Enabling Active Distribution Systems’ Participation in Tertiary Frequency Regulation Through Coalitional Game Theory-Based Reserve Allocation

The integration of distributed energy resources (DERs) into the power grid has made it important for distribution systems to participate in frequency regulation. Regulatory authorities (e.g., Federal Electricity Regulatory Commission in the United States) are recommending that DERs participate in energy and reserve markets, and a mechanism is needed to facilitate this at the distribution level. Though a single distribution system may not have sufficient reserves for tertiary frequency regulation, stacked reserves from multiple distribution systems can be utilized in frequency regulation. This paper proposes a coalitional game theory-based approach for reserve allocation, enabling DERs to participate in tertiary frequency regulation. The proposed two-stage approach involves computing worthiness index (WI) and power loss reduction (PLR) characteristic functions in the first stage and equivalent Shapley values in the second stage. The Shapley values are then used to determine distribution factors for reserve allocation among DERs. We demonstrate the effectiveness of the proposed method through case studies on IEEE 13-node, IEEE 34-node, and IEEE 123-node distribution test systems. The method is shown to be effective in allocating reserves and determining active power set-points of DERs, and can enable efficient participation of DERs in tertiary frequency regulation.


PCAR i
Priced capacity available for reserve of the i th DER.

UCAR i
Unpriced capacity available for reserve of the i th DER.TUCAR Total unpriced capacity available for reserve.

P ci
Sellable capacity of the i th DER.

P ei
Market clearing capacity of the i th DER.P R Total power reserve to be allocated from a distribution system.The integration of distributed energy resources (DERs) into the power grid has revolutionized the operation of power systems, presenting both challenges and benefits [1].DERs offer numerous advantages, including enhanced grid resilience, reduced carbon emissions, and the potential for localized energy production.However, their integration also introduces a host of challenges that must be addressed to ensure the reliable and efficient operation of the power grid.Among these challenges, frequency control and regulation emerge as crucial aspects that demand careful consideration and innovative solutions [2], [3].Frequency regulation problems, such as frequency deviation, arise when there is an imbalance between generation and load, triggered by a variety of factors, including faults, substantial load changes, generator tripping, and grid islanding [4].To address these frequency deviation concerns, frequency regulating devices are employed.In traditional vertically integrated utility systems, operators rely on optimal power flow (OPF) solutions to determine the operating points of individual generators, optimizing generation costs while adhering to network and reserve constraints.However, in the context of deregulated power systems, tertiary frequency regulation schemes strive to maximize social welfare by allocating spinning reserves from participating generators or DERs involved in primary and secondary frequency regulation [5].While the individual impact of a single DER on frequency control may appear negligible, the collective potential of a fleet of DERs offers an opportunity for their meaningful engagement.Nonetheless, allocating impactful reserves from DERs poses a considerable challenge, demanding flexible and efficient solutions.
Frequency regulation and control in power systems are generally classified into primary, secondary, and tertiary categories [6].A primary frequency control is the first line of defense that responds to a disturbance quickly (within milliseconds).If primary frequency control is unable to fully compensate for the disturbances, secondary frequency control schemes kick in, adjusting the power generated by the generating units to reinstate frequency to its nominal value.After executing the secondary frequency control, the resulting operating points of different generators may not be economically optimal.The tertiary frequency control schemes can then be initiated by system operators to minimize cost of generation through solving an optimal power flow while satisfying the reserve requirement [6].The tertiary frequency control/regulation schemes used in traditional power systems, on the other hand, may not be directly applicable to DER-based distribution networks.In order for distribution systems to participate in tertiary frequency regulation, an efficient approach for allocating reserves among DERs and determining their active power set points is required.
DERs and other distributed resources have been extensively employed for frequency regulation and various ancillary services.In [7], a bilateral transactive energy coordination framework has been introduced to manage grid participants' trading activities in the evolving power grid with a high deployment of DERs and proactive customers, aiming to optimize bilateral transactions and enhance the grid's effectiveness, including ancillary services such as frequency regulation.In [8], a distributed computational approach has been proposed for a distribution system market, enabling proactive agents and responsive loads to coordinate bilateral energy transactions, optimize social welfare, and ensure secure grid operation, including ancillary services like frequency regulation.An optimized energy management strategy for photovoltaic (PV) systems with energy storage has been introduced in [9], which uses constant-byhours power references to participate in day and intraday electricity markets while mitigating the stochastic nature of PV production, offering potential for efficient utilization of energy storage systems for ancillary services and frequency regulation.In [10], an overview of demand response and transactive energy has been presented, highlighting their potential to enhance power system reliability and economic operation, especially in managing the supply and demand sides of DERs, while promoting real-time, autonomous, and decentralized decision-making.This makes them valuable for ancillary services and frequency regulation in a changing and distributed resource-rich environment.In [11], a framework was introduced for DERs in distribution networks to provide primary frequency response, enhancing power system reliability and frequency regulation.The framework includes designed power-frequency droop slopes to ensure a guaranteed characteristic at the distribution feeder head while adhering to fairness objectives for DERs' locations.In [12], a robust decentralized secondary frequency control design for islanded microgrids was presented, which leverages measurements from DERs and communication networks to enhance resiliency and coordination, with additional cyber intrusion detection mechanisms, adding value for ancillary services and frequency regulation in modern electric power systems.While existing literature has sufficiently addressed primary and secondary frequency regulation with DERs, a research gap exists in the utilization of DERs for reserve allocation for tertiary frequency regulation.
Several approaches for tertiary frequency regulation and control in transmission systems and microgrids have been reported in the literature.One such approach, outlined in [13], focuses on achieving optimal tertiary frequency control while taking into account market-based regulation.Another strategy, presented in [14], utilizes model predictive control (MPC) to activate tertiary frequency control reserves.Additionally, a mixed integer linear programming-based optimization tool, introduced in [15], offers a means to activate tertiary frequency control reserves at the transmission level.In [16], a load frequency control has been proposed from the perspective of restructured power systems.In [17], the traditional automatic generation control (AGC) has been modified and implemented from the perspective of deregulated power systems.In [18], a data-oriented technique for estimating secondary and tertiary reserves is introduced and evaluated on an actual system.The challenges of power system reserve configuration in a grid with increasing new energy sources and complex ultra-high-voltage grid structures have been analyzed in [19] by establishing an optimization model for demand-side response participation in reserve configuration, ensuring system stability and reliability through decomposition coordination and unit allocation.The paper [20] has focused on optimizing a local energy community's participation in providing tertiary frequency reserves using electric vehicles and a battery storage system, with a two-stage scheduling approach for maximizing profits and demonstrating the impact of control parameters on real-time profitability.Despite the development and utilization of numerous techniques and algorithms for tertiary frequency regulation in microgrids and transmission systems, the task of enabling active distribution systems to contribute reserves remains a persistent challenge.
The utilization of coalitional game theoretic methodologies in power and energy systems has experienced a notable upsurge in recen years, owing to their capacity to assign unique payoffs to game players, taking into account their individual contributions.To facilitate power exchange between microgrids and the main grid, a coalitional game theory-based energy management system has been introduced in [21].
In another work [22], a coalitional game theory-based approach has been proposed to determine the optimal sizes and locations of distributed energy resources.Furthermore, for the active participation of distribution systems in secondary frequency regulation, a coalitional game theory-based approach has been put forth in [23].In the pursuit of reliability enhancement and power loss reduction within active distribution systems and microgrids, a game-theoretic approach based on computing the locational marginal price at each bus has been presented in [24].This approach provides economic incentives to individual players in the game when improvements in system reliability and reductions in power loss are achieved.
This paper proposes a two-stage coalitional game theoretic approach for the allocation of reserves among DERs for tertiary frequency regulation.In the initial stage, characteristic functions, namely the worthiness index (WI) and power loss reduction (PLR), are computed for various possible coalitions of participating DERs.The WI represents the value or importance of DERs in maximizing social welfare, while the PLR quantifies the reduction in power loss resulting from the inclusion of DERs.In the second stage, equivalent Shapley values and corresponding distribution factors are determined.These values play a crucial role in the allocation of reserves among DERs and the determination of their active power set-points.The proposed approach is substantiated and validated through case studies conducted on several test systems, showcasing its effectiveness and efficiency.The main contributions of this paper are summarized as follows: • Proposing a framework for participation of DERs in tertiary frequency regulation.The proposed framework will help system operators to allocate reserve among different DERs in an effective and efficient manner.
• Developing a coalitional game theoretic framework to allocate reserve among different DERs for tertiary frequency regulation.The coalitional game theoretic approaches have the ability to uniquely assign payoffs among players of the game.
• Proposing two types of characteristic functions, namely the worthiness index (WI) and power loss reduction (PLR), of each coalition for the coalitional game model under consideration.In order to maximize the social welfare and benefit, three factors priced capacity available for reserves (PCAR), reserve bid price (RBP), and performance index (PI) are proposed to compute the WI of each DER.
• Proposing a distribution factor based on the Shapley value to allocate reserves among the participating DERs.By considering the marginal contribution of each DER, the Shapley value ensures a fair and equitable distribution of reserves.The active power set points of each DER is then calculated using the proposed distribution factor.
The subsequent sections of this paper are organized as follows.Section II outlines the presentation of tertiary frequency regulation as a coalitional game.In Section III, the concepts of coalitional game theory, including the Shapley value, are elucidated.Section IV provides an explanation of the proposed approach for tertiary frequency regulation.Detailed case studies conducted on the modified IEEE 13-node, IEEE 34-node, and IEEE 123-node distribution test systems are presented in Section V. Finally, Section VI offers concluding remarks to wrap up the paper.

II. TERTIARY FREQUENCY REGULATION, COALITIONAL GAME, AND MARKET FRAMEWORK
Frequency fluctuation in a power system occurs when there is a mismatch between electricity generation and consumption.Under such scenarios, frequency control/regulation devices come into play to bring the system frequency back to its nominal value and ensure that it stays within a specified range of the nominal frequency [25], [26].Conventionally, the frequency control is usually categorized into primary, secondary, and tertiary frequency control.The primary frequency control refers to the localized control of a generator's power according to its droop control characteristic.The aim of the primary frequency control is to re-establish the balance between generation and consumption (load plus losses) in the system at a frequency that may be different from the nominal frequency [25]-i.e., the primary frequency control may stabilize the frequency but it may not be able to restore it to the nominal value.Whereas the primary frequency control simply tries to maintain system stability by ensuring balance between generation and load, the secondary frequency control restores the frequency to the nominal value within seconds to a few minutes [27], [28].
The tertiary frequency regulation scheme economically allocates reserves considering the technical constraints for frequency control and response.If the reserve or part of the reserve is used by the secondary frequency control after a disturbance or imbalance between generation and load, the tertiary frequency regulation restores the reserve to the desired level [6].In other words, the tertiary frequency control is used to relieve secondary reserve, following the loss of large generator or load, or to correct prediction inaccuracies [25].The tertiary frequency control is triggered manually in response to system operators' requests.
Under vertically integrated monopolistic structure of power systems, the tertiary frequency regulation is responsible for optimal power sharing among the generators to minimize the overall cost or to maximize the net social welfare [5].In case of electric utilities operating under partial or fully deregulated environment, the operating reserve capacity is treated as a commodity like energy in an electricity market [29].The co-optimization of energy and reserve dispatch has been investigated in [30] in the setting of a pool-based market, where the modeling of lost opportunity has also been presented using a general approach.An offering mechanism of aggregators in the energy and reserve power markets has been investigated in [31].A new formulation of independent clearing of energy and reserve market has been introduced in [32], which also considers lost opportunity cost of generating units for clearing reserve market.The co-optimization of energy and reserve has been proposed in [33] with the consideration of two compensation mechanisms: uplift payments and lost opportunity costs.A mathematical model for battery energy storage system (BESS) participation in reserve market has been developed in [34].A bidding strategy for the participation of BESS in the reserve market has been presented in [35].
As mentioned earlier, in the context of tertiary frequency regulation, the allocation of reserves primarily focuses on economic considerations and operational constraints.Tertiary reserve allocation is mainly used to compensate for the reserves that were initially allocated during primary and secondary frequency regulation to address disturbances or imbalances in the system, especially following the loss of large generators or loads.Tertiary reserves are also activated manually in response to system operator requests.
In this paper, the allocation of reserves and determination of power set-points for DERs are conceptualized as a coalitional game, with the participating DERs being considered as the players.To incentivize DERs to engage in tertiary frequency regulation, they are given the opportunity to submit bid prices for reserves.Additionally, DERs provide information regarding their sellable capacity to a virtual aggregator located at the substation, which then relays this information to the system operators.The overall process of reserve optimization is illustrated in Figure 1, with the specific focus of this paper being the final step of the process (highlighted within a green dashed rectangle).Figure 2 depicts the framework of the proposed coalitional game theoretic approach for the allocation of reserves in an active distribution system for tertiary frequency regulation.Utilizing the data received from DERs and considering their historical performance, the distribution system operator calculates two characteristic functions, namely the worthiness index (WI) and power loss reduction (PLR), for each coalition.These characteristic functions are essential for ensuring a fair distribution of reserves among DERs.
One of the integral aspects of the reserve allocation is the development of a market framework that establishes clear and equitable compensation mechanisms and cost allocation for DERs.Within this framework, DERs are encouraged to engage actively in tertiary frequency regulation, benefiting from transparent and fair compensation mechanisms.DER compensation can originate from various sources, including the electric utility and system operator, ensuring that these entities fairly remunerate DERs in alignment with the associated costs.Moreover, as part of the broader financial ecosystem, the electric utility or system operator, upon compensating DERs, may ultimately pass on these costs to electricity customers.This cost allocation is based on the additional expenses incurred due to the reserve allocation ancillary service.The costs incurred for reserve allocation impact electricity customers, as these costs are factored into electricity pricing.By preserving financial equilibrium and transparency, this market framework fosters the sustainable integration of DERs into the power grid, all while ensuring that the financial aspects of their participation are clearly defined and justly executed.

III. COALITIONAL GAME THEORY AND SHAPLEY VALUE
In the field of game theory, games are typically classified into two main categories: (a) coalitional games and (b) non-coalitional games.In non-coalitional games, players operate independently and compete against each other to maximize their individual utilities.On the other hand, coalitional games allow players to form alliances or coalitions with one another to enhance both individual and coalitional utilities [36].As players join coalitions to maximize their individual utilities, any coalition must yield utilities that are equal to or greater than the utilities obtained by individual players.Non-coalitional games primarily focus on maximizing the individual utilities of players, while coalitional games aim to improve the collective utility of the coalition.Considering that the regulation of system frequency requires the combined effort of all DERs, coalitional games are more suitable for addressing the tertiary frequency regulation problem at hand [37].Consequently, the subsequent discussions will focus solely on coalitional games.The structure of a coalitional game encompasses essential components that shape its dynamics and outcomes: 1) Players' Ensemble: Within this game, a finite set of players, denoted as N, takes center stage as the grand coalition.Each player holds a distinct role, contributing to the formation of coalitions and influencing the collective outcomes.2) Characteristic Function: At the heart of the coalitional game lies the characteristic function, denoted as V (S).This function establishes a connection between coalitions and their associated values.It captures the essence of coalitional dynamics by mapping every potential coalition of players, drawn from the grand coalition, to a set of coalitional values.Symbolically, it can be expressed as: Here, 2 N denotes the power set of N, representing the collection of all possible subsets of players.The characteristic function captures the diverse nature of coalitions and assigns them corresponding values, thus encapsulating the intricate interplay between cooperation and competition.Notably, the characteristic function adheres to the fundamental condition that the value of an empty coalition (φ) is zero, expressed as V (φ) = 0. Various solution approaches are employed in coalitional games to distribute the overall payout or incentive among individual players.These approaches include the Shapley value, the core, the Nucleolus, and the Nash-bargaining solution.Each approach offers a unique method for allocating rewards or benefits fairly among the participating players.

A. THE CORE OF A COALITIONAL GAME
In the field of game theory, the core denotes a set of potential allocations within a coalitional game where no further improvements can be achieved through alternative coalitions.It represents a collection of payout assignments that ensures no individual player or subgroup within the coalition has any incentive to leave the existing coalition and form a new one.The core can be mathematically defined as 5226 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
follows [38]: Within the core, the total sum of allocated payouts corresponds to the value of the grand coalition, ensuring that all players collectively receive their deserved portion.Additionally, for any subset of players forming a coalition, the sum of their assigned payouts is guaranteed to be at least as large as the value of that particular coalition.This condition promotes stability and discourages the formation of sub-coalitions with the aim of obtaining greater benefits.However, this stability requirement can be too strict and sometimes make the core empty.That's why alternative solution approaches, such as the Shapley value, are often used instead.

B. THE SHAPLEY VALUE
The Shapley value represents a solution approach within coalitional game theory that enables the allocation of total earnings among individual players in a scenario where all participants engage in the game.The computation of the Shapley value for a given coalitional game can be described as follows [39].
Here, n = |N| denotes the total number of players involved in the game.The Shapley value formula calculates the contribution of player j by summing over all possible coalitions S in which j is a member.The expression considers the difference between the characteristic function value of the coalition S and the value of the coalition obtained by excluding player j from S. Various factorial terms are used to weigh the contributions appropriately.The Shapley value possesses several significant properties, which are outlined below [36]: 1) Efficiency: The Shapley value ensures an equitable distribution of the value generated by the grand coalition among the individual players.It embodies the principle that the sum of their Shapley values equals the total value of the coalition.Symbolically, it can be expressed as follows: This property guarantees that each player receives a fair share of the collective value generated by their cooperation.2) Individual Rationality: The Shapley value ensures that participating players are rewarded with a share greater than or equal to their individual contribution.
It guarantees that when a player joins a coalition, their Shapley value exceeds their standalone value.Mathematically, it can be stated as follows: This property incentivizes players to participate in coalitions and discourages them from remaining outside due to the assurance of a fair reward.3) Symmetry: The Shapley value exhibits symmetry, meaning that players who contribute equally to coalitions are assigned equal values.If two players, denoted as i and j, make identical contributions to every coalition they join, their respective Shapley values are equal: This symmetry property ensures that players with equal contributions are treated fairly and receive comparable rewards for their participation.4) Dummy Player: A player who does not significantly impact the value of any coalition is referred to as a ''dummy player''.The Shapley value assigns a value of zero to such players.If a player, denoted as i, fails to influence the value of any coalition they join, their Shapley value is zero: This property prevents dummy players from gaining unjustified rewards, ensuring that only players who make meaningful contributions receive positive Shapley values.5) Linearity: The Shapley value exhibits linearity, meaning that the value assigned to the sum of characteristic functions is equal to the sum of the individual Shapley values corresponding to each characteristic function.
For two characteristic functions, V 1 and V 2 , of a coalitional game, the linearity property is expressed as follows: This property allows for the decomposition of the overall value into individual contributions, facilitating the analysis and understanding of the distribution of value among the players in complex coalitional settings.

IV. THE PROPOSED APPROACH
In this section, a coalitional game theoretic approach is introduced for allocating reserves among DERs in a distribution system.The approach consists of two stages.In the first stage, the characteristic functions, namely WI and PLR, are computed.Subsequently, Shapley values are determined, and in the second stage, reserves are allocated based on these values.Additionally, we propose several metrics, including priced capacity available for reserve (PCAR), unpriced capacity available for reserve (UCAR), and Distribution Factor (DF), which are described in detail in this section.These metrics play a crucial role in ensuring a fair distribution of reserves among DERs.
The WI of each DER reflects its value or importance in the allocation of reserves to maximize social welfare.The calculation of WI for each DER takes into account three factors: priced capacity available for reserve (PCAR), reserve bid price (RBP), and performance index (PI).
The available capacity for reserve (i.e., after energy market clearance) from the i th DER can be calculated as follows.
Available Capacity for Reserve = P ci − P ei (9) where P ci is total capacity of the i th DER available for selling and P ei is the accepted capacity of the i th DER after clearing energy market.Since the i th DER has accepted P ei through market clearing and its local demand is already met, this reserve can be considered unpriced and is, therefore, named as unpriced capacity available for reserve (UCAR).However, it is to be noted that the DER will get paid whenever the reserve is used for secondary frequency control.The total unpriced capacity available for reserve (TUCAR) of a distribution system is calculated by adding UCAR of each DER behind the substation of the distribution system.(10) If the reserve command received from the distribution system operator, denoted as P R , exceeds TUCAR, then an additional reserve needs to be allocated from P ei .Under such circumstances, the DER has to be paid lost opportunity cost for the additional reserve.Thus, the capacity available for an additional reserve after deducting the power being supplied to critical loads is named as priced capacity available for reserve (PCAR) and is computed as follows.
where α c is the critical load factor denoting the fraction of power being supplied by each DER to critical loads.This fraction of power cannot be allocated for reserves.
For the case of P R being higher than TUCAR, one of the main functions of tertiary frequency controller or regulator is to allocate P R − TUCAR optimally among the participating DERs.
The worthiness index (WI) of the i th DER is determined by the following expression: where PCAR i is PCAR of the i th DER, RBP i is reserve bid price of the i th DER, and PI i is the performance index of the i th DER.
The performance history consisting of the values of committed power and supplied power of DERs is utilized to determine the performance index (PI) of each DER.The errors between committed power and supplied power are utilized in order to compute the coefficient of determination (also referred to as R-squared score), which is, here, the performance index (PI) of each DER.The expression for the performance index of the i th DER is as follows.
where R 2 i is the R-squared score of the i th DER, ESS i is the sum of squares of errors between committed power and supplied power of the i th DER, and TSS i is the total sum of squares of committed power of the i th DER.
The WI of each DER serves as the initial characteristic function in the proposed coalitional game, while PLR acts as the second characteristic function.PLR is determined by the disparity between the active power loss of the system when including a specific coalition of DERs and the power loss when no DERs are present.
Using the characteristic functions associated with each participating DER and their coalitions, Shapley values (ψ 1,i and ψ 2,i ) are computed using equation (3).Subsequently, the normalized Shapley values corresponding to each characteristic function are determined as follows.
By utilizing the normalized Shapley values obtained from equations ( 14) and ( 15), the equivalent Shapley value for the i th DER is calculated in the following manner.
Next, the distribution factor (DF) for the i th DER is determined using equation (17).These distribution factors play a role in distributing the difference between P R and TUCAR (i.e., P R -TUCAR) among the participating DERs.(17) 5228 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The active power set point for the i th DER is subsequently determined in the following manner.
The proposed method to compute reserves allocated and set points of DERs for tertiary frequency regulation can be summarized as follows.
1) Gather information on lines, loads, DERs, etc. in the system.2) Retrieve the values of P c , P e , and RBP for each DER.
3) Calculate the performance index (PI) for each DER based on historical data of committed power and supplied power.4) Determine the priced capacity available for reserve (PCAR) and the worthiness index (WI) for each DER. 5) Compute two types of characteristic functions, namely the worthiness index (WI) and power loss reduction (PLR), for each possible coalition by considering all potential combinations of DERs.6) Calculate the Shapley values using Equation (3) and normalize them using Equations ( 14) and (15).7) Determine the equivalent Shapley values using Equation ( 16) and calculate the distribution factors using Equation ( 17).8) Determine the allocated reserves and set points of DERs using Equation (18).The flowchart illustrating the proposed approach is depicted in Figure 4.

V. CASE STUDIES AND DISCUSSIONS
In order to showcase the efficiency and effectiveness of the proposed approach and solution algorithm, numerical simulations are conducted using modified versions of the IEEE 13-node, IEEE 34-node, and IEEE 123-node distribution test systems.These case studies serve to demonstrate the practical application and performance of the proposed method.
The core of these case studies focuses on detailed power flow analyses.These analyses are essential because the tested systems are naturally unbalanced, and accurate calculations of AC power flow are critical for reliable results.To achieve this, an integrated environment that combines OpenDSS and MATLAB is utilized in this work.OpenDSS, developed by the Electric Power Research Institute (EPRI) and available as open-source software, accurately calculates the AC power flow within these systems [40].

A. SYSTEM DESCRIPTIONS
The IEEE 13-node distribution test system represents a compact feeder operating at a voltage level of 4.16 kV.It has a significant load capacity and features various components.At the substation, there is a voltage regulator consisting of three 1-phase units connected in a star configuration.The system is characterized by the presence of shunt capacitor banks, an in-line transformer, and both spot and distributed loads, which may exhibit imbalances.According to the data from [41], the total active load and reactive load of this system amount to 3.577 MW and 1.725 MVAr, respectively.Additionally, three DERs have been introduced into the system, as illustrated in Figure 5.
The IEEE 34-node system represents a real distribution system located in Arizona, USA.It operates at a nominal voltage of 24.9 kV and is characterized as a long feeder with a relatively light load.The system features shunt capacitor banks, two in-line voltage regulators, an in-line transformer, and a combination of spot and distributed loads, which may exhibit imbalances, as described in [41].In this study, three 3-phase DERs have been integrated into the system, specifically at nodes 844, 890, and 834, while a single-phase DER has been added to phase 1 of node 822, as depicted in Figure 6.
The IEEE-123 node system represents a 4.16 kV distribution system.It consists of a combination of overhead and underground line segments with different phasing configurations.The system includes shunt capacitor banks, four voltage regulators, and unbalanced spot loads with diverse load types such as PQ (constant power), constant current, and constant impedance.According to [41], the total active load and reactive load of this system are reported as 3490 kW and 1925 kVAr, respectively.In this study, ten 3-phase DERs have been introduced into the system at various nodes, as depicted in Figure 7.

B. PARAMETERS, IMPLEMENTATION, AND RESULTS
The proposed approach initiates by gathering data pertaining to the sellable capacity (P c ), energy market clearing capacity (P e ), and reserve bid price (RBP) for each DER.Utilizing these values along with the critical load factor (α c ), the priced capacity available for reserve (PCAR) is computed for each DER.The performance index (PI) is then determined for each DER based on the historical data comprising committed power and supplied power.In the case of the modified IEEE 13-node system, the sellable capacities of the DERs are reported as 250 kW, 350 kW, and 450 kW.The energy market clearing capacities for the considered timestamp are 220 kW, 350 kW, and 400 kW, respectively, for the DERs.The reserve bid prices for the DERs are set at $10/kW, $20/kW, and $12/kW.These parameter values are detailed in Table 1.For a critical load factor (α c = 0.5), PCAR of each DER obtained using (11) are 110 kW, 175 kW, and 200 kW, which are also shown in Table 1.The values of PI of each DER are 0.9998, 0.9009, and 0.8272, which are R-squared scores computed based on past history of committed power and supplied power of DERs.Based on (12), the PI of each DER has a direct impact on its worthiness index (WI).Since DER-3 has the least PI value, 0.8272, compared to DER-1, its WI (from Table 4) is 13.79, which is higher than WI of DER-1 by only 2.8, even though its capacity is almost double.Likewise, the various parameters of the DERs for the modified IEEE 34-node system and IEEE 123-node system are presented in Table 2 and Table 3, respectively.
As mentioned in Section III, a coalitional game is defined by a finite set of players and characteristic functions.The proposed approach considers two types of characteristic functions: WI and PLR.These characteristic functions are defined for all possible coalitions of DERs.The PLR of a coalition represents the difference in power loss between the system without any DERs and the system with the DERs in that coalition.For the modified IEEE 13-node system, Table 4 presents the possible coalitions of DERs along with the corresponding values of WI and PLR.For instance, DER-1 has a WI value of 10.99 and a PLR value of 8.59 kW.DER-2 has a WI value of 7.88 and a PLR value of 3.79 kW.When both DER-1 and DER-2 form a coalition, the WI value is 18.88 and the PLR value is 12.53 kW.Similarly, for the modified IEEE 34-node system, Table 5 illustrates the possible coalitions and their corresponding values of WI and PLR.It's worth noting that a similar table can be constructed for the modified IEEE 123-node system, although it is not provided here.
The characteristic functions presented in Table 4 and 5 are utilized to calculate Shapley values using equation (3).These Shapley values are then normalized based on equations ( 14) and (15).Subsequently, the equivalent Shapley values are computed using equation (16).Finally, the distribution factor (DF) of each DER is determined using equation (17).For the modified IEEE 13-node system, the distribution factors of DERs 1, 2, and 3 are 0.2729, 0.2185, and 0.5086, respectively.The allocated reserves for the DERs (assuming P R = 100 kW) are 34.09kW, 8.28 kW, and 57.63 kW.These values can be found in Table 6.Similarly, for the modified IEEE 34-node system, the distribution factors of DERs 1, 2, 3, and 4 are 0.2949, 0.2141, 0.3499, and 0.1410, respectively.The allocated reserves for the DERs (assuming P R = 100 kW) are 28.85 kW, 16.42 kW, 20.50 kW, and 34.23 kW.These values are displayed in Table 7. Furthermore, for the modified IEEE 123-node system, the distribution factors and allocated reserves (assuming P R = 100 kW) for the DERs can be found in Table 8.

C. COMPARISON
The allocated reserves, obtained through the proposed approach, are compared with those obtained using a DERcapacity-based approach in terms of total reserve cost.The DER-capacity-based approach relies solely on the computation of distribution factors based on the sellable capacities of DERs, lacking the comprehensive considerations offered by coaltional theory-based approach.Table 6, Table 7, and Table 8 provide insights into the distribution factors and allocated reserves of DERs, obtained through both the DER-capacity-based approach and our proposed approach, for the modified IEEE 13-node, IEEE 34-node, and IEEE 123-node systems, respectively.Notably, the outcomes within the modified IEEE 13-node system showcase the remarkable superiority of our proposed approach.It yields  a remarkably low total reserve cost of merely $174.57,showcasing a substantial advantage over the DER-capacitybased approach, which incurs a significantly higher cost  of $511.82 obtained by the DER-capacity-based approach.Consistently, the modified IEEE 123-node system aligns with this positive trend, as our approach diminishes the total reserve cost to $274.94, in contrast to the alternative approach's cost of $381.82.These results firmly establish the effectiveness of our proposed approach in achieving substantial cost reductions and optimizing the allocation of reserves in these systems.Figure 8 illustrates the comparison of total reserve costs obtained using the proposed approach and the DER-capacity-based approach for all the considered test systems.The results demonstrate that implementing the proposed reserve allocation approach leads to a substantial decrease in total reserve costs.This reduction in costs carries significant advantages for both aggregators and system operators.
To show the benefit of the proposed approach from the perspective of DER owners, the individual utility function is defined for the i th DER as follows: where RBP i is the reserve bid price, AR i is the allocated reserve, UCAR i is the unpriced capacity available for reserve, PI i is the performance index, and P ci is the total sellable capacity of the i th DER.
Based on the values of allocated reserves, the individual utilities of DERs are calculated using both proposed and capacity-based approaches for all test systems under consideration, which are shown in Tables 6, 7, and 8. Figures 9, 10, and 11 show histograms of the individual utilities of DERs for IEEE 13-node, 34-node, and 123-node systems.From the figures, we can see that the individual utilities of DERs have less variance (or more closeness) for the proposed approach.The standard deviation of individual utilities of DERs for the proposed approach is 0.0371 and that for the capacity-based approach is 0.6316 in case of IEEE 13-node system.Similarly, the standard deviation of individual utilities of DERs for the proposed approach is   0.1214 and that for the capacity-based approach is 0.3232 in case of IEEE 34-node system; and the standard deviation of individual utilities of DERs for the proposed approach is 0.1924 and that for the capacity-based approach is 0.5040 in case of IEEE 123-node system.The more close (or less variant) individual utilities of DERs imply the similar level of satisfaction of DERs when the reserves are allocated based on the proposed approach.This occurs because the reserves are allocated based on Shapley values, which consider the individual DERs' marginal contributions in the allocation process.

D. COMPUTATIONAL NEEDS AND SCALABILITY OF THE PROPOSED APPROACH
The proposed approach is implemented on a personal computer equipped with a 64-bit Intel i5 core processor operating at 3.15 GHz, 8 GB of RAM, and running on the Windows operating system.The execution times of the proposed approach for the IEEE 13-node, IEEE 34-node, and IEEE 123-node systems are recorded as 0.46 seconds, 0.57 seconds, and 65.43 seconds, respectively.These results demonstrate that the proposed approach is suitable for both day-ahead (DA) and 15-minute ahead scheduling of reserves, given its efficient execution time.The execution times indicate that the proposed approach will still be applicable for DA scheduling of reserves even in the case of large distribution systems with many DERs.
The scalability of the proposed solution algorithm is evident through its successful application to various real-world distribution systems, as demonstrated in the preceding subsections.These case studies reinforce its adaptability to diverse scenarios.Importantly, the proposed reserve allocation problem is not a real-time operational problem that requires frequent execution.Consequently, high-speed computation is not a primary concern in this context.Modern computers are perfectly capable of handling this task for most practical-sized systems.However, in situations where resources are limited or specific needs arise, the algorithm remains flexible.It can accommodate simpler and approximate methods for power flow analysis, such as DC power flow.This flexibility ensures the proposed method remains a reliable and practical solution, even when resources are scarce.

VI. CONCLUSION
In this paper, we have proposed a coalitional game theorybased approach to evaluate the potential participation of active distribution systems in tertiary frequency regulation.By considering two distinct characteristic functions, the worthiness index (WI) and power loss reduction (PLR), we accurately captured the coalition dynamics among distributed energy resources (DERs).The computation of equivalent Shapley values and distribution factors in the second stage ensured an equitable distribution of reserves among DERs and facilitated the determination of their active power set-points.The proposed approach incorporated crucial variables and parameters of DERs, including sellable capacity, market clearing capacity, performance index, and reserve bid price.The performance index, derived from historical data on committed and supplied power of DERs, provided valuable insights into their past behavior and capabilities.Through comprehensive case studies conducted on the modified IEEE 13-node, IEEE 34-node, and IEEE 123-node distribution test systems, we empirically demonstrated the effectiveness and efficiency of our approach.The results revealed improvements in reserve allocation, benefiting both system operators and individual DERs.Importantly, our findings highlighted the superiority of our approach over the traditional DER-capacity-based method in terms of total reserve costs.The use of Shapley values, which consider the marginal contributions of each DER, played a pivotal role in achieving these cost reductions.This cost-effectiveness offers significant advantages to aggregators and system operators, enabling them to optimize reserve allocation strategies.

ACKNOWLEDGMENT
This article was prepared as an account of work sponsored by an agency of the United States Government.Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information,apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof.The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
ψ 1,i Shapley value of the i th DER corresponding to the first characteristic function.ψ 2,i Shapley value of the i th DER corresponding to the second characteristic function.ψ norm 1,i Normalized Shapley value of the i th DER corresponding to the first characteristic function.ψ norm 2,i Normalized Shapley value of the i th DER corresponding to the second characteristic function.ψ I. INTRODUCTION

FIGURE 1 .
FIGURE 1. Layout of the reserve optimization process.

FIGURE 2 .
FIGURE 2. Layout of the proposed coalitional game theoretic approach.

FIGURE 3 .
FIGURE 3. Allocation of reserves for the i th DER.

FIGURE 4 .
FIGURE 4. Flowchart of the proposed approach.

FIGURE 9 .
FIGURE 9.Histograms showing the distribution of individual utilities of DERs in case of the IEEE 13-node system for both proposed approach (Left) and capacity-based approach (Right).

FIGURE 10 .
FIGURE 10.Histograms showing the distribution of individual utilities of DERs in case of the IEEE 34-node system for both proposed approach (Left) and capacity-based approach (Right).

FIGURE 11 .
FIGURE 11.Histograms showing the distribution of individual utilities of DERs in case of the IEEE 123-node system for both proposed approach (Left) and capacity-based approach (Right).

VOLUME 12, 2024 5233
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TABLE 1 .
Parameters of DERs in the modified IEEE 13-node system.

TABLE 2 .
Parameters of DERs in the modified IEEE 34-node system.

TABLE 3 .
Parameters of DERs in the modified IEEE 123-node system.

TABLE 4 .
Characteristic functions: worthiness index (WI) and power loss reduction for the modified IEEE 13-node system.

TABLE 5 .
Characteristic functions: worthiness index (WI) and power loss reduction for the modified IEEE 34-node system. of $425.00. the modified IEEE 34-node system exhibits a noticeable disparity, with our approach resulting a total reserve cost of $353.11,surpassing the figure

TABLE 6 .
Distribution factors, allocated reserves, and individual utilities of DERs for the modified IEEE 13-node system.

TABLE 7 .
Distribution factors, allocated reserves, and individual utilities of DERs for the modified IEEE 34-node system.

TABLE 8 .
Distribution factors, allocated reserves, and individual utilities of DERs for the modified IEEE 123-node system.
FIGURE 8. Bar graphs showing comparison of total reserve costs.