Novel Frequency Regulation Scenarios Generation Method Serving for Battery Energy Storage System Participating in PJM Market

Abstract

As one of the largest frequency regulation markets, the Pennsylvania-New Jersey-Maryland Interconnection (PJM) market allows extensive access of Battery Energy Storage Systems (BESSs). The designed signal regulation D (RegD) is friendly for use with BESSs with a fast ramp rate but limited energy. Designing operating strategies and optimizing the sizing of BESSs in this market are significantly influenced by the regulation signal. To represent the inherent randomness of the RegD signal and reduce the computational burden, typical frequency regulation scenarios with lower resolution are often generated. However, due to the rapid changes and energy neutrality of the RegD signal, generating accurate and representative scenarios presents challenges for the methods based on shape similarity. This paper proposes a novel probability-based method for generating typical regulation scenarios. The method relies on the joint probability distribution of two features with a 15-min resolution, extracted from the RegD signal with a 2 s resolution. The two features can effectively portray the characteristic of RegD signal and its influence on BESS operation. Multiple regulation scenarios are generated based on the joint probability distributions of these features at first, with the final typical scenarios chosen based on their probability distribution similarity to the actual distribution. Utilizing regulation data from the PJM market in 2020, this paper validates and analyzes the performance of the generated typical scenarios in comparison to existing methods, specifically K-means clustering and the forward scenarios reduction method.

1. Introduction

To achieve the carbon-neutral target set for 2050, the deployment of Battery Energy Storage Systems (BESSs) has been highly encouraged to accelerate energy transition [1]. Apart from assisting the integration of renewable energy sources such as wind and solar [2], BESSs can support modern power systems by providing a range of applications, including energy arbitrage [3], peak demand reduction [4], frequency regulation [5], voltage support [6], transmission and distribution upgrade deferral [7], and resilience enhancement [8]. However, the broader adoption of BESS technologies is primarily constrained by its low return on investment due to the direct revenue generated, though the above services can be modest in comparison to the initial high upfront costs.

The economic valuation of various services provided by BESSs is investigated in [9,10,11,12], where frequency regulation is regarded as the most profitable among the array of applications. Additionally, stacking frequency regulation with other services in the process of utilizing BESSs enables the achievement of multiple revenue streams. This strategy is recommended in the above research to improve the investment feasibility of BESSs. As highlighted in [9], which assessed the economic viability of BESSs’ various applications including energy arbitrage, regulation, reserve, load following, peak shaving, and ramp products, stacking frequency regulation with other services can significantly reduce the payback period from 25 years to as few as 12.9 years. Study [11] presents a similar conclusion, indicating that the returns from frequency regulation substantially surpass those from energy arbitrage and spinning reserves. Furthermore, the feasibility of technique and economics by using BESSs to provide frequency regulation with energy arbitrage and reliability is explored in [12], which shows that both grid reliability and revenue are enhanced.

The Pennsylvania-New Jersey-Maryland Interconnection (PJM) operates the largest fast frequency regulation (FFR) market globally, allowing for significant integration of BESSs [13]. Compared to other frequency regulation markets, such as the Midwest Independent Transmission System Operator (MISO) and the New York Independent System Operator (NYISO) in the USA, which use the automatic generation control (AGC) signal for all resources, PJM has specifically designed a frequency regulation signal called regulation D (RegD) with a two-second resolution for fast-ramping assets like BESSs [14]. On this basis, extensive research about BESSs participating in the FFR market is executed based on this signal [12,13,15,16,17]. For instance, a day-ahead operating strategy for a grid-connected BESS is presented in [17], which aims to maximize the revenue from energy arbitrage and frequency regulation. Another operating strategy for BESSs providing energy arbitrage, peak shaving, and frequency regulation is presented in [18], where a three-stage framework is proposed to consider the day-ahead and real-time action of BESSs. To accurately estimate the revenue from frequency regulation and make precise day-ahead decisions, the uncertainty of RegD signals in both amplitude and direction should be considered, where both [17,18] utilize typical frequency regulation scenarios and their associated probabilities to describe the uncertainty of RegD signals. Additionally, some attention has been dedicated to the optimization of BESSs’ sizing to provide multiple services including frequency regulation [19,20,21]. Accurately estimating the revenue from frequency regulation is fundamental to making reliable investment decisions. Especially for BESSs providing multiple services, it needs to carefully design its rated energy and power to balance the benefits generated from the provided services. The accuracy and efficiency of sizing optimization also depend on the selected typical frequency regulation scenarios. A sizing optimization framework presented in [19] is designed for BESSs to provide energy arbitrage and frequency regulation, which estimates the revenue based on the deterministic frequency regulation scenario, which can easily lead to inaccurate estimations of frequency-related benefits. In [20], a customer-side BESSs’ sizing optimization framework that integrates energy arbitrage, peak shaving, and FFR is proposed. It employs the stochastic optimization method, where typical scenarios and their probabilities are utilized to represent the uncertainty of frequency regulation signals, thereby ensuring the accuracy and reliability of the investment decision.

Typically, a frequency regulation scenario is referred to the RegD signals lasting for a day. Various methods for selecting typical regulation scenarios are discussed in [5,17,18,19,20,21,22]. One approach, as outlined in [19], involves subjectively choosing daily regulation signals per month for BESSs’ sizing optimization. However, this method overlooks the inherent uncertainty of regulation scenarios, potentially resulting in sub-optimal BESS sizing outcomes. In contrast, the objective-based typical scenario generation methods are demonstrated in [5,17,18,20,22]. The K-means clustering method is employed in [5,22] to generate a subset including four typical scenarios along with their respective probabilities. Similarly, a subset of multiple typical regulation scenarios with probabilities is established using the forward scenario reduction (FSR) algorithm, as demonstrated in [17,18,20]. In these studies, typical regulation scenarios are usually generated based on one year of historical data, which can better reflect the uncertainty surrounding regulation scenarios compared to [19]. However, doubts persist. Figure 1a illustrates quick and irregular fluctuations in RegD signals, prompting an exploration of the effectiveness of the K-means method for generating typical scenarios based on shape similarity. Crucially, there is currently a lack of validation for the effectiveness of typical frequency regulation scenarios generated through various methods. Moreover, to reduce the computational burden, the final typical regulation scenarios are generated by sampling at a longer resolution, such as 15 min or 1 h [12,19,21]. On the one hand, this sampling method may change the characteristics of the RegD signal, specifically its energy neutrality within 15-min intervals. On the other hand, random sampling may alter the typical scenarios, rendering them unrepresentative. Thus, an effective method that can generate typical frequency regulation scenarios with a longer resolution but keep the characteristics of RegD signal are required.

Aiming at this gap, two features with a lower resolution are presented in our previous research [20] for equivalently replacing the original RegD signal. Based on the two features, this paper proposes a novel method for generating typical frequency regulation scenarios serving the operation and planning of BESSs providing FFR in the PJM market. The contributions are summarized as follows:

• The proposed typical frequency regulation scenarios, composed of two features with a 15-min resolution, are generated based on their joint probability distribution. The two features, calculated based on the RegD signal and BESS parameters, capture the influence of the RegD signal on BESS operation and the characteristic of energy neutrality during 15 min of the RegD signal;
• The effectiveness of typical regulation scenarios generated based on K-means clustering and FSR in prior research is validated, which is from the perspective of accuracy in estimating regulation revenue and the consistency of probability distribution.

2. Market Rule

This section introduces the characteristics of the RegD signal, the market rule of BESSs participating in FFR in the PJM market, and outlines the mathematical model employed for optimizing the size of BESSs participating in FFR.

2.1. RegD Signal

In different ancillary service markets, the rules of frequency regulation vary in terms of regulation signal and payment mechanisms. This paper targets BESSs participating in the PJM market. To maintain the system frequency, the regulation signal will be sent to the regulation resource. Unlike other markets, such as MISO and NYISO, which directly use AGC signals for all resources, PJM has developed two distinct signals, known as Regulation A (RegA) and RegD. These signals are designed to efficiently manage the area control error (ACE) by leveraging resources with different characteristics effectively. These signals are crucial for maintaining grid stability and ensuring efficient operation.

RegA is typically designed for traditional generation resources with slow ramping rates but a large capacity. This signal compensates for larger and longer fluctuations in the power system, necessitating resources that can sustain output for extended periods.

Conversely, RegD is a fast and dynamic signal, which is tailored for resources with rapid ramp rates that can instantaneously respond to changes in demand, often exemplified by BESSs. Additionally, the RegD signal is thoughtfully designed to maintain conditional energy neutrality within 15 min. Energy neutral means that the amount of power output used to increase the grid frequency provided by a RegD resource matches the amount used to decrease the grid frequency by the same resource; that is, the mean value of the RegD signal during 15 min is 0. Conditional energy neutrality refers to the PJM market prioritizing managing ACEs while only achieving energy neutrality under appropriate system conditions. Thus, it is specifically suitable for BESSs with limited energy.

Since the PJM market operates on an hourly clearing basis, three hourly regulation signals are randomly selected and exhibited in Figure 1a. It can be found that the RegD signal, which changes rapidly with a 2 s resolution, ranges between −1 and 1. When the RegD signal is above 0, the BESS is required to release power to support the grid. Conversely, when the signal is below 0, the BESS needs to absorb power from the grid, thereby charging the BESS. Figure 1b shows the mean value of RegD signals over a 15 min interval, indicating that most values fall between −0.3 and 0.3, which is consistent with the concept of conditional energy neutrality.

2.2. Frequency Regulation Market Rule

The FFR market operates on a day-ahead basis, requiring BESS operators to submit bidding proposals in advance. These proposals specify both the participation hours and the corresponding bidding capacities for the upcoming operating day. For the given hour h, the revenue $R_h^{FR}$, as defined in (1), depends on the regulation clearing price ${\sigma }_h^{FR}$, the bidding capacity $C_h^{bid}$, and the performance score ${\delta }_h$. Correlation, delay, and precision are three aspects to evaluate whether the BESS performs well in tracking the RegD signal [14]. The BESS usually achieves a perfect score in the correlation and delay aspects [23]. The precision score, as defined in (2), is usually focused on in the process. It reflects the discrepancy between the actual power $P_{t,h}^{FR}$ from the BESS and the power it should exchange according to its bidding capacity and the current RegD signal, $g_{t,h}$.

$$R_h^{FR}={\sigma }_h^{FR}{C}_h^{bid}{\delta }_h$$

(1)

$${\delta }_h=1-\frac{1}{T}{\displaystyle \sum _{t=1}^T\left|\frac{P_{t,h}^{FR}+C_h^{bid}{g}_{t,h}}{C_h^{bid}{g}_{t,h}}\right|}$$

(2)

2.3. Framework of Optimizing BESS Sizing

In the profit-oriented BESS-invested project, it usually takes the net revenue, net present value, or the investment return rate as the objective function in the planning model. This section employs a stochastic optimization model presented in [20], with the objective function of maximizing the expected net revenue, as shown in (3). It is achieved by subtracting investment costs from the expected regulation revenue. In this process, the accurate estimation of regulation revenue, $R_h^{FR}$, and the degradation cost, $C_h^{de}$, is the foundation for accurate and reliable decision making. Both of them are highly related to the frequency regulation scenarios, which are full of uncertainty. The uncertainty of RegD signals is modeled by multiple typical scenarios and their corresponding probability.

$$max{\displaystyle \sum _{s\in S}{p}_s}{\displaystyle \sum _{h\in 24}(R_{h,s}^{FR}}-C_{h,s}^{de})-C^{Inv}$$

(3)

where $C_{h,s}^{de}$ is the BESS degradation cost at hour h in the scenario s incurred by participating in FFR. $C^{Inv}$ is the initial investment cost, which is related to the rated power and size of the BESS. s denotes the index of typical regulation scenarios, and p(s) is the corresponding probability.

In this section, the linear degradation model, consistent with [24], is utilized. It is regarded as a linear function of BESS charging/discharging power.

$$C_{h,s}^{de}={\sigma }^{de}{\displaystyle \sum _{t=1}^T\left(\left|P_{t,h,s}^c{\eta }^c\right|+\left|P_{t,h,s}^d/{\eta }^d\right|\right)}\Delta t$$

(4)

where ${\sigma }^{de}$ represents the degradation cost. $P_{t,h,s}^c$ and $P_{t,h,s}^d$ are the charging power and discharging power of the BESS at time t of hour h in scenario s. ${\eta }^c$ and ${\eta }^d$ are the charging efficiency and discharging efficiency of the BESS. $\Delta t$ is the selected resolution of the optimization problem.

In the optimization model, utilizing RegD signal with a 2 s resolution would impose a significant computational burden on BESSs’ sizing optimization. It is customary to sample RegD signals at a lower resolution, such as sampling in 1 h. However, random sampling will change the selected typical scenarios, making it not representative. Adopting the average value of the RegD signal over an hour reduces the action of the BESS due to the offset between positive and negative RegD signals. Consequently, this approach will lead to an underestimation of the degradation costs associated with the BESS. Additionally, taking 1 h as the sampling time neglects the characteristic inherent of conditional energy neutral.

Consequently, two features with a 15-minute resolution, denoted as $g^I$ and $g^{II}$, are introduced in [20]. Specifically, $g^I$ in (5) reflects the impact of the RegD signal on the state of charge (SoC) of the BESS, and portrays the conditional energy-neutral characteristics of the RegD signals. $g^{II}$ in (6) is associated with the degradation costs incurred by the BESS participating in FFR.

$$g_{t,h}^I={\displaystyle \sum _{t_1\in t}\left(g_{t_1,h}^{+}{\eta }^c+g_{t_1,h}^{-}/{\eta }^d\right)}\Delta t_1/\Delta t$$

(5)

$$g_{t,h}^{II}={\displaystyle \sum _{t_1\in t}\left(\left|g_{t_1,h}^{+}{\eta }^c\right|+\left|g_{t_1,h}^{-}/{\eta }_d\right|\right)}\Delta t_1/\Delta t$$

(6)

where $g_{t_1,h}^{+}$ and $g_{t_1,h}^{-}$ denote positive and negative RegD signals. $\Delta t_1$ and $\Delta t$ denote the resolutions, which are 2 s and 15 min, respectively.

The following constraints are included in the model to decide the time, bidding capacity, and actual power of the BESS in a day.

$$P_{t,h,s}=(1-{\gamma }_{h,s})P_{t,h,s}^{FR}+{\gamma }_{h,s}{P}_{t,h,s}^R$$

(7)

$$0\le C_{h,s}^{bid}\le (1-{\gamma }_{h,s})P^{rate}$$

(8)

$$P_{t,h,s}-{\gamma }_{h,s}{P}^{rate}\le P_{t,h,s}^{FR}\le P_{t,h,s}+{\gamma }_{h,s}{P}^{rate}$$

(9)

$$-P^{rate}(1-{\gamma }_{h,s})\le P_{t,h,l}^{FR}\le P^{rate}(1-{\gamma }_{h,s})$$

(10)

$$-P^{rate}{\gamma }_{h,s}\le P_{t,h,s}^R\le P^{rate}{\gamma }_{h,s}$$

(11)

where $P_{t,h,s}$ is the power from the BESS at time t of hour h in the scenario s. ${\gamma }_{h,s}$ is a binary variable indicating whether the BESS provides frequency regulation service at hth hour in the sth scenario. When it equals 0, it represents the BESS providing frequency regulation service. $P_{t,h,s}^{FR}$ is the exchanged power from the BESS for supporting frequency regulation. $P_{t,h,s}^R$ is the power of the BESS to recover its energy. $P^{rate}$ is the rated power of the BESS.

The constraints (12)–(17) are added to ensure that the security operation of the BESS is in the rated power and SoC limitation in the operation process.

$$P_{t,h,s}=P_{t,h,s}^c+P_{t,h,s}^d$$

(12)

$$-P^{rate}\le P_{t,h,s}\le P^{rate}$$

(13)

$$0\le P_{t,h,s}^d\le P^{rate}$$

(14)

$$-P^{rate}\le P_{t,h,s}^c\le 0$$

(15)

$$SO{C}^{lb}\le SO{C}^{ini}-{\displaystyle \sum _{h=1}^H{\displaystyle \sum _{t=1}^T\frac{[P_{t,h,s}^c{\eta }^c+P_{t,h,s}^d/{\eta }_d]\Delta t}{E^{rate}}}}\le SO{C}^{ub}$$

(16)

$${\displaystyle \sum _{h=1}^H{\displaystyle \sum _{t=1}^T[P_{t,h,s}^c{\eta }^c+P_{t,h,s}^d/{\eta }^d]\Delta t}=0}$$

(17)

where $SO{C}^{ini}$ is the initial SoC of the BESS. In the BESSs’ operating process, it cannot surpass the limitation [SoC^lb, SoC^ub] to protect the BESSs’ lifetime. T equals 4 since the resolution is 15 min. $E^{rate}$ is the rated energy of the BESS.

3. Frequency Regulation Scenarios Generation

3.1. Regulation Scenario Analysis

Instead of the regulation scenario comprising the original RegD signal with a 2 s resolution, the frequency regulation scenario in this paper consists of two features, $g^I$ and $g^{II}$, each with a 15-minute resolution as introduced in Section 2.3. The improved scenario can be directly applied in designing day-ahead operation strategies and BESS sizing optimization, ensuring precise optimization results and reducing computational complexity. As shown in Figure 1a, the original RegD signals display irregular shapes, which makes it challenging to select typical scenarios directly based on shape similarity.

Therefore, compared with generating typical scenarios based on the original RegD signal and then calculating the two features, this paper prefers to directly generate typical frequency regulation scenarios based on the probability distribution of these two features.

When generating typical scenarios based on the probability distribution, $g^I$ and $g^{II}$ need to exhibit independence in their temporal sequences and show consistent behavior across the same statistical distribution. To validate the applicability of the probability distributions derived from all signals, the Pearson correlation coefficient [25] is initially employed to ascertain the presence of time correlation in $g^I$. Subsequently, the consistency of the probability distributions of $g^I$ at different moments of the day is examined based on their probability density functions. An identical examination is also applied to $g^{II}$.

Based on the RegD signals obtained from the PJM market in 2020, the values of two features are first calculated. Due to the 15-minute resolution, there are 96 groups of $g^I$ and $g^{II}$ each day, with each group containing data from 366 days within the year. Following this, Pearson’s correlation coefficient and the probability density functions of $g^I$ and $g^{II}$ are calculated for 96 groups, as visually represented in Figure 2 and Figure 3.

Figure 2a,b present the Pearson correlation coefficient for $g^I$ and $g^{II}$, elucidating their temporal correlations between 96 moments in a day. A Pearson correlation coefficient greater than 0.5 is generally considered to indicate a moderate to strong correlation between variables. Notably, the values are primarily concentrated in the range of [0 to 0.2], indicative of a very weak correlation. This suggests that the impact of a signal at the current moment on the signal at a subsequent moment is exceedingly minimal. Figure 3a illustrates the probability density functions of the 96 groups, showcasing similar probability distributions. Analogous observations are applicable to $g^{II}$ in Figure 3b. Consequently, the probability distribution derived from the entire year’s data is deemed suitable for each moment.

3.2. Proposed Method for Generating Frequency Scenarios

In this section, an algorithm is proposed for directly generating typical scenarios composed of $g^I$ and $g^{II}$, which are based on the joint probability distribution of the two features. By using the joint probability distribution, the inherent correlation between $g^I$ and $g^{II}$ can be captured, leading to the typical scenarios better representing the uncertainty of RegD signal. Concrete steps are as follows:

(1)

To obtain the probability distribution of $g^I$ and $g^{II},$ the historical RegD signal lasting for a year (366 days) is employed in this section. Based on (5) and (6), the original RegD signal with a resolution of 2 s lasting for 15 min is converted into two features {$g^I,g^{II}$}. Finally, 366 × 96 groups of {$g^I,g^{II}$} data are obtained.

(2)

A cluster analysis is executed for $g^I$ and $g^{II}$ separately using the K-means method. For $g^I$, set the number of cluster center as M. Then, partition 366 × 96 groups of $g^I$ data into M clusters based on the minimum Euclidean distance. For each cluster, calculate its centroid associated with probability according to (18) and (19).

$$G_m^I=\frac{1}{Num\left(g_m^I\right)}{\displaystyle \sum _{k=1}^{Num\left(g_m^I\right)}{g}_{m,k}^I},\forall m\in M$$

(18)

$$p_m^I=\frac{Num\left(g_m^I\right)}{366\times 96}$$

(19)

where, $G_m^I$ and $p_m^I$ are the centroid and probability of cluster m, respectively. $g_m^I$ is a set that includes all data in cluster m. Num(·) is utilized for counting the number of $g^I$ in cluster m.

Similarly, set the number of cluster centers for $g^{II}$ as N. The centroids $G_n^{II}$ associated with probabilities $p_n^{II}$ of the N clusters are calculated according to (20) and (21).

$$G_n^{II}=\frac{1}{Num\left(g_n^{II}\right)}{\displaystyle \sum _{k=1}^{Num\left(g_n^{II}\right)}{g}_{n,k}^{II}},\forall n\in N$$

(20)

$$p_n^{II}=\frac{Num\left(g_n^{II}\right)}{366\times 96}$$

(21)

(3)

Calculate the discrete joint probability of centroids $G_m^I$ and $G_n^{II}$ based on the clustering analysis for $g^I$ and $g^{II}.$ Concretely, when a group of data {$g_m^I$, $g_n^{II}$} belongs to set m and n at the same time, it will be assigned into the set m × n. Consequently, the joint probabilities $p_{m,n}$ are calculated based on the number of the intersections of $g_m^I$ and $g_n^{II}$, as shown in (22).

$$p_{m,n}=\frac{Num\left(g_m^I\cap g_n^{II}\right)}{366\times 96}$$

(22)

(4)

J regulation scenarios are generated based on the joint probability distribution of $G_m^I$ and $G_n^{II}$. Each scenario consists of 96 groups of {$G_m^I,G_n^{II}$}, representing 96 moments in a day.

(5)

To select the typical scenarios that have the most consistent probability distribution with the actual one, a cluster analysis is performed for each generated scenario. Here, $G_{m,j}^I$ and $G_{n,j}^{II}$ are utilized to represent the centroids of $g^I$ and $g^{II}$ of the generated scenario j. $p_{m,j}^I$ and $p_{n,j}^{II}$ denote their probabilities. To measure the difference in probability distribution, an index E is proposed in (23). Equal weight is assigned to the two features to measure their difference. And, it is necessary to eliminate the difference in dimension of $G_m^I$ and $G_n^{II}$ by normalization.

$$\begin{array}{ll}E=& {\displaystyle \sum _{m\in M}{\left(Nor\left(G_{m,j}^I\right)p_{m,j}^I-Nor\left(G_m^I\right)p_m^I\right)}^2}\\ & +{\displaystyle \sum _{n\in N}{\left(Nor\left(G_{n,j}^{II}\right)p_{n,j}^{II}-Nor\left(G_n^{II}\right)p_n^{II}\right)}^2},j\in J\end{array}$$

(23)

where, Nor($·$) denotes the normalization step.

Finally, S scenarios with the smallest difference E will be employed as the typical regulation scenarios.

4. Case Study

The regulation information, lasting for an entire year in 2020, is employed in this research [26,27] for validating the effectiveness of the proposed method. It is worth mentioning that effective typical regulation scenarios should accurately reflect the uncertainty of the RegD signals. Additionally, these typical scenarios are primarily used for day-ahead operation strategy design or BESSs’ sizing optimization. Therefore, the regulation revenues calculated based on the typical scenarios should closely match the revenues calculated using the actual RegD signals. Given the above considerations, the effectiveness of typical regulation scenarios generated by the proposed method is calculated from two aspects in this section. The first involves comparing the monthly revenue derived from typical regulation scenarios to the actual revenue. The second experiment focuses on assessing the consistency of the probability distribution of the typical regulation scenarios in relation to the actual distribution, to determine whether the generated scenarios accurately reflect the real uncertainty. Additionally, the results based on the proposed method are compared with the results based on FSR and K-means clustering.

4.1. Typical Regulation Scenarios Generation

In this section, the typical regulation scenarios based on the proposed method are generated according to the steps in Section 3.2. The parameters involved in Section 3.2 are listed in

Table 1. Adopted values of parameters in Section 3.2.

Parameters	M	N	J	S
Values	10	5	1000	10

.

Firstly, $g^I$ and $g^{II}$ are calculated based on (5) and (6). As Figure 3b shows, the values of $g^I$ are mainly located in the range of [−1 to 1], and the values of $g^{II}$ are mainly located in the range of [20 to 110]. When a battery participates in the FFR market in rated power, even a small change in $g^I$ will cause a significant variation in SoC, while the impact of $g^{II}$ on the degradation cost is not as substantial as $g^I$. Thus, M and N of centroids for $g^I$ and $g^{II}$ are separately set as 10 and 5 for striking a balance between accuracy and computational complexity.

The clustering results of the two features are depicted in Figure 4a,b, and the calculated joint probability distribution according to (19) is presented in Figure 4c. Notably, a significant portion of $g^I$ values are concentrated around 0, aligning with the criterion of conditional energy neutrality. Furthermore, a clear relationship emerges: larger $g^{II}$ values correspond to $g^I$ with a large absolute value, whereas $g^{II}$ exhibits a broader distribution in the case of $g^I$ around 0.

Using the joint probability distribution presented in Figure 4c, 1000 scenarios are generated as a basis for the following typical scenarios selection. Subsequently, a cluster analysis is conducted for these scenarios, and S-typical regulation scenarios with equal probabilities are selected based on (20) with the smallest E. Here, S is set to 10, which is the usual number in the current research for calculating the regulation revenue. Three of these regulation scenarios are illustrated in Figure 5.

In addition, 10 typical regulation scenarios, along with their respective probabilities, are generated based on FSR and K-means clustering. Subsequently, the $g^I$ and $g^{II}$ values of ten scenarios are computed based on (5) and (6). Three typical scenarios based on the two methods are depicted in Figure 6 and Figure 7, respectively. It can be observed that the value of $g^I$ in Figure 7a is closer to 0, while $g^{II}$ in Figure 7b, compared with the one using the proposed method and FSR in Figure 5b and Figure 6b, exhibits lower values.

4.2. Comparison of Monthly Revenue

To validate the effectiveness of typical regulation scenarios, the most straightforward approach is to compare the regulation revenue based on the generated typical scenarios with the real revenue. Here, the monthly revenue is taken as a comparison index. Firstly, the real monthly revenue is calculated as the reference based on the real RegD signal. Subsequently, the expected monthly revenue is optimized based on the stochastic optimization model (3), where the generated typical scenarios and their corresponding probability are employed. Here, the selected BESS is the lithium iron phosphate battery (LFP). This choice is driven by the well-known safety, thermal stability, long cycle life, and environmentally friendly characteristics of the LFP. The related parameters of LFP are listed in

Table 2. Parameters of lithium ferro-phosphate (LI-LFP) battery.

Parameters	Efficiency (%)	Range of SOC (%)	Degradation Cost per MWh ($/MWh)
Values	95	15–90	18.75

.

Taking January as an illustrative example, the actual monthly regulation revenue is derived by summing the daily regulation revenue, which is calculated based on the actual RegD signal and price. Then, the estimated monthly revenues are calculated based on the generated typical scenarios by the proposed method, K-means clustering, and FSR. Here, to illustrate the effectiveness of the proposed method, the sizes of BESSs are changed for sensitivity analysis. As shown in

Table 3. BESS Size Configurations for Sensitivity Analysis.

	Size Group
	1	2	3	4	5
BESS Size	0.4 MW, 1 MWh	0.8 MW, 1 MWh	1.2 MW, 1 MWh	1.6 MW, 1 MWh	2 MW, 1 MWh

, the rated energy of BESS is set at a constant value of 1 MWh, while the rated power gradually increases as 0.4, 0.8, 1.2, 1.6, and 2 MW for observing the impact of the BESS with a high power size and high energy size on regulation revenue. The rated power is changed due to the rated power having a more significant impact on regulation revenue.

Figure 8 illustrates the monthly revenues for five distinct sizes of BESSs, and the percentage error (PE) between the actual monthly revenue and the estimated revenue based on scenarios generated by different methods is presented in

Table 4. Percentage error (PE) of estimated monthly revenue based on different BESS sizes.

PE	Size Group
	1	2	3	4	5
Proposed method	3.24	2.34	0.57	−1.77	−3.89
FSR	−16.4	−16.39	−16.54	−16.56	−16.28
K-means clustering	30.07	27	21.88	20.29	18.8

. Notably, the monthly regulation revenue of ten typical scenarios based on the proposed method falls within the interquartile range (box), with the final expected revenue denoted by the red triangles. Obviously, the proposed method consistently outperforms other methods across all BESS size groups, with the smallest PE not exceeding ±4%. For BESSs with a high energy size, there is a tendency to overestimate the monthly revenue, whereas for BESSs with a high power size, the proposed method tends to underestimate the monthly revenue. Overall, the proposed method offers a significant advantage in achieving an accurate monthly revenue. Subsequently, the revenues derived from the typical scenarios generated by FSR, while generally lower than the actual value, are noteworthy. In contrast, the monthly revenue based on the typical scenarios generated by K-means clustering consistently shows an overestimation for all size groups of BESSs. Moreover, the magnitude of this error tends to increase with the size of the BESS.

4.3. Cluster Analysis

To explore the factors contributing to the aforementioned results, a cluster analysis is performed on the generated typical scenarios based on three methods. The emphasis is on assessing their consistency with the distribution of real RegD signals presented in Figure 2a,b. To mitigate the impact of varying initial centroids on K-means clustering results, the clustered outcomes of the real RegD signals are employed as the initial centroids for the cluster analysis. The cluster results of g^I and g^II of three methods are illustrated in Figure 9, Figure 10 and Figure 11.

According to the outcomes of the cluster analysis, the proposed method demonstrates the most consistent distribution with the actual one, followed by the scenarios generated by FSR. It is noteworthy that the cluster center for $g^{II}$ based on FSR tends to be higher than the actual value, contributing to increased degradation costs and reduced profits, as shown in Figure 8. Scenarios generated by K-means clustering tend to have smaller $g^I$ and $g^{II}$ values, and show substantial differences in distribution compared to the real data. This often leads to an overestimation of BESS revenue and results in sub-optimal outcomes. This is because the generated typical scenarios by K-means clustering are the mean values of all the samples in each cluster. Considering RegD signal ranges [−1, 1], averaging will lead to the positive and negative signal canceling out. Thus, the RegD signals in typical scenarios generated by K-means clustering are easily lower than the real value and more centered around zero, further leading to the generated $g^I$ and $g^{II}$ values being smaller than the actual values. Additionally, as shown in Figure 1a, the frequent fluctuations of the RegD signal cause significant shape differences, making K-means clustering, a shape similarity-based method, challenging when generating regulation scenarios.

5. Conclusions

This paper proposes a novel method for generating typical frequency regulation scenarios for BESSs participating in FFR service in the PJM market, which utilizes the joint probability distribution of two features extracted from the RegD signal. The two features effectively reflect the characteristics of the RegD signal and its impact on BESSs’ operation. Simulation results demonstrate that the monthly regulation revenue derived from the proposed typical scenarios most closely approximates the actual revenue. Typical scenarios generated by K-means clustering tend to overestimate regulation revenue, while those generated by FSR exhibit a tendency to underestimate it. The cluster analysis results reveal that the distribution of scenarios generated by K-means clustering does not align with the real scenarios. This underscores the inadequacy of K-means clustering, based on shape similarity, for the generation of regulation scenarios. The proposed method, rooted in joint probability distribution analysis, emerges as a robust approach for effectively capturing the uncertainty inherent in regulation scenarios. This approach provides more reliable and realistic insights, facilitating the determination of BESSs’ optimal size and an operating strategy that maximizes the revenue in the context of fast frequency regulation service. In other frequency regulation markets, such as MISO and NYISO, the regulation signals differ from the RegD signal utilized in the PJM market. Consequently, the adopted two features, $g^I$ and $g^{II}$, need to be redesigned to align with the specific signal properties of each market for serving the subsequent probability analysis. This topic will be further investigated in our future work.