Quantitative Frequency Security Assessment of Modern Power System Considering All the Three Indicators in Primary Frequency Response

Abstract

The primary frequency response scale is deteriorating in the modern power system due to the high penetration of different power devices. Frequency security assessments are essential for the operation or stability-checking of the power system. Firstly, this paper establishes the Unified Transfer Function Structure (UTFS) of power systems with highly penetrated wind turbines. Based on the UTFS, this paper analyzes the three indicators of the primary frequency responses. Secondly, to better assess the security of the frequency, the secondary frequency drop (SFD) is avoided, with the frequency response parameters of the wind turbines calculated. Moreover, considering all three indicators of the primary frequency response, this paper proposes a frequency security margin index (FSMI). The FSMI divides the system stability margin into three levels, quantitively and linearly representing the frequency response capability of different power devices. Finally, to show the effectiveness and practicability of the FSMI, this paper establishes a simulation model with high wind energy penetration, including four machines and four zones in DigSILENT. Based on the FSMI, the frequency stability margins in different typical operating scenarios are divided into three zones: “Absolut secure”, “Secure” and “Relative secure”. The FSMI also shows the dominant frequency stability problem and the risk of system frequency instability for each zone. Considering the checking principles, the frequency stability margin is equivalently expanded by calculating the energy storage’s minimum frequency response capacity.

1. Introduction

1.1. Background and Literature Review

1.1.1. Background

For the sustainable development of energy, the penetration rate of renewable generation devices in the power system has gradually increased. On the other hand, the primary frequency response scale is deteriorating [1,2,3]. The participation of wind turbines and energy storage systems (ESS) in frequency responses can effectively improve frequency stability. Still, there are many new frequency stability problems, such as the secondary frequency droop (SFD) caused by the over-response of wind turbines [4,5,6]. Considering the multiple types of frequency response resources, quantitative assessments of the frequency security margin have played an important role in scheduling operation strategies and in maintaining the reliability of the power system.

1.1.2. The Frequency Response Model for Frequency Security Assessment

Frequency security assessments of primary frequency responses are based on the precise system frequency response model [7,8,9]. At present, the simplified mathematical methods for dynamic frequency responses mainly include the Average System Frequency Response Model (ASFR) and the Unified Transfer Function Structure (UTFS) [10,11,12,13,14,15,16,17]. Reference [13] proposes a modified SFR model, which makes a good overall estimation of the frequency. However, it does not take multiple types of frequency response resources into consideration. Due to the spatiotemporal characteristics of wind turbines and the decoupling parts of power electronic devices, the parameters in the ASFRs of modern power systems cannot be aggregated in the same manner as in a traditional power system [14]. Considering different generations, including wind turbines, references [15,16] propose a frequency response model to analyze a frequency stability-improving method. However, the frequency analytical formula is too complex to describe the dominant frequency stability problem quantitatively and linearly. frequency analyses based on ASFRs make it challenging to demonstrate the frequency response characteristics of various power devices intuitively. Reference [17] finds that the common-mode frequency (CMF) is more accurate in representing the global frequency in a power system with a high penetration of electronic power devices. The UTFS can approximate the CMF by simplifying the calculation of the frequency response parameters of the different power devices. It is worth noting that the new energy storage systems, such as electric spring installations in [18,19], account for a relatively small proportion, and when analyzing system frequency responses based on UTFS, it does not take them into account.

1.1.3. The Research on System Frequency Security Assessment

With the more renewable generations connecting to AC power systems, finding the relationship between frequency security boundaries and dominant frequency stability problems is a key topic for power system planning and operation [20]. Dynamic frequency security assessments are concentrated on three indicators: the rate of the change of frequency (f_RoCoF), the frequency nadir (f_N) and the steady-state frequency (f_ss) [21]. When the over-response of wind turbines causes a SFD problem, the frequency recovery process deteriorates further, and there is no specific principle for assessing the SFD [22]. References [23,24,25] analyze the mechanism between the SFD and the wind turbine control strategies. They find that the appropriate control strategy of wind turbines can effectively avoid SFD. Most studies do not consider all three indicators simultaneously to improve the frequency stability or assess the frequency security margins. Considering the stability of f_N, reference [26] proposes a kind of control strategy for DFIG to improve the frequency stability. Considering f_RoCoF and f_N, reference [27] estimates the distance to the frequency security limits. Reference [28] establishes four response indexes to represent the frequency dynamic process. Partial research has considered the cumulative effect of the primary frequency response. For example, considering the cumulative effect, reference [29] analyzes critical disturbances and control strategies, and reference [30] estimates the penetration levels of renewable power devices. The security indexes proposed above can quantitively assess the frequency security margin from only one or two aspects. They have the problem of linearizing the frequency response capacity of multi-type power devices or clearly reflecting the security degree of the frequency. In addition, the frequency security assessment is affected by the randomness of the output of renewable generations such as wind turbines, which puts forward higher requirements for real-time performance [31]. References [32,33] find that online monitoring and management methods can effectively provide data support for frequency security assessments based on a simplified frequency response model.

1.2. Contribution and Organization

1.2.1. Contribution and Novelty

Given the above, the research on system frequency assessment mainly focuses on three kinds of problems: the frequency response model considering multiple types of frequency response resources, the SFD caused by the over-response of wind turbines and the frequency security margin index considering the three indicators.

Considering the above three research gaps, the main contribution of this paper is a frequency security margin index (FSMI) to assess the frequency security in primary frequency responses quantitively. Moreover, this paper calculates the frequency response parameters of wind turbines to avoid the SFD to better assess the frequency security. The novelty of FSMI is as follows:

• The FSMI linearizes the frequency support capacity of different power devices and divides the frequency security margin into three levels (Absolute secure, Secure and Relative secure).
• The boundaries of FSMI are related to the resistance of the system frequency to power disturbances.
• Based on the FSMI, the frequency stability margin can be equivalently expanded in different typical operating scenarios by calculating the minimum capacity of ESS.

1.2.2. Organization of This Paper

The remainder of this paper is as follows. The UTFS model of the power system with a high penetration of wind turbines is established, and the three leading indicators are presented based on UTFS in Section 2. The frequency response parameters of wind turbines are calculated, which can avoid the SFD and make full use of the rotor kinetic energy in Section 3. The frequency security margin index (FSMI) is proposed, and the key parameters’ definitions are explained in Section 4. Case studies are presented to verify the effectiveness and practicability of the index in Section 5. Finally, the conclusions are presented in Section 6.

2. The Analysis of Frequency Security Based on UTFS

This section analyzes the UTFS model, considering the frequency response characteristics of different power devices. Based on this, this paper calculates the frequency stability indicators in the primary frequency response process, which provide the theoretical basis for the following parts.

2.1. The UTFS of Modern Power System

The common-mode frequency (CMF) can more effectively reflect the frequency response of a modern power system. The CMF response expression of the power system can be calculated as follows:

$$\left\{\begin{array}{l}\Delta w_s=\Delta P_d\cdot \left(\frac{1}{2H_{eq}s+D_{eq}+\frac{K_{eq}}{1+T_{eq}s}}\right)\\ \Delta P_m=\Delta w_s\cdot \frac{K_{eq}}{1+T_{eq}s}\end{array}\right.$$

(1)

where Δw_s represents the frequency deviation, ΔP_d is the load deviation, ΔP_m is the frequency response power and H_eq, D_eq, K_eq and T_eq represent the equivalent UTFS parameters of the power system. H_eq is the identical inertia time constant, D_eq is the equivalent damping coefficient, K_eq is the regulation coefficient and T_eq is the equal adjustment time constant. The UTFS mode is shown in Figure 1a.

UTFS mode focuses on the CMF, and the UTFS parameters solution is a parameter optimization process. The CMF of the system is decided by all the power devices participating in the frequency response. The optimization process can be divided into sub-optimization problems of the n power generation equipment of the system, as shown in Figure 1b. The process is as follows [15]:

$$\left\{\begin{array}{c}\underset{H_i,D_i,K_i}{\min }{\displaystyle {\int }_{t_0}^{t_s}\left(\Delta P_{di}\left(t\right)-\Delta P_{di}^{\prime }\left(t\right)\right)dt}\hfill \\ s.t.\Delta w\left(s\right)=\Delta P_{di}\left(s\right)\cdot G_i\left(s\right)\hfill \\ \hfill \Delta w\left(s\right)=\Delta P_{di}^{\prime }\left(s\right)\cdot G_i^{\prime }\left(s\right)\hfill \\ \hfill G_i\left(s\right)=\frac{1}{2H_{i}s+D_i+\frac{K_i}{1+T_{eq}s}}\hfill \end{array}\right., i=1,2,\dots ,n$$

(2)

where ΔP_di(s), ΔP_di′(s) is the disturbance power shared by the power generation and under the UTFS, the G_i(s), G_i′(s) is the UTFS model and the approximate UTFS model of the i power generation equipment.

The UTFS parameters in Equation (2) can be linearized to obtain the equivalent UTFS of the power system:

$$\left\{\begin{array}{l}{\displaystyle \sum H_i=H_{eq}}\\ {\displaystyle \sum D_i=D_{eq}}\\ {\displaystyle \sum K_i=K_{eq}}\end{array}\right.$$

(3)

Based on this, this paper can get the Laplace transform of the frequency response:

$$\Delta f\left(s\right)=f_s\cdot \frac{w_n^2}{D_{eq}+K_{eq}}\cdot \frac{1+T_{eq}s}{s^2+2\xi w_{n}s+w_n^2}\cdot \frac{\Delta P_d}{s}$$

(4)

where f_s is the system reference frequency (50 Hz). The w_n and ξ are the equivalent natural frequency and the damping ratio of the system, which can be calculated as follows:

$$\left\{\begin{array}{l}{w}_n^2=\frac{D_{eq}+K_{eq}}{2H_{eq}{T}_{eq}}\\ \xi =\frac{2H_{eq}+D_{eq}{T}_{eq}}{2\left(D_{eq}+K_{eq}\right)}\cdot w_n\end{array}\right.$$

(5)

2.2. Three Indicators in Primary Frequency Response

This paper proposes a frequency security boundary disturbance ΔP_dmax, which represents the maximum fault disturbance of the system under the frequency stability limits. The system frequency stability limits also refer to the rate of the change in frequency (f_RoCoF), the frequency nadir (f_N) and the steady-state frequency (f_ss) that vary within the stable margin during the primary frequency response process. The actual specific stability requirements of the power system determine the security principles of the above three frequency indicators.

This paper can get the maximum value of the f_RoCoF in the time domain by taking Laplace’s Initial value theorem to Equation (4):

$${\left.f_{RoCoF\max }=\frac{df}{dt}\right|}_{t=0}=f_s\cdot \frac{\Delta P_d}{2H_{eq}}$$

(6)

If the security principle of f_RoCoF is λ₁, the maximum system power disturbance ΔP_d1 under λ₁ is as follows:

$$\Delta P_{d1}={\lambda }_1\cdot \frac{2H_{eq}}{f_s}$$

(7)

This paper can get the f_N in the time domain by taking an inverse Laplace transformation to Equation (4):

$$f_N=f_s\cdot (1+\frac{1}{K_{eq}+D_{eq}}\left[1+\alpha e^{-\xi w_n{t}_{nadir1}\sin \left(w_t{t}_n{}_{adir1}+\varphi \right)}\right]\Delta P_d)$$

(8)

where:

$$\left\{\begin{array}{l}\alpha =\sqrt{\frac{1-2T_{eq}\xi w_n+T_{eq}^2{w}_n^2}{1-{\xi }^2}}\\ w_t=w_n\sqrt{1-{\xi }^2}\\ \varphi ={\tan }^{-1}\left(\frac{w_t{T}_{eq}}{1-\xi w_n{T}_{eq}}+{\tan }^{-1}\left(\frac{\sqrt{1-{\xi }^2}}{\xi }\right)\right)\end{array}\right.$$

(9)

If the security principle of f_N is (f_s + λ₂), the maximum system power disturbance ΔP_d2 under λ₂ is as follows:

$$\Delta P_d{}_2={\lambda }_2\cdot \frac{K_{eq}+D_{eq}}{f_s\cdot \left(1+\alpha e^{-\xi w_n{t}_{nadir1}\sin \left(w_t{t}_n{}_{adir1}+\varphi \right)}\right)}$$

(10)

This paper can get the f_ss in the time domain by taking the Laplace terminal value theorem to Equation (4):

$$f_{ss}=f_s\cdot \left(1-\frac{\Delta P_d}{D_{eq}+K_{eq}}\right)$$

(11)

If the security principle of f_ss is (f_s + λ₃), the maximum system power disturbance ΔP_d3 under λ₃ is as follows:

$$\Delta P_{d3}={\lambda }_3\cdot \frac{K_{eq}+D_{eq}}{f_s}$$

(12)

Considering all three frequency security indicators comprehensively, the frequency security boundary ΔP_dmax is as follows:

$$\Delta P_{dmax}=\min \left\{\Delta P_{d1},\Delta P_{d2},\Delta P_{d3}\right\}$$

(13)

The minimum value ΔP_dmax among the three frequency security boundaries represents the dominant frequency stability problem. When the wind turbines do not participate in the frequency response, the UTFS parameters of wind turbines’ G_wind(s) equals 0 in Figure 1. The equivalent UTFS parameters of the power system in (3) are represented by S₀(H_eq0, D_eq0, K_eq0). Therefore, the ΔP_dmax in (13) is represented by ΔP_dmax0.

3. The Analysis of Frequency Security without SFD

The G_wind in Figure 1 varies with the frequency response control parameters of the wind turbines, which will affect the setting of the frequency security boundary ΔP_dmax. Meanwhile, it should avoid the SFD phenomenon for assessing frequency security better. In this section, the paper calculates the virtual inertial and virtual droop control parameters of the wind turbines, which avoids the SFD while making full use of the frequency response capability of the wind turbines at different wind speeds. Based on this, it analyzes the frequency security margin.

3.1. The Frequency Response Parameters of Wind Turbines

By analyzing the mechanism between the SFD and wind turbine control strategy based on UTFS, the root cause of the SFD is the secondary power drop caused by the wind turbine speed recovery strategy. Therefore, the key to avoiding SFD is to set the appropriate frequency response parameters, mainly determined by the available rotational kinetic energy and the frequency response time.

The mechanical power P_m of DFIG captured by a wind turbine is as follows [18]:

$$\left\{\begin{array}{l}{P}_m=\frac{1}{2}\rho \pi R^2{v}^3{C}_P(\lambda ,\beta )\\ C_P=0.22\left(\frac{116}{{\lambda }_i}-0.4\beta -5\right)e^{-12.5/{\lambda }_i}\\ \frac{1}{{\lambda }_i}=\frac{1}{\lambda +0.08\beta }-\frac{0.035}{{\beta }^3+1}\\ \lambda =\frac{{\omega }_{\mathrm{r}}R}{v}\end{array}\right.$$

(14)

where ρ, R, v, C_P, β and w_r are the air density, blade length, wind speed, wind power coefficient, pitch angle and rotor speed, respectively.

When the DFIGs participate in the frequency response with the virtual inertial and virtual droop control, the power reference P_ref is as follows [18]:

$$\left\{\begin{array}{l}{P}_{ref}=P_{MPPT}-\left(k_H\cdot \frac{df}{dt}+k_D\cdot df\right)\\ P_{MPPT}\left(v,w_r\right)=\left\{\begin{array}{ll}{k}_{opt}{w}_r^3\hfill & w_r<w_{r\max }\hfill \\ P_{max}\hfill & w_r>w_{r\max }\hfill \end{array}\right.\\ k_{opt}=\frac{1}{2}\rho \pi \frac{R^4}{{\lambda }^3}{C}_P\left({\lambda }_{opt},\beta =0\right)\end{array}\right.$$

(15)

where P_MPPT, k_H and k_D are the power reference of MPPT, virtual inertial control gain and virtual droop control gain. P_max is the maximum allowable power reference in MPPT, and w_rmax is the top w_r in MPPT.

To take full advantage of the frequency response capability of a wind turbine, the relationship between P_ref and available rotational kinetic energy ΔE_k is as follows:

$${\displaystyle {\int }_0^{t_1}{P}_{ref}(v,w_r)-P_m(v,w_r)}dt=\frac{1}{2}J\left(w_{{}_{r0}}^2-w_{r1}^2\right)=\Delta E_k$$

(16)

where J is the inertia constant of the wind turbine and when w_r drops from w_r0 to w_r1, it corresponds to time t₁ at point C, as shown in Figure 2. Where the different solid lines refer to the mechanical power P_m(v, w_r) of DFIG captured in different wind speed v. The dotted dash line refers to the P_MPPT in Equation (15) with wind speed v₂. The dash line refers to the power reference P_ref in Equation (15) with wind speed v₂.

At point A (t₀, w_r0), the df = 0 and the wind turbines participate in frequency responses autonomously with the virtual inertial control in Equation (15). The response amplitude depends on the virtual inertial control gain k_H; at point C (t₁, w_r1), the df/dt = 0 and the wind turbines participate in an infrequency response with the virtual droop control in Equation (15). The response amplitude depends on the virtual droop k_D; at point D (w_r2), the P_m(v, w_r) = P_ref, which means the system frequency returns to steady-state and the rotor speed of the wind turbines is recovering.

When the DFIGs adopt a virtual inertial and drop control strategy, the P_ref can be approximated by a slash line BC, as seen in Figure 2 [7]. This paper assumes the v is constant from t₀ to t₁, so the P_ref in Equation (16) can be approximated as follows:

$${\displaystyle {\int }_0^{t_1}{P}_{ref}(v,w_r)dt\approx }{\displaystyle {\int }_0^{t_1}\frac{1}{2}\left(P_{ref}(v,w_{r0})+P_{ref}(v,w_{r1})\right)dt}=\frac{1}{2}\left(P_{ref}(v,w_{r0})+P_m(v,w_{r1})\right)t_1$$

(17)

The P_m(v, w_r) satisfies the differential increment theorem, and it assumes the w_r goes down with time linearly. P_m(v, w_r) can be approximated by Taylor’s series yields:

$$\left\{\begin{array}{l}{\displaystyle {\int }_0^{t_1}{P}_m\left(v,w_r\right)dt=}{\displaystyle {\int }_0^{t_1}{P}_m\left(v,w_{r0}\right)+\Delta P_m(v,w_r)dt}\\ \Delta P_m(v,w_r)=\frac{1}{2}\rho \pi R^2{v}^3\cdot \Delta C_p(\lambda ,\beta )=\frac{1}{2}\rho \pi R^2{v}^3\left(\frac{\partial C_p}{\partial t}{|}_{t=t_0}\cdot t+\frac{1}{2}\cdot \frac{{\partial }^2{C}_p}{\partial t^2}\cdot t^2\right)\\ \frac{\partial C_p}{\partial t}{|}_{t=0}=\frac{\partial C_p}{\partial \lambda }\frac{\partial \lambda }{\partial w_r}\frac{\partial w_r}{\partial t}{|}_{t=0}=\frac{R}{v}\frac{\partial C_p}{\partial \lambda }\frac{\partial w_r}{\partial t}{|}_{t=0}\\ \frac{{\partial }^2{C}_p}{\partial t^2}{|}_{t=0}={\left(\frac{R}{v}\right)}^2\frac{{\partial }^2{C}_p}{\partial {\lambda }^2}{\left(\frac{\partial w_r}{\partial t}\right)}^2{|}_{t=0}+\frac{R}{v}\frac{\partial C_p}{\partial \lambda }\frac{{\partial }^2{w}_r}{\partial t^2}{|}_{t=0}\end{array}\right.$$

(18)

Based on Equations (16)–(18), it can calculate the power reference P_m(v,w_r0) at time t₀:

$$P_{ref}(v,w_{r0})=\frac{2}{t_1}\left(\Delta E_k+{\displaystyle {\int }_0^{t_1}{P}_m(v,w_r)}dt\right)-P_m(v,w_{r1})$$

(19)

To avoid the secondary power disturbance at w_r1, P_ref(v, w_r1) should be equal to P_m(v, w_r1). When the frequency drops to f_N, corresponding to time t_N, |df| has the maximum value, and df/dt has the opposite sign in the right neighborhood. So, when t₁ equals t_N in (16), P_ref(v, w_r1) is more significant than P_m(v, w_r1) in the neighborhood of time t_N. This can help avoid secondary power disturbances effectively.

Therefore, the P_ref(v, w_r0) and P_ref(v, w_r1) are determined by k_H*df/dt at time t₀ and k_D*df at time t₁(t_N). And the amplitudes of k_H*df/dt and k_D*df is shown the curve A, B and C–E in Figure 2. Based on this, it can calculate the gains of k_H and k_D:

$$\left\{\begin{array}{l}{k}_{Hn}=\frac{P_{ref}{}_{\max n}-P_{MPPT}(v_n,w_{r0})}{f_{RoCoF}{{}_{\max }}_{T0}}t=t_0\\ k_{Dn}=\frac{P_m\left(v_n,w_{r1}\right)-P_{MPPT}(v_n,w_{r1})}{\Delta f_{NT0}}t=t_N\end{array}\right.$$

(20)

where v_n is the wind speed of wind farm n; k_Hn and k_Dn are the frequency response gains of wind farm n; P_refmaxn equals to P_ref(v, w_r0) in Equation (19) of wind farm n; f_RoCoFmaxT0 and ∆f_NT0 are the maximum f_RoCoF and ∆f_N is calculated using Equations (6) and (8) based on S₀.

3.2. The Analysis of ΔP_dmax Considering the Frequency Response of Wind Turbines

When DFIGs participate in frequency responses based on Equation (20), the UTFS parameters G_wind(s) can be calculated using Equation (2). It can figure the equivalent UTFS parameters S₁(H_eq1, D_eq1, K_eq1) using Equation (3). Based on this, this paper recalculates the frequency security boundary ΔP_dmax1 using Equations (6)–(13).

The frequency response gains of k_H and k_D are determined by P_refmaxn and P_m(v_n, w_r), which are related to v_n. The v_n of the wind farm n is in dynamic change, so the frequency security margin ΔP_dmax1 varies with v_n in a particular range. When the wind speed of all wind farms in the power system reaches the maximum wind speed v_max allowed by the wind turbine operation, ΔP_dmax1 has the full value. When the wind speed of all wind farms in the power system reaches the minimum wind speed v_min the wind turbine allows, ΔP_dmax1 has the minimum value. In this state, the wind turbine cannot participate in frequency response, so G_wind(s) equals 0, and ΔP_dmax1 equals ΔP_dmax0.

The existing research has shown that accurate wind speed prediction can be achieved at a minute level [4]. ΔP_dmax1 can be calculated according to the v₁–v_n predicted by the system control center in real time.

4. The Analysis of the FSMI

4.1. The Frequency Response Control of ESS

To simplify the analysis, this paper uses electrochemical energy storage to verify the application prospect of FSMI. The energy storage system (ESS) participates in primary frequency response in two main ways: virtual frequency response and emergency power response. This paper is concerned with frequency security in the primary frequency response process, so we assume that the ESS is fully responsive in this process.

When the virtual frequency response control strategy is adopted, the power reference P_ESS of ESS is coupled to the system frequency f. The UTFS parameters of ESS can be calculated using Equation (2). The equivalent UTFS parameters S₂(H_eq2, D_eq2, K_eq2) and ΔP_dmax2 can be calculated using Equation (3) and Equations (6)–(13).

When the emergency power response control strategy is adopted, the system control center will calculate the minimum output of ESS by considering all kinds of stability constraints comprehensively. To reduce the impact of power disturbances on the frequency, the ESS tends to operate at a particular time point, such as t₀. This can be regarded as an equivalent reduction in power disturbance, and it does not affect the system-equivalent UTFS parameters. When the system frequency comes into steady-state, the ESS will gradually drop out from operation with the assistance of the synchronous power generators. The ΔP_dmax2 can be calculated as follows:

$$\Delta P_{d\max 2}=\Delta P_{d\max 1}+P_{ESS}$$

(21)

4.2. The Three Levels of FSMI

Based on the above analysis, this paper proposes the system frequency security margin index FSMI to analyze the frequency stability quantitatively under the different levels of a power disturbance. The FSMI mainly consists of three elements: η, ΔP_dmax0 − ΔP_dmax2 and k₀ − k₂. The η in FSMI represents the stability of the system frequency. The higher the η value is, the more stable the frequency is. ΔP_dmax0 − ΔP_dmax2 play an auxiliary role in determining the dominant frequency stability problem under the different operating scenarios of the system. According to the different frequency response strategies of multi-type devices, the FSMI can be divided into three zones for better understanding, as shown in Figure 3; the k₀ − k₂ refers to the decline rate of η value in the three zones, which are related to the resistance of system frequency stability. The lower the k value is, the more stable the frequency is, as shown in Figure 3.

The FSMI diagram of the power system can be determined in three steps:

Firstly, based on whether the wind turbine or ESS participates in the primary frequency response, the FSMI divides the system operating state into three typical scenarios: traditional frequency response, DFIG frequency response and ESS frequency response. The frequency security boundary ΔP_dmax of each scenario can be calculated based on Equation (13).

Secondly, we calculate the η₀ − η₂ of the three typical scenarios based on ΔP_dmax0 − ΔP_dmax2. After normalization, the expression of η is shown in Equation (22). The k₀ − k₂ refers to the decline rate of the η value in the three scenarios. When there is no power disturbance, η equals 100%; when the power disturbance equals ΔP_dmax2, η equals 0; when the power disturbance is greater than ΔP_dmax2, the system frequency will lose control and become unstable.

$$\begin{array}{l}\eta =\left\{\begin{array}{l}1-\frac{k_0\cdot \Delta P_d }{{\eta }_{\max }}, \Delta P_d<\Delta P_{d\max 0} \\ 1-\frac{k_0\cdot \Delta P_{d\max 0}}{{\eta }_{\max }}-\frac{k_1\left(\Delta P_d-\Delta P_{d\max 0}\right)}{{\eta }_{\max }} , \Delta P_{d\max 0}<\Delta P_d<\Delta P_{d\max 1}\\ 1-\frac{k_0\cdot \Delta P_{d\max 0}+k_1\left(\Delta P_{d\max 1}-\Delta P_{d\max 0}\right)}{{\eta }_{\max }}-\frac{k_2\left(\Delta P_d-\Delta P_{d\max 1}\right)}{{\eta }_{\max }}, \Delta P_{d\max 0}<\Delta P_d<\Delta P_{d\max 2} \end{array}\right.\\ {\eta }_{\max }=k_0\cdot \Delta P_{d\max 0}+k_1\cdot \left(\Delta P_{d\max 1}-\Delta P_{d\max 0}\right)+k_2\cdot \left(\Delta P_{d\max 2}-\Delta P_{d\max 1}\right)\end{array}$$

(22)

Thirdly, this paper refines the values of k in the three typical scenarios, as shown in Equation (23). The values k₀ − k₂ are related to the dominant frequency stability problem based on Equations (6)–(13). Moreover, there may be different dominant frequency stability problems in the three typical scenarios. The k relates the ΔP_d to the η of the FSMI, reflecting the resistance of the system frequency.

$$k_{0-2}=\left\{\begin{array}{l}\frac{f_s}{2H_{eq}}, \Delta P_{d\max }=\Delta P_{d1}\\ f_s\frac{1+\alpha e^{-\xi w_n{t}_{nadir1}\sin \left(w_t{t}_n{}_{adir1}+\varphi \right)}}{K_{eq}+D_{eq}}, \Delta P_{d\max }=\Delta P_{d2}\\ \frac{f_s}{K_{eq}+D_{eq}}, \Delta P_{d\max }=\Delta P_{d3}\end{array}\right.$$

(23)

In the above analysis, ESS takes the virtual frequency response control strategy. When taking the emergency power response at t₀, k₂ is as follows:

$$k_2=\frac{1}{P_{ESS}}-\frac{k_0\cdot \Delta P_{d\max 0}+k_1\left(\Delta P_{d\max 1}-\Delta P_{d\max 0}\right)}{P_{ESS}}$$

(24)

To specify the relative distance of the system frequency security margin, the FSMI divides the margin into three levels: Absolute secure, Secure and Relative secure, as shown in Figure 3. Therefore, the (ΔP_dmax0, η₀) is the boundary between the “Absolute secure” and “Secure”. When ΔP_d is less than ΔP_dmax0, the frequency can be stabilized by the primary frequency response only with the synchronous generators. Considering the capability of other power devices, the frequency stability is “absolutely secure” and the (ΔP_dmax1, η₁) is the boundary between “Secure” and “Relative secure”. When ΔP_d is less than ΔP_dmax1, the frequency can be stabilized with the assistance of wind turbines. Considering the varying wind speed, the frequency stability is “Secure”. When ΔP_d is less than ΔP_dmax2, the frequency can be stabilized with the assistance of ESS. The system has no other frequency capability, so the frequency stability is “relatively stable”. When ΔP_d is bigger than ΔP_dmax2, the frequency stability is unstable. The k₀ − k₂ represents the frequency security resistance on the power disturbance of three different levels.

It is worth noting that ΔP_dmax1 is strongly correlated with the wind speed v, so the setting of index FSMI should be updated regularly according to the wind speed v of each wind farm in the system.

5. Numerical Case Results

This paper establishes a simulation model with high wind energy penetration, including four machines and four zones in DigSILENT, as shown in Figure 4.

Area 1 and Area 2 are the traditional four-machine and two-zone areas. The steady-state output power of the synchronous unit G1–G4 is 845 MW. The G1 is the balancing unit and Area 3 and Area 4 are DFIGs-ESS farms. Several DFIG_0.69 kV_1.5 MW teams aggregate WT1–WT4; the details are shown in Figure 4 and

Table 1. Operation parameters of WT1–WT4.

	WT1	WT2	WT3	WT4
Number of units	100	100	50	50
Wind speed v (m/s)	6	7	8	9
Initial rotor speed wr0 (p.u.)	0.73	0.84	0.98	1.2

. Several Battery_10 kV_30 MVA units aggregate ES1–ES2 and the emergency power disturbance ΔP_L is a step load change in L1. The rated frequency of the system is 50 Hz, and the base capacity is 100 MW. The allowable minimum and maximum rotor speeds are 0.7 p.u and 1.2 p.u, and the converter overload limit power is set to 1.1 p.u.

The frequency security indexes f_RoCoF, f_N, and f_ss are related to the protection control strategy of the actual local power grid. In this paper, the security margins λ₁, λ₁, and λ₂ in Section 2.2 are set as −0.5 Hz/s, −0.5 Hz and −0.2 Hz, respectively.

5.1. The Response Parameters of Wind Turbines

When the wind turbines do not participate in the frequency response, the equivalent UTFS parameter matrix S₀(H_eq0, D_eq0, K_eq0) is S₀(21.5, 13, 39.5), calculated using Equations (2) and (3) with T_eq equaling 10. When ΔP_L is set as 0.2 p.u., the comparison between frequency responses based on the UTFS frequency domain and DIgSILENT time domain is shown in Figure 5. The relative error of simulation is less than 5%, so the UTFS model can effectively quantify the frequency response of the power system with a high penetration of wind energy.

The dominant frequency stability problem can be analyzed using Equations (6)–(13). In this case, it is that the f_N reaches the (50-λ₂), and the ΔP_dmax0 is 0.2 p.u. in Equation (13). With the ΔP_dmax0 = 0.2 p.u., the f_RoCoFmaxT0, ∆f_NT0 in Equation (20) are 0.23 and 0.5, respectively. The k_H1–k_H4 and k_D1–k_D4 of WT₁–WT₄ can be calculated using Equation (20), as shown in

Table 2. Frequency response gains of each wind farm.

	WT1	WT2	WT3	WT4
kHn	0.24	1.06	2.44	5.82
kDn	0.225	0.338	0.467	0.572

.

When the wind turbines participate in primary frequency responses with the virtual inertial and droop gains in Table 2, the frequency response for different power disturbances ΔP_L is shown in Figure 6. When ΔP_L is set as 0.2 p.u., the f_N increases from 49.5 Hz to 49.69 Hz. When ΔP_L is set as 0.4 p.u., the system’s frequency is unstable, and there is severe SFD.

When ΔP_L is set as 0.3 p.u., the f_N reaches 49.51 Hz, and the output of WT1–WT4 is shown in Figure 7a. When the f_N comes to 49.51 Hz, the rotor speed w_r of WT1–WT3 drops to around 0.7 p.u. in Figure 7b. The results show that the proposed control strategy avoids the SFD while entirely using the wind turbines’ frequency regulation capabilities at different wind speeds. Because the converter overload limit power is set to 1.1 p.u., and the highest output of WT4 is 82.5 MW in Figure 7a, the rotor’s kinetic energy is released slowly. When the system frequency drops to the lowest point (11.78 s, 49.51 Hz), the w_r of WT4 drops to 0.79 instead of 0.7.

5.2. The FSMI of the Power System

When the WT1–WT4 participate in frequency responses with the response gains in Table 2, the equivalent UTFS parameter matrix S₁(26.8, 27.5, 61.5) can be calculated using Equations (2) and (3) with T_eq equaling 10. Based on Equations (6)–(13), the dominant frequency stability problem is that the f_N reaches 49.5 Hz, and the ΔP_dmax1 is 0.294 p.u. In this paper, the ESS takes the emergency power response control strategy at t₀, and the P_ESS is set to 0.11 p.u. based on Equation (21). Based on Equations (22)–(24), the parameters of the system frequency security index η can be calculated, as shown in

Table 3. The parameters of FSMI.

	Absolut Secure	Secure	Relative Secure
ΔPdmax0–2	0.20 p.u.	0.29 p.u.	0.40 p.u.
η0–2	0.424	0.2523	0
k0–2	2.88	1.72	2.69

and Figure 8.

The operating status of synchronous power generators G1–G4 determines the “Absolute secure” zone. It is not affected by environmental factors, and it usually does not change quickly. The “Secure” zone is determined by the frequency response capability of WT1–WT4, which is constrained by the wind speed. The “Relative secure” zone is determined by the response capacity of ESS, which is generally constrained by the economics of ESS.

As shown in Figure 8 and Table 3, in the “Absolute secure” zone, even when the WT1–WT4 do not participate in frequency responses, the primary frequency response can meet the requirements of the three security margins. The boundary ΔP_dmax0 and decline rate k₀ of “Absolute secure” is bigger than the others. The traditional frequency response capability is the maximum, but the frequency stability resistance is the minimum. In the “Secure” zone, with a certain wind speed (v₁–v₄), the ΔP_dmax1 is 0.29 p.u. The frequency response capability of wind turbines is 0.09 p.u., and the k₁ is 1.72. In the “Relative secure” zone, the ΔP_dmax2 is 0.40 p.u. and the k₂ is 2.69 p.u. Even though wind turbines’ frequency response capabilities are minimal, the effect of the frequency stable resistance is the best of the three levels. The capability of ESS is variable. The boundary ΔP_dmax2 is effectively extended by setting the capacity of ESS. Considering the typical operation model of the system and the wind speed in different DFIG farms, the ESS frequency response strategies can be analyzed in different typical scenarios.

5.3. Equivalently Expand the Frequency Security Margin

5.3.1. The FSMI of Two Scenarios

This paper sets two different typical operating states of the system according to the frequency response capacity of wind turbines. Scenario 1: when v₁–v₄ are 9 m/s, w_r1–w_r4 are around 1.2 p.u. and the “Secure” zone gets the maximum; Scenario 2: when v₁–v₄ are 5.9 m/s, w_r1–w_r4 are around 0.7 p.u. and the “Secure” zone gets the minimum. The ESS takes the emergency power response control strategy, and the frequency response capability is 0.11 p.u. in both Scenarios.

The equivalent UTFS parameter matrix in Scenario 1 is S₀(21.5, 13, 39.5) and S₁(36.7, 35.4, 75.8), which can be calculated using Equations (2) and (3). The ΔP_dmax0, ΔP_dmax1 and ΔP_dmax2 are 0.2, 0.37 and 0.48, respectively. The k₀, k₁ and k₂ in Scenario 1 are 2.88, 1.34 and 1.78, respectively, which can be calculated using Equation (23). Based on Equations (6)–(13), the dominant frequency stability problem in both zones is that as ΔP_d increases, the value of f_N reaches the security margin. Because there is no frequency response from wind turbines in Scenario 2, the S₀ is S₀(21.5, 13, 39.5) and the S₁ equals 0. The ΔP_dmax0, ΔP_dmax2 are 0.2 and 0.31, respectively. The k₀, k₂ in Scenario 2 are 2.88 and 5.23, respectively. The FSMIs of the two Scenarios are shown in Figure 9. They can clearly and quantitatively assess the frequency security margins of the two Scenarios.

5.3.2. The Analysis of FSMI in Two Scenarios

In Scenario 1, the ΔP_dmax0 > ΔP_dmax1 − ΔP_dmax0 > ΔP_dmax2 − ΔP_dmax1, which means the frequency response ability of the traditional power device is the strongest and the frequency response ability of the ESS is the weakest. In “Absolute secure”, the system frequency is absolutely stable and is not affected by environmental factors such as wind speed. In “Secure”, the frequency can remain stable, with the support of renewable power devices such as wind turbines. In “Relative secure”, the frequency stability can be maintained by relying on the frequency response ability of ESS. When the operating state of the system is determined, ΔP_dmax0 − ΔP_dmax2 will also be determined. The relative magnitude of k₀ − k₂ reflects the resistance of the system frequency to the fault disturbance. The k₀ > k₂ > k₁, which means the resistance of the system frequency stability in “Relative secure” is the maximum and in “Absolute secure” is the minimum. In other words, when 0.37 < ΔP_d, it is more likely to cause cascading failures in “Absolute secure”.

In Scenario 2, there is no “Secure”. Considering the analysis steps of Scenario 1, the frequency response ability of the traditional power device is the strongest, and the frequency response ability of the ESS is the weakest. The resistance of the system frequency stability in “Relative secure” is the minimum. When 0.2 < ΔP_d, the frequency stability of the system is determined by the frequency response margin of the energy storage. Considering k, and compared with Scenario 1, the risk of cascading failures is more likely.

Affected by complex environmental factors, the frequency response capacity of wind turbines is constantly fluctuating, which results in the boundary of the “Secure” zone not being fixed. The ΔP_dmax1 and k₁ in FSMI can effectively assess the supporting effects of wind turbines’ frequency responses on security margin. As shown in Figure 9, the response capacity of wind turbines varies from 0 to 0.17 p.u. as the wind speed changes. As the wind speed increases, the frequency stability resistance increases (k₁ decreases). Meanwhile, when the response capacity of ESS is fixed, the boundary of “Relative secure” (0.11 p.u) will be determined. However, there are different k₂ and a greater k₂ means a greater risk.

5.3.3. Application of FSMI in Two Scenarios

A code on security and stability for power systems requires that when a fault disturbance occurs, the frequency should be recovered within the allowable range by the primary frequency response [20]. It also requires the primary frequency response to meet the frequency stability checking, such as the N − 1 principle. With the construction of a modern power system, there is the risk of an active power drop beyond the “Secure” zone boundary in the N-1 principle checking. It cannot determine the frequency response capability of the wind turbines and the boundary of the “Secure” zone. However, the boundary of the “Relative secure” zone can be equivalently extended by the frequency response of ESS.

For example, this paper checks the N − 1 principle by the wind turbines’ off-grid disturbance. In Scenario 1, the boundary of “Relative secure” reaches a maximum of 0.48 p.u., but the maximum output of wind turbines is 0.6 p.u. from WT4. In Scenario 2, there is no “Secure” zone, and the maximum production of wind turbines is 0.25 p.u. from WT1. The boundary of “Relative secure” is 0.31 p.u. To meet the N-1 principle, it needs an additional frequency response capacity of ESS 0.12 p.u. in Scenario 1, and it allows the 0.06 p.u. reduction in the frequency response capacity in Scenario 2. The frequency response under the different power faults of the two Scenarios is shown in Figure 10.

6. Conclusions and Research Prospect

6.1. Conclusions

To better assess the frequency security, this paper avoids the SFD with the response parameters of wind turbines. Based on this and considering all three indicators, this paper proposes a frequency security index FSMI, which divides the frequency security into three levels. Each level can linearly reflect the dominant frequency problem. As seen in the analyses of the case studies, the FSMI can quantitively assess the effect of wind turbines and ESS on primary frequency responses. The main results are as follows:

• Based on the UTFS, this paper analyzes the mechanism between the SFD and the wind turbine control strategies. For assessing the frequency security margin better, the SFD is avoided with the frequency response parameters of wind turbines while making full use of the rotor kinetic energy.
• A frequency security index FSMI is proposed, considering all the frequency stability indicators (f_RoCoF, f_N, f_s) in the primary frequency response. The FSMI linearly divides the system frequency security margin into three levels, and it can reflect the frequency stability resistance and the frequency response capability of the different power devices.
• Based on FSMI, it can quantitively assess the frequency security margin of the different typical operating Scenarios and equivalently expand the security margin. Meanwhile, the minimum frequency response capacity of ESS is calculated for different frequency stability checking principles.

6.2. Research Prospects

This paper focuses on the linear and quantitative assessment of the system frequency security, considering three indicators. In practice, the system frequency is strongly related to the actual operating condition of the modern power system, and this paper uses partial simplification to ensure a better analysis. Therefore, the research prospects are as follows:

• The external characteristics of each generating device directly affects the system frequency and this paper simplifies the detailed control modeling of the wind turbines and ESS. Taking various types of complex control strategies into the frequency response model is one of the next research focuses.
• The system frequency assessment is affected by the complex operating boundaries of the actual system. The paper only verifies the feasibility and validity of the FSMI through the typical Scenarios. We will further improve the FSMI with the actual data of the modern power system.
• In order to simplify the analysis, only electrochemical energy storage is used to illustrate the application prospect of FSMI. Combining the different frequency response characteristics of multi-type energy storage systems with FSMI is the focus of our next study.