Frequency is a crucial index for measuring power quality, representing the balance of active power in power systems (He and Wen, 2021). With the increasing penetration of renewable energy sources, the inertia of power systems is decreasing and the effective maintenance of the frequency nadir (f _nadir ) and quasi-steady state frequency (f _ss ) consequently becomes challenging, posing a threat to system stability.

Therefore, system operators all over the world are focused on setting a series of frequency response (FR) services. Among FR energy sources, energy storage systems (ESSs) installed in distribution networks have been widely used (GB/T 30370-2013, 2013; Rana et al., 2023). The National Grid in Britain has set various dynamic frequency control products (AEMO, 2023), the Australian Energy Market Operator (AEMO) has proposed a Contingency Frequency Control Ancillary Service (FCAS) and a Regulation FCAS (National Grid ESO, 2019), and in Guangdong, China, a LiFePO₄ (LFP) battery is also used as a frequency control product (Wang et al., 2023). However, the design of the aforementioned ESSs relies entirely on simulation analysis. Systematic methods for system operators to evaluate the frequency support ability of an ESS and calculate the main parameters of an ESS have not been proposed.

ESSs can function both as generators and loads. Existing research mainly focuses on the construction of the ESS FR model. In these studies, the classical FR model proposed by Anderson and Mirheydar (1990) has been widely used. Based on the classical model, researchers have developed an ESS transfer function model (Aik, 2006; Yang et al., 2022). In Chen et al. (2016), the penetration rate of an ESS is considered to improve the FR model. However, ESS FR models based on the classical FR model only consider the reheat turbines of synchronous generators (SGs); thus, they are not suitable for systems with other types of gas/hydraulic turbines. To avoid this limitation, generic FR models have been proposed by Gao et al. (2021), Ju et al. (2021), and Zhang et al. (2021). In Ju et al. (2021) and Gao et al. (2021), the FR of an SG is described as an nth-order function, and in Zhang et al. (2021), all generation sources are presented as lead-lag functions, and the FR of the system can be described as the classical FR model. Nevertheless, generic FR models present the system frequency characteristics in an aggregated manner, making it difficult to distinguish the FR of an ESS.

To precisely evaluate the frequency support ability of an ESS, many ESS control strategies have been proposed. An ESS management strategy was proposed by Ben Elghali et al. (2019) to determine the optimal capacity of an ESS based on system frequency, and an ESS shaping strategy was introduced by Jiang et al. (2021) to maintain the f _nadir with the optimal cost of an ESS (Mustafa and Altinoluk H, 2023) and aging minimization (Wang et al., 2020). In Xiong et al. (2021), first-order functions were used to size an ESS based on the rate of change of frequency (RoCoF) to avoid dealing with high-order transfer functions. Recently, ESS control schemes employing robust control (Xiong et al., 2020), grid-tied inverter design (Xiong et al., 2016), self-adaptive control (Wu et al., 2020), predictive models based on the uncertainty of renewable sources (Zarei and Ghaffarzadeh, 2024), and ESS generation (Baker et al., 2017; Zarei and Ghaffarzadeh, 2024) have been used to design ESSs. However, these methods are only suitable for specific power grids, limiting their broader applicability. Moreover, the control strategies always ignore the capacity limit and droop limit of an ESS and regard the frequency response output of an ESS as a step change, resulting in significant errors in evaluating the frequency support ability of an ESS.

In this paper, an ESS equivalent aggregated model (EAM) is introduced and a new method named the Energy Storage Designing Method (ESDM) based on an EAM is proposed. An EAM consists of a multistep model named FM to maintain the f _nadir and a model named QM to maintain the f _ss . For both FM and QM, which include a first-order system FR model and a first-order ESS FR model, it is convenient for system operators to evaluate and analyze the frequency support ability of an ESS and lay the foundation of ESS sizing. Since renewable sources such as wind farms and photovoltaic (PV) panels always work in Maximum Power Point Tracking (MPPT) mode (Bai et al., 2015; Mohanty et al., 2016) and are strongly related to the weather, and the participation of renewable sources in frequency modulation is not mandatory at present (Guangfu, 2020; Guangfu, 2022), SGs and ESSs are still the main resources for frequency regulation. Therefore, the proposed ESDM can effectively calculate the capacity and the droop of an ESS based on a historical event and therefore accurately maintain the f _nadir and f _ss of the power system.

When there is an imbalance in the active power of the power system, the system’s primary frequency response (PFR) can be described in Figure 1, and it can also be described by the classical swing equation as shown in (Eq. 1). 2 H ∂ Δ f t ∂ t + D Δ f t = Δ P m − Δ P d , (1) where H [s] is the inertia constant, D [p.u.] is the equivalent damping factor, ΔP _m [p.u.] is the mechanical power deviation from generators, and ΔP _d [p.u.] is the power disturbance. During a frequency event, the system frequency must have a nadir. Due to the monotone decreasing and converging of the step response of the first-order system, if only the f _nadir is considered, there must be a first-order power function with a minimum value that is equal to the f _nadir as shown in Figure 2A. Similar to the f _nadir , the f _ss can also be described as a first-order function as shown in Figure 2B.

In Figure 2, t _nadir1 is the time at which the system reaches the frequency nadir at the maximum rate of the change of frequency (RoCoF _max ), while t _nadir is the time at which the system reaches the f _nadir .

Therefore, the system equivalent FR (SEFR) model is depicted in Figure 3. If K = K ₁ , SEFR can be used to predict the f _nadir after a frequency event, with ∆f = ∆f _nadir at t = ∞. Similarly, when K = K ₂ , SEFR is used to forecast the f _ss with ∆f = f _ss at t = ∞. According to Figure 3, the SEFR model can be shown as follows: Δ f s Δ P d s = 1 2 H s + D + K (2)

Assuming that the load disturbance during a frequency event undergoes a step change, with an amplitude of ΔP _d , the time-domain expression of the system frequency can be obtained by solving (Eq. 3). Δ f s = 1 2 H s + D + K ⋅ Δ P d s Δ f * t = L − 1 Δ f s = Δ P d D + K 1 − e − D + K 2 H t (3)

Δf ^* (t) is the per unit system frequency. It is clear from (Eq. 3) that Δf ^* (t) is an increasing function, so its maximum value can be calculated as shown in (Eq. 4). Δ f max * = lim t → ∞ Δ f * t = Δ P d D + K (4)

For a historical frequency event, the f _nadir and f _ss can be acquired from system operators so that K ₁ and K ₂ can be easily calculated. K 1 = Δ P d f N f N − f n a d i r − D K 2 = Δ P d f N f N − f s s − D , (5) where f _N is the base of system frequency (i.e., 50 Hz or 60 Hz).

As for the f _ss , if only PFR is considered, SEFR can accurately symbolize the f _ss because both the actual value and SEFR value are calculated when the time approaches infinity, i.e., t = ∞. To analyze the accuracy of the SEFR in representing the f _nadir , a parameter named E is introduced to symbolize the error between the actual f _nadir and the SEFR value at t _nadir . E can be shown as E = Δ f n a d i r * − Δ f * t n a d i r = Δ f n a d i r * e − Δ P d 2 H Δ f n a d i r * t n a d i r Δ f n a d i r * = f N − f n a d i r f N . (6)

According to (1), RoCoF _max can be described as (7), and if frequency continues to fall at RoCoF _max , t _nadir1 can be calculated as follows: R o C o F max = Δ P d 2 H , (7) t n a d i r 1 = Δ f n a d i r * R o C o F max . (8)

A parameter named φ is proposed to describe the relationship between t _nadir1 and t _nadir , so that E can be described as t n a d i r 1 = φ ⋅ t n a d i r E = Δ f n a d i r * e − 1 φ , (9) where φ is a constant and φ ≤ 1.

According to Gao et al. (2021), t _nadir usually falls in 8.5 s–10 s, and in many areas, RoCoF _max can be very large (Xiong et al., 2021); thus, φ can be very large so that E can be very small.

The parameters of an ESS are always designed based on the maximum power disturbance (ΔP _dmax ), which means the utilization ratio of an ESS will be quite low, and an ESS with a large droop and capacity is not energy-efficient. Since ΔP _dmax is a small probability event, an ESS designed based on ΔP _dmax is not flexible in dealing with normal ΔP _d .

The modified four-generator two-area (4G2A) system with PV penetration and an line commutated converter based High Voltage Direct Current (LCC-HVDC) connection is used for simulation in this section, as shown in Figure 6.

G1–G4 represent synchronous generators; P _L7 and P _L8 are the equivalent loads at bus 7 and bus 9, respectively; and C ₇ and C ₈ represent reactive compensations. A grid-connected ESS is connected to bus 10. Grid-connected PVs, named PV1 and PV2, are connected to bus 1 and 6. The power rating of each synchronous generator is 900 MVA, and the capacity of LCC-HVDC is 800 MVA, resulting in the power rating of the receiving system (Area 2) being 2600 MVA. The parameters of the simulation system are from Kundur (1994). The mechanical power gain factor is 1 p.u., the power generated by the high-pressure turbine is 0.4 p.u., the reheat time constant is 8 s, and the equivalent damping factor is 0. The system frequency characteristics are listed in Table 1.

This paper proposes a method for calculating the capacity and droop of an ESS based on historical frequency events to maintain the f _nadir and f _ss . The proposed method is convenient and accurate for system operators to evaluate the frequency support ability of an ESS and design ESSs. Furthermore, an ESS based on the ESDM proves to be energy-efficient. Given that all parameters are provided by system operators, the method holds significant practical applications. Moreover, the proposed method serves as a foundation for ESS sizing and control of distribution network ESSs.