mathrm{T}$ represents the number of hours of power reserve reallocation.

5. Modeling of Real Studies for the Reallocation of Power Reserves

To validate the technical-economic methodology, case studies from real-time operation are considered, where power reserve reallocations are applied for primary frequency control with real data [37], as shown in

Table 2. List of economic merit and power reserves according to the PFC dynamic factor(±).

Power Plant	Technology	Variable Cost	DF(±) PFC-10s	DF(±) PFC-5min	GAPR(±) PFC-10s (MW)	GAPR(±) PFC-5min (MW)
G1	Wind	0	0.46	0.46	5	5
G2	Solar	0	0.46	0.98	11	11
G3	Solar	0	0.13	0.63	3	12
G4	Hydro	0	0.22	1.0	3	12
G5	Wind	0	0.46	0.47	18	18
G6	Solar	0	0.84	0.99	27	27
G7	Hydro	0	0.89	1.0	31	35
G8	Hydro	0	0.90	1.0	32	36
G9	Solar	0	0.55	0.56	37	37
G10	Solar	0	0.77	0.64	47	45
G11	Hydro	0	0.24	0.96	11	45
G12	Solar	0	0.85	0.85	62	62
G13	Hydro	53,3	0.18	0.99	8	39
G14	Hydro	60.4	0.55	1.0	10	19
G15	Coal	60.7	0.35	0.46	9	17
G16	Coal	61.2	0.51	1.20	8	6
G17	Hydro	64.4	0.35	1.0	20	52
G18	Coal	66.0	0.85	1.0	14	16
G19	Hydro	66.2	0.73	1.0	19	26
G20	Coal	66.6	0.45	1.45	20	32
G21	Hydro	68.4	0.75	1.0	14	18
G22	Gas	69.9	0.56	0.67	37	44
G23	Gas	70.0	0.62	1.0	23	37
G24	Gas	72.0	0.54	0.63	17	20
G25	Gas	76.0	0.50	0.63	27	36
G26	Gas	80.1	0.5	0.63	14	18
G27	Gas	89.3	0.36	1.0	12	14
G28	Gas	104.7	0.84	1.0	18	22
G29	Gas	105.6	0.43	0.7	23	37
G30	Gas	108.6	0.85	1.0	17	20
G31	Gas	121.0	0.31	1.0	4	18
G32	Gas	121.1	0.20	0.24	2	16
G33	Gas	147.1	1.0	1.0	18	21
G34	Diesel	149.8	0.56	0.67	19	22
G35	Diesel	174.2	0.54	0.63	17	20
G36	Diesel	179.8	0.50	0.63	28	36
G37	Coal	181.6	0.28	0.60	5	10
G38	Diesel	182.5	0.50	0.63	28	36
G39	Coal	186.5	0.01	0.33	1	5
G40	Diesel	193.8	0.59	0.80	20	27
G41	Diesel	194.4	0.50	0.63	14	18
G42	Diesel	210.5	0.84	1.0	19	22
G43	Diesel	259.6	0.85	1.0	17	20
G44	Diesel	307.4	0.43	0.70	23	37
G45	Diesel	319.1	0.80	1.0	14	18

. The importance of this technical-economic methodology is its easy validation in any real or study power system, as long as the initial conditions of variable generation costs (VC), marginal cost changes (MgC) of the system, gross active power reserve, and dynamic response factors for the times of 10 s and 5 min, respectively, are used.

The regulations governing the ancillary services market aim to ensure optimal performance of electricity generation resources in terms of power quality and security. However, the high costs associated with the ancillary services market differ with the generation resources used, mainly due to the immediacy of the system operator or simply due to the lack of mathematical models that allow minimizing the total cost of power reserves. It is evident that in several Latin American countries, this ancillary services market is premature in several of its lines of research, with frequency control being one of the most striking.

Currently, real-time power reserve reallocations are solved with inefficient methods that increase the overall cost of system operation. The most usual method is called the “Maximum Power” method, which corresponds to allocating the reallocation of power reserves in those plants with a lower variable generation cost. The second method corresponds to the use of power plants with higher variable generation costs, called the “Technical Minimum” method. Both methods are based on the extreme use of plants in the economic merit list for the reallocation of power reserves, causing unnecessary displacement of the marginal cost and an exponential increase in the overall operating cost. In addition, there is a third method called “Random Direct Instruction”, which uses the generation plants randomly from the economic merit list to reallocate the power reserves of the primary frequency control.

However, the fourth method is named “Supra-/Infra-Marginal” and corresponds to the proposed work as a novel and efficient solution, because it uses plants from the economic merit list as candidates, named supramarginal (MgS) and inframarginal (MgI) due to the closeness between the variable cost of generation and the real-time marginal cost of the system. This method allows minimizing the total cost of reallocated reserves through the opportunity cost generated for each plant as the real marginal cost of the system shifts towards the variable generation cost of the candidate plants. Next, the validation of the model using traditional methods of reserve reallocation and the proposed methodology is presented, as shown in

Table 3. Most prominent methodology for real-time PFC(±) power reserve reallocations.

Method	Abbreviation	Status	Cost Reserve
Maximum Power	MP	Current	Non-Optimal
Technical Minimum	TM	Current	Non-Optimal
Random Direct Instruction	RDI	Current	Local Optimal
Supra/Infra-Marginal	MgS	Proposal	Global Optimal
	MgI	Proposal	Global Optimal

.

5.1. Real Case Study of PFC(±)10s for a Power Reserve Deficit

For this real case study, power reserve reallocation is required for real-time primary frequency control, which has a response time of 10 s in the PFC(±)10s. That is, if the real-time marginal cost of the system has a value of 80.1 USD/MWh corresponding to plant G26 (Gas) and at that same instant plants G20 (Coal), G21 (Hydro), G22 (Gas), G23 (Gas), G24 (Gas) and G25 (Gas) present technical problems for one hour and cannot commit their gross power reserve of 138 MW in total for the PFC(±)-10s, according to Table 2. Therefore, the system operator must reallocate power reserves in real time. To achieve power reserve reallocation, both traditional methods and the proposed method can be used, as indicated in Table 3. However, the total costs of power reserve reallocations differ considerably depending on the method used. These case studies validate the proposed methodology, as they demonstrate a reduction in total costs compared to traditional methods.

5.1.1. Maximum Power Method

This method applies power reserve reallocations with the use of plants with lower variable generation cost and that are generating at their maximum power capacity; therefore, the variable generation cost is lower than the real-time marginal cost of the system (80.1 USD/MWh). To replace a total gross reserve of 138 MW of the failed plants G20, G21, G22, G23, G24, and G25, the model equations of the technical methodology, according to Equation (1) and the economic methodology of Equation (5) must be considered. Subsequently, the plants with variable generation cost equal to zero are selected, with the highest $\mathrm{G}\mathrm{A}\mathrm{P}\mathrm{R}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}$ reserve capacities and their associated dynamic factor $\mathrm{D}\mathrm{F}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}$, according to Table 2. Therefore, the total reserve deficit of each fault plant, G20, G21, G22, G23, G24, and G25, is expressed as follows:

Calculation of the Total Net Power Reserve of Faulty Power Plants G20, G21, G22, G23, G24, and G25:

• G20 (Coal) = 20 × 0.45 = 9 MW
• G21 (Hydro) = 14 × 0.75 = 11 MW
• G22 (Gas) = 37 × 0.56 = 21 MW
• G23 (Gas) = 23 × 0.62 = 14 MW
• G24 (Gas) = 17 × 0.54 = 9 MW
• G25 (Gas) = 27 × 0.50 = 14 MW

Therefore, the total net reserve shortfall in the PFC(±)-10s equals 78 MW. In other words, reserve replacement using the maximum power methodology will be used. Calculation of the Total Net Power Reserve of the Plants to be Reallocated G1 to G12:

• G1 (Wind) = 5 × 0.46 = 2 MW
• G2 (Solar) = 11 × 0.46 = 5 MW
• G3 (Solar) = 3 × 0.13 = 0 MW
• G4 (Hydro) = 3 × 0.22 = 1 MW
• G5 (Wind) = 18 × 0.46 = 8 MW
• G6 (Solar) = 27 × 0.84 = 23 MW
• G7 (Hydro) = 31 × 0.89 = 28 MW
• G8 (Hydro) = 32 × 0.90 = 29 MW
• G9 (Solar) = 37 × 0.55 = 20 MW
• G10 (Solar) = 47 × 0.77 = 36 MW
• G11 (Hydro) = 11 × 0.24 = 3 MW
• G12 (Solar) = 62 × 0.85 = 53 MW

Therefore, to satisfy the 78 MW net power reserve deficit of the faulted plants G20 through G25, the candidate plants G1 through G12, which have a total net power reserve capacity of 208 MW, are considered. In other words, the solution for this methodology consists of a mathematical expression that allows minimizing the total costs in the net power reserve reallocations, in those plants that offer the lowest reallocation cost.

• $\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}10\mathrm{s}}$ = (80.1) × (G1 + G2 + G3 + G4 + G5 + G6 + G7 + G8 + G9 + G10 + G11 + G12)
• Subject to constraints.
• G1 + G2 + G3 + G4 + G5 + G6 + G7 + G8 + G9 + G10 + G11 + G12 ≤ 78 MW
• 0 ≤ G1 (Wind) ≤ 2 MW
• 0 ≤ G2 (Solar) ≤ 5 MW
• 0 ≤ G3 (Solar) ≤ 0 MW
• 0 ≤ G4 (Hydro) ≤ 1 MW
• 0 ≤ G5 (Wind) ≤ 8 MW
• 0 ≤ G6 (Solar) ≤ 23 MW
• 0 ≤ G7 (Hydro) ≤ 28 MW
• 0 ≤ G8 (Hydro) ≤ 29 MW
• 0 ≤ G9 (Solar) ≤ 20 MW
• 0 ≤ G10 (Solar) ≤ 36 MW
• 0 ≤ G11 (Hydro) ≤ 3 MW
• 0 ≤ G12 (Solar) ≤ 53 MW

From the results, the following net power reserves are obtained for each power plant, along with the lowest total cost of net power reserve reallocation.

• G1 (Wind) = 2 MW
• G2 (Solar) = 5 MW
• G3 (Solar) = 0 MW
• G4 (Hydro) = 1 MW
• G5 (Wind) = 8 MW
• G6 (Solar) = 23 MW
• G7 (Hydro) = 28 MW
• G8 (Hydro) = 11 MW
• G9 (Solar) = 0 MW
• G10 (Solar) = 0 MW
• G11 (Hydro) = 0 MW
• G12 (Solar) = 0 MW
• Total Cost $\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}10\mathrm{s}}$ = 6248 USD

5.1.2. Technical Minimum Method

This method involves the reallocation of power reserves by utilizing plants with higher variable generation costs that are operating at their minimum power or are out of service. Consequently, the variable generation cost exceeds the system’s marginal cost in real-time (80.1 USD/MWh). To replace a total gross reserve of 138 MW of the failed plants G20, G21, G22, G23, G24, and G25, the model equations of the technical methodology, according to Equation (1) and the economic methodology of Equation (4), must be considered. Subsequently, the plants with the highest variable generation costs are selected, along with the highest $\mathrm{G}\mathrm{A}\mathrm{P}\mathrm{R}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}$ reserve capacities and their associated dynamic factor $\mathrm{D}\mathrm{F}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}$, as shown in Table 2. Therefore, the total reserve deficits of each fault plant G20, G21, G22, G23, G24, and G25 are expressed as follows:

Calculation of the Total Net Power Reserve of Faulty Power Plants G20, G21, G22, G23, G24, and G25:

• G20 (Coal) = 20 × 0.45 = 9 MW
• G21 (Hydro) = 14 × 0.75 = 11 MW
• G22 (Gas) = 37 × 0.56 = 21 MW
• G23 (Gas) = 23 × 0.62 = 14 MW
• G24 (Gas) = 17 × 0.54 = 9 MW
• G25 (Gas) = 27 × 0.50 = 14 MW

Therefore, the total net reserve shortfall in the PFC(±)-10s equals 78 MW. In other words, reserve replacement will be conducted using the technical minimum methodology. Calculation of the Total Net Power Reserve of the Plants to be Reallocated G35 to G45:

• G35 (Diesel) = 17 × 0.54 = 9 MW
• G36 (Diesel) = 28 × 0.50 = 14 MW
• G37 (Coal) = 5 × 0.28 = 1 MW
• G38 (Diesel) = 28 × 0.50 = 14 MW
• G39 (Coal) = 1 × 0.01 = 0 MW
• G40 (Diesel) = 20 × 0.59 = 12 MW
• G41 (Diesel) = 14 × 0.50 = 7 MW
• G42 (Diesel) = 19 × 0.84 = 16 MW
• G43 (Diesel) = 17 × 0.85 = 14 MW
• G44 (Diesel) = 23 × 0.43 = 10 MW
• G45 (Diesel) = 14 × 0.80 = 11 MW

Therefore, to satisfy the deficit of the 78 MW of net power reserve of the failing plants from G20 to G25, the candidate plants G35 to G45 are considered, which have a total net power reserve capacity of 108 MW. In other words, the solution for this methodology consists of a mathematical expression that allows minimizing the total costs in the net power reserve reallocations, in those plants that offer the lowest reallocation cost.

• $\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}10\mathrm{s}}$ = (94.1G35 + 99.7G36 + 101.5G37 + 102.4G38 + 106.4G39 + 113.7G40 + 114.3G41 + 130.4G42 + 179.5G43 + 227.3G44 + 239.0G45)
• Subject to constraints.
• G35 + G36 + G37 + G38 + G39 + G40 + G41+ G42 + G43 + G44 + G45 ≤ 78 MW
• 0 ≤ G35 (Diesel) ≤ 9 MW
• 0 ≤ G36 (Diesel) ≤ 14 MW
• 0 ≤ G37 (Coal) ≤ 1 MW
• 0 ≤ G38 (Diesel) ≤ 14 MW
• 0 ≤ G39 (Coal) ≤ 0 MW
• 0 ≤ G40 (Diesel) ≤ 12 MW
• 0 ≤ G41 (Diesel) ≤ 7 MW
• 0 ≤ G42 (Diesel) ≤ 16 MW
• 0 ≤ G43 (Diesel) ≤ 14 MW
• 0 ≤ G44 (Diesel) ≤ 10 MW
• 0 ≤ G45 (Diesel) ≤ 11 MW

From the results, the following net power reserves are obtained for each power plant, along with the lowest total cost of net power reserve reallocation.

• G35 (Diesel) = 9 MW
• G36 (Diesel) = 14 MW
• G37 (Coal) = 1 MW
• G38 (Diesel) = 14 MW
• G39 (Coal) = 0 MW
• G40 (Diesel) = 12 MW
• G41 (Diesel) = 7 MW
• G42 (Diesel) = 16 MW
• G43 (Diesel) = 5 MW
• G44 (Diesel) = 0 MW
• G45 (Diesel) = 0 MW
• Total Cost $\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}10\mathrm{s}}$= 8926 USD

5.1.3. Random Direct Instruction Method

This method applies power reserve reallocations by randomly selecting plants from the economic merit list. That is, the candidate plants are those that are generating at maximum power, minimum power, and close to marginal cost in real time. In other words, the cost of the reserve of each of these plants varies according to their economic position in the random selection with respect to the real-time marginal cost of the system (80.1 USD/MWh). However, to replace the total gross reserve of 138 MW of the failing plants G20, G21, G22, G23, G24, and G25, the model equations of the technical methodology, according to Equation (1) and the economic methodology of Equations (4) and (5), respectively, must be considered. Subsequently, the plants with the highest $\mathrm{G}\mathrm{A}\mathrm{P}\mathrm{R}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}$ reserve capacities and their associated dynamic factor $\mathrm{D}\mathrm{F}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}$, are randomly selected, as shown in Table 2. Therefore, the total reserve deficits of each fault plant, G20, G21, G22, G23, G24, and G25, are expressed as follows:

Calculation of the Total Net Power Reserve of Faulty Power Plants G20, G21, G22, G23, G24, and G25:

• G20 (Coal) = 20 × 0.45 = 9 MW
• G21 (Hydro) = 14 × 0.75 = 11 MW
• G22 (Gas) = 37 × 0.56 = 21 MW
• G23 (Gas) = 23 × 0.62 = 14 MW
• G24 (Gas) = 17 × 0.54 = 9 MW
• G25 (Gas) = 27 × 0.50 = 14 MW

Therefore, the total net reserve shortfall in the PFC(±)-10s equals 78 MW. In other words, reserve replacement with random direct instruction methodology will be used. Calculation of the Total Net Power Reserve of the Plants to be Reallocated from G7–G10, G16–G19, and G42–G45:

• G7 (Hydro) = 31 × 0.89 = 28 MW
• G8 (Hydro) = 32 × 0.90 = 29 MW
• G9 (Solar) = 37 × 0.55 = 20 MW
• G10 (Solar) = 47 × 0.77 = 36 MW
• G16 (Coal) = 8 × 0.51 = 4 MW
• G17 (Hydro) = 20 × 0.35 = 7 MW
• G18 (Coal) = 14 × 0.85 = 12 MW
• G19 (Hydro) = 19 × 0.73 = 14 MW
• G42 (Diesel) = 19 × 0.84 = 16 MW
• G43 (Diesel) = 17 × 0.85 = 14 MW
• G44 (Diesel) = 23 × 0.43 = 10 MW
• G45 (Diesel) = 14 × 0.80 = 11 MW

Therefore, to satisfy the shortfall of the 78 MW of net power reserve of the failing plants from G20 to G25, the candidate plants from G7–G10, G16–G19, and G42–G45 are considered, which have a total net power reserve capacity of 201 MW. In other words, the solution for this methodology consists of a mathematical expression that allows minimizing the total costs in the net power reserve reallocations, in those plants that offer the lowest reallocation cost.

• $\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}10\mathrm{s}}$ = (80.1G7 + 80.1G8 + 80.1G9 + 80.1G10 + 18.9G16 + 15.7G17 + 14.1G18 + 13.9G19 + 130.4G42 + 179.5G43 + 227.3G44 + 239.0G45)
• Subject to constraints.
• G7 + G8 + G9 + G10 + G16 + G17 + G18 + G19 + G42 + G43 + G44 + G45 ≤ 78 MW
• 0 ≤ G7 (Hydro) ≤ 28 MW
• 0 ≤ G8 (Hydro) ≤ 29 MW
• 0 ≤ G9 (Solar) ≤ 20 MW
• 0 ≤ G10 (Solar) ≤ 36 MW
• 0 ≤ G16 (Coal) ≤ 4 MW
• 0 ≤ G17 (Hydro) ≤ 7 MW
• 0 ≤ G18 (Coal) ≤ 12 MW
• 0 ≤ G19 (Hydro) ≤ 14 MW
• 0 ≤ G42 (Diesel) ≤ 16 MW
• 0 ≤ G43 (Diesel) ≤ 14 MW
• 0 ≤ G44 (Diesel) ≤ 10 MW
• 0 ≤ G45 (Diesel) ≤ 11 MW

From the results, the following net power reserves are obtained for each plant, along with the lowest total cost of net power reserve reallocation.

• G7 (Hydro) = 0 MW
• G8 (Hydro) = 29 MW
• G9 (Solar) = 12 MW
• G10 (Solar) = 0 MW
• G16 (Coal) = 4 MW
• G17 (Hydro) = 7 MW
• G18 (Coal) = 12 MW
• G19 (Hydro) = 14 MW
• G42 (Diesel) = 0 MW
• G43 (Diesel) = 0 MW
• G44 (Diesel) = 0 MW
• G45 (Diesel) = 0 MW
• Total Cost $\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}10\mathrm{s}}$ = 3833 USD

5.1.4. Supra-/Infra-Marginal Method

This method corresponds to the proposal of this work and is applied in the reallocation of power reserves with the use of plants from the economic merit list categorized as supramarginal, when the variable generation cost of each plant is higher than the marginal cost of the system, and inframarginal, when the variable generation cost is lower than the marginal cost. In other words, the supramarginal and inframarginal candidate plants are those that are generating in the vicinity of the real-time marginal cost. In other words, the reserve cost of each of these plants becomes an opportunity cost as the economic shift in the real-time system marginal cost (80.1 USD/MWh) approaches each candidate plant. However, to replace the total gross reserve of 138 MW of the failing plants G20, G21, G22, G23, G24, and G25, the model equations of the technical methodology, according to Equation (1) and the economic methodology of Equations (4) and (5), respectively, must be considered. Subsequently, the supramarginal and inframarginal candidate plants are selected close to the real-time marginal cost, with the highest $\mathrm{G}\mathrm{A}\mathrm{P}\mathrm{R}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}$ reserve capacities and their associated dynamic factor $\mathrm{D}\mathrm{F}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}10\mathrm{s}}$, as shown in Table 2. Therefore, the total reserve shortfall of each fault plant, G20, G21, G22, G23, G24, and G25, is expressed as follows:

Calculation of the Total Net Power Reserve of Faulty Power Plants G20, G21, G22, G23, G24, and G25:

• G20 (Coal) = 20 × 0.45 = 9 MW
• G21 (Hydro) = 14 × 0.75 = 11 MW
• G22 (Gas) = 37 × 0.56 = 21 MW
• G23 (Gas) = 23 × 0.62 = 14 MW
• G24 (Gas) = 17 × 0.54 = 9 MW
• G25 (Gas) = 27 × 0.50 = 14 MW

Therefore, the total net reserve shortfall in the PFC(±)-10s equals 78 MW. In other words, reserve replacement using the supra-/infra-marginal methodology will be used. Calculation of the Total Net Power Reserve of the Plants to be Reallocated from G14–G19 and G27–G32:

• G14 (Hydro) = 10 × 0.55 = 6 MW
• G15 (Coal) = 9 × 0.35 = 3 MW
• G16 (Coal) = 8 × 0.51 = 4 MW
• G17 (Hydro) = 20 × 0.35 = 7 MW
• G18 (Coal) = 14 × 0.85 = 12 MW
• G19 (Hydro) = 19 × 0.73 = 14 MW
• G27 (Gas) = 12 × 0.36 = 4 MW
• G28 (Gas) = 18 × 0.84 = 15 MW
• G29 (Gas) = 23 × 0.43 = 10 MW
• G30 (Gas) = 17 × 0.85 = 14 MW
• G31 (Gas) = 4 × 0.31 = 1 MW
• G32 (Gas) = 2 × 0.20 = 0 MW

Therefore, to meet the shortfall of the 78 MW of net power reserve of the failing plants from G20 to G25, the candidate plants from G14–G19 and G27–G32 are considered, which have a total net power reserve capacity of 90 MW. In other words, the solution for this methodology consists of a mathematical expression that allows minimizing the total costs in the net power reserve reallocations, in those plants that offer the lowest reallocation cost.

• $\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}10\mathrm{s}}$ = (19.71G14 + 19.4G15 + 18.9G16 + 15.7G17 + 14.1G18 + 13.9G19 + 9.2G27 + 24.6G28 + 25.5G29 + 28.5G30 + 40.9G31 + 41.0G32)
• Subject to constraints.
• G14 + G15 + G16 + G17 + G18 + G19 + G27 + G28 + G29 + G30 + G31 + G32 ≤ 78 MW
• 0 ≤ G14 (Hydro) ≤ 6 MW
• 0 ≤ G15 (Coal) ≤ 3 MW
• 0 ≤ G16 (Coal) ≤ 4 MW
• 0 ≤ G17 (Hydro) ≤ 7 MW
• 0 ≤ G18 (Coal) ≤ 12 MW
• 0 ≤ G19 (Hydro) ≤ 14 MW
• 0 ≤ G27 (Gas) ≤ 4 MW
• 0 ≤ G28 (Gas) ≤ 15 MW
• 0 ≤ G29 (Gas) ≤ 10 MW
• 0 ≤ G30 (Gas) ≤ 14 MW
• 0 ≤ G31 (Gas) ≤ 1 MW
• 0 ≤ G32 (Gas) ≤ 0 MW

From the results, the following net power reserves are obtained for each power plant, along with the lowest total cost of net power reserve reallocation.

• G14 (Hydro) = 6 MW
• G15 (Coal) = 3 MW
• G16 (Coal) = 4 MW
• G17 (Hydro) = 7 MW
• G18 (Coal) = 12 MW
• G19 (Hydro) = 14 MW
• G27 (Gas) = 4 MW
• G28 (Gas) = 15 MW
• G29 (Gas) = 10 MW
• G30 (Gas) = 3 MW
• G31 (Gas) = 0 MW
• G32 (Gas) = 0 MW
• Total Cost $\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}10\mathrm{s}}$ = 1472 USD

5.2. Real Case Study of PFC(±)-5min for Power Reserve Deficits

For this real case study, power reserve reallocation is required for real-time primary frequency control, which has a response time of 5 min in the PFC(±)5min. That is, if the real-time marginal cost of the system has a value of 147.1 USD/MWh corresponding to plant G33 (Gas) and at that same instant plants G14 (Hydro), G15 (Coal), G16 (Coal), G17 (Hydro), G18 (Coal) and G19 (Hydro) present technical problems for one hour and cannot commit their gross power reserve of 136 MW in total for the PFC(±)-5min, according to Table 2. Therefore, the system operator must reallocate power reserves in real time. To achieve power reserve reallocation, both traditional methods and the proposed method can be used, as indicated in Table 3. However, the total costs of power reserve reallocations differ considerably depending on the method used. These case studies validate the proposed methodology, as they demonstrate a reduction in total costs compared to traditional methods.

5.2.1. Maximum Power Method

This method applies the reallocation of power reserves with the use of plants with lower variable generation cost and that are generating at their maximum power capacity; therefore, the variable generation cost is lower than the marginal cost of the system in real time (147.1 USD/MWh). To replace a total gross reserve of 136 MW of the failed plants G14, G15, G16, G17, G18, and G19, the model equations of the technical methodology, according to Equation (2) and the economic methodology of Equation (5), must be considered. Subsequently, the plants with variable generation cost equal to zero are selected, with the highest $\mathrm{G}\mathrm{A}\mathrm{P}\mathrm{R}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ reserve capacities and their associated dynamic factor $\mathrm{D}\mathrm{F}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$, as shown in Table 2. Therefore, the total reserve deficit of each fault plant G14, G15, G16, G17, G18, and G19 is expressed as follows:

Calculation of the Total Net Power Reserve of Faulty Power Plants G14, G15, G16, G17, G18, and G19:

• G14 (Hydro) = 19 × 1.00 = 19 MW
• G15 (Coal) = 17 × 0.46 = 8 MW
• G16 (Coal) = 6 × 1.20 = 7 MW
• G17 (Hydro) = 52 × 1.00 = 52 MW
• G18 (Coal) = 16 × 1.00 = 16 MW
• G19 (Hydro) = 26 × 1.00 = 26 MW

Therefore, the total net reserve shortfall in the PFC(±)5min equals 128 MW. In other words, reserve replacement using the maximum power methodology will be used. Calculation of the Total Net Power Reserve of the Plants to be Reallocated G1 to G12:

• G1 (Wind) = 5 × 0.46 = 2 MW
• G2 (Solar) = 11 × 0.98 = 11 MW
• G3 (Solar) = 12 × 0.63 = 8 MW
• G4 (Hydro) = 12 × 1.00 = 12 MW
• G5 (Wind) = 18 × 0.47 = 8 MW
• G6 (Solar) = 27 × 0.99 = 27 MW
• G7 (Hydro) = 35 × 1.00 = 35 MW
• G8 (Hydro) = 36 × 1.00 = 36 MW
• G9 (Solar) = 37 × 0.56 = 21 MW
• G10 (Solar) = 45 × 0.64 = 29 MW
• G11 (Hydro) = 45 × 0.96 = 43 MW
• G12 (Solar) = 62 × 0.85 = 53 MW

Therefore, to satisfy the deficit of the 128 MW of net power reserve of the failing plants from G14 to G19, the candidate plants G1 to G12 are considered, which have a total net power reserve capacity of 285 MW. In other words, the solution for this methodology consists of a mathematical expression that allows minimizing the total costs in the net power reserve reallocations, in those plants that offer the lowest reallocation cost.

• $\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ = (147.1) × (G1 + G2 + G3 + G4 + G5 + G6 + G7 + G8 + G9 + G10 + G11 + G12)
• Subject to constraints.
• G1 + G2 + G3 + G4 + G5 + G6 + G7 + G8 + G9 + G10 + G11 + G12 ≤ 128 MW
• 0 ≤ G1 (Wind) ≤ 2 MW
• 0 ≤ G2 (Solar) ≤ 11 MW
• 0 ≤ G3 (Solar) ≤ 8 MW
• 0 ≤ G4 (Hydro) ≤ 12 MW
• 0 ≤ G5 (Wind) ≤ 8 MW
• 0 ≤ G6 (Solar) ≤ 27 MW
• 0 ≤ G7 (Hydro) ≤ 35 MW
• 0 ≤ G8 (Hydro) ≤ 36 MW
• 0 ≤ G9 (Solar) ≤ 21 MW
• 0 ≤ G10 (Solar) ≤ 29 MW
• 0 ≤ G11 (Hydro) ≤ 43 MW
• 0 ≤ G12 (Solar) ≤ 53 MW

From the results, the following net power reserves are obtained for each power plant, along with the lowest total cost of net power reserve reallocation.

• G1 (Wind) = 2 MW
• G2 (Solar) = 11 MW
• G3 (Solar) = 8 MW
• G4 (Hydro) = 12 MW
• G5 (Wind) = 8 MW
• G6 (Solar) = 27 MW
• G7 (Hydro) = 35 MW
• G8 (Hydro) = 25 MW
• G9 (Solar) = 0 MW
• G10 (Solar) = 0 MW
• G11 (Hydro) = 0 MW
• G12 (Solar) = 0 MW
• Total Cost $\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ = 18,829 USD

5.2.2. Technical Minimum Method

This method applies power reserve reallocations with the use of plants with higher variable generation costs that are operating at their minimum power or are out of service; therefore, the variable generation cost exceeds the real-time marginal cost of the system (147.1 USD/MWh). To replace a total gross reserve of 136 MW of the failed plants G14, G15, G16, G17, G18, and G19, the model equations of the technical methodology, according to Equation (2) and the economic methodology of Equation (4), must be considered. Subsequently, the plants with the highest variable generation cost are selected, with the highest $\mathrm{G}\mathrm{A}\mathrm{P}\mathrm{R}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ reserve capacities and their associated dynamic factor $\mathrm{D}\mathrm{F}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$, according to Table 2. Therefore, the total reserve deficits of each fault plant, G14, G15, G16, G17, G18, and G19, are expressed as follows:

Calculation of the Total Net Power Reserve of Faulty Power Plants G14, G15, G16, G17, G18, and G19:

• G14 (Hydro) = 19 × 1.00 = 19 MW
• G15 (Coal) = 17 × 0.46 = 8 MW
• G16 (Coal) = 6 × 1.20 = 7 MW
• G17 (Hydro) = 52 × 1.00 = 52 MW
• G18 (Coal) = 16 × 1.00 = 16 MW
• G19 (Hydro) = 26 × 1.00 = 26 MW

Therefore, the total net reserve shortfall in the PFC(±)5min equals 128 MW. In other words, reserve replacement will be conducted using the technical minimum methodology. Calculation of the Total Net Power Reserve of the Plants to be Reallocated G34 to G45:

• G34 (Diesel) = 22 × 0.67 = 15 MW
• G35 (Diesel) = 20 × 0.63 = 13 MW
• G36 (Diesel) = 36 × 0.63 = 23 MW
• G37 (Coal) = 10 × 0.60 = 6 MW
• G38 (Diesel) = 36 × 0.63 = 23 MW
• G39 (Coal) = 5 × 0.33 = 2 MW
• G40 (Diesel) = 27 × 0.80 = 22 MW
• G41 (Diesel) = 18 × 0.63 = 11 MW
• G42 (Diesel) = 22 × 1.00 = 22 MW
• G43 (Diesel) = 20 × 1.00 = 20 MW
• G44 (Diesel) = 37 × 0.70 = 26 MW
• G45 (Diesel) = 18 × 1.00 = 18 MW

Therefore, to satisfy the deficit of the 128 MW of net power reserve of the failing plants from G20 to G25, the candidate plants G34 to G45 are considered, which have a total net power reserve capacity of 201 MW. In other words, the solution for this methodology consists of a mathematical expression that allows minimizing the total costs in the net power reserve reallocations, in those plants that offer the lowest reallocation cost.

• $\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ = (2.7G34 + 27.1G35 + 32.7G36 + 34.5G37 + 35.4G38 + 39.4G39 + 46.7G40 + 47.3G41 + 63.4G42 + 112.5G43 + 160.3G44 + 172.0G45)
• Subject to constraints.
• G34 + G35 + G36 + G37 + G38 + G39 + G40 + G41+ G42 + G43 + G44 + G45 ≤ 128 MW
• 0 ≤ G34 (Diesel) ≤ 15 MW
• 0 ≤ G35 (Diesel) ≤ 13 MW
• 0 ≤ G36 (Diesel) ≤ 23 MW
• 0 ≤ G37 (Coal) ≤ 6 MW
• 0 ≤ G38 (Diesel) ≤ 23 MW
• 0 ≤ G39 (Coal) ≤ 2 MW
• 0 ≤ G40 (Diesel) ≤ 22 MW
• 0 ≤ G41 (Diesel) ≤ 11 MW
• 0 ≤ G42 (Diesel) ≤ 22 MW
• 0 ≤ G43 (Diesel) ≤ 20 MW
• 0 ≤ G44 (Diesel) ≤ 26 MW
• 0 ≤ G45 (Diesel) ≤ 18 MW

From the results, the following net power reserves are obtained for each power plant, along with the lowest total cost of net power reserve reallocation.

• G34 (Diesel) = 15 MW
• G35 (Diesel) = 13 MW
• G36 (Diesel) = 23 MW
• G37 (Coal) = 6 MW
• G38 (Diesel) = 23 MW
• G39 (Coal) = 2 MW
• G40 (Diesel) = 22 MW
• G41 (Diesel) = 11 MW
• G42 (Diesel) = 13 MW
• G43 (Diesel) = 0 MW
• G44 (Diesel) = 0 MW
• G45 (Diesel) = 0 MW
• Total Cost $\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ = 4617 USD.

5.2.3. Random Direct Instruction Method

This method applies power reserve reallocations by randomly selecting plants from the economic merit list. That is, the candidate plants are those that are generating at maximum power, minimum power, and close to marginal cost in real time. In other words, the cost of the reserve of each of these plants varies according to their economic position in the random selection with respect to the real-time marginal cost of the system (147.1 USD/MWh). However, to replace the total gross reserve of 136 MW of the failing plants G14, G15, G16, G17, G18, and G19, the model equations of the technical methodology, according to Equation (2) and the economic methodology of Equations (4) and (5), respectively, must be considered. Subsequently, the plants with the highest $\mathrm{G}\mathrm{A}\mathrm{P}\mathrm{R}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ reserve capacities and their associated dynamic factor $\mathrm{D}\mathrm{F}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ are randomly selected, as shown in Table 2. Therefore, the total reserve deficit of each fault plant, G14, G15, G16, G17, G18, and G19, is expressed as follows.

Calculation of the Total Net Power Reserve of Faulted Power Plants G14, G15, G16, G17, G18, and G19:

• G14 (Hydro) = 19 × 1.00 = 19 MW
• G15 (Coal) = 17 × 0.46 = 8 MW
• G16 (Coal) = 6 × 1.20 = 7 MW
• G17 (Hydro) = 52 × 1.00 = 52 MW
• G18 (Coal) = 16 × 1.00 = 16 MW
• G19 (Hydro) = 26 × 1.00 = 26 MW

Therefore, the total net reserve shortfall in the PFC(±)5min equals 128 MW. That is, reserve replacement with random direct instruction methodology will be used. Calculation of the Total Net Power Reserve of the Plants to be Reallocated from G7–G10, G31–G32, G34–G35, and G42–G45:

• G7 (Hydro) = 35 × 1.00 = 35 MW
• G8 (Hydro) = 36 × 1.00 = 36 MW
• G9 (Solar) = 37 × 0.56 = 21 MW
• G10 (Solar) = 45 × 0.64 = 29 MW
• G31 (Gas) = 18 × 1.00 = 18 MW
• G32 (Diesel) = 16 × 0.24 = 4 MW
• G34 (Diesel) = 22 × 0.67 = 15 MW
• G35 (Hydro) = 20 × 0.63 = 13 MW
• G42 (Diesel) = 22 × 1.00 = 22 MW
• G43 (Diesel) = 20 × 1.00 = 20 MW
• G44 (Diesel) = 37 × 0.70 = 26 MW
• G45 (Diesel) = 18 × 1.00 = 18 MW

Therefore, to satisfy the shortfall of the 128 MW of net power reserve of the failing plants from G14 to G19, the candidate plants from G7–G10, G31–G32, G34–G35, and G42–G45 are considered, which have a total net power reserve capacity of 257 MW. In other words, the solution for this methodology consists of a mathematical expression that allows minimizing the total costs in the net power reserve reallocations, in those plants that offer the lowest reallocation cost.

• $\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ = (141.7G7 + 147.1G8 + 147.1G9 + 147.1G10 + 26.1G31 + 26.0G32 + 2.7G34 + 27.1G35 + 63.4G42 + 112.5G43 + 160.3G44 + 172.0G45)
• Subject to constraints.
• G7 + G8 + G9 + G10 + G31 + G32 + G34 + G35 + G42 + G43 + G44 + G45 ≤ 128 MW
• 0 ≤ G7 (Hydro) ≤ 35 MW
• 0 ≤ G8 (Hydro) ≤ 36 MW
• 0 ≤ G9 (Solar) ≤ 21 MW
• 0 ≤ G10 (Solar) ≤ 29 MW
• 0 ≤ G31 (Gas) ≤ 18 MW
• 0 ≤ G32 (Diesel) ≤ 4 MW
• 0 ≤ G34 (Diesel) ≤ 15 MW
• 0 ≤ G35 (Diesel) ≤ 13 MW
• 0 ≤ G42 (Diesel) ≤ 22 MW
• 0 ≤ G43 (Diesel) ≤ 20 MW
• 0 ≤ G44 (Diesel) ≤ 26 MW
• 0 ≤ G45 (Diesel) ≤ 18 MW

From the results, the following net power reserves are obtained for each power plant, along with the lowest total cost of net power reserve reallocation.

• G7 (Hydro) = 0 MW
• G8 (Hydro) = 0 MW
• G9 (Solar) = 7 MW
• G10 (Solar) = 29 MW
• G31 (Gas) = 18 MW
• G32 (Gas) = 4 MW
• G34 (Diesel) = 15 MW
• G35 (Diesel) = 13 MW
• G42 (Diesel) = 22 MW
• G43 (Diesel) = 20 MW
• G44 (Diesel) = 0 MW
• G45 (Diesel) = 0 MW
• Total Cost $\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ = 9907 USD

5.2.4. Supra-/Infra-Marginal Method

This method corresponds to the proposal of this work and is applied in power reserve reallocations with the use of plants from the economic merit list categorized as supramarginal, when the variable generation cost of each plant is higher than the marginal cost of the system, and inframarginal, when the variable generation cost is lower than the marginal cost. In other words, the supramarginal and inframarginal candidate plants are those that are generating in the vicinity of the real-time marginal cost. In other words, the reserve cost of each of these plants becomes an opportunity cost as the economic shift in the real-time system marginal cost (147.1 USD/MWh) approaches each candidate plant. However, to replace the total gross reserve of 136 MW of the failing plants G14, G15, G16, G17, G18, and G19, the model equations of the technical methodology, according to Equation (2) and the economic methodology of Equations (4) and (5), respectively, must be considered. Subsequently, the supramarginal and inframarginal candidate plants are selected close to the real-time marginal cost, with the highest $\mathrm{G}\mathrm{A}\mathrm{P}\mathrm{R}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ reserve capacities and their associated dynamic factor $\mathrm{D}\mathrm{F}{(\pm )}_{\mathrm{n}\text{-}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}^{\mathrm{P}\mathrm{F}\mathrm{C}\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$, according to Table 2. Therefore, the total reserve deficit of each fault plant, G14, G15, G16, G17, G18, and G19, is expressed as follows.

Calculation of the Total Net Power Reserve of Faulty Power Plants G14, G15, G16, G17, G18, and G19:

• G14 (Hydro) = 19 × 1.00 = 19 MW
• G15 (Coal) = 17 × 0.46 = 8 MW
• G16 (Coal) = 6 × 1.20 = 7 MW
• G17 (Hydro) = 52 × 1.00 = 52 MW
• G18 (Coal) = 16 × 1.00 = 16 MW
• G19 (Hydro) = 26 × 1.00 = 26 MW

Therefore, the total net reserve shortfall in the PFC(±)5min equals 128 MW. In other words, reserve replacement will be conducted using the supra-/infra-marginal methodology. Calculation of the Total Net Power Reserve of the Plants to be Reallocated from G27–G32 and G34–G39:

• G27 (Gas) = 14 × 1.00 = 14 MW
• G28 (Gas) = 22 × 1.00 = 22 MW
• G29 (Gas) = 37 × 0.70 = 26 MW
• G30 (Gas) = 20 × 1.00 = 20 MW
• G31 (Gas) = 18 × 1.00 = 18 MW
• G32 (Diesel) = 16 × 0.24 = 4 MW
• G34 (Diesel) = 22 × 0.67 = 15 MW
• G35 (Diesel) = 20 × 0.63 = 13 MW
• G36 (Diesel) = 36 × 0.63 = 23 MW
• G37 (Coal) = 10 × 0.60 = 6 MW
• G38 (Diesel) = 36 × 0.63 = 23 MW
• G39 (Coal) = 5 × 0.33 = 2 MW

Therefore, to satisfy the shortfall of the 128 MW of net power reserve of the failing plants from G14 to G19, the candidate plants from G27 to G32 and G34 to G39 are considered, which have a total net power reserve capacity of 186 MW. In other words, the solution for this methodology consists of a mathematical expression that allows minimizing the total costs in the net power reserve reallocations, in those plants that offer the lowest reallocation cost.

• $\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ = (57.8G27 + 42.4G28 + 41.5G29 + 38.5G30 + 26.1G31 + 26.0G32 + 2.7G34 + 27.1G35 + 32.4G36 + 34.5G37 + 35.4G38 + 39.4G39)
• Subject to constraints.
• G27 + G28 + G29 + G30 + G31 + G32 + G34 + G35 + G36 + G37 + G38 + G39 ≤ 128 MW
• 0 ≤ G27 (Gas) ≤ 14 MW
• 0 ≤ G28 (Gas) ≤ 22 MW
• 0 ≤ G29 (Gas) ≤ 26 MW
• 0 ≤ G30 (Gas) ≤ 20 MW
• 0 ≤ G31 (Gas) ≤ 18 MW
• 0 ≤ G32 (Diesel) ≤ 4 MW
• 0 ≤ G34 (Diesel) ≤ 15 MW
• 0 ≤ G35 (Diesel) ≤ 13 MW
• 0 ≤ G36 (Diesel) ≤ 23 MW
• 0 ≤ G37 (Coal) ≤ 6 MW
• 0 ≤ G38 (Diesel) ≤ 23 MW
• 0 ≤ G39 (Coal) ≤ 2 MW

From the results, the following net power reserves are obtained for each power plant, along with the lowest total cost of net power reserve reallocation.

• G27 (Gas) = 0 MW
• G28 (Gas) = 0 MW
• G29 (Gas) = 4 MW
• G30 (Gas) = 20 MW
• G31 (Gas) = 18 MW
• G32 (Gas) = 4 MW
• G34 (Diesel) = 15 MW
• G35 (Diesel) = 13 MW
• G36 (Diesel) = 23 MW
• G37 (Coal) = 6 MW
• G38 (Diesel) = 23 MW
• G39 (Coal) = 2 MW
• Total Cost $\mathrm{T}{\mathrm{C}}_{\mathrm{P}\mathrm{F}\mathrm{C}(\pm )\text{-}5\mathrm{m}\mathrm{i}\mathrm{n}}$ = 3755 USD

6. Results and Analysis of Power Reserve Reallocation Methodologies

The results of actual case studies, which involve real-time power reserve reallocations with dynamic primary frequency control responses for 10 s and 5 min, are presented below. Four methodologies of power reserve reallocations are used in each case study: The first is the “maximum power” methodology, and the second is the “technical minimum” methodology. These methodologies are the most commonly used by system operators due to the ease of selecting candidate plants in terms of reallocation time. However, they cause a severe displacement of the marginal cost, an increase in the total cost of reserve reallocation, and an overall increase in the operational cost. Third, there is the “random direct instruction” methodology, which enables the system operator to reallocate power reserves at a lower total cost than the other methodologies of “maximum power” and “technical minimum”. Unfortunately, this methodology is not as effective because the random selection may be concentrated in the candidate plants farthest away from the real-time marginal cost. Finally, in fourth place is the proposed “supra-/infra-marginal” methodology, which allows for reducing the total costs of power reserve reallocations compared to other methodologies, by selecting candidate plants that move in the vicinity of the marginal reserve in real time.

6.1. Analysis of the Real Case Study of PFC(±)10s

For this real case study, the four methodologies of power reserve reallocation for primary frequency control with dynamic response at 10 s showed differentiated costs, as shown in Figure 8. For the case of power reserve reallocation with the “maximum power” methodology, the total cost is USD6248, with the use of candidate plants with zero variable cost, highlighting that the highest net reserve contribution is for plant G7 (Hydro) = 28 MW, G6 (Solar) = 23 MW, and G8 (Hydro) = 11 MW. However, with the “technical minimum” methodology, candidate plants with the highest variable generation costs are used, highlighting the plants with the highest net reserve contribution for G36 (Diesel) = 14 MW, G38 (Diesel) = 14 MW and G42 (Diesel) = 16 MW, with a total cost of 8926 USD, higher than the rest of the other methodologies. While in the “random direct instruction” methodology, the total reserve costs decrease considerably to USD 3833, and the highest net power reserve contributions correspond to the G8 (Hydro) = 29 MW, G19 (Hydro) = 14 MW, and G9 (Solar) = 12 MW plants. Finally, with the proposed “supra-/infra-marginal” methodology, the total cost of reserve reallocation is USD1472, which shows that this methodology allows obtaining the minimum cost of power reserve reallocation compared to the other methodologies. In addition, the highest net power reserve contributions for the “supra-/infra-marginal” methodology correspond to the G28 (Gas) = 15 MW, G19 (Hydro) = 14 MW, and G18 (Coal) = 12 MW plants.

6.2. Analysis of the PFC(±)5min Real Case Study

For this real case study, the four methodologies for power reserve reallocation in primary frequency control with dynamic response at 5 min showed different costs, as shown in Figure 9. For the case of power reserve reallocation with the “maximum power” methodology, the total cost is USD 18,829 and corresponds to the highest cost with respect to the other methodologies, due to the fact that the candidate plants are far from the marginal cost in real time (147.1 USD/MWh), despite the fact that the use of these candidate plants have zero variable cost, highlighting that the highest net reserve contribution is for the G7 (Hydro) = 35 MW, G6 (Solar) = 27 MW and G8 (Hydro) = 25 MW plants. However, with the “technical minimum” methodology, candidate plants with the highest variable generation costs are used, highlighting the plants with the highest net reserve contribution for G36 (Diesel) = 23 MW, G38 (Diesel) = 23 MW and G40 (Diesel) = 22 MW, with a total cost of 4617 USD, well below with respect to the “maximum power” methodology, due to the selection of plants of this methodology are close to the marginal cost in real time. While using the “random direct instruction” methodology, the total reserve costs increase to USD 9907, and the largest contributions of net power reserves correspond to the G10 (Solar) = 29 MW, G42 (Diesel) = 22 MW, and G43(Diesel) = 20 MW plants. This “random direct instruction” methodology does not guarantee the minimum cost of reallocation of power reserves, due to the random nature of the candidate plants that are far from the marginal cost in real-time. Finally, with the proposed “supra/infra-marginal” methodology, the total cost of reserve reallocation is USD3755, which shows that this methodology allows for obtaining the minimum cost of power reserve reallocation compared to the other methodologies. In addition, the highest net power reserve contributions correspond to the G36 (Diesel) = 23 MW, G38 (Diesel) = 23 MW, and G30 (Gas) = 20 MW plants.

7. Conclusions of Results and Future Work

In this work, the main advances and contributions were discussed, identifying advantages, disadvantages, and what still needs to be developed in the real-time ancillary services market, using real case studies for power reserve reallocations with the use of conventional electric generation and the relevance of integrating solar–wind renewable generation in the primary frequency control. The technical-economic methodology used for real-time power reserve reallocations allows for reducing costs, even with the participation of conventional generation that holds a monopolistic role in this market. However, renewable generation has been able to displace conventional generation for the most part and successfully integrate into the ancillary services market and the energy market. Furthermore, the development of this technical-economic model for there al-time ancillary services market enables remunerating and making profitable all generation plants that have an opportunity cost as the real-time marginal cost shifts towards the variable generation cost of each plant.

7.1. Validation of the Technical-Economic Model with Analysis and Discussion of Results

The technical-economic model for power reserve reallocations for primary frequency control is reliable and flexible. It allows the system operator to make decisions, overlapping traditional methods such as the maximum power method, the technical minimum method, and even the random direct instruction method. It is important to highlight that the capacity of each generation plant to respond with the power reserve in a dynamic response time of 10 s and 5 min varies according to the type of technology. However, with the proposed supra/infra-marginal methodology for the actual PFC(±)10s case study that considers a gross power reserve shortfall of 138 MW and a real-time marginal cost of 80.1 USD/MWh, the total costs of power reserve reallocations are equivalent to USD1472, reducing by approximately 60 to 80% the total costs compared to the other methodologies. While for the actual PFC(±)5min case study that considers a gross power reserve shortfall of 136 MW and a real-time marginal cost of 147.1 USD/MWh, the total costs of power reserve reallocations equal USD1472, reducing by approximately 20–80% the total costs compared to the other methodologies. Therefore, the opportunity cost is key because it depends on the real-time marginal cost. As the penetration of solar–wind renewable generation expands, the variable costs of generation will equal the real-time marginal cost, allowing for reallocations of power reserves at zero cost.

7.2. Future Work and Opportunities for Improvement to the Technical-Economic Model of Power Reserve Reallocations

Frequency control remains a constant challenge for system operators. Establishing the technical and economic balances that allow the integration of conventional generation and renewable generation for an efficient energy transition makes it even more complex in the so-called ancillary services market. However, the security of the system, which reduces the marginal cost and operational cost, must be ensured through the optimal dispatch and withdrawal of plants linked to the dedicated energy market that supplies demand according to marginal theory and the market for ancillary services for frequency control. The proposed model effectively addresses the problem of reallocating power reserves at the minimum cost. However, the opportunity costs of the plants for the reallocation of reserves are subject to conventional generation through a monopolistic structure and the negligent use of the system’s inertia, which conditions the plants to operate out of economic order at minimum power. The proposed model in this situation is not optimal when selecting these plants, which operate far from the real-time marginal cost and are not classified as candidate plants in the supra-/infra-marginal methodology due to their high cost. Additionally, several system operators in South America implement a forced power reserve reallocation to justify excess generation at a minimum power level that disagrees with the proposed model. Finally, it is important to define, as future work, that the proposed model is able to monitor and avoid monopolistic actions with the use of thermal power plants not justified in the energy market, and a technical-economic model with a backup regulation for the market of ancillary services in the primary control of frequency in real time.