Development of an equivalent system frequency response model based on aggregation of distributed energy storage systems

Energy storage systems (ESSs) installed in distribution networks have been widely adopted for frequency regulation services due to their rapid response and flexibility. Unlike existing ESS design methods which focus on control strategies, this paper proposes a new method based on an ESS equivalent aggregated model (EAM) for calculating the capacity and the droop of an ESS to maintain the system frequency nadir and quasi-steady state frequency using low-order functions. The proposed method 1) uses first-order functions to describe the frequency response (FR) of synchronous generators (SGs); 2) ignores the control strategies of SGs, making the method systematic and allowing it to avoid analyzing complex high-order functions; and 3) is suitable for low inertia systems. The applicability and accuracy of the method is demonstrated using a modified four-generator two-area (4G2A) system.

1 Introduction

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.

2 System equivalent frequency response model

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).

$2H\frac{\partial \Delta f\left(t\right)}{\partial t}+D\Delta f\left(t\right)=\Delta P_m-\Delta P_d,(1)$

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.

Figure 1

Figure 1. System PFR.

Figure 2

Figure 2. (A) Representation of f_nadir using the step response of the first-order system, and (B) Representation of f_ss using the step response of the first-order system.

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 = K₁, SEFR can be used to predict the f_nadir after a frequency event, with ∆f = ∆f_nadir at t = ∞. Similarly, when K = K₂, 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:

$\frac{\Delta f\left(s\right)}{\Delta P_d\left(s\right)}=\frac{1}{2Hs+D+K}(2)$

Figure 3

Figure 3. SEFR model.

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).

$\left\{\begin{array}{l}\Delta f\left(s\right)=\frac{1}{2Hs+D+K}\cdot \frac{\Delta P_d}{s}\\ \Delta f^{*}\left(t\right)=L^{-1}\left[\Delta f\left(s\right)\right]=\frac{\Delta P_d}{D+K}\left(1-e^{-\frac{D+K}{2H}t}\right)\end{array}\right.(3)$

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).

$\Delta f_{\max }^{*}=\underset{t\to \infty }{\lim }\Delta f^{*}\left(t\right)=\frac{\Delta P_d}{D+K}(4)$

For a historical frequency event, the f_nadir and f_ss can be acquired from system operators so that K₁ and K₂ can be easily calculated.

$\left\{\begin{array}{l}{K}_1=\frac{\Delta P_d{f}_N}{f_N-f_{nadir}}-D\\ K_2=\frac{\Delta P_d{f}_N}{f_N-f_{ss}}-D\end{array}\right.,(5)$

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

$\left\{\begin{array}{l}E=\Delta f_{nadir}^{*}-\Delta f^{*}\left(t_{nadir}\right)=\Delta f_{nadir}^{*}{e}^{-\frac{\Delta P_d}{2H\Delta f_{nadir}^{*}}{t}_{nadir}}\\ \Delta f_{nadir}^{*}=\frac{f_N-f_{nadir}}{f_N}\end{array}\right..(6)$

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:

$RoCo{F}_{\max }=\frac{\Delta P_d}{2H},(7)$

$t_{nadir1}=\left|\frac{\Delta f_{nadir}^{*}}{RoCo{F}_{\max }}\right|.(8)$

A parameter named φ is proposed to describe the relationship between t_nadir1 and t_nadir, so that E can be described as

$\left\{\begin{array}{l}{t}_{nadir1}=\phi \cdot t_{nadir}\\ E=\Delta f_{nadir}^{*}{e}^{-\frac{1}{\phi }}\end{array}\right.,(9)$

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.

3 The proposed EAM

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.

3.1 The proposed FM

A new model named FM is proposed to calculate the parameters of an ESS based on different levels of ΔP_d and different required frequency deviations as shown in Figure 4A.

Figure 4

Figure 4. (A) The proposed FM, and (B) The proposed QM.

V_si and δ_si, respectively, represent the equivalent capacity and droop of an ESS for addressing frequency events with a power disturbance level ΔP_di, and Δf_i is the system-required frequency maximum deviation at ΔP_di.

The principle of the ESS FM model is that different levels of power disturbances have different occurrence probabilities. For example, a% of disturbance lies in 0 to ΔP_d1, b% of disturbance lies in ΔP_d1 to ΔP_d2, and others lie in ΔP_d2 to ΔP_dmax. Therefore, according to the range of power disturbances that need to be addressed, system operators can design the V_si and δ_si of an ESS using the FM model, as depicted in Figure 5, and choose the appropriate combinations of V_si and δ_si based on their economic or technical needs.

Figure 5

Figure 5. FR of ESS dealing with f_nadir.

The Δf_max shown in Figure 5 can be selected as load-shedding frequency deviation to deal with ΔP_dmax of the power system, and the frequency response characteristic of an ESS at ΔP_di should be divided into three parts to deal with different disturbance levels according to FM. The product of V_si and δ_si can be described as (10).

$\left\{\begin{array}{l}\frac{f_N\Delta P_{di}}{D+V_{si}{\delta }_{si}+K_1}\le \Delta f_i\\ V_{si}{\delta }_{si}\ge \frac{f_N\Delta P_{di}}{\Delta f_i}-D-K_1\iff V_{si}{\delta }_{si}\Delta f_i+\frac{f_N\Delta P_{di}}{\Delta f_{nadiri}}\Delta f_i\\ \ge f_N\Delta P_{di}\Rightarrow V_{si}{\delta }_{si}\ge f_N\Delta P_{di}\left(\frac{1}{\Delta f_i}-\frac{1}{\Delta f_{nadiri}}\right),\end{array}\right.(10)$

where Δf_i is the knee point of FR of an ESS and can also be illustrated as the system-required frequency maximum deviation at ΔP_di which can be selected by system operators. Δf_nadiri symbolizes the frequency deviation at ΔP_di from a historical frequency event which can be easily acquired from system operators. In applications, system operators can select the Δf_i based on their economic or technical needs of an ESS and the stability of the power grid.

3.2 The proposed QM

As the proposed FM model does not consider detailed governor-turbine dynamics, it cannot be used to represent frequency dynamics after the nadir. To address this limitation, the QM model is proposed to characterize the f_ss, as illustrated in Figure 4B.

The product of an ESS’s capacity, V_m, and droop, δ_m, can be calculated as

$\frac{\Delta P_{d\max }{f}_N}{D+K_2+V_m{\delta }_m}\le \Delta f_{ss\max }\Rightarrow V_m{\delta }_m\ge \frac{\Delta P_{d\max }{f}_N}{\Delta f_{ss\max }}-\left(D+K_2\right).(11)$

System operators always set up a rigorous limitation of f_ss deviation (Δf_ss), so the calculation of V_m and δ_m can be based on the ΔP_dmax, where the Δf_ssmax is the required maximum Δf_ss.

3.3 The proposed EAM

The EAM includes the FM model and the QM model to deal with the f_nadir and f_ss, as mentioned above. The timing of switching between FM and QM depends on ξ and the time t_nadir. The t_nadir can be acquired from system operators and is smaller when an ESS takes part in FR; thus, it is suitable that the moment of switching should be greater than t_nadir. ξ is introduced to measure the f_ss without the QM mode’s participation.

$\xi =\frac{f_N\Delta P_{di}}{D+K_2+V_{si}{\delta }_{si}}(12)$

3.4 Constraint condition in the ESDM

This section compares the energy efficiency of ESS designs based on different levels of ΔP_d and ΔP_dmax to establish the constraint conditions of the ESDM. If a power system experiences a disturbance ΔP_dm, according to the ESDM, the capacity and droop of an ESS are V_sm and δ_sm, respectively. The output power of an ESS, P_m1, is given by (Eq. 13).

$P_{m1}=V_{sm}{\delta }_{sm}\frac{\Delta P_{dm}{f}_N}{D+K_1+V_{sm}{\delta }_{sm}}=\frac{\Delta P_{dm}{f}_N}{\frac{\Delta P_{dm}{f}_N}{\Delta f_{nadir1}{V}_{sm}{\delta }_{sm}}+1}(13)$

If V_smδ_sm < V_smaxδ_smas (the product of V_sm and δ_sm is based on ΔP_dm), an ESS designed through the ESDM is more energy-saving.

4 Simulation results

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.

Figure 6

Figure 6. Modified four-generator two-area (4G2A) system.

G1–G4 represent synchronous generators; P_L7 and P_L8 are the equivalent loads at bus 7 and bus 9, respectively; and C₇ and C₈ 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.

Table 1

Table 1. Simulation Scenario.

4.1 Installed PV capacity of 33.3%

In scenario I, the power ratings of PV1 and PV2 are both 450 MVA. Furthermore, 90% of ΔP_d is below 0.037 p.u., and the system’s ΔP_dmax is 0.046 p.u.

According to (10), if Δf₁ is selected as 0.2 Hz, and Δf_max is 0.3 Hz, V_si, δ_si should satisfy V_s1δ_s1 ≥ 1.613 and V_s2δ_s2 ≥ 0.708. According to (12), ξ = 0.172 Hz, and according to (11), V_mδ_m ≥ 3.067. The simulation results are shown in Figure 7.

Figure 7

Figure 7. (A) FR of Scenario I at ΔP_d = 0.046, and (B) FR of Scenario I at ΔP_d = 0.037.

It can be seen in Figure 7 that FM and QM can accurately describe the f_nadir and f_ss, respectively. The orange curve in Figure 7 shows that the ESDM effectively maintains f_nadir and f_ss. Considering that Δf_ss is smaller than Δf_ssmax when ΔP_d = 0.037, the ESS will not switch to f_ss maintaining mode.

4.2 Installed PV capacity of 66.7%

In scenarios II and III, G1 is replaced with PV1 and PV2, both with capacities of 900 MVA.

4.2.1 Scenario II

In scenario II, 90% of ΔP_d is below 0.037 p.u., and the system’s ΔP_dmax is 0.0468 p.u.

According to (10), if Δf₁ is selected as 0.2 Hz, and Δf_max is 0.3 Hz, V_si, δ_si should satisfy V_s1δ_s1 ≥ 1.733 and V_s2δ_s2 ≥ 2.251. According to (12), ξ = 0.167 Hz, and according to (11), V_mδ_m ≥ 3.704. Taking ΔP_dmax as an example, simulation results are shown in Figure 8.

Figure 8

Figure 8. (A) FR of Scenario II at Switching moment = 10 s, and (B) FR of Scenario II at Switching moment = 7 s.

Figure 8 shows different switching times and combinations of capacity and droop of an ESS. It can be seen that FM and QM can accurately describe the f_nadir and f_ss, respectively. Additionally, the orange curve in Figure 8 shows that the ESDM effectively evaluates the frequency support ability of an ESS and maintains f_nadir and f_ss.

4.2.2 Scenario III

In scenario III, 40% of ΔP_d is below 0.037 p.u., 50% of ΔP_d lies between 0.037 p.u. and 0.05 p.u., and ΔP_dmax is 0.06 p. u. According to (10), if Δf₁ is selected as 0.2 Hz, Δf₂ is selected as 0.5 Hz, and Δf_max is 0.8 Hz, V_si, δ_si should satisfy V_s1δ_s1 ≥ 1.609, V_s2δ_s2 ≥ 1.608, and V_s3δ_s3 ≥ 2.654.

Eq. 12 yields ξ = 0.21 Hz for ΔP_d2 and ξ = 0.276 Hz for ΔP_d3, indicating that the ESS should be in f_ss maintaining mode and V_mδ_m ≥ 7.595 according to (11).

Figure 9 demonstrates that the ESDM maintains f_nadir and f_ss not only at ΔP_dmax but also at various ΔP_d levels (as shown in Figure 9A). For instance, in Figure 9A, the f_nadir is larger than 59.5 Hz but lower than 59.8 Hz, which means that ΔP_d is larger than 0.036 and smaller than 0.05. Therefore, the ESS should be switched to f_ss maintaining mode for added assurance.

Figure 9

Figure 9. (A) FR of Scenario III at ΔP_d = 0.047, and (B) FR of Scenario I at ΔP_d = 0.06.

4.3 Discussion

From Figures 7–9, it is evident that the ESS based on EAM is conservative at the f_nadir but exhibits some error at the f_ss. That is because of the neglect of the coupling relationship between active power and voltage in the model. With an increase in power disturbance, the active power support increases, leading to higher line losses and reduced load voltage. Taking the system load surge as an example, the active power of the system increases so that the load voltage decreases. As for the constant impedance load, active power is positively correlated with the voltage. Consequently, the actual power disturbance is lower than expected. With the frequency support provided by an ESS and SGs, the system frequency is recovered and the load voltage therefore increases. The increasing voltage increases the power disturbance, leading to tiny errors in maintaining the f_ss (as observed by the red lines (59.84 Hz) in 10; the error of 0.01 Hz is smaller than the dead-band of 0.015 Hz (GB/T 40595-2021, 2021)). In simulation scenarios, D is set as zero but cannot be zero in reality. As for FM and QM models used for the ESS calculation, D is not one of the input parameters according to (10) and (11), and all input parameters are from system operators, so D will not influence the accuracy of the models.

5 Conclusion

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.

Data availability statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author contributions

SD: Methodology, project administration, supervision, writing–original draft, and writing–review and editing. JZ: Data curation, investigation, methodology, validation, writing–original draft, and writing–review and editing. LY: Methodology, project administration, supervision, and writing–original draft. ZC: Methodology, project administration, supervision, and writing–review and editing.

Funding

The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. The research is supported by State Key Laboratory of HVDC (Grant No. SKLHVDC-2022-KF-02).

Conflict of interest

Authors SD and LY were employed by China Southern Power Grid Co., Ltd.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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