Flexibility Evaluation Method for Aggregated Multi-Type Energy Storage Stations in Multi-Market Environments

Abstract

This study addresses the unclear regulation capability boundaries and resource dispatching challenges caused by the heterogeneity and market volatility of dispersed energy storage resources. A flexibility evaluation method for aggregated multi-type energy storage stations in multi-market environments is proposed. The approach first constructs an operation model to maximize revenues from both energy and frequency regulation (FR) markets, which informs the capacity allocation between the two. Next, a Minkowski-based aggregation method for heterogeneous resources is proposed to characterize the physical boundaries of aggregated energy storage in the power–energy–time space. Finally, peak regulation and frequency regulation response rates are defined as indicators to quantitatively analyze the dynamic relationship between price signals and the flexibility potential of energy storage. The results demonstrate that that the method can clearly delineate the physical boundaries of the energy storage system and effectively evaluate the flexibility of the energy storage system. This indicates that the method can guide grid operators in optimizing resource allocation and improving system flexibility.

1. Introduction

1.1. Background and Motivation

In the context of the construction of new power systems, with the large-scale integration of renewable energy and the growing demand for electricity, energy storage technology, with its millisecond-level response speed and precise bidirectional regulation capability, has become one of the core technologies for ensuring the balance of electricity supply and demand, as well as enhancing the flexibility and stability of power systems [1]. Energy storage systems not only provide regulation during power supply–demand imbalances but also effectively address grid fluctuations caused by weather changes, unforeseen events, and other factors [2]. With the continuous deepening of electricity system reform, energy storage is gradually transitioning from being driven by directives to becoming an independent market entity, participating in more market environments for flexible resource scheduling and optimization [3].

However, despite the huge potential of energy storage technology in grid regulation, its actual flexibility in practical applications is still deeply influenced by factors such as market incentive mechanisms, capacity coupling of storage technologies, and the technological differences among energy storage resources within the system [4,5]. In particular, when energy storage resources within a province are decentralized and diversified, the differences in physical characteristics between energy storage resources and market fluctuations create complex scheduling problems. These issues lead to unclear boundaries for the flexibility of energy storage resources, which further increases the difficulty of grid scheduling and limits the efficient utilization of flexibility resources. Therefore, how to clarify the flexibility boundaries of these dispersed resources and achieve effective scheduling in a multi-market environment has become a key issue to be addressed in current power system scheduling.

1.2. Literature Review

The issue of multi-market participation and scheduling optimization for energy storage systems has received widespread attention. Existing research focuses on joint optimization in energy, reserve, and frequency regulation markets, analyzing capacity allocation, power regulation, and revenue stacking. Some studies [6] have proposed a model for joint optimization in energy and reserve markets, showing how price differences drive capacity distribution, improving system efficiency. Another study [7] examined hybrid energy storage arbitrage strategies, highlighting the influence of multi-energy transaction paths. Further studies analyzed energy storage’s multi-market participation strategies, with other research [8] improving reserve market participation through decision-focused learning, and another [9] validating the contribution of energy storage in frequency regulation.

In addition to participation strategies, the differentiated technical characteristics of various energy storage types play a crucial role in their market response and scheduling value. One study [10] highlighted that multi-type energy storage can meet the flexibility needs of high-penetration renewable systems, while another [11] compared the characteristics and applications of various technologies. Another study [12] focused on the advantages of energy storage in frequency regulation, and another [13] showed how hybrid systems improve regulation capabilities and reduce costs.

Building on these technical characteristics, some studies have explored coordinated optimization of multi-energy storage resources, using multi-time scale optimization, rolling scheduling, and hierarchical control. One [14] proposed a multi-time scale rolling optimization method for integrated energy systems with battery and hydrogen storage, while another [15] introduced a coordinated control strategy for hybrid systems to reduce costs and improve flexibility. A different study [16] constructed a multi-layer scheduling strategy for shared energy storage, and another [17] demonstrated how joint optimization of multi-energy storage and multi-energy resources can enhance system reliability.

Overall, while existing research has expanded from single energy storage scheduling to coordinated optimization, gaps remain. These include the lack of comprehensive assessment of the aggregated flexibility of multi-energy storage at the regional level, the focus on individual profit-maximizing behavior rather than synergies, and the absence of clear, quantitative indicators for measuring flexibility across markets. Addressing these gaps will require the development of a flexibility evaluation method for aggregated multi-type energy storage stations in multi-market environments to guide grid scheduling and market design.

1.3. Contributions

To address the above issues, this study proposes a flexibility evaluation method for aggregated multi-type energy storage stations in multi-market environments. First, a profit-maximization model considering the technical characteristics of multi-type energy storage is constructed to characterize the capacity allocation logic between energy peak regulation and frequency regulation markets. Second, Minkowski theory is applied to achieve the mathematical aggregation of heterogeneous resources, further revealing the physical boundary of provincial aggregated energy storage in the power differences energy differences time space. Finally, key indicators such as peak regulation contribution rate and frequency regulation contribution rate are defined, and the dynamic mapping relationship between price signals and energy storage flexibility potential is quantitatively analyzed.

The main contributions of this study are as follows:

(1)

This study proposes a framework to quantify the aggregated flexibility of multi-type energy storage systems, considering their roles in both energy peak shaving and frequency regulation markets. It offers a comprehensive view of how different energy storage technologies contribute to grid flexibility.

(2)

This study uses Minkowski theory to mathematically aggregate heterogeneous energy storage systems, providing a unified model that defines the physical boundaries of the provincial aggregated energy storage system. This approach overcomes the limitations of traditional single-unit scheduling models and more accurately represents the actual storage capacity.

(3)

Indicators, such as peak regulation contribution rate and frequency regulation contribution rate, are introduced to quantify the dynamic mapping between price signals and the flexibility of energy storage. This analysis highlights how market price fluctuations influence resource allocation in multi-market environments.

This study provides a new framework for grid operators to identify the flexibility potential of dispersed energy storage resources and optimize resource scheduling. Furthermore, through the quantitative evaluation of energy storage flexibility in a multi-market environment, this study provides a theoretical basis for multi-market resource coordination, which can effectively improve grid scheduling efficiency and enhance the stability and economic performance of the power system.

2. Multi-Market Collaborative Optimization of Multi-Type Energy Storage Operation

2.1. Technical Characteristics and Functional Differences of Multi-Type Energy Storage

In the flexible regulation resources of new power systems, multi-type energy storage resources exhibit significant functional differences in grid dispatch due to differences in energy density, power response time, and cycle life.

Lithium iron phosphate-based electrochemical energy storage, due to its high technological maturity and flexible energy-to-power ratio, has become the main resource for energy market arbitrage and frequency regulation services. Compressed air energy storage (CAES), as a long-duration energy storage system, demonstrates greater economic resilience in inter-period energy migration and peak shaving due to its high energy-to-power ratio and lower deep discharge degradation costs. On the other hand, power-type resources like flywheel energy storage (FES), although having smaller energy storage capacity, offer millisecond-level response speeds and almost no cycling losses, making them indispensable for tracking high-frequency regulation signals and maintaining system frequency stability.

The differences in technical characteristics directly determine the decision preferences of energy storage under different price signals. Energy-type energy storage is more inclined to charge during low-price periods and discharge during high-price periods, using its larger capacity reserves to capture price differences. Power-type energy storage, on the other hand, tends to stay in the medium SoC range, reserving sufficient power capacity to obtain frequency regulation market capacity compensation and high-frequency mileage revenue.

2.2. Multi-Market Optimization Model for Energy Storage

In the context of multi-market collaborative operation, energy storage, as a rational market entity, makes bidding decisions based on different market price signals. The market mechanism in this study refers to the PJM electricity market design, where peak regulation incentives are reflected through price mechanisms in the energy market, guiding energy storage to perform peak shaving and valley filling operations using the price differences during peak and valley periods, which are divided based on grid load.

2.2.1. Objective Function

As a rational market participant, an independent energy storage station aims to maximize its total daily revenue through inter-temporal arbitrage and provision of ancillary services under physical constraints. The objective function is expressed as:

$$F=\max {\displaystyle \sum _{t=1}^T(}{R}_{n,t}^{\mathrm{En}}+R_{n,t}^{\mathrm{Fr}}-C_{n,t}^{life})$$

(1)

where $R_{n,t}^{\mathrm{En}}$ and $R_{n,t}^{\mathrm{Fr}}$ are energy arbitrage revenue and frequency regulation compensation for energy storage n in period t. $C_{n,t}^{life}$ is aging cost of energy storage n in time period t.

$$R_{n,t}^{\mathrm{En}}=(1+\alpha ){\gamma }_t^{\mathrm{En}}{P}_{n,t}^{\mathrm{net}}\Delta t$$

(2)

$$P_{n,t}^{\mathrm{net}}=P_{n,t}^{\mathrm{dis}}-P_{n,t}^{\mathrm{cha}}$$

(3)

where $\alpha$ is the PF subsidy discount coefficient for peak-valley period electricity price; ${\gamma }_t^{\mathrm{En}}$ is the electricity price for period t; $P_{n,t}^{\mathrm{net}}$, $P_{n,t}^{\mathrm{dis}}$, $P_{n,t}^{\mathrm{dis}}$ are the actual output net power, discharge power, and charging power of the energy storage in period t; $\Delta t$ is the time interval.

Due to the significant differences in degradation mechanisms among different types of energy storage technologies, it is difficult to describe them using a unified degradation cost model. For PSH, CAES, and FES, the degradation cost and cycling loss caused by deep charge/discharge cycles within an intraday scheduling horizon are relatively small and can be approximately neglected. In contrast, BESS, as a major resource participating in the frequency regulation market, is subject to frequent and rapid power adjustments, which may significantly accelerate battery aging [18]. Therefore, a degradation cost model is introduced for BESS and incorporated into the objective function to reflect the impact of battery aging on market bidding and operational decisions.

$$N^{\mathrm{cycle}}={\mathsf{\beta }}_0\cdot D^{-{\mathsf{\beta }}_1}\cdot e^{{\mathsf{\beta }}_2\cdot (1-D)}$$

(4)

$$C_n^N={\displaystyle \frac{C^{\mathrm{cl}}}{N^{\mathrm{cycle}}}}$$

(5)

$$C_{n,t}^{\mathrm{life}}=\left\{\begin{array}{l}{C}_n^N(S_{n,t+1})-C_n^N(S_{n,t}),\\ 0,\end{array}\right.\begin{array}{c}{S}_{n,t}>S_{n,t+1}\\ S_{n,t}<S_{n,t+1}\end{array}$$

(6)

where $N^{\mathrm{cycle}}$ represents the cycle life at a given depth of discharge; $D$ is the depth of discharge; ${\mathsf{\beta }}_0$, ${\mathsf{\beta }}_1$, and ${\mathsf{\beta }}_3$ are the reference cycle number, depth decay index, and nonlinear correction coefficient of the BESS, respectively. $C_n^N$ represent the aging cost induced by a single cycle; $C^{\mathrm{cl}}$ represents investment and construction costs of energy storage; $S_{n,t}$ is SoC of the energy storage n in period t.

$$R_{n,t}^{\mathrm{Fr}}=R_{n,t}^{\mathrm{cap}}+R_{n,t}^{\mathrm{mile}}$$

(7)

$$R_{n,t}^{\mathrm{cap}}={\gamma }_t^{\mathrm{cap}}{C}_{n,t}^{\mathrm{fr}}$$

(8)

$$R_{n,t}^{\mathrm{mile}}={\gamma }_t^{\mathrm{mile}}(r_t^{\mathrm{mile}}{C}_{n,t}^{\mathrm{fr}}-err{o}_{n,t})$$

(9)

$$erro=\left(\left|r_t^{+}{C}_{n,t}^{\mathrm{fr}}-P_{n,t}^{\mathrm{fr},\mathrm{dis}}\right|+\left|r_t^{-}{C}_{n,t}^{\mathrm{fr}}-P_{n,t}^{\mathrm{fr},\mathrm{cha}}\right|\right)$$

(10)

where $R_{n,t}^{\mathrm{cap}}$ and $R_{n,t}^{\mathrm{mile}}$ represent the revenue from FR capacity and FR mileage, respectively; ${\gamma }_t^{\mathrm{cap}}$ and ${\gamma }_t^{\mathrm{mile}}$ represent the price of FR capacity and FR mileage, respectively; $C_{n,t}^{\mathrm{fr}}$ is the FR capacity bid by the energy storage in period t; $err{o}_{n,t}$ is the error of the FR response of the energy storage in period t; $P_{n,t}^{\mathrm{fr},\mathrm{dis}}$ and $P_{n,t}^{\mathrm{fr},\mathrm{cha}}$ represent the upward and downward FR power of the energy storage in period t; $r_t^{\mathrm{mile}}$, $r_t^{+}$, and $r_t^{-}$ represent the mileage value of the AGC signal, the upward signal and the downward signal in period t respectively.

2.2.2. Constraints

(1)

Multi-Market Coordination Constraints

In this study, discharging power is defined as positive. The baseline power represents the scheduled output in the energy market, while the total power reflects the actual output after responding to FR signals. To ensure that the awarded capacity remains within physical limits and accurately describes the coupling relationship between markets, the following constraints are established:

$$P_{n,t}^{\mathrm{net}}=P_{n,t}^{\mathrm{base}}+P_{n,t}^{\mathrm{fr},\mathrm{dis}}-P_{n,t}^{\mathrm{fr},\mathrm{cha}}$$

(11)

$$-P_n^{\max }\le P_t^{\mathrm{base}}\le P_n^{\max }$$

(12)

$$P_{n,t}^{\mathrm{base}}+C_{n,t}^{\mathrm{fr}}\le P_n^{\max }$$

(13)

$$0\le C_{n,t}^{\mathrm{fr}}\le P_n^{\max }$$

(14)

where $P_{n,t}^{\mathrm{base}}$ is the baseline power of the energy storage for the period t; $P_n^{\max }$ is the upper limit of the charging and discharging power of the energy storage n.

In practical operation, system operators impose strict requirements on regulation accuracy. Resources with persistent response deviations exceeding limits may lose market eligibility. Therefore, a response accuracy parameter is introduced with a minimum threshold to ensure compliance with market access requirements:

$$K_{n,t}^{\mathrm{fr}}=1-{\displaystyle \frac{\left|r_t^{+}{C}_{n,t}^{\mathrm{fr}}-P_{n,t}^{\mathrm{fr},\mathrm{dis}}\right|+\left|r_t^{-}{C}_{n,t}^{\mathrm{fr}}-P_{n,t}^{\mathrm{fr},\mathrm{cha}}\right|}{r_t^{\mathrm{mile}}{C}_{n,t}^{\mathrm{fr}}}}$$

(15)

$$K^{\min }\le K_{n,t}^{\mathrm{fr}}\le 1$$

(16)

where $K_{n,t}^{\mathrm{fr}}$ is the accuracy parameter of the energy storage for the FR signal over period t; $K^{\min }$ is the minimum entry response accuracy required in the FR market.

(2)

Energy Storage Constraints

A unified parameterized modeling approach is adopted for multi-type energy storage to characterize differences in power response and energy shifting capabilities. The constraints include:

$$0\le P_{n,t}^{\mathrm{cha}}\le P_n^{\max }$$

(17)

$$0\le P_{n,t}^{\mathrm{dis}}\le P_n^{\max }$$

(18)

$$-R_n^{\max }\le P_{n,t}^{\mathrm{net}}-P_{n,t-1}^{\mathrm{net}}\le R_n^{\max }$$

(19)

$$S_{n,t}=S_{n,t-1}+(P_{n,t}^{\mathrm{cha}}{h}_n-P_{n,t}^{\mathrm{dis}}/h_n)/E_n^{cap}$$

(20)

$$S_n^{\min }\le S_{n,t}\le S_n^{\max }$$

(21)

$$S_{n,T}=S_{n,0}$$

(22)

where $R_n^{\max }$ is the climb rate of the energy storage n; $S_n^{\max }$ and $S_n^{\min }$ are the upper and lower limits of SoC of the energy storage n, respectively; $h_n$ is the efficiency of the energy storage n; $E_n^{cap}$ is the capacity of the energy storage n. At the beginning and end of each operating day of energy storage, the SoC remains consistent. The subsequent modeling process in this study will conduct detailed characterization of the technical parameters of various energy storage types, represent their differences through parametric methods, and lay the foundation for the assessment of the energy storage’s flexibility.

3. Aggregation of Energy Storage Stations and Evaluation Indicators

3.1. Physical Boundary Aggregation Method

At the grid level, the flexibility of energy storage is not a simple summation of individual parameters, but rather the aggregation of feasible constraint sets in space. For a provincial grid containing multiple storage types such as pumped storage (PSH), battery energy storage systems (BESS), CAES, and FES, Minkowski theory is applied to characterize the aggregated physical boundary.

Let the power output trajectory and energy state trajectory of each independent energy storage unit n within T time periods form a high-dimensional convex polyhedron set, which is jointly defined by the constraints such as power limits, SoC evolution, ramp rate, and capacity coupling as described in Chapter Two. Then, the overall physical boundary space set of the provincial aggregated energy storage can be expressed as:

$$S_{sys}=S_1\oplus S_2\oplus \dots \oplus S_N$$

(23)

As there is no direct power coupling relationship among the various energy storages at the physical level, based on the decomposition property of linear programming on sets, the solution for the global revenue maximization objective is equivalent to the vector sum of the optimization results of each sub-unit within their respective physical feasible spaces.

In the P-E space, the physical boundary is represented as a closed area with a width of $E_{sys}^{cap}$ and a height of $P_{sys}^{\max }$. It is defined as the maximum charging and discharging power that the current energy state can provide for storage. This boundary does not take into account the profit-seeking intentions of the energy storage, but only depicts the flexible resources that can be mobilized theoretically. Its expression is as follows:

$$\left\{\begin{array}{l}{P}_{sys}^{\max \_cha}={\displaystyle \sum _{n=1}^N\max (P_n^{\max },(E_n^{cap}{S}_n^{\max }-E_{n,t})/\Delta t)}\\ P_{sys}^{\max \_dis}={\displaystyle \sum _{n=1}^N\max (P_n^{\max },(E_{n,t}-E_n^{cap}{S}_n^{\min })/\Delta t)}\end{array}\right.$$

(24)

$$E_{sys}^{cap}={\displaystyle \sum _{n=1}^N(S_n^{\max }-S_n^{\min })E_n^{cap}}$$

(25)

where $P_{sys}^{\max \_cha}$ and $P_{sys}^{\max \_dis}$ are the maximum charging and discharging power of the provincial aggregated energy storage in the corresponding energy state; $E_{sys}^{cap}$ represents the total adjustable energy of the provincial aggregated energy storage.

It should be noted that the proposed provincial aggregation model is developed from a planning-oriented flexibility assessment perspective. The objective is to estimate the potential regulation space of existing provincial energy storage resources based on their technical parameters and market signals, rather than to conduct network-constrained economic dispatch under a specific operating condition. Therefore, transmission power flow constraints are not explicitly considered in the current model.

3.2. Evaluation Indicators for Flexibility

To characterize the flexibility contribution of the energy storage entity in a multi-market environment, this section constructs flexibility evaluation indicators covering the dimensions of peak regulation and frequency regulation, aiming to quantitatively assess the actual supporting capacity of the energy storage resources under complex constraints and their potential to respond to market demands.

3.2.1. Peak Regulation Indicators

Peak regulation capability is characterized from two aspects: peak shaving rate and valley filling rate. These are defined as the ratio of actual charging/discharging power to rated power during peak and valley periods, respectively. Their expressions are as follows:

$${\eta }_{n,t}^{peak}={\displaystyle \frac{P_{n,t}^{dis}}{P_n^{\max }}},t\in T_{n,peak}$$

(26)

$${\eta }_{n,t}^{valley}={\displaystyle \frac{P_{n,t}^{cha}}{P_n^{\max }}},t\in T_{n,valley}$$

(27)

where ${\eta }_{n,t}^{peak}$ is the peak shaving rate; ${\eta }_{n,t}^{valley}$ is the valley filling rate; $T_{n,peak}$ and $T_{n,valley}$ are the time period sets corresponding to peak and valley, respectively.

Additionally, the total peak regulation response rate over a typical day is defined as follows:

$${\eta }_n^{tf}={\displaystyle \frac{1}{T_{n,tf}}}({\displaystyle \sum _{t\in T_{n,peak}}\frac{P_{n,t}^{dis}}{P_n^{\max }}}+{\displaystyle \sum _{t\in T_{n,valley}}\frac{P_{n,t}^{cha}}{P_n^{\max }}})$$

(28)

where ${\eta }_n^{tf}$ is the total peak regulation response rate; $T_{n,tf}$ is the time period sets of peak regulation, which is the union of peak and valley periods.

3.2.2. Frequency Regulation Indicators

Frequency regulation capability is quantified using the FR capacity response rate, defined as the ratio of awarded regulation capacity to rated power:

$${\eta }_{n,t}^{fr}={\displaystyle \frac{C_{n,t}^{\mathrm{fr}}}{P_n^{\max }}},t\in T_{n,fr}$$

(29)

where ${\eta }_{n,t}^{fr}$ is the FR capacity response rate; $T_{n,fr}$ is the time period set for which the frequency regulation capacity was awarded for energy storage.

Similarly, the total frequency regulation response rate over the entire operation cycle is defined to evaluate overall performance:

$${\eta }_n^{fr}={\displaystyle \frac{1}{T_{n,fr}}}{\displaystyle \sum _{t\in T_{n,fr}}\frac{C_{n,t}^{\mathrm{fr}}}{P_n^{\max }}}$$

(30)

4. Case Study

4.1. Case Setup

To validate the proposed method, a typical daily operation scenario of a provincial power grid is simulated. The scheduling horizon is 24 h with a 15 min interval, resulting in 96 time periods. The system load ranges from 60 GW to 98 GW. Periods with load above 85 GW and below 65 GW are defined as peak and valley periods, respectively. The discount coefficient $\alpha$ for peak and off-peak electricity prices is 0.18. The energy market prices, regulation market prices, and regulation signals for the typical day were obtained from the 2024 historical data published on the PJM official website using the K-means clustering method, and the representative typical day corresponds to the scenario with the highest occurrence probability among the clustering results. The maximum entry response accuracy of the frequency regulation market is referenced from the literature [19]. In this case study, the market prices and regulation signals are assumed to be known in advance, since the focus of this study is to evaluate the planning-level flexibility potential of aggregated multi-type energy storage under a representative market scenario. The impacts of market price uncertainty and regulation signal uncertainty on energy storage scheduling will be further investigated using robust optimization or stochastic programming in future work. Additionally, the example configuration included 10 multi-type energy storage stations with heterogeneous technical parameters [11,20]. The detailed physical parameters are shown in

Table 1. Parameters of various types of energy storages.

Num	Type	Capacity (MWh)	Rated Power (MW)	Climb Rate (MW/h)	Efficiency	Soc Limits
1	PSH	1200	300	20	0.88	0.9/0.1
2	PSH	1000	250	20	0.88	0.9/0.1
3	PSH	800	200	15	0.88	0.9/0.1
4	PSH	700	150	15	0.88	0.9/0.1
5	BESS	400	200	100	0.95	0.9/0.1
6	BESS	300	150	120	0.95	0.9/0.1
7	BESS	200	100	120	0.95	0.9/0.1
8	CAES	800	150	30	0.75	0.9/0.1
9	CAES	600	120	20	0.75	0.9/0.1
10	FES	50	100	300	0.90	0.9/0.1

. The model established in this study was solved by calling the Gurobi solver in MATLAB R2024b.

4.2. Typical Operation Analysis

The coordinated operation situation of multi-type energy storage stations in multiple markets on typical day is shown in Figure 1 and Figure 2. In the typical day, the output and SoC trajectories of energy storage stations reflect the differentiated and coordinated characteristics of their physical properties and market positioning.

As shown in Figure 1 and Figure 2, PSH has a large energy capacity and is the main support for energy storage output. Its SoC curve exhibits long-term fluctuations. During the early morning low-price period, it performs large-scale peak shaving charging. In the daytime peak period, it carries out stable energy shifting. This supports the system’s basic peak regulation needs. In contrast, BESS has a smaller capacity but shows more frequent, minor oscillations in its SoC curve. This reflects its advantage of rapid power response. It helps maintain peak regulation benchmarks and actively participates in the frequency regulation market to capture high mileage revenues. CAES lies between the two. It demonstrates balanced medium- to long-term regulation and moderate-speed response. Notably, FES has high-frequency, micro-amplitude fluctuations in its SoC. It rarely participates in large-scale energy shifting. Instead, it provides instantaneous power compensation and fine frequency regulation.

Figure 3 visually presents the performance of various energy storages in peak regulation reduction (peak shaving/valley filling) and frequency regulation. Observations reveal that pumped storage resources exhibit an extremely high response rate during peak shaving and valley filling periods, confirming their role as a stabilizing factor in the energy balance over large time periods. Similarly, BESS also plays a significant supporting role in peak shaving and valley filling. In sharp contrast, CAES and FES have a peak shaving response rate generally below 0.5 during peak and valley periods; during non-peak core periods, these resources maintain a high frequency regulation response rate almost throughout the entire period, with the orange area dominating.

This dynamic evolution of response rates essentially reveals the deep complementary advantages of multi-energy storage resources in the energy and power dimensions: large-capacity energy resources tend to achieve high shares and long-term peak regulation responses, leveraging large-scale reserves to secure the time-shifted benefits of certain energy certainty; while high-power resources ensure system frequency safety through their sensitive instantaneous power output capabilities, maximizing the utilization of flexibility resources in multiple markets.

4.3. Aggregated Physical Boundary Analysis

Figure 4 illustrates the physical boundaries of provincial aggregated energy storage, obtained through Minkowski aggregation. In this figure, the blue translucent regions represent the full-dimensional physical constraint boundaries of the aggregated energy storage system over the scheduling period, forming a dynamically evolving, time-varying capacity tunnel. This set-based boundary description is consistent with existing studies on multi-type energy storage flexibility modeling, in which the feasible region of heterogeneous storage resources is characterized by considering the coupling among power, energy, and operational constraints [10,11]. The red solid line depicts the system’s operational trajectory, derived from the optimization process. Its projection on the P-t rear wall represents the aggregated energy storage output, showing the power output logic in response to the market environment. The projection on the E-t bottom surface reflects the corresponding energy migration. Furthermore, the operation trajectory remains fully enclosed within this tunnel, thereby confirming the physical feasibility of the optimization scheme.

Figure 5 further illustrates the physical constraint boundary of the aggregated system in the P-E plane. The envelope line forms a distinct asymmetric convex polygon. In the middle section of the energy axis, the power boundary is constrained by the total rated power of the aggregated energy storage system, creating a horizontal power saturation zone. As the system’s energy state approaches either the low or full energy limits, the adjustable power range contracts linearly due to individual energy storage units’ SoC limitations and energy conservation constraints, forming an energy-limited zone. This phenomenon is also consistent with the flexibility degradation caused by SoC constraints reported in energy storage operation and hybrid energy storage scheduling studies [13]. Since the aggregation consists of energy storage units with varying energy-to-power ratios, the boundary slope exhibits a non-uniform multi-segmented feature, reflecting the gradual degradation of flexibility as each energy storage unit’s capacity depletes. The multi-segmented boundary also reflects the heterogeneous technical characteristics of PSH, BESS, CAES, and FES, which have different advantages in long-duration energy shifting and fast power regulation [11,20]. The scattered points within the envelope represent the actual operating states, and the instantaneous distance between these points and the physical boundary quantifies the available regulation margin the system can use after executing the planned scheduling instructions.

4.4. Sensitivity Analysis of Price Signals

To further investigate the impact of price signals on the flexibility of energy storage, this section presents a quantitative analysis of the response characteristics of provincial aggregated energy storage under varying price fluctuation coefficients. Figure 6 illustrates the sensitivity patterns of the total peak regulation response rate and the total frequency regulation response rate for each scenario.

The energy storage system exhibits distinct “price-oriented” and “stock game” characteristics. In the peak regulation dimension (as shown in Figure 6 above), the total peak regulation response rate increases significantly with rising subsidies, reaching a peak of 0.481 at certain subsidy levels. This suggests that higher subsidy intensity can effectively stimulate the deeper flexibility potential of energy storage. However, as the frequency regulation price coefficient increases, the peak regulation response rate at the same subsidy level shows a notable contraction. This highlights the “crowding out” effect, where high returns in the frequency regulation market compete for peak regulation resources. In the frequency regulation dimension (as shown in Figure 6 below), the response rate follows a completely opposite trend. It peaks at 0.933 in the low subsidy and high frequency price range, then gradually decreases as the subsidy increases.

This complementary evolution trend reveals that the total flexibility of provincial aggregated energy storage is constrained by its own SoC boundaries and ramp rate limits. Enhancing response capability in a multi-market environment does not simply result in an additive increase but rather in the reconfiguration of resources driven by varying price signals. To ensure the system’s basic peak regulation demand is met, reasonable price thresholds must be set to balance the interplay between the frequency regulation and peak regulation markets.

5. Conclusions

To address the challenges of unclear regulatory capacity boundaries and difficulties in resource scheduling caused by the diverse physical characteristics and market fluctuations of dispersed energy storage resources in provincial power grids, this paper proposes a flexibility evaluation method for aggregated multi-type energy storage stations in multi-market environments.

The research demonstrates that the physical characteristics of various energy storage systems dictate their differentiated roles in different markets: large-capacity resources focus on the long-term evolution of SoC to support peak regulation, while high-power resources ensure frequency stability through rapid, high-frequency responses. The three-dimensional physical boundary and response rate indicators clearly illustrate how the complementary advantages of energy and power dimensions enable multiple energy storage systems to maximize their flexibility potential within the physical constraints. Moreover, sensitivity analysis reveals the “stock game” nature of resource allocation under multi-market incentives. Therefore, establishing a price coordination mechanism and setting reasonable compensation weights are essential strategies for guiding multiple energy storage resources to effectively participate in grid regulation and balance the regulation demands across different markets.

Future work will further extend the proposed framework by incorporating transmission network constraints to evaluate the deliverable flexibility of distributed energy storage under specific grid topologies, nodal power injections, and congestion conditions. In addition, robust optimization or stochastic programming will be introduced to describe the uncertainty of market prices and frequency regulation signals, thereby improving the applicability of the flexibility evaluation method in real market environments.