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Article

Impact of Penalty Structures on Virtual Power Plants in a Day-Ahead Electricity Market

by
Youngkook Song
1,
Myeongju Chae
1,
Yeonouk Chu
1,
Yongtae Yoon
1 and
Younggyu Jin
2,*
1
Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Republic of Korea
2
Power System Economics Laboratory, Department of Electrical Engineering, Jeju National University, Jeju 62343, Republic of Korea
*
Author to whom correspondence should be addressed.
Energies 2024, 17(23), 6042; https://doi.org/10.3390/en17236042
Submission received: 6 October 2024 / Revised: 20 November 2024 / Accepted: 30 November 2024 / Published: 1 December 2024
(This article belongs to the Special Issue Distributed Energy Resources in Transactive Energy Systems—2nd Edition)
Browse Figures
Figure 1
<p>Classification of penalty structures based on the penalty scope.</p> "> Figure 2
<p>Structure of penalty coefficient and penalty cost.</p> "> Figure 3
<p>Flowchart to determine an optimal bidding strategy using stochastic programming.</p> "> Figure 4
<p>SMP and its errors over 24 h in the day-ahead and real-time markets.</p> "> Figure 5
<p>Forecasted power generation of a VPP for 24 h.</p> "> Figure 6
<p>Impact of penalty coefficient on revenue in Case 3, Case 7, Case 11, and Case 12.</p> "> Figure 7
<p>Impact of penalty coefficient on penalty costs in Case 3, Case 7, Case 11, and Case 12.</p> "> Figure 8
<p>Comparison of total revenue concerning both the penalty rate and penalty coefficient.</p> "> Figure 9
<p>Impact of penalty coefficient on power deviation in Case 1–12.</p> "> Figure 10
<p>Curtailed power in all simulation cases.</p> "> Figure 11
<p>Impact of tolerance band on revenue in Case 9.</p> "> Figure 12
<p>Effect of tolerance band on total revenue.</p> ">
Versions Notes

Abstract

:
The rapid increase in distributed energy resources has augmented the significance of virtual power plants (VPPs), which are essential for the aggregation and management of variable renewable energy resources (RERs). The inherent variability and uncertainty of RERs necessitate the implementation of deviation penalties to address the discrepancies between the awarded bids and actual generation, which is crucial in maintaining market stability and ensuring reliable grid operations. Therefore, this study proposes a framework for deviation penalty structures, categorizing penalties based on three factors: the penalty scope, penalty rate, and penalty coefficient. The simulation results show that the penalty scope significantly influences the revenue of VPPs, with over-generation penalty structures typically yielding higher profitability. Conversely, dual-sided penalty structures result in lower total revenues compared to one-sided penalty structures. For instance, when the penalty price coefficient is set to 0.1, the total revenue of a dual-sided penalty structure is approximately 62.26% lower than that of a one-sided penalty structure during the morning period. The results also demonstrate that deviation penalty structures have a direct impact on power deviations and curtailment behavior. Finally, we offer recommendations for the design of an effective penalty structure aimed at assisting policymakers and distributed system operators (DSOs) in structuring market mechanisms, which not only facilitate the integration of RERs but also enhance their economic viability within electricity markets.

1. Introduction

1.1. Renewable Energy Sources and Virtual Power Plants

As technology advances and industries become more sophisticated, the electricity demand has increased, alongside a rapid increase in electricity generation. Notably, the electricity consumption from data centers, artificial intelligence, and the cryptocurrency sector is projected to double by 2026, exceeding 1000 terawatt-hours—comparable to the annual consumption of Japan [1]. The increase in power generation has also led to a significant rise in carbon emissions, contributing to a climate crisis that includes global warming [2]. To address the climate crisis driven by increased power generation and consumption, global initiatives such as RE100 have underlined the importance of and interest in environmentally friendly energy resources, or renewable energy resources (RERs) [3]. Consequently, a substantial portion of traditional power generation is replaced by solar, wind, tidal, and hydroelectric power [4].
However, unlike conventional energy sources, RERs present challenges in controlling and predicting power generation. The inherent difficulty in forecasting the output of renewable energy leads to critical issues within the power grid [5]. Renewable energy generation, in contrast to traditional power generation managed by a limited number of operators, involves a decentralized model, where numerous distributed energy resource (DER) operators contribute power to the grid. This decentralized structure significantly increases the operational complexity and places a greater burden on the system operators responsible for managing and coordinating these distributed energy resources. As a result, the concept of a virtual power plant (VPP) has been introduced, wherein DERs are aggregated and managed as a single, coordinated power generator [6,7,8]. Like a traditional power generator, the VPP functions as a unified power generation entity in grid operation and participates in the electricity market.

1.2. Bidding in the Electricity Market

In the electricity market, bidding is a fundamental concept designed to ensure a stable power supply and transparent price determination. To maintain grid stability and optimize operations, DSOs impose various constraints and penalties during the bidding process. These measures include rules to prevent market disruption and deviation penalties closely related to the revenues of VPPs [9]. In this context, a deviation penalty is applied when a power generator either overproduces or underproduces electricity compared to its bid amount. Such mechanisms are crucial as they help to ensure a reliable electricity supply by encouraging accurate bidding and production behaviors among generators.
Numerous studies have been conducted on bidding strategies to optimize the market operations and enhance the system reliability [10,11,12,13,14]. There are two primary bidding strategies: single-segment bidding and multi-segment bidding. Single-segment bidding involves submitting a single price–power pair, which is simple but has limited flexibility. In contrast, multi-segment bidding allows for multiple price–power pairs, providing greater control over the bids and potentially optimizing the financial and operational outcomes. The choice between these strategies depends on the specific conditions and requirements of each electricity market [15,16,17]. This study primarily investigates multi-segment bidding strategies because they provide valuable insights into market dynamics and operational strategies.

1.3. Research Review and Contribution

Some previous studies have explored the role of deviation penalties in market bidding models, assessing their economic impacts. For example, Zhang et al. developed an optimal bidding model for VPPs using shared energy storage, focusing on how deviation penalties affect profit distribution among renewable energy operators [18]. Ma et al. proposed a strategy for the maximization of the benefits from battery energy storage systems in the hour-ahead market while minimizing penalties [19]. Jeong et al. explored a “DeepBid” strategy that employed deep reinforcement learning to optimize bidding under price and generation uncertainties [20]. Özcan et al. examined the effects of applying uniform penalty coefficients for both over- and under-generation in offshore wind farms, yielding similar outcomes across different scenarios [21]. Peng et al. developed a wind–thermal combined bidding model considering the conditional value at risk but used fixed penalty coefficients, without analyzing the specific impact of penalties on the overall revenue [22]. Song et al. introduced a hydrogen–wind–photovoltaic system into the flexible ramping market with a bi-level bidding strategy to account for interactions with market players [23]. Their analysis fixed the penalty prices based on the bid or market clearing prices, focusing solely on power deviations, without exploring penalty pricing mechanisms. Wang et al. proposed a bidding strategy model for VPPs, integrating green certificates and carbon trading to maximize the economic benefits [24]. They included a fixed power output error and penalty coefficient, without analyzing penalty pricing.
However, these studies primarily focused on the financial implications of fixed or variable penalty prices and did not explore the structural implications of penalty mechanisms on bidding behavior and market dynamics. This gap highlights a critical oversight, as penalties not only serve as financial adjustments but also strategically influence bidding decisions and market interactions.
The key contributions of this study are outlined below.
  • Framework for Deviation Penalty Structure: We propose deviation penalty structures specifically designed for VPPs in day-ahead electricity markets. This framework categorizes penalties into three dimensions: the penalty scope, penalty rate, and penalty coefficient. This classification serves as a foundational basis for distribution system operators (DSOs) to design penalty structures that enhance strategic planning and operational efficiency.
  • Analysis of the Impact of Penalty Structures on the VPP’s Revenue: This study examines the effects of various penalty structures on the revenue of the VPP. It includes a comprehensive analysis of how tolerance band settings affect revenue outcomes, thereby allowing the VPP to optimize its operational strategies under different market conditions.
  • Relationship between Penalty Structures and Power Generation of RERs: We investigate how various penalty structures affect the curtailment of RERs and the deviation from the scheduled power output. This analysis establishes a direct connection between penalty structures and operational decision-making.
  • Recommendations for Design of Effective Penalty Structures: In light of our findings, we propose strategic recommendations for DSOs to design an appropriate penalty structure. These recommendations aim to align with the market demand for operational flexibility while fostering more efficient and sustainable interactions within the market.
The remainder of this paper is organized as follows. Section 2 reviews the existing frameworks for electricity markets that incorporate deviation penalties. In Section 3, we propose a systematic approach to categorizing these deviation penalties and introduce an optimization model aimed at maximizing the revenue of the VPP in accordance with the proposed penalty structures. Section 4 presents simulation results that reflect the impact of each deviation penalty structure on various objective values. Section 5 offers recommendations for the design of penalty structures based on the simulation results, followed by concluding remarks in Section 6.

2. Electricity Market Frameworks

2.1. Existing Electricity Market with Deviation Penalties

Bidding procedures in electricity markets vary among independent system operators (ISOs), with each market employing an optimized structure tailored to its unique characteristics and regulatory environment. Table 1 presents a comparative summary of bidding systems in major electricity markets, highlighting key factors, such as the maximum number of bidding segments and the price caps and floors. Typically, nine to twelve bidding segments are allowed for by ISOs such as the Pennsylvania–New Jersey–Maryland Interconnection (PJM), California Independent System Operator (CAISO), New York Independent System Operator (NYISO), Midcontinent Independent System Operator (MISO), and Korea Power Exchange (KPX). Price caps are strategically employed to prevent monopolization by specific power producers and excessive increases in generation costs. For example, PJM has established a price cap of USD 1000/MWh without a specific price floor; in contrast, CAISO implements a soft cap of USD 1000/MWh under normal conditions, which escalates to a hard cap of USD 2000/MWh during power shortages. KPX operates with a unique price cap of 0/kWh and incorporates renewable energy certificate (REC) prices into its price floor.
Each market addresses energy price volatility through its operational framework, which includes institutional mechanisms designed to ensure stable operations for power producers. The regulations and policies associated with these markets are carefully tailored to the region’s electricity demand and supply conditions. Most ISOs have adopted multi-segment bidding as a strategy to improve system stability and facilitate effective participation from power producers. Additionally, price caps are implemented to mitigate the risk of monopoly and prevent excessively high prices, thereby fostering a more balanced operational environment within the market.

2.2. Deviation Penalty in VPP’s Revenue

The revenue of the VPP consists of three principal components. The first component pertains to energy settlement payments, which are earned from the sale of energy produced and supplied to the electricity market. The second one involves revenue derived from REC sales. The third component, which is the focus of this study, is the deviation penalty. While energy settlement payments and REC sales directly contribute to the operator’s income, deviation penalties occur due to discrepancies between the awarded bid quantity and actual generation, resulting in financial losses.
Due to the inherent variability, it is challenging to precisely control the amount of energy generated from RERs. Thus, grid operators impose penalties on VPPs to ensure the reliability of grid operations. For instance, in the NYISO market, the penalty is calculated as outlined in Equation (1): the difference between the awarded bid quantity at time t , P a w a r d   t , and the actual generation at time t , P ( t ) , multiplied by the penalty rate, ρ p e n _ c a p , which is based on the capacity price. Penalties are incurred when this difference exceeds the allowable tolerance, P t o l , which is set at 3% of the installed capacity. This means that if the real-time actual generation is less than 97% or exceeds 103% of the awarded bid quantity, a penalty is applied based on the difference. Similarly, CAISO also imposes penalties for generation discrepancies that exceed the allowable tolerance. The penalty is calculated by multiplying the excess generation by the ex-post price, and the allowable margin is set as the smaller value between 3% of the maximum output and 5MW.
P e n a l t y _ 1 = P a w a r d   t − P ( t ) × ρ p e n _ c a p ,             i f     P a w a r d   t − P ( t ) > P t o l 0 ,                                                                                                           o t h e r w i s e
KPX implements a distinctive penalty structure that exclusively penalizes instances of over-generation, as delineated in Equation (2). The permissible tolerance is established at 12% of the generator’s capacity. A notable characteristic of this system is that the penalty rate is contingent upon the SMP. When the SMP is positive, the penalty rate is calculated in accordance with the SMP at time t , ρ p e n _ s m p ( t ) . Conversely, when the SMP is negative, the penalty rate is determined based on the REC price, ρ p e n _ r e c . This adaptive mechanism adjusts the penalties in response to market price fluctuations, playing a crucial role in risk management for the VPP.
P e n a l t y _ 2 = P t − P a w a r d   t − P t o l × ρ p e n _ s m p t ,       i f   P ( t ) − P a w a r d   t > P t o l   a n d   ρ s m p ≥ 0   P ( t ) − P a w a r d   t − P t o l × ρ p e n _ r e c ,                   i f     P ( t ) − P a w a r d   t > P t o l   a n d   ρ s m p < 0 0 ,                                                                                                                 o t h e r w i s e

3. Penalty Structures in Optimization Problem

As mentioned in Section 2.2, deviation penalties are intended to impose disadvantages on power producers when they either under-generate or over-generate. In Section 3, we will examine various penalty structures and develop an optimization formulation to obtain an optimal bidding strategy considering the penalties, while also analyzing the relationship between the penalty structures and the resulting bidding outcomes.

3.1. Structure of Deviation Penalties

The structure of deviation penalties varies according to the penalty scope, penalty rate, and penalty coefficient. Thus, deviation penalties can be classified based on the specific features of each factor.

3.1.1. Classification Based on Penalty Scope

Deviation penalties in electricity markets are structured based on the penalty scope—whether the actual power generation is in a surplus or deficit relative to the bid quantity. Each of these structures has distinctive characteristics and can be categorized as follows.
  • Over-Generation Penalty Structure (OPS): Penalties are applied only when the actual generation exceeds the bid quantity, which is referred to as surplus generation. This structure, as shown in Equation (3), is designed to prevent power generation beyond the forecasted amount.
  • Under-Generation Penalty Structure (UPS): This structure imposes penalties when the actual generation falls short of the bid quantity, leading to a deficit in supply. As outlined in Equation (4), its aim is to mitigate the risk of an insufficient power supply that may fail to meet the market demand.
  • Dual-Sided Penalty Structure (DPS): Penalties under this structure are imposed for deviations in both directions, whether there is a surplus or deficit in generation, as given in Equation (5). In other words, this approach ensures that penalties are applied regardless of overproduction or underproduction compared to the bid.
P d e v _ O P S t = P ( t ) − P d a t                                                 i f     P d a t ≤ P ( t ) ≤ P m a x 0 ,                                                                                                                               o t h e r w i s e
P d e v _ U P S t = P d a t − P ( t )                                                   i f     P m i n ≤ P ( t ) ≤ P d a t 0 ,                                                                                                                               o t h e r w i s e
P d e v _ D P S t = P ( t ) − P d a t                                               i f     P m i n ≤ P ( t ) ≤ P m a x 0 ,                                                                                                                             o t h e r w i s e
where P d e v t represents the deviation quantity at time t , which is the difference between the actual generation, P ( t ) , and the awarded bid quantity from the day-ahead market at time t , P d a t ; and P m i n and P m a x denote the minimum and maximum generation capacities, respectively.
The deviation quantity can be further classified based on the direction of the deviation. P d e v _ u p ( t ) refers to the deviation caused by over-generation (DOG) at time t , where the actual generation exceeds the bid quantity. On the other hand, P d e v _ d w ( t ) refers to the deviation caused by under-generation (DUG) at time t , where the actual generation falls short of the bid quantity. These can be represented as follows in Equations (6) and (7):
P d e v _ u p ( t ) = P ( t ) − P d a t                                       i f     P d a t ≤ P ( t ) ≤ P m a x 0 ,                                                                                                                   o t h e r w i s e
P d e v _ d w ( t ) =   P d a t − P ( t )                                         i f     P m i n ≤ P ( t ) ≤ P d a t 0 ,                                                                                                                   o t h e r w i s e
The concept of allowable tolerance, P t o l , refers to the acceptance of a specified range of deviation. Without such tolerance, power producers are at a high risk of incurring penalties due to the inevitable uncertainties of RERs. Consequently, a measure is established wherein penalties are not enforced within a defined range, as delineated in Equation (8). The tolerance band, ε b a n d , is selected based on the degree of uncertainty in the generation forecasts or market conditions.
P t o l = P m a x × ε b a n d
In summary, the classification of penalty structures based on the penalty scope is illustrated in Figure 1. A deviation occurs when the difference between the actual generation and the awarded bid quantity exceeds the tolerance. Furthermore, as described in Equations (3)–(5), penalties can be categorized as follows: OPS, which penalizes only when the generation is higher than the awarded bid quantity; UPS, which penalizes only when the generation is lower than the awarded bid; and DPS, which penalizes deviations in both directions.

3.1.2. Classification Based on Penalty Rate

The penalty rate serves as a weighting factor in the computation of the penalty cost at time t , ρ p e n ( t ) , associated with the portion of the deviation that exceeds P t o l . There are two main approaches to determining the penalty rate, namely the SMP-based penalty rate structure (SPRS) and the REC-based penalty rate structure (RPRS), which are outlined in Table 2. In both structures, the penalty price coefficient, ξ , is employed to adjust the penalty rate.
The coefficient ξ is pivotal in adjusting the penalty cost after the penalty rate is determined. ξ moderates revenue fluctuations caused by sudden market price changes, allowing for the rational adjustment of penalty costs. To establish ξ , it is essential to consider the variations in the SMP, V s m p ( t ) , and REC price, V r e c , as detailed in Equations (9)–(11). In Equation (11), λ and μ are weighting factors for the SMP and REC price fluctuations, respectively. These factors enable a tailored adjustment to the penalty cost by emphasizing either the SMP or REC volatility, depending on the market conditions and the operational objectives of the VPP. This study assumes that the REC prices do not fluctuate on an hourly basis, allowing for the simplified and more predictable calculation of ξ when relying on REC-based penalty adjustments.
V s m p ( t ) = ρ d a _ s m p ( t ) − ρ r t _ s m p t ρ r t _ s m p t
V r e c = ρ d a _ r e c − ρ r t _ r e c ρ r t _ r e c
ξ = λ   ×   V s m p ( t ) + μ   ×   V r e c

3.1.3. Classification Based on Penalty Coefficient

The penalty coefficient at time t , σ t , is a variable that is multiplied by the designated penalty rate, significantly influencing the total penalty cost. A higher penalty coefficient results in a correspondingly larger penalty cost for the operator. The structure of the penalty coefficient can be designed as follows.
  • Linear Penalty Coefficient Structure (LPCS): The penalty coefficient increases linearly with the deviation at a rate determined by the slope k p e n . As the deviation grows, the penalty coefficient rises, leading to higher penalty costs. This linear relationship is represented in Equation (12).
  • Fixed Penalty Coefficient Structure (FPCS): In this structure, the penalty coefficient remains constant at c p e n , regardless of the deviation, as shown in Equation (13).
σ L P C S t =                       P d e v t − P t o l × k p e n                                                       i f     P d e v t − P t o l ≥ 0 0 ,                                                                                                                   o t h e r w i s e
σ F P C S t =                           c p e n                                                       P d e v t − P t o l ≥ 0                 0 ,                                                                                 o t h e r w i s e
Figure 2 illustrates how the penalty coefficient can be classified. In Figure 2a, two types of penalty coefficient structures are presented. The LPCS increases proportionally to the amount by which the generation exceeds P t o l , while the FPCS maintains a constant value, regardless of the degree of deviation in generation. Figure 2b demonstrates how the penalty cost varies with σ t . In the case of the LPCS, the penalty cost quadratically increases when P d e v t exceeds P t o l . On the other hand, for the FPCS, since σ t is a constant value, the penalty cost linearly increases.

3.1.4. Case Analysis Based on Penalty Structures

Based on the penalty structures presented in Section 3.1.1, Section 3.1.2 and Section 3.1.3, a set of twelve cases has been generated, as summarized in Table 3. Each case is defined by three key factors: the penalty scope, penalty rate, and penalty coefficient. Analyses of these cases will offer valuable insights into how different penalty structures can affect the operational outcomes.

3.2. Problem Formulation Integrating the Penalty Structure

This model incorporates the penalty structures described in Section 3.1 into the optimization problem to determine an optimal bidding strategy for the VPP. The primary objective is to maximize the total revenue of the VPP, which encompasses energy settlement payments, penalty costs, and REC sales. This section also presents details of the bidding constraints and provides mathematical procedures to solve the optimization problem.

3.2.1. Objective Function

The objective function is the total revenue of the VPP in Equation (14), which consists of three elements: energy settlement payments ( R e n e r g y t ), the REC sales revenue ( R r e c t ), and the penalty cost ( R p e n a l t y t ), as described in Equation (15).
Maximize ⁡ R t o t a l t
R t o t a l t = R e n e r g y t + R r e c t − R p e n a l t y t
Energy settlement payments are calculated based on both the day-ahead SMP, ρ d a _ s m p ( t ) , and the real-time SMP, ρ r t _ s m p t , as described in Equation (16). In this study, we consider only the bids submitted in the day-ahead market. The portion corresponding to the awarded bid quantity from the day-ahead market is priced according to ρ d a _ s m p ( t ) , while the difference between the actual generation and the awarded bid quantity is settled at ρ r t _ s m p t . The REC sales revenue is calculated as the product of actual generation and the REC price, as represented in Equation (17). Penalty costs occur when the deviation quantity exceeds the allowable tolerance, as shown in Equation (18). P d e v t , σ t , and ρ p e n ( t ) vary according to the penalty structure. The total penalty volume is determined by the difference between the deviation quantity and the allowable tolerance. The penalty cost is calculated by multiplying the total penalty volume by the penalty coefficient and the penalty rate.
R e n e r g y t = P d a t × ρ d a _ s m p ( t ) + P ( t ) − P d a t × ρ r t _ s m p t
R r e c t = P ( t ) × ρ r e c
R p e n a l t y t = P d e v t − P t o l × σ t × ρ p e n ( t )                                       i f     P t o l ≤ P d e v t 0 ,                                                                                                                   o t h e r w i s e

3.2.2. Constraints

The bidding process is structured into up to N s e g segments, with each segment specifying a pair of a generation quantity and price. Bidding rules are implemented in the model such that the bid price, ρ b i d s , t , and bid quantity, P b i d s , t , for each subsequent segment must be greater than those in the previous segment, as shown in Equations (19) and (20). This rule not only aligns bid quantity increments with potential revenue increases from energy settlements during periods of peak demand, but it also ensures that each bid is clearly distinguishable by avoiding ambiguities in the allocation of bid quantities at the same price across different segments. As specified in Equation (21), only a minimum bid price, ρ m i n b i d , is applied to prevent market price disruptions and ensure profitability. Additionally, the bid quantity must not exceed the maximum generation capacity or fall below the minimum generation capacity, as represented in Equations (22) and (23).
ρ b i d s + 1 , t > ρ b i d s , t               f o r   s   ∈ 1 , 2 , … N s e g − 1
P b i d s + 1 , t > P b i d s , t             f o r     s   ∈ 1 , 2 , … N s e g − 1
ρ b i d s , t ≥ ρ m i n b i d ( t )           f o r     ∀ s
P b i d s , t ≤ P m a x           f o r       ∀ s
P b i d s , t ≥ P m i n             f o r       ∀ s  
In a multi-segment bidding approach, the awarded bid quantity from the day-ahead market at time t , denoted as P d a t , is determined based on the day-ahead SMP ρ d a _ s m p ( t ) . Equation (24) defines this relationship, where P d a t is assigned the bid quantity P b i d s d a , t from the highest bidding segment s d a with a bid price lower than or equal to the day-ahead SMP. If no such segment exists, P d a t is set to zero.
P d a t = P b i d s d a , t i f   t h e r e   e x i s t s   s d a   s u c h   t h a t     s d a = max s ∈ 1 , 2 , … N seg ⁡ s ρ d a _ s m p ( t ) ≥ ρ b i d s , t 0                                                                                                                               o t h e r w i s e                                                                                                                                      
To mitigate potential penalties, the VPP can strategically reduce the output below the available generation through power curtailment. Therefore, the actual generation, P ( t ) , is calculated by subtracting the curtailed generation, P c u r t t , from the available generation, P a v a i l t , as shown in Equation (25). Importantly, P c u r t t should not be negative, as it represents a reduction in output, as given in Equation (26).
P t = P a v a i l t − P c u r t t        
P c u r t t ≥ 0  

3.3. Optimization Algorithm

When formulating an optimization problem to determine a bidding strategy, it is essential to consider the inherent uncertainties in both ρ d a _ s m p ( t ) and ρ r t _ s m p t . For VPPs, the unpredictability of RERs further complicates the difficulties during the bidding process. To consider such uncertainties, stochastic programming (SP) has been widely used in studies dealing with VPP trading and bidding [30,31,32,33]. In this study, we also employ the SP method to handle these uncertainties by considering potential scenarios and determining the optimal bidding inputs that maximize the expected value under uncertainty. The specific procedures of the SP are presented in Figure 3.
First, the penalty structure is determined based on the cases in Table 3. The input data for the model include the expected day-ahead SMP ( ρ ^ d a _ s m p ( t ) ), real-time SMP ( ρ ^ r t _ s m p t ), and forecasted generation ( P ^ t ), along with their associated errors of ε ρ ^ d a _ s m p t , ε ρ ^ r t _ s m p t , and ε P ^ ( t ) . These inputs form the foundation for the algorithm to generate a range of scenarios, equipping the VPP with strategies to handle various market conditions effectively. For each scenario, the probability of ρ d a _ s m p , γ 1 ( t ) , ρ r t _ s m p , γ 2 ( t ) , and P γ 3 t is determined as described in Equation (27). γ denotes the z-score as the probabilities for each scenario are assumed to follow a normal distribution, with the forecasted value as the expected value and the standard deviation based on the associated errors. Γ denotes the set of all possible z-score scenarios. Scenario α is constructed by combining these values as Equation (28). γ 1 , γ 2 , and γ 3 represent scenario indices corresponding to the uncertainties in the day-ahead SMP, real-time SMP, and generation output, respectively.
x γ t = x ^ t + ε x ^ t × γ   γ ∈ Γ         f o r         x ∈ ρ d a _ s m p , ρ r t _ s m p , P
α = ρ d a _ s m p , γ 1 t , ρ r t _ s m p , γ 2 t , P γ 3 t γ 1 , γ 2 , γ 3 ∈ Γ
Once a scenario is defined, the probability of the occurrence of each variable, π x γ t , and their simultaneous occurrence, π α , is calculated using Equations (29) and (30). δ is a variable used to calculate the probability over a continuous probability graph.
π x γ t     P ( x γ t + ε x ^ t × γ − δ ≤ X ≤ x γ t + ε x ^ t × γ + δ )
π α =   ∏ γ 1 , γ 2 , γ 3 ∈ Γ π x γ t
Then, the expected total revenue across all scenarios is maximized, as expressed in Equation (31), where N s c e n is the total number of scenarios, and R   t o t a l , α t is the total revenue at time t for the scenario α .
Maximize ⁡ ∑ α = 1 N π α R   t o t a l , α t
The constraints in Equations (16)–(26) guide the optimization process. Piecewise linearization is applied in the design of the LPCS in Equation (12), where the penalty costs increase in a quadratic curve as P d e v t exceeds P t o l . By breaking down the quadratic cost function into linear segments, this technique significantly reduces the computational complexity, facilitating a more feasible and efficient model. For Equation (18), which involves the calculation of the R p e n a l t y t , the McCormick envelope method is used to handle the product of bounded variables within explicit constraints. This scenario presents a bilinear programming problem; thus, the McCormick envelope method is employed to approximate the non-linear terms with linear inequalities, effectively transforming the problem into a linear programming format. This approach allows us to approximate complex non-linearities with a linear model, improving both the solvability and computational efficiency [34].
Once the number of generated scenarios reaches the specified number, N s c e n , the scenario generation process is complete. The algorithm then solves a linear convex optimization problem to obtain the optimal solution.
The vector Y represents the expected values derived from the results of each scenario α , weighted by the probabilities of each scenario α occurring, as defined in Equation (32). Y α denotes the results of scenario α , which are computed by applying the bid quantities and bid prices determined through the bidding strategy optimization model to the corresponding scenario α . Y includes components such as energy settlement payments, penalty costs, and various aspects of power generation and deviations. These values reflect the optimal bidding strategy and the expected value across all scenarios.
Y = ∑ α = 1 N s c e n π α Y α = E Y Y = R e n e r g y t , R p e n a l t y t , R r e c t , P d a   t , P d e v _ u p t , P d e v _ d w t , P c u r t t , P t

4. Simulation Environment & Results

4.1. Simulation Environment

Simulations are conducted to analyze the impact of the penalty structures in Section 3.1 on the revenue, power deviation, and curtailed generation of the VPP. The results are obtained from the optimization model in Section 3.2 and the algorithm in Section 3.3 implemented by the CPLEX solver.
The SMP data for both the day-ahead and real-time markets are taken from the KPX database, covering the period from March to June 2024. The SMP data are represented as box plots in Figure 4. The errors ε ρ ^ d a _ s m p t and ε ρ ^ r t _ s m p t are defined as the mean absolute errors of both the day-ahead and real-time SMP, respectively, indicating the average magnitude of the deviations from the actual values. Figure 5 presents the forecasted 24 h generation profile of a VPP in Jeju, Korea, on 17 March 2024.
The parameters used in the simulation are listed in Table 4. The errors in generation are set uniformly at 20% of the forecasted generation for all periods. The minimum bid price, ρ m i n b i d ( t ) , is set as a conservatively lower value than the minimum day-ahead SMP. It is assumed that the REC prices do not fluctuate on an hourly basis due to their less dynamic nature compared to the SMP. Specifically, the REC price is set at the average value of the SMP during the same period.

4.2. Simulation Results

4.2.1. VPP’s Revenue and Deviation Quantity

Figure 6 represents the VPP’s revenue—its energy settlement payments, penalty costs, REC sales revenue, and total revenue—across different values of the penalty price coefficient for Cases 3, 7, 10, and 11 during the morning ( t 6), afternoon ( t 12), and evening ( t 20), when ε b a n d is 20%. Figure 7 represents only the penalty cost extracted from Figure 6 for a more detailed analysis.
In Case 3, since the penalty scope corresponds to the OPS, the operator primarily engages in overbidding. Given the simplicity of this strategy, which focuses exclusively on overbidding, the revenue is less affected by fluctuations in the real-time SMP and penalty price coefficient. This case achieves the highest total revenue as the operator can curtail the generation output after overbidding. The second-highest total income happens in Case 7 as the UPS causes the operator to focus on underbidding. Case 11 employs the DPS, which imposes penalties for over- and under-generation, so the operator cannot perform extreme bidding like in Case 3 and Case 7. Therefore, when the penalty price coefficient is set to 0.1, Case 11 (DPS) generates significantly less total revenue compared to Case 3 (OPS): 62.26% lower in the morning, 6.96% lower in the afternoon, and 2.94% lower in the evening.
Figure 6a, which illustrates Case 3 at t 6, shows that the REC sales revenue is significantly lower than the energy settlement payment. This is because, at t 6, the operator adopts aggressive overbidding as the expected day-ahead SMP is higher than the expected real-time SMP and the REC price. The proportion of the REC sales revenue is only 7.71%, compared to an average of 31.93% in other periods. In Figure 7a, corresponding to Case 3, only a minor penalty is incurred at t 20 as the difference between the two SMP values is not as large as in earlier periods, and the errors are low. Thus, engaging in aggressive overbidding at this time poses a greater risk.
In Cases 7, 11, and 12, the total revenue decreases significantly as the penalty price coefficient increases, as illustrated in Figure 6b–d. Notably, the penalty price coefficient affects only the total revenue and does not consistently lead to an increase in penalty costs. Moreover, if the day-ahead SMP is considerably higher than the real-time SMP, the influence of the penalty price coefficient on the control of penalties is limited, as demonstrated in the results at t 12.
For cases with the same penalty scope, the reduction in revenue due to an increase in the penalty price coefficient is more pronounced in cases with the LPCS compared to the FPCS, as demonstrated in Figure 6c,d. For example, the average total revenue in Case 11 is KRW 1,853,221, compared to KRW 1,386,640 in Case 12. The reason for these differences is that the penalty costs significantly rise as the penalty coefficient increases linearly, even for a low penalty volume.
Figure 8 compares the total revenue for each case under varying penalty rates and penalty coefficient structures with ξ = 0.1 and ε b a n d = 20%. Overbidding strategies are consistently employed in the cases with the OPS, specifically Cases 1–4. Since the risk of penalty costs associated with receiving large awarded bid quantities is low, these cases are relatively unaffected by changes in the penalty price and penalty coefficient.
In contrast, the penalty coefficient structure significantly impacts the total revenue for both the UPS and DPS (e.g., Case 5–12). For example, at t 6 in Figure 8a, the FPCS generates higher revenue than the LPCS, exhibiting the most substantial discrepancies. This period is characterized by a higher expected day-ahead SMP than the expected real-time SMP, making it advantageous to receive larger awarded bids. However, due to insufficient generation during this period, the discrepancy between the awarded bid quantity and actual generation can be substantial, leading to higher penalty costs.
Figure 8b illustrates the differences in the total revenue between the RPRS and SPRS. The RPRS consistently yields slightly higher revenues than the SPRS, as the risk of incurring penalty costs under the RPRS is lower. Figure 8c further emphasizes the interplay between the penalty rates and penalty coefficient structures. Compared to Figure 8a, the differences in the total revenue become more pronounced when both the penalty rate and penalty coefficient structures are varied simultaneously. This highlights the compounding effect of these factors on the revenue outcomes of VPP operations, particularly under the UPS and DPS, where the penalty costs are more sensitive to deviations in generation.
Figure 9 shows the power deviations caused by over-generation and under-generation for all cases during the morning ( t 6), afternoon ( t 12), and evening ( t 20), across different values of the penalty price coefficient when ε b a n d is equal to 20%. Since over-generation and under-generation cannot occur simultaneously within a single scenario, the expected values of the deviations are calculated across all scenarios using Equation (32), where the power deviation from under-generation is defined as a negative value.
When the penalty scope is the OPS as in Cases 1–4, the operators receive a larger awarded bid quantity than their actual generation. Thus, as shown in Figure 9, the DUG is significantly larger than the DOG in these cases. Additionally, as the penalty price coefficient increases, the penalty costs rise, leading to a reduction in the DOG, while the DUG—unaffected by the penalty—remains primarily unchanged. For ξ ≤ 0.01 , over-generation occurs only at t 20 where the lower values of the prediction errors at t 20 allow for more expected revenue. At t 6, the DUG is consistently lower than at other times because the expected day-ahead SMP is higher than the expected real-time SMP, making it advantageous to receive more large bids. This overbidding during a low-generation period (1.02 MWh) results in a higher deviation caused by under-generation but still leads to favorable energy settlement payments. When comparing Case 1 (SPRS) and Case 3 (RPRS), the DOG is enormous in Case 3 because the REC price is lower than the expected real-time SMP. Additionally, comparing the LPCS cases (Case 1 and Case 3) with the FPCS cases (Case 2 and Case 4) reveals that the DOG decreases significantly under the LPCS. However, the DUG remains relatively unchanged across these cases.
In Cases 5–8, where the penalty scope is the UPS, underbidding is chosen to avoid larger awarded bids. As a result, unlike in Cases 1–4, the DUG is smaller and the DOG is significantly larger. However, at t 6 for ξ ≤ 1 , where the expected day-ahead SMP is higher than the expected real-time SMP, under-generation becomes more pronounced compared to other periods, making it advantageous to make larger bids and, consequently, resulting in an increased DUG. In Case 5, substantial under-generation occurs at t 12 as the expected day-ahead SMP is about 1.9 times higher than the expected real-time SMP. The greater DUG observed in Case 5 is due to the combination of the SPRS and LPCS. In Cases 6 and 8, which use the FPCS, the penalties are based on different price structures, yet the impact of the penalty coefficient remains significant. These cases display higher DUGs than Cases 5 and 7 due to the lower penalty costs.
In Cases 9–12, where the penalty scope is the DPS, the penalty costs are settled on the DOG and DUG. When the penalty price coefficient is low (e.g., ξ = 0.001), the impact of the penalties is minimal, leading to results similar to those observed in the UPS cases. However, as ξ increases, the influence of the penalties becomes more balanced between the DUG and DOG, particularly in cases using the LPCS, where the penalty costs rise rapidly with an increase in ξ .
The results of Case 9 show a similar pattern as those of Case 5 at low penalty price coefficients (e.g., ξ = 0.001 or 0.01). Similarly, Case 11 follows the same pattern as Case 7. Furthermore, when the penalty price coefficient is low, Cases 10 and 12—where the penalty coefficient is fixed—demonstrate patterns similar to Cases 6 and 8. This observation indicates that the DPS penalty scope’s influence is reduced at low values of ξ , causing the system’s behavior to align more closely with the UPS cases.

4.2.2. Curtailed Generation and Tolerance Band

From the VPP’s perspective, curtailing generation to avoid penalty costs is crucial. Figure 10 illustrates the curtailed generation for cases under the OPS and DPS with ε b a n d   = 20% and ξ = 1. In contrast, curtailing operations are precluded in the UPS due to its structural characteristics.
The curtailment stands out in Cases 9 and 11 during the periods from t 14 to t 19. These cases share two important characteristics. First, they both employ the DPS, which applies penalties for deviations in both over-generation and under-generation. Second, they utilize the LPCS, causing the penalty costs to escalate rapidly with increasing deviations. This combination prompts more substantial output adjustments by the operator to mitigate penalties. During the period in which curtailment occurs, Case 9 exhibits higher power curtailment than Case 11. However, between t 11 and t 13, as the REC price is higher than the real-time SMP, Case 11 experiences higher curtailment than Case 9. Although the REC revenue is high, the penalties in Case 11 outweigh this benefit, necessitating greater curtailment.
The next highest curtailment levels occur in Cases 1 and 3 with the OPS and LPCS. Thus, it can be concluded that curtailment is predominantly observed in cases where the LPCS is applied. In Cases 1 and 3, where penalties are applied only for over-generation, curtailment occurs during periods of high generation and a real-time SMP, making it beneficial to maximize the revenue in the real-time market through energy settlements. However, curtailment is less frequent than in Cases 9 and 11, as there is no penalty for under-generation, reducing the need for aggressive output adjustments. For example, between t 10 and t 13, when the expected real-time SMP is low, no curtailment is needed due to the low probability of incurring penalty costs. Furthermore, OPS cases exhibit similar curtailment patterns regardless of the penalty rate structures.
While most curtailment under the OPS is relatively small, there are notable periods of significant curtailment (e.g., t 19, t 22– t 24). During periods with high ε ρ ^ d a _ s m p t (e.g., t 19, t 23, t 24), the curtailment increases substantially due to the higher likelihood of incurring significant penalty costs. However, at t 22, significant curtailment occurs despite the low ε ρ ^ d a _ s m p t . This anomaly is caused by reduced overbidding in response to a high expected real-time SMP, resulting in lower awarded bids. Paradoxically, this increases the likelihood of significant penalty costs, driving curtailment during this period.
Figure 11 illustrates the variation in total revenue for Case 9, with ξ = 1, with respect to the tolerance band. As the tolerance band increases, a notable decrease in the penalty cost is observed. However, the influence of the tolerance band extends beyond the penalty costs, affecting other revenue components, such as the energy settlement payments and REC sales revenue. This highlights that a higher tolerance band provides greater flexibility to the bidding strategy to minimize penalties while enhancing the overall revenue streams. Consequently, the total revenue varies considerably with changes in the tolerance band. The increase in total revenue with a larger tolerance band underscores the importance of its setting in maximizing the VPP’s profit. This clearly demonstrates that adjusting the tolerance band has a direct effect on the revenues.
Figure 12 compares the total revenue for Cases 1, 5, and 9 during periods of high generation and small differences between the expected day-ahead and real-time SMPs—specifically at t 6, t 20, t 21, and t 22—under varying tolerance bands. Case 1 exhibits minimal sensitivity to changes in the tolerance band across all the analyzed periods. This is particularly evident at t 6, where the revenue remains almost unaffected by variations in the tolerance band. The risk of overbidding is mitigated at these periods due to the low generation levels, high expected day-ahead SMP, and minor SMP prediction errors. As a result, the likelihood of incurring penalty costs is low and the tolerance band has little impact on the revenue.
Case 5 shows slightly greater sensitivity to the tolerance band than Case 1 but remains less sensitive than Case 9. This difference arises because an underbidding strategy is predominantly employed in Case 5. By avoiding significant deviations above the generation levels, the UPS limits the influence of the tolerance band on the revenue. Notably, there are instances where Case 5 generates higher revenue than Case 1. For example, at t 22, when the expected day-ahead SMP is lower than the expected real-time SMP, the total revenue of the OPS is lower than that of the UPS. In this situation, an underbidding strategy proves more profitable than an overbidding strategy, enabling Case 5 to capitalize on the favorable real-time market conditions. This reversal in revenue patterns contrasts with other periods, where the OPS typically yields higher returns.
Case 9, characterized by the DPS, demonstrates the highest sensitivity to the tolerance band among the three cases. This sensitivity arises because penalties are applied in both directions, amplifying the impact of the tolerance band on the total revenue. As the tolerance band increases, the total revenue increases due to the reduced penalty costs. However, the marginal improvement in revenue diminishes with larger tolerance bands, as the generation capabilities impose an upper limit on the penalty costs.

5. Discussion

Based on the findings in Section 4.2, we offer recommendations for the design of an effective penalty structure aimed at assisting policymakers and DSOs in structuring market mechanisms, which not only facilitate the integration of RERs but also enhance their economic viability within electricity markets. These guidelines are categorized by their impact on the VPP revenues, electricity market stability, energy efficiency, and market operations’ controllability, which are outlined in Table 5.
The choice of penalty scope significantly affects the VPP revenues. The OPS is generally preferable in terms of maximizing the VPP revenue, especially when the expected day-ahead SMP is lower than the expected real-time SMP. Conversely, underbidding approaches for the UPS may prove more profitable in scenarios where the expected real-time SMP exceeds the day-ahead SMP. Across non-OPS structures, the choice of the penalty rate and coefficient, notably the LPCS and SPRS, often results in lower revenues than their alternatives. Furthermore, broader tolerance bands consistently lead to an increase in revenues across all penalty structures, although the marginal improvements diminish as the tolerance band increases.
For markets where stability is paramount, penalty structures like the DPS are particularly beneficial as they minimize the DOG and DUG, reducing overall deviations. This is especially important in markets with significant DER penetration, where stability requires close adherence to the balance in supply and demand. While the OPS specifically targets DOG reduction, it often leads to a substantial DUG and is better suited for systems with supplementary generation resources. In contrast, the UPS may result in an increased DOG but can be advantageous for systems with large energy storage capabilities. Additionally, the LPCS and SPRS tend to produce lower deviations compared to the FPCS and RPRS, making them preferable for maintaining market stability.
From an energy efficiency perspective, minimizing unnecessary power curtailments is a priority, as curtailments represent wasted electricity. Non-curtailment structures, such as the UPS or FPCS, are ideal if significant energy storage systems exist. Due to its broader penalty scope, the DPS generally enforces more curtailments compared to the OPS. Within the DPS framework, the choice between the RPRS and SPRS is heavily influenced by external conditions like the real-time SMP, highlighting the complex relationship between penalty structures and environmental factors.
For DSOs, the ability to adjust market operations dynamically based on penalty coefficients is crucial in maintaining operational flexibility. The OPS offers limited flexibility as changes in penalty rates or coefficients have minimal impact on revenues or deviations, reducing its adaptability. In contrast, the UPS and DPS provide excellent controllability, with the DPS standing out due to its heightened sensitivity to penalty structure adjustments, particularly the tolerance bands. The DPS allows for fine-tuned operational control, making it a highly flexible option for dynamic market environments. However, while the penalty structures indirectly influence revenues through energy settlement payments and REC sales, they do not directly control the penalty costs. This limitation restricts the ability of DSOs to manage the penalty costs directly, highlighting the need for a balanced approach that considers both the indirect and direct impacts of penalty structures on market operations.
While this study holds substantial engineering value and provides partial guidance for the design of penalty mechanisms, it could delve deeper into the real-world dynamics and practical implementation challenges. Based on the findings, future research could further deepen our understanding by quantitatively analyzing the sensitivity of specific penalty coefficients and their interactions with other penalty variables. This could provide a clearer perspective on the optimal penalty settings under particular conditions, enhancing the applicability of penalty systems across various policy environments. By exploring these aspects in detail, future studies could refine the penalty structures and contribute to creating optimized mechanisms that better align with the dynamic conditions of renewable energy markets, facilitating the more robust integration of renewables into the power system.

6. Conclusions

This study has thoroughly analyzed the impact of penalty structures on VPPs in a day-ahead electricity market, considering the unique characteristics and challenges of renewable energy sources. By categorizing penalties into three factors—the penalty scope (OPS, UPS, DPS), penalty rate (SPRS, RPRS), and penalty coefficient (LPCS, FPCS)—we have developed a comprehensive framework for the assessment of how different penalty structures influence the VPP revenues and operational behaviors.
Our analysis focused on three key revenue components—energy settlement payments, the REC sales revenue, and the penalty costs—and their combined effect on the total revenue. The findings reveal that the choice of penalty scope significantly impacts the VPP revenues. The OPS generally yields the highest revenues by minimizing deviations in over-generation, thus encouraging overbidding strategies that maximize returns. Conversely, the UPS and DPS, which penalize under-generation and power deviations in both directions, tend to yield lower revenues than the OPS. These two structures show a notable decline in revenue as the penalty coefficients increase, particularly when the LPCS is applied. In contrast, stable revenue outcomes are observed for the FPCS. Additionally, the SPRS incurs higher penalty costs than the RPRS, highlighting the critical role of penalty rate structures in shaping profitability.
Beyond revenue analysis, we examined power deviations, curtailments, and their implications for penalty structure recommendations. Penalty structures are ideal if significant energy storage systems are available to store energy. Moreover, due to its broader penalty scope, the DPS typically enforces more curtailments than the OPS. Furthermore, the DPS is influenced heavily by external conditions like the real-time SMP, which underscores the complex interplay with environmental factors. From an operational flexibility perspective, dynamically adjusting market operations based on the penalty coefficients is essential for DSOs. The OPS exhibits the least flexibility as changes in penalty rates or coefficients have a minimal impact on revenues or deviations. In contrast, the UPS and DPS offer greater controllability due to their responsiveness to changes in penalty structures, with the DPS being notably flexible owing to its sensitivity to tolerance band adjustments.
In conclusion, this research contributes significantly to understanding the complexities surrounding penalty structures in electricity markets. By providing strategic guidelines for DSOs in structuring penalty costs, this study could help to optimize various operational and economic objectives. Through a detailed framework and comprehensive analysis, it offers valuable insights for DSOs and policymakers to design market mechanisms that support the efficient and reliable integration of renewable energy resources into the power grid.

Author Contributions

Conceptualization, Y.S. and Y.J.; methodology, Y.S. and Y.J.; formal analysis, Y.S. and Y.J.; investigation, Y.S., M.C. and Y.J.; writing—original draft preparation, Y.S., M.C., Y.C. and Y.J.; writing—review and editing, Y.Y.; visualization, Y.S.; supervision, Y.J. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by the 2024 scientific promotion program funded by Jeju National University.

Data Availability Statement

The data described in section “Simulation Environment” is available at https://kpx.or.kr (accessed on October 2024).

Conflicts of Interest

The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of the data; in the writing of the manuscript; or in the decision to publish the results.

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Figure 1. Classification of penalty structures based on the penalty scope.
Figure 1. Classification of penalty structures based on the penalty scope.
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Figure 2. Structure of penalty coefficient and penalty cost.
Figure 2. Structure of penalty coefficient and penalty cost.
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Figure 3. Flowchart to determine an optimal bidding strategy using stochastic programming.
Figure 3. Flowchart to determine an optimal bidding strategy using stochastic programming.
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Figure 4. SMP and its errors over 24 h in the day-ahead and real-time markets.
Figure 4. SMP and its errors over 24 h in the day-ahead and real-time markets.
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Figure 5. Forecasted power generation of a VPP for 24 h.
Figure 5. Forecasted power generation of a VPP for 24 h.
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Figure 6. Impact of penalty coefficient on revenue in Case 3, Case 7, Case 11, and Case 12.
Figure 6. Impact of penalty coefficient on revenue in Case 3, Case 7, Case 11, and Case 12.
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Figure 7. Impact of penalty coefficient on penalty costs in Case 3, Case 7, Case 11, and Case 12.
Figure 7. Impact of penalty coefficient on penalty costs in Case 3, Case 7, Case 11, and Case 12.
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Figure 8. Comparison of total revenue concerning both the penalty rate and penalty coefficient.
Figure 8. Comparison of total revenue concerning both the penalty rate and penalty coefficient.
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Figure 9. Impact of penalty coefficient on power deviation in Case 1–12.
Figure 9. Impact of penalty coefficient on power deviation in Case 1–12.
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Figure 10. Curtailed power in all simulation cases.
Figure 10. Curtailed power in all simulation cases.
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Figure 11. Impact of tolerance band on revenue in Case 9.
Figure 11. Impact of tolerance band on revenue in Case 9.
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Figure 12. Effect of tolerance band on total revenue.
Figure 12. Effect of tolerance band on total revenue.
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Table 1. Bidding systems in electricity markets [25,26,27,28,29].
Table 1. Bidding systems in electricity markets [25,26,27,28,29].
ISODA/RTNumber of SegmentsPrice CapPrice Floor
PJM [25]O/O10USD 1000/MWhNot Implemented
CAISO [26]O/O10Soft Cap—USD 1000/MWhNot Implemented
Hard Cap—USD 2000/MWh
NYISO [27]O/O12 m i n b i d ,   m a x 1000 ,   m i n 2000 , r e f e r e n c e Minimum of Generation Cost
MISO [28]O/O9USD 1000/MWhMinimum of Generation Cost
KPX [29]O/O10KRW 0/kWh− (REC Price)
Table 2. Penalty rate-based structures.
Table 2. Penalty rate-based structures.
Penalty Rate StructurePenalty RatePenalty Cost
SPRSSMP-Based ρ p e n _ S M P ( t ) = ρ s m p ( t ) ∗   ξ
RPRSREC-Based ρ p e n _ R E C ( t ) = ρ r e c ∗   ξ
Table 3. Case variations based on penalty structures.
Table 3. Case variations based on penalty structures.
CasePenalty ScopePenalty RatePenalty Coefficient
1OPSSPRSLPCS
2FPCS
3RPRSLPCS
4FPCS
5UPSSPRSLPCS
6FPCS
7RPRSLPCS
8FPCS
9DPSSPRSLPCS
10FPCS
11RPRSLPCS
12FPCS
Table 4. Parameters in simulation.
Table 4. Parameters in simulation.
ParameterValueParameterValue
ε P ^ ( t ) P ^ t × 0.2 [MWh] k p e n 1 [1/MW]
N s e g 10 c p e n 1
ρ m i n b i d ( t ) ρ d a _ s m p , m i n ( t ) × (−2) [KRW/MWh] P m a x 60 [MW]
ρ r e c 76 [KRW/MWh] P m i n 0 [MWh]
Γ [−3, −2, −1, 0, 1, 2, 3] δ 0.5
Table 5. Comparison of penalty structures across objectives.
Table 5. Comparison of penalty structures across objectives.
ObjectivePenalty Rate StructurePenalty RatePenalty Cost
Maximizing VPP Profit- OPS > UPS > DPS
- UPS > OPS > DPS
(Depends on SMP)		SPRS > RPRSLPCS > FPCS
Enhancing Market Stability- DPS > OPS > UPS
- DPS > UPS > OPS
(Depends on System)		SPRS > RPRSLPCS > FPCS
Improving Energy EfficiencyUPS > OPS > DPS(Depends on SMP)FPCS > LPCS
Ensuring Operational FlexibilityDPS > UPS > OPSSPRS > RPRSLPCS > FPCS
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Song, Y.; Chae, M.; Chu, Y.; Yoon, Y.; Jin, Y. Impact of Penalty Structures on Virtual Power Plants in a Day-Ahead Electricity Market. Energies 2024, 17, 6042. https://doi.org/10.3390/en17236042

AMA Style

Song Y, Chae M, Chu Y, Yoon Y, Jin Y. Impact of Penalty Structures on Virtual Power Plants in a Day-Ahead Electricity Market. Energies. 2024; 17(23):6042. https://doi.org/10.3390/en17236042

Chicago/Turabian Style

Song, Youngkook, Myeongju Chae, Yeonouk Chu, Yongtae Yoon, and Younggyu Jin. 2024. "Impact of Penalty Structures on Virtual Power Plants in a Day-Ahead Electricity Market" Energies 17, no. 23: 6042. https://doi.org/10.3390/en17236042

APA Style

Song, Y., Chae, M., Chu, Y., Yoon, Y., & Jin, Y. (2024). Impact of Penalty Structures on Virtual Power Plants in a Day-Ahead Electricity Market. Energies, 17(23), 6042. https://doi.org/10.3390/en17236042

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