Frequency Dynamics of Power Systems with Inertial Response Support from Wind Generation

Abstract

Inertial response support from wind turbine generators has become a priority requirement in most grid codes to improve the frequency response and frequency stability margins of power systems. However, the interaction between MPPT and inertial controllers may significantly degrade the power system dynamics. Therefore, there is a need to comprehensively understand the electromechanical dynamics of power systems with high penetration of wind generation. In this context, this work proposes a simplified dynamic model to assess the electromechanical dynamics of modern power systems with inertial response support from wind generation. The proposed simplified model allows simple analyses of the intrinsic and extrinsic aspects of wind generation that directly affect the system frequency dynamics and the dynamics of wind turbine generators. As a secondary contribution, this work also provides a comprehensive assessment of intrinsic and extrinsic aspects of wind generation that significantly affect the electromechanical dynamics of power systems with inertial response support from wind generation.

1. Introduction

The wind generation installed capacity in the world reached 433 GW and 906 GW at the end of 2015 and 2022, respectively, which represents an increase of 209.2% in this period [1]. China and the USA are the countries with the highest wind generation installed capacity and reached 365.4 and 144.2 GW at the end of 2022, respectively. Asia, Europe, and the Americas, continents with the highest wind generation installed capacity, reached 436.8 GW, 255.5 GW, and 234.4 GW at the end of 2022, respectively [1]. An average growth of 15% per year is expected for the world installed capacity of wind generation in the period from 2022 to 2027 [1]. This large increase in the wind generation, expected for the coming years, may aggravate operational and control problems caused by wind turbine generators (WTGs) [1].

The degradation of power system frequency dynamics, caused by the penetration of wind generation (WG) and photovoltaic generation, is one of the most relevant challenges of modern power systems [2], which may lead to generator shutdown and load shedding by the frequency relays. System operators have proposed grid codes that require inertial response support from WTGs [3] in order to improve the frequency dynamics and frequency stability margins of power systems with relevant penetration of renewable generation [4,5]. Unlike primary frequency control, inertial control mainly improves the rate of change of frequency (RoCoF) and the frequency nadir [6].

Several control approaches to provide inertial response support have been proposed to satisfy the grid codes and improve the dynamics and reliability of power systems [7,8,9]. The inertial response support from WTGs that operate at the maximum power point (MPP) is one of the most employed approaches [10,11]. The use of a proper virtual inertia value is essential to improve the inertial response and prevent the degradation of power system dynamics, which can lead to generator shutdown and load shedding [12]. On the other hand, the inertial response can lead WTGs to instability due to large deviations of the wind turbine speed [13].

Some works have suggested that the higher the value of the virtual inertia, the better the frequency behavior of the system [14]. However, the action of the classical MPPT control, typically used in commercial WTGs, associated with high values of virtual inertia, can significantly degrade the system frequency dynamics [15,16]. Thus, the typical electromechanical dynamics of power systems with high penetration of WG need to be comprehensively understood and evaluated to improve the inertial support provided by WTGs [4,17].

Simplified analytical formulations for the dynamic analyses of WTGs that support inertial response can be employed by wind farm and power system operators to assist the virtual inertia tuning and provide a comprehensive understanding of the impact of the inertial support on WTGs and system frequency response. Simplified system frequency response models have been proposed to assess the impact of the inertial support provided by WG on the dynamics of the power system frequency [18,19,20]. However, the models proposed in [18,20] do not consider the intrinsic dynamics of the WTG and, consequently, they cannot be employed to evaluate the WTG dynamics during the inertial contribution. In [21], a simplified system frequency response model is proposed to optimize the virtual inertia of WG, minimizing the second frequency dip caused by the wind turbine speed recovery. However, the model proposed in [21] does not consider the wind turbine speed dynamics and the influence of the MPPT on the system frequency dynamics.

Methodologies based on simplified models are proposed in [13,19] to determine the virtual inertia and the maximum speed deviation of the wind turbine. These methodologies determine the virtual inertia in such a way as to avoid stability loss of the WTG and to optimize the WTG inertial contribution. However, the approach proposed in [13] disregards the WTG dynamics in the formulation that describes the system frequency. Thus, such a formulation does not describe the detrimental coupling between the MPPT and inertial controllers on the system frequency response. Therefore, the formulation proposed in [13] is suitable only for power systems with low penetration of WG. Although the model proposed in [19] considers the coupling between the MPPT and inertial control, it is valid only for small perturbations, since the formulation is based on a linearized model.

Considering the drawbacks of the aforementioned models, the proposition of new simplified models is required to bridge the gaps in the dynamic assessment of power systems with high penetration of WG. In addition, the analysis of power systems with high penetration of WG requires the modeling of the electromechanical dynamics of WTGs, as addressed in [22].

In the context of tools to assess the dynamics of modern power systems, this work proposes a simplified dynamic model to evaluate the electromechanical dynamics of systems with WG providing inertial response support. The proposed simplified model can assess the intrinsic and extrinsic aspects of WG that directly affect the system frequency dynamics and the electromechanical dynamics of WTGs. Aspects such as the insertion level of WG and the type of control approach employed in the WTG can be easily evaluated with the proposed simplified model. The antagonistic interaction between the inertial and MPPT controllers, which degrades the system frequency response, can also be easily assessed by the proposed model. The simplified model is composed of four differential equations and accurately describes the electromechanical dynamics of the equivalent WTG and frequency response of power systems. As an innovative contribution over other simplified models, the proposed model considers the nonlinear characteristics of the wind turbine mechanical power and MPPT control. A comprehensive assessment of the intrinsic and extrinsic aspects of WG that significantly affect the electromechanical dynamics of power systems with inertial support from WTGs is also another relevant contribution of this work. The carried-out analyses evaluate how the wind speed, value of virtual inertia, load variation magnitude, and WTG control mode affect the frequency dynamics of power systems.

Power systems with high penetration of renewable generation have more complex dynamics as compared with conventional power systems based on synchronous generators. Therefore, there is a need for understanding these new dynamics and their impact on power system dynamics and control. The proposed model, performed analyses, and obtained results provide a deeper and comprehensive understanding of the dynamics of power systems with inertial response support from WTGs. The proposed simplified model and the performed analyses are useful for the planning and operation of power systems with high penetration of WG [23], tuning of inertial controllers [24], and evaluation of new control approaches.

The paper is structured as follows. Section 2 addresses the typical electromechanical dynamics of type-4 WTGs operating with inertial control based on virtual synchronous generators (VSG). Section 3 presents the formulation of the proposed simplified model. The accuracy validation of the proposed model and the dynamic assessment of power systems with inertial response support from WG are presented in Section 4. Finally, Section 5 presents the conclusions about the proposed model and the performed dynamic assessment.

2. Electromechanical Dynamics of WTGs Operating with Inertial Controller

Among the different types of WTGs, the type-3 and type-4 WTGs are the dominant topology commercially used to provide inertial response support, due to their static converters and control loops that allow the implementation of inertial controllers [25]. Inertial response support may be provided by WTGs operating in the PQ control mode or Qf/Vf control mode [24,26]. In the PQ control mode, the WTG control loops regulate the active and reactive powers and in the Qf control mode the WTG control loops regulate the frequency and reactive power at the WTG output. PQ and Qf control modes are also known as grid-following control and grid-forming control, respectively [24,26]. In this section, type-4 WTG with its inertial controller operating in PQ and Qf control modes is briefly addressed, since it is not an innovative contribution of this work.

The control diagram of a type-4 WTG operating in the PQ control mode with inertial controller is shown in Figure 1 [8], where$H_v$ is the virtual inertia constant, ${\omega }_t$ is the wind turbine speed, and k_opt is the gain of the classical MPPT employed in commercial WTGs. The virtual inertia moment may be written as a function of the inertia constant as $J_v={2S}_b{H}_v/{\omega }_b^2$, where $S_b$ and ${\omega }_b$ are, respectively, the base power and base frequency of the WTG. The power reference assigned to the control loop of the WTG is generated by the MPPT and inertial controllers. The simultaneous action of these two controllers couples the wind turbine speed dynamics and system frequency dynamics. The control loops of the WTG are implemented in the dq reference frame using typical PI controllers [8].

The control diagram of a type-4 WTG operating in the Qf control mode with inertial response support from VSG is shown in Figure 2 [17]. The MPPT control generates the power reference (P_v,m) assigned to the WTG, also coupling the system frequency dynamics and wind turbine speed dynamics. The frequency $\left({\omega }_v\right)$ and reactive power at the WTG output are regulated by the control loops of the inverter, which includes the VSG, and the rectifier control loop regulates the DC-link voltage [25,27]. The controllers of the inverter and rectifier are detailed addressed in [27] and [28], respectively.

The mechanical dynamics of conventional synchronous generators, which correspond to the generator speed and inertial response, are emulated by a low-order VSG, as addressed in [24,26]. The employed VSG and MPPT are detailed in Figure 3, where ${\omega }_v$ is the VSG angular speed, ${\theta }_v$ is the virtual mechanical phase of the VSG, and $D_v$ is the virtual damping coefficient of the VSG. The PWM algorithm of the WTG uses the angle ${\theta }_v$ to control the frequency at the inverter output [27,28]. A low-pass filter is employed to provide mechanical damping to the VSG [29]. In Figure 3, $T_d$ and ${\omega }_{v,d}$ are the time constant and output of the low-pass filter, respectively.

Different kinds of generators (i.e., synchronous and asynchronous machines) are employed in commercial type-3 and type-4 WTGs. However, the different kinds of generators present the same mechanical dynamics, described by Newton’s second law of motion ($J_t{\omega }_{t}d{\omega }_t/dt=P_{t,m}-P_{v,e},$ also presented in (4) and (A9) and known as the swing equation) [13,16,18,30,31]. The inertial response support of the WTGs is mainly determined by the inertial controller and its parameters [26]. Different kinds of inertial controllers are also employed in commercial WTGs, as previously addressed. However, despite the difference between inertial controllers (i.e., grid-forming and grid-following inertial controllers), the inertial response and mechanical dynamics of the WTGs due to the inertial controller action are quite similar, as compressively assessed in [26]. In this work, the VSG control was chosen due to its grid-forming capability, which is highly desired in modern power systems. Considering that the different generators and different inertial controllers present equivalent mechanical dynamics [18,26], the proposed model can be employed to describe the mechanical dynamics of generic commercial type-3 and type-4 WTGs, if proper equivalent parameters for the generator and inertial controller are adopted.

Typical Electromechanical Dynamics

This subsection presents an analysis of the typical electromechanical dynamics of WTGs operating with inertia control. The analysis of these dynamics is relevant to show the natural coupling of the WTG dynamics and frequency dynamics, caused by the MPPT control.

The analysis assumes that at t = t₀ the system is subjected to a load increase, and the WTG operates at the MPP with MPPT control. The typical frequency behavior of a generic power system, which is presented in Figure 4, is commonly divided into three periods according to the dominant dynamics: (T1) inertial response, where the generators release the kinetic energy from the rotating masses; (T2) action of the primary frequency control; (T3) action of the secondary frequency control.

The inertial response from the WTG is proportional to the rate of change of the frequency and, therefore, its contribution is negligible in period T3. During periods T1 and T2, the inertial controller contributes to the inertial response, releasing or absorbing energy from the system, which results in the absorption or release of kinetic energy of the wind turbine. This dynamic behavior is represented by the transient power balance of the WTG presented in Equation (1). In such a formulation, the dynamics of the DC-link voltage were neglected, since they are very fast compared to the electromechanical dynamics of the inertial controller and wind turbine.

$$\Delta P_{v,e}\cong \underset{\mathrm{Group}\mathrm{of}\mathrm{terms}1}{\underbrace{∆P_{ref}-J_v{\omega }_v\frac{d{\omega }_v}{dt}}}\cong \underset{\mathrm{Group}\mathrm{of}\mathrm{terms}2}{\underbrace{\mathsf{\Delta }{P}_{t,m}-J_t{\omega }_t\frac{d{\omega }_t}{dt}}}.$$

(1)

In (1), ${\omega }_t$ is the wind turbine speed, ${\omega }_v$ is the VSG angular speed, $J_v$ is the virtual moment of inertia of the inertial controller, $J_t$ is the inertia moment of the wind turbine−generator set, $∆P_{t,m}$ is the mechanical power variation of the wind turbine, $\Delta P_{v,e}$ is the active power variation at the WTG output, and $∆P_{ref}$ is the variation in power reference of the VSG. In the formulations presented in this work, the variables without the upper bar are in the international system of units (SI), and the variables with the upper bar are in the per-unit system (p.u.). The active power at the WTG output (P_v,e) during the inertial response is mainly determined by the action of the inertial and MPPT controllers. In Equation (1), the active power variation at WTG output due to the inertial controller action is given by $-J_v{\omega }_{v}d{\omega }_v/dt,$ and the term ${\mathsf{\Delta }P}_{ref}=K_{opt}\left({\omega }_t^3-{\omega }_{MPP}^3\right),$where ω_MPP is the wind turbine speed at MPP, is the active power variation caused by the MPPT controller action. The wind turbine inertial power ${(J}_t{\omega }_{t}d{\omega }_t/dt)$ and the mechanical power variation of the wind turbine (ΔP_t,m) during the inertial response are described by the group of terms 2 in Equation (1). The active power variation in the WTG (ΔP_v,e) during the inertial response support is almost equal to the variation in the mechanical power and inertial power of the wind turbine, as shown in the transient power balance and also addressed in [17].

The transient power balance described in Equation (1) can be better understood by analyzing the wind turbine mechanical power and WTG output power illustrated in Figure 5. The dynamics of the wind turbine speed, shown in Figure 6, are also considered in this analysis. The electromechanical dynamics of the WTG are mainly affected by the inertial and MPPT controllers, as comprehensively addressed in the remainder of this section.

After a load variation, the inertial control increases the active power supplied by the WTG, resulting in a positive power variation (ΔP_v,e). A positive ΔP_v,e provides decelerating energy (E_des,1) to the wind turbine, resulting in the turbine deceleration and negative variation of the turbine mechanical power (ΔP_t,m). The decelerating energy provided to the wind turbine is equal to the kinetic energy variation of the wind turbine (ΔE_k), which corresponds to the area between P_v,e and P_t,m, illustrated in Figure 5.

Analyzing Equation (1) and Figure 5 and Figure 6, it is possible to observe that the inertial power ($J_t{\omega }_{t}d{\omega }_t/dt$) and $\mathsf{\Delta }{P}_{t,m}$ are negative in the period between t₀ and t₂. Thus, considering that $\Delta P_{v,e}$ =$\mathsf{\Delta }{P}_{t,m}-J_t{\omega }_{t}d{\omega }_t/dt$, as showed in Equation (1), it is possible to conclude that only part of the wind turbine inertial power is delivered to the grid by the inertia controller. It is worth remarking that this analysis is valid for MPPT operation, since in de-loaded operation in the over-speed region, a wind turbine deceleration increases the mechanical power ($\mathsf{\Delta }{P}_{t,m}>0$). The increase of $P_{t,m}$, in the de-loaded operation, increases the power delivered to the grid ($P_{v,e}$) by the inertial controller.

The MPPT control also affects the inertial response support of the WTG. As the wind turbine slows down, the MPPT control reduces the power reference of the WTG ($∆P_{ref})$, reducing ΔP_v,e to accelerate the wind turbine towards the MPP speed. During the wind turbine speed recovery, the MPPT and inertial controllers have antagonistic actions. The detrimental interaction of the MPPT reduces the inertial response support of the WTG, which is equivalent to reducing the virtual inertia (J_v) assigned to the inertial controller.

Besides reducing the inertial response support of the WTGs, the MPPT control makes $\Delta P_{v,e}$ negative before the occurrence of the frequency nadir (t_n). The decrease of $P_{v,e}$, which can be observed in Figure 5, after t₁, reduces the frequency nadir, degrading the system frequency response. The wind turbine deceleration ceases when P_v,e becomes equal to P_t,m, at time t₂.

The output power P_v,e becomes smaller than P_t,m after t₂, providing accelerating energy (E_accel,1) to the wind turbine. After time t_n, the RoCoF becomes positive and, consequently, the inertial controller reduces the magnitude of P_v,e. Thus, the MPPT and inertial controllers accelerate the wind turbine towards the MPP, as shown in Figure 6. This synergetic action of both controllers leads to an overshoot in the wind turbine speed (ω_t > ω_MPP). After ω_t becomes higher than ω_MPP (Δω_t > 0), the MPPT acts to decelerate the turbine, increasing P_v,e, while the inertial control reduces P_v,e, accelerating the wind turbine. This antagonistic action of the MPPT and inertial controllers lasts until time t₄, when P_v,e becomes equal to P_t,m again. The performed analysis shows that the interaction of the MPPT and inertial controllers alternates between synergetic and antagonistic actions during the dynamics of the system frequency. Such alternance supply accelerating and decelerating energy to the wind turbine (E_des and E_accel), leading to an oscillatory response to the wind turbine speed and output power of the WTG.

It is worth remarking that the WTG would naturally oscillate around the MPP only due to the MPPT control action. However, the interaction between the MPPT and inertial controllers increases the oscillation of the output power and wind turbine speed. The results presented in Section 4 show that the amplitude of these oscillations varies as a function of the load variation magnitude, virtual inertial of the VSG, and WTG operating point.

Based on the term $P_{v,e}\cong K_{opt}\left({\omega }_t^3-{\omega }_{MPP}^3\right)-J_v{\omega }_{v}d{\omega }_v/dt$presented in Equation (1), it is possible to characterize the antagonistic or synergetic interaction of the inertia and MPPT controllers as a function of the speed deviation of the wind turbine (Δω_t = ω_t − ω_MPP) and system frequency derivative ($d{\omega }_v/dt).$ The Cartesian space $d{\omega }_v/dt$ versus Δω_t, illustrated in Figure 7, characterizes the antagonistic and synergetic interactions of the MPPT and inertial controllers. As previously addressed, the antagonistic interaction reduces the effectiveness of the inertial support provided by the inertial control, and the synergetic interaction increases the system oscillations.

3. Simplified Model Proposed to Assess the Electromechanical Dynamics

The formulation of the simplified model proposed to assess the electromechanical dynamics of modern power systems with WG providing inertial response is addressed in this section. The proposed model describes the electromechanical dynamics of large generic equivalent power systems, composed of an equivalent WTG, representing a set of wind farms, an equivalent synchronous generator, representing the set of conventional synchronous generators, and an equivalent load.

The simplified model is composed of four nonlinear differential equations. Simplifying hypotheses are adopted to provide a low-order model that accurately describes the dynamics of power systems. The adopted simplifying assumptions are typically employed in dynamic analyses of power systems. Dynamics significantly faster than the system frequency dynamics are disregarded, since they do not significantly affect the dynamics of interest [29]. The switching dynamics of the WTG static converter, the DC link dynamics of the WTG, the LCL filter dynamics of the WTG, and the damping component of the VSG are neglected.

The mechanical dynamics of the equivalent synchronous generator and equivalent WTG, considering the previously mentioned assumptions, are described by [29]:

$$\frac{J_{sg}{\omega }_{base}^2}{S_{sg,base}}{\overline{\omega }}_{sg}\frac{d{\overline{\omega }}_{sg}}{dt}=∆{\overline{P}}_{sg,m}-∆{\overline{P}}_{sg,e},$$

(2)

$$\frac{J_v{\omega }_{base}^2}{S_{t,base}}{\overline{\omega }}_v\frac{d{\overline{\omega }}_v}{dt}=∆{\overline{P}}_{v,m}-∆{\overline{P}}_{v,e},$$

(3)

$$\frac{J_t{\omega }_{t,base}^2}{S_{t,base}}{\overline{\omega }}_t\frac{d{\overline{\omega }}_t}{dt}=∆{\overline{P}}_{t,m}-∆{\overline{P}}_{v,e},$$

(4)

where ${\overline{\omega }}_{sg}$ and ${\overline{\omega }}_v$ are, respectively, the angular frequencies of the equivalent synchronous generator and equivalent WTG, $J_{sg}$ and $J_v$ are the inertia moments of the equivalent synchronous generator and equivalent WTG, respectively, $∆{\overline{P}}_{sg,m}$ and $∆{\overline{P}}_{v,m}$ are, respectively, the mechanical power of the equivalent synchronous generator and virtual mechanical power of the WTG, $∆{\overline{P}}_{sg,e}$ and $∆{\overline{P}}_{v,e}$ are the output power of the equivalent synchronous generator and equivalent WTG, respectively, ${\overline{\omega }}_t$ is the wind turbine speed, $J_t$ is the inertia moment of the equivalent wind turbine, $∆{\overline{P}}_{t,m}$ is the mechanical power of the wind turbine, ${\omega }_{base}$ and ${\omega }_{t,base}$ are, respectively, the base frequencies of the system and wind turbine, and $S_{sg,base}$ and $S_{t,base}$ are the base powers of the equivalent synchronous generator and equivalent WTG, respectively.

Equation (2) is represented on the base power of the equivalent synchronous generator and Equations (3) and (4) are on the base power of the equivalent WTG. Therefore, Equations (2)–(4) are rewritten in the base power of the power system ($S_{sys,base}$) as:

$$\frac{J_{sg}{\omega }_{base}^2}{S_{sg,base}}{\overline{\omega }}_{sg}\frac{d{\overline{\omega }}_{sg}}{dt}\frac{S_{sg,base}}{S_{sys,base}}=∆{\overline{P}}_{sg,m}-∆{\overline{P}}_{sg,e},$$

(5)

$$\frac{J_v{\omega }_{base}^2}{S_{t,base}}{\overline{\omega }}_v\frac{d{\overline{\omega }}_v}{dt}\frac{S_{t,base}}{S_{sys,base}}=∆{\overline{P}}_{v,m}-∆{\overline{P}}_{v,e},$$

(6)

$$\frac{J_t{\omega }_{t,base}^2}{S_{t,base}}{\overline{\omega }}_t\frac{d{\overline{\omega }}_t}{dt}\frac{S_{t,base}}{S_{sys,base}}=∆{\overline{P}}_{t,m}-∆{\overline{P}}_{v,e}.$$

(7)

The relation $d_w=S_{t,base}/S_{sys,base}$ is the index of replacement of conventional generation by WG, also known as the WG penetration level. Considering that the base power of the system is given by:

$$S_{sys,base}=S_{t,base}+S_{sg,base}.$$

(8)

The term $S_{sg,base}/S_{sys,base}$, in Equation (5), can also be written as a function of $d_w,$ as:

$$1-d_w=\frac{S_{sg,base}}{S_{sys,base}}.$$

(9)

Using the index $d_w$ and replacing the inertia moment $J_{sg}$ and $J_v$ by the inertia constant H_sg and H_v, the model (5)–(7) is rewritten as:

$${2H}_{sg}\left(1-d_w\right){\overline{\omega }}_{sg}\frac{d{\overline{\omega }}_{sg}}{dt}=∆{\overline{P}}_{sg,m}-∆{\overline{P}}_{sg,e},$$

(10)

$$d_{w}2H_v{\overline{\omega }}_v\frac{d{\overline{\omega }}_v}{dt}=∆{\overline{P}}_{v,m}-∆{\overline{P}}_{v,e},$$

(11)

$$d_w{J}_t\frac{{\omega }_{t,base}^2}{S_{t,base}}{\overline{\omega }}_t\frac{d{\overline{\omega }}_t}{dt}=∆{\overline{P}}_{t,m}-∆{\overline{P}}_{v,e}.$$

(12)

The frequencies ${\overline{\omega }}_{sg}$ and ${\overline{\omega }}_v,$during the dominant inertial response (i.e., period T1 in Figure 4), may present slightly different instantaneous values. However, the mean value of these frequencies over a time window of about hundreds of ms can be considered equal in analyses of system frequency dynamics. Thus, considering ${\overline{\omega }}_{sg}={\overline{\omega }}_v=\overline{\omega },$Equations (10) and (11) can be rewritten as:

$${2H}_{sg}\left(1-d_w\right)\overline{\omega }\frac{d\overline{\omega }}{dt}=∆{\overline{P}}_{sg,m}-∆{\overline{P}}_{sg,e},$$

(13)

$$d_{w}2H_v\overline{\omega }\frac{d\overline{\omega }}{dt}=∆{\overline{P}}_{v,m}-∆{\overline{P}}_{v,e}.$$

(14)

Isolating the term $d\overline{\omega }/dt$ in Equation (13) and replacing in Equation (14), results in Equation (15).

$$\frac{∆{\overline{P}}_{v,m}-∆{\overline{P}}_{v,e}}{d_{w}2H_v}=\frac{∆{\overline{P}}_{sg,m}-∆{\overline{P}}_{sg,e}}{{2H}_{sg}\left(1-d_w\right)}.$$

(15)

Considering the system load variation $∆{\overline{P}}_L$, given by Equation (16), it is possible to isolate the term $∆{\overline{P}}_{sg,e}$ and replace it in Equation (15), which results in Equation (17).

$$∆{\overline{P}}_L=∆{\overline{P}}_{sg,e}+∆{\overline{P}}_{v,e}.$$

(16)

$$\frac{∆{\overline{P}}_{v,m}-∆{\overline{P}}_{v,e}}{d_{w}2H_v}=\frac{∆{\overline{P}}_{sg,m}-(∆{\overline{P}}_L-∆{\overline{P}}_{v,e})}{{2H}_{sg}\left(1-d_w\right)}.$$

(17)

The output power of the equivalent WTG can be written from Equation (17) as:

$$∆{\overline{P}}_{v,e}=\frac{{2H}_{sg}\left(1-d_w\right)∆{\overline{P}}_{v,m}+{2H}_v{d}_w\left(∆{\overline{P}}_{sg,m}-∆{\overline{P}}_L\right)}{\left(2H_{sg}\left(1-d_w\right)+{2H}_v{d}_w\right)}.$$

(18)

Replacing Equation (18) in Equation (12), it is possible to describe the dynamics of the wind turbine speed as:

$$d_w{J}_t\frac{{\omega }_{t,base}^2}{S_{t,base}}{\overline{\omega }}_t\frac{d{\overline{\omega }}_t}{dt}=∆{\overline{P}}_{t,m}+\frac{{2H}_v{d}_w\left(∆{\overline{P}}_{sg,m}-∆{\overline{P}}_L\right)-{2H}_{sg}\left(1-d_w\right)∆{\overline{P}}_{v,m}}{\left(2H_{sg}\left(1-d_w\right)+{2H}_v{d}_w\right)}.$$

(19)

The system frequency dynamics can be obtained by isolating $∆{\overline{P}}_{sg,e}$ in Equation (13) and $∆{\overline{P}}_{v,e}$ in Equation (14), and replacing both equations in Equation (16), which results in:

$$\left(2H_{sg}\left(1-d_w\right)+{2H}_v{d}_w\right)\overline{\omega }\frac{d\overline{\omega }}{dt}=∆{\overline{P}}_{sg,m}+{∆\overline{P}}_{v,m}-\left(∆{\overline{P}}_{sg,e}+∆{\overline{P}}_{v,e}\right).$$

(20)

The term $(∆{\overline{P}}_{sg,e}+∆{\overline{P}}_{v,e})$ in Equation (20) can be replaced by $∆{\overline{P}}_L$, presented in Equation (16), which results in Equation (21). It is possible to observe in Equation (21) that the system frequency dynamics depend on $∆{\overline{P}}_{v,m}$, which represents the power reference provided by the MPPT control as a function of wind turbine speed (${\overline{P}}_{v,m}=k_{opt}{\overline{\omega }}_t^3$). This aspect consistently shows a coupling between the dynamics of the wind turbine speed and the system frequency. Therefore, the degradation of the inertial response, caused by such coupling, can be assessed using the proposed model.

$$\left(2H_{sg}\left(1-d_w\right)+{2H}_v{d}_w\right)\overline{\omega }\frac{d\overline{\omega }}{dt}=∆{\overline{P}}_{sg,m}+{∆\overline{P}}_{v,m}-∆{\overline{P}}_L.$$

(21)

The proposed model can represent different kinds of conventional generation units, such as steam units and hydraulic units, different kinds of MPPT control, and different de-loaded approaches for the WTG. These control approaches are related to the power variation$∆{\overline{P}}_{sg,m}$and $∆{\overline{P}}_{v,m.}$ Considering the conventional MPPT control for the WTG, the power variation $∆{\overline{P}}_{v,m}$ is given by:

$$∆{\overline{P}}_{v,m}=k_{opt}\left({\overline{\omega }}_t^3-{\overline{\omega }}_{MPP}^3\right),$$

(22)

where k_opt is the optimal gain of the MPPT control and ${\overline{\omega }}_{MPP}$ is the wind turbine speed at MPP. The conventional generation considered in this work is a steam unit, whose mechanical power is described by the simplified model (23) and (24) [30].

$$\frac{d\overline{x}}{dt}=\frac{1}{T_{CH}}\left(\Delta u-\overline{x}\right),$$

(23)

$$\frac{d\Delta {\overline{P}}_{sg,m}}{dt}=\frac{1}{T_{CH}}\left(\overline{x}-\Delta {\overline{P}}_{sg,m}-F_{HP}{T}_{RH}\frac{d\overline{x}}{dt}\right).$$

(24)

In Equations (23) and (24), $\Delta u$ is the control signal corresponding to the primary frequency control of the synchronous generator, $T_{CH}$ is the reheater time constant, $F_{HP}$ is the fraction of total power generated by the turbine high-pressure section, and $\overline{x}$ is an auxiliary variable. The secondary frequency control was not considered in the proposed study since its impact is negligible on the dynamics of the inertial control and primary frequency control.

The set of nonlinear differential Equations (19), (21), (23) and (24) composes the final simplified model, which describes the system frequency dynamics and dynamics of the equivalent WTG providing inertial response support. This proposed model can be used in different contexts to assess the impact of WG integration on the system frequency dynamics. The model also allows the inclusion of the primary frequency control in WG. However, this work is focused only on the inertial response support.

The mechanical power deviation of the wind turbine, employed in Equation (19), is given by

$$∆{\overline{P}}_{t,m}={\overline{P}}_{t,m}-{\overline{P}}_{t,m,MPP}.$$

(25)

The mechanical power ${\overline{P}}_{t,m}$ is given by the wind turbine aerodynamic model, presented in the Appendix A, and ${\overline{P}}_{t,m,MPP}$ is the mechanical power at the MPP (i.e., ${\overline{P}}_{{\left.t,m,MPP\right|}_{{\overline{\omega }}_{MPP}}})$.

Algorithm 1 describes the steps to perform dynamic analyses with the proposed model. The wind turbine speed (${\overline{\omega }}_t$), system frequency ($\overline{\omega }=\overline{f})$, and mechanical power of the synchronous equivalent generator $(\Delta {\overline{P}}_{sg,m})$ are the state variables of the model. The output power of the equivalent WTG ($∆{\overline{P}}_{v,e}$), wind turbine mechanical power ($∆{\overline{P}}_{t,m}$), and mechanical power of the VSG ($∆{\overline{P}}_{v,m}$), also variables of the model, are determined by Equations (18), (22) and (25). The wind speed, V_w, and systema load variation, $∆{\overline{P}}_L$, are the input variables of the model. The virtual inertia, H_v, and the penetration index of WG, d_w, are the main parameters assigned to the WG. The proposed model can also be represented in the form of block diagram, as illustrated in Figure 8. The antagonistic action of the MPPT ($∆{\overline{P}}_{v,m}$) and inertial controllers, which affects the output power of the WTG, $∆{\overline{P}}_{v,e},$ is more evident in such block diagram.

Algorithm 1: Dynamic analyses with the proposed model

Chose the desired virtual inertia for the WG, Hv, and the penetration index of WG, dw. Chose the wind speed of interest, Vw, and the magnitude of the system load variation, $∆{\overline{P}}_L$. Chose the parameters of the equivalent synchronous generator and equivalent WTG (i.e., wind turbine parameters in (A1)–(A4) presented in the Appendix A). Calculate the initial operating point of the equivalent WTG, ${\overline{\omega }}_t\left(0\right)={\overline{\omega }}_{MPP}$ and ${\overline{P}}_{t,m}\left(0\right)={\overline{P}}_{t,m,MPP}$. Solve Equations (19), (21), (23) and (24) using any generic differential equation solver. Assess the system dynamics by means of the state variables and variables $∆{\overline{P}}_{v,e}$, $∆{\overline{P}}_{v,m},$and $∆{\overline{P}}_{t,m}.$

The MPPT control, nonlinearity of the mechanical power curve of the wind turbine, and virtual inertia are the main intrinsic aspects of WG that affect the dynamics of WTGs and power system frequency. The wind speed and load variation magnitude are extrinsic aspects that also affect the system frequency dynamics. The additional power provided by the inertial control is almost proportional to the virtual inertia and magnitude of the load variation. However, the detrimental impact of the MPPT control is nonlinear as a function of the wind turbine speed deviation. There is also a nonlinearity in the mechanical power of the wind turbine as a function of the wind speed. The nonlinearities of the wind turbine mechanical power and MPPT control action affect the frequency dynamics of power systems. These practical aspects are addressed in the practical analyses presented in the next section.

It is worth remarking that the performed analyses are also valid for type-3 WTG with VSG, since the wind turbines of type-3 and type-4 WTGs with inertial control have similar mechanical dynamics [18,26], as addressed in Section 2.

4. Results and Discussions

The effectiveness and accuracy of the proposed model and the system dynamics are evaluated with nonlinear time-domain simulations. Initially, the performed analyses compare a high-order model of the test system with the simplified proposed model to validate the proposed model. In the sequence, the proposed model is employed to comprehensively assess the typical electromechanical dynamics of power systems with high penetration level of WG with inertial response support. These analyses consider different aspects that affect the system dynamics, such as wind speed, virtual inertia value, control mode of the WTG, and load variation magnitude.

The employed test system is composed of one equivalent wind farm (WF) and one equivalent steam generation unit with a conventional synchronous generator (SG), as shown in Figure 9. Both generation units have nominal power of 900 MVA in the base scenario, which results in a WG penetration of 50%.

The equivalent test system, inspired by the Kundur test system [30], highlights the dynamics of multiple WFs operating simultaneously and the impact of the MPPT and inertial control on the system frequency dynamics of bulk power systems. The equivalent SG represents the system multiple generators, as typically considered in frequency dynamics studies [20]. WFs, composed of multiple WTGs operating in parallel, are typically represented by equivalent models in frequency regulation studies [32]. Such equivalent models can accurately describe the mechanical dynamics and system frequency dynamics [32]. The computational effort and modeling complexity inherent to dynamic analyses can be considerably reduced with the use of accurate equivalent models of WFs [33]. An equivalent model, for example, is employed in [17] to represent a 900 MW WF in a dynamic analysis of a bulk power system.

A 47th-order model of the test system is employed in the performed analyses only to validate the proposed simplified model. The high-order model has also been used in the dynamic assessment presented in [22] and it is concisely described in the Appendix A. Such a model was implemented in the dq reference frame [34] and describes with high accuracy the electromechanical dynamics of the equivalent WTG and SG, as compared with the classical linear models employed in frequency studies of bulk power systems [18].

The system model, represented by ordinary differential equations, was implemented in the Matlab^® software (version 9.13) using programming code. The ODE23tb solver was employed in the programming interface of the Matlab^® to perform the time-domain simulations.

4.1. Accuracy Validation of the Proposed Simplified Model

The accuracy of the proposed simplified model is evaluated in this section considering the 47th-order model previously mentioned. The nonlinear simulations consider the occurrence of a load step in the test system at t = 1 s. The WF operates with H_v = 10 p.u.s and wind speed of 9 m/s, which results in an MPP speed of 0.9 p.u. and active power of 0.154 p.u. in the system base power. The equivalent conventional generator operates with an initial active power of 0.2 p.u. and H_sg = 5.6 p.u.s. The simplified model is evaluated in two scenarios: (1) small load variation of 1%; (2) large load variation of 3%.

Figure 10 presents the electromechanical dynamics of the test system considering the simplified model and the 47th-order model for scenarios 1 and 2. As illustrated in Figure 10a, the simplified model describes the system frequency with high accuracy in the two scenarios. The same high accuracy is observed in the wind turbine speed shown in Figure 10b. The output power of the equivalent WTG (${\overline{P}}_{v,\mathrm{e}}$), shown in Figure 10c, presents a relevant error during the first 3 s of the dynamic response. Such an error occurs because the simplified model does not represent the electromagnetic dynamics of the WTG, since the model is predominantly composed of mechanical equations. Thus, the proposed simplified model is not suitable to access electromagnetic dynamics of WTGs.

4.2. Impact of the Inertial Controller on the Frequency Nadir and Frequency Oscillations

The inertial controller of the WG changes the dynamic behavior of the system frequency, affecting the RoCoF and frequency nadir. In addition, the simultaneous action of the inertial and MPPT controllers causes frequency oscillations that may result in a relevant second frequency dip after load variations. Such a phenomenon is comprehensively evaluated in this section, based on the analysis of the system frequency dynamics and dynamics of the equivalent WTG. The analysis is performed with the proposed simplified model considering different wind speeds and virtual inertia values.

Figure 11a shows the system frequency dynamics for a wind speed of 9 m/s and virtual inertias of 5, 10, and 15 p.u.s for the equivalent WTG. The conventional generator operates with an initial active power of 0.2 p.u. and H_sg = 5.6 p.u.s. This analysis shows that, unlike the behavior of conventional generation, the significant increase in the virtual inertia resulted in a negligible improvement of the frequency nadir and a considerable increase in the amplitude of the frequency oscillations.

Figure 11b shows the system frequency dynamics for the wind speed of 12 m/s and virtual inertias of 5 p.u.s, 10 p.u.s, and 15 p.u.s. It is possible to observe that the increase in the virtual inertia of the WG also causes an increase in the amplitude of the frequency oscillations. However, unlike the results presented in Figure 11a, the frequency nadir slightly decreases with the increase of the virtual inertia. The increase in the inertia in the conventional synchronous generation improves the frequency nadir, while the increase in the virtual inertia of the WG has degraded the frequency nadir. This detrimental dynamic behavior is caused by the nonlinearity of the mechanical power curve of the wind turbine (${\overline{P}}_{t,m}$) and the action of the MPPT controller ($∆{\overline{P}}_{v,m}=k_{opt}{\overline{\omega }}_t^3$). The powers ${\overline{P}}_{t,m}$ and ${\overline{P}}_{v,m}$ present higher variation for higher wind speed. As shown in Equation (21), ${∆\overline{P}}_{t,m}$ and ${∆\overline{P}}_{v,m}$ affect the system frequency dynamics and degrade the frequency nadir.

The increase in the frequency oscillations is caused by the increase in the inertial contribution of the equivalent WTG, as can be seen in the wind turbine speed and WTG output power shown in Figure 12. As shown in Figure 12a, the higher the inertial contribution, the higher the wind turbine speed deviation and MPPT control action ($∆{\overline{P}}_{v,m}=k_{opt}\left({\overline{\omega }}_t^3-{\overline{\omega }}_{MPP}^3\right)$). The MPPT control action reduces the WTG output power to restore the wind turbine speed, which reduces the minimum output power, as can be observed in Figure 12b.

The MPPT control action starts to accelerate the wind turbine towards the MPP speed at t = 4.35 s, before the frequency nadir, which occurs at t = 6.55 s for H_v = 10 p.u.s and V_w = 12 m/s. Up to the frequency nadir, at t = 6.55 s, the inertial and the MPPT controllers have antagonistic actions, where the inertial controller provides decelerating energy to the wind turbine and the MPPT controller provides accelerating energy. However, during the positive rate of change of the system frequency, after t = 6.55 s, both controllers present synergetic actions, accelerating simultaneously the wind turbine, which causes an overshoot in the wind turbine speed. The higher the virtual inertia, the higher the overshoot of the wind turbine speed. This overshoot results in oscillations in the wind turbine speed and, due to the coupling between the wind turbine speed and system frequency, such oscillations propagate to the system frequency.

Figure 13 shows the system frequency nadir as a function of wind speed and virtual inertia. The increase in the virtual inertia increases the frequency nadir for wind speeds below 10.4 m/s and reduces the frequency nadir for wind speeds above 10.4 m/s. Such behavior is caused by the nonlinearities of the wind turbine mechanical power as a function of the wind speed. Although there is an increase in the frequency nadir for some operational conditions, this increase is not significant. As the characteristic of the wind turbine mechanical power (${\overline{P}}_{t,m}$) plays an essential role in this behavior, different mechanical models of wind turbines may result in different behaviors. The wind turbine model presented in [35], which is different from the model adopted in this work [36], for example, does not increase the frequency nadir with the increase of the virtual inertia over the entire operational range of the wind turbine.

4.3. Impact of the Control Mode of the WG on the System Dynamics

The impact of the WG control mode on the system frequency dynamics is evaluated in this section considering different control modes. Three control modes typically employed in commercial WTG are evaluated: (1) MPPT control; (2) de-loaded control with de-loaded power curve; (3) de-loaded control with fixed power reference. The equivalent WTG operates with H_v = 10 p.u.s and a wind speed of 9 m/s in the three scenarios.

The de-loaded operation with a de-loaded power curve is based on Equation (22), where the optimal gain is replaced by a sub-optimal gain as a function of a desired de-loading factor [37]. On the other hand, the de-loaded operation with a fixed power reference uses a constant power reference for the WTG (i.e.,${\overline{P}}_{v,m}={\overline{P}}_{ref,const}$, which implies that $∆{\overline{P}}_{v,m}=0$). This fixed power reference for the WTG, also employed in [17], decouples the wind turbine speed dynamics from the system frequency dynamics. A high de-loaded margin of 50% has been employed to highlight the difference between the dynamics of the MPPT and de-loaded control modes.

Figure 14 presents the system frequency for the three control modes, where scenarios 1, 2, and 3 resulted in frequency nadirs of 59.873 Hz, 59.882 Hz, and 59.898 Hz, respectively. The improvement of the frequency nadir in scenario 2 occurs because the MPPT gain (k_opt) is reduced to provide a de-loaded margin, reducing the amplitude of the coupling term $∆{\overline{P}}_{v,m}$ in Equation (21). In scenario 3, the frequency nadir improvement occurs due to the decoupling between the dynamics of the wind turbine speed and system frequency, caused by $∆{\overline{P}}_{v,m}=0$.

4.4. Impact of the WG Penetration Level on System Dynamics

The analysis presented in this subsection comprehensively shows the impact of the WG penetration level on the dynamics of the system frequency and equivalent WTG. WG penetrations of 20%, 40%, and 60% are considered in the analysis. The three scenarios consider wind speed of 9 m/s and virtual inertia of 10 p.u.s for the equivalent WTG operating with classical MPPT control.

The increase in the WG penetration level has significantly degraded the frequency nadir, steady-state frequency error, and electromechanical oscillations, as shown in Figure 15a. The reduction in the frequency nadir and the increase in the steady-state frequency error occur due to the absence of primary frequency control in the WG. The oscillations have been caused by the interaction between the MPPT and inertial controllers. Therefore, inertial control is necessary but not sufficient to provide a suitable performance to the system. The minimum speed of the wind turbine, shown in Figure 15b, reduces with the increase of the WG insertion level due to the increase in the frequency dip of the system.

5. Conclusions

This work proposes a simplified model to assess the electromechanical dynamics of power systems with WG providing inertial response support. The proposed model allows the performance of simple and fast analyses that can provide an in-depth insight into the dynamics of the system frequency and equivalent WTG for different penetration levels of WG, wind speeds, virtual inertia values, control approaches, and load disturbances. The simplified model incorporates the antagonistic interaction between the MPPT and inertial controllers, which causes a relevant detrimental impact on the system frequency. The performed validation has shown that the proposed model, composed of four differential equations, describes the power system frequency dynamics with high accuracy.

As a second contribution, the work presents a comprehensive analysis of the main intrinsic and extrinsic aspects of WG that significantly affect the electromechanical dynamics of power systems with WG providing inertial response support. The impact of wind speed, value of virtual inertia, WG penetration level, WTG control mode, and load variation magnitude on the system dynamics is comprehensively assessed. The results have shown that several aspects degrade the RoCoF, frequency nadir, and overall dynamic response of the system frequency.

The interaction between the MPPT and inertial controller couples the dynamics of the wind turbine speed and system frequency, degrading the frequency nadir and frequency oscillations. A significant increase in the virtual inertia results in a negligible improvement of the frequency nadir and a considerable increase in the amplitude of the frequency oscillations. Such detrimental behavior does not occur in conventional SGs, in which the increase in natural inertia significantly improves the frequency nadir. The higher the virtual inertia, the higher the amplitude of the frequency and wind turbine speed oscillations. The higher the load variation amplitude, the higher the amplitude of the frequency oscillations, which may result in a relevant second frequency dip. The increase in the wind speed, depending on the wind turbine aerodynamic characteristics, may slightly degrade the frequency dynamics, due to the increase in the nonlinearity of the mechanical power of the wind turbine. All addressed detrimental dynamics are mainly caused by the interactions of the MPPT and inertial controller and nonlinearities of the mechanical power curve of the wind turbine. In comparison with the MPPT control mode, the de-loaded control mode of WTGs with fixed power reference improves the frequency nadir and the frequency oscillations, due to the decoupling between the dynamics of the wind turbine speed and system frequency. The performed analyses have shown that inertial control is necessary but not sufficient to provide a suitable performance to power systems, since the increase in the penetration level of WG with inertial response support may significantly degrade the frequency nadir, steady-state frequency error, and frequency oscillations.

The performed analyses and the proposed model are useful to formulate new control approaches and to assist power system operators in the planning and operation of power systems. They also can be employed by engineers to design preventive and corrective actions to mitigate the negative detrimental interactions between the conventional MPPT and inertial controller, improving the electromechanical oscillations and frequency nadir, as well as extending the inertial response support of WG.

The incorporation of an equivalent photovoltaic generation in the simplified model is a direction of future prospective research, with emphasis on the impact of this kind of generation on the inertial response of power systems and the electromechanical dynamics of WG. Wind and photovoltaic generation without inertial response support will be incorporated in the equivalent model in such prospective research, considering a large set of operational scenarios in a Monte Carlo approach. The determination of complementary regression models from the proposed simplified model is also one of the future directions of this research.