1 Introduction

1.1 Background

The depletion of fossil energy sources and their detrimental environmental impact have led to an increasing demand for renewable energy sources [1]. This growing demand is expected to drive the dominance of renewable energy in power systems in the future. Figure 1 provides an overview of the installed capacity and the change in installed capacity of renewable energy sources over a 1-year period (2021–2020) [2]. Solar energy has emerged as the most favored source of electricity generation among renewables in recent years, as depicted in the figure. Globally, the installed capacity of solar photovoltaic (PV) reached 767 GW in 2020 and further increased to 942 GW in 2021. This substantial growth highlights the increasing prominence of solar-based electricity generation in grids worldwide.

Fig. 1
figure 1

Installed capacity a difference between 2021 and 2020 b in 2021

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Solar energy differs from conventional energy sources in terms of electricity generation. While synchronous generators utilize conventional energy sources to generate alternating current (AC) electricity, PV panels generate direct current (DC) electricity from solar energy. As a result, while a synchronous generator plant can directly connect to the grid, a PV panel plant requires an inverter to convert DC electricity into AC electricity for grid connection. Moreover, unlike conventional plants, PV plants (PVPs) lack a rotating structure.

PVPs are commonly operated using maximum power point tracking (MPPT) techniques [3, 4] which are not used to provide frequency support to the grid. The absence of a rotating structure in PVPs leads to a reduction in the total moment of inertia within the grid [5]. This decrease in total moment of inertia poses a significant challenge for grid stability [6]. In other words, PVPs lack inertial response [7, 8], resulting in a reduced overall frequency response of the grid [9, 10]. To address this issue, the concept of a virtual moment of inertia can be employed in the grid to mitigate the lack of frequency response in the presence of PVPs. This virtual moment of inertia acts as a substitute to compensate for the absence of inertial response from PVPs and helps maintain grid stability.

1.2 Literature Review

In the literature, studies have been conducted for grids with high PVP penetration. Some of these studies cover frequency support by PVP. Furthermore, the literature shows the effect of PVPs on the frequency at various penetration levels. Some of these studies concentrate solely on frequency stability, while others briefly examine the effect on frequency. The outstanding studies on the effect of PVPs with different penetration levels are summarized below.

In [11], the study investigates the effectiveness of PVPs in offering diverse forms of frequency support. The researchers employ a power reserve and an offline maximum power point tracking (MPPT) method. The paper presents the results of different frequency support techniques across varying levels of PVP penetration. Additionally, the study introduces a technique to sustain power reserve, which significantly reduces the frequency bottom point and the rate of frequency change. In [12], the study focuses on examining the impact of PVPs on the electrical grid in northern Chile. The research investigates how power converters in PVPs can function as virtual synchronous generators through the implementation of a specific controller. A comparison is made between the impact of these plants and PVPs utilizing conventional converter controllers. The suggested models and controllers are evaluated and verified using an actual PVP connected to the electrical grid in northern Chile. The study's results indicate that PVPs with virtual synchronous power converters can effectively regulate frequency and power oscillations. In [13], the focus is on PVPs that incorporate two distinct power converters. The first power converter utilizes synchronous power controllers (SPCs), enabling harmonious operation with the grid. The second power converter employs conventional power converter controllers. The study compares the performance of PVPs with SPCs to those with conventional power converter controllers. The comparison involves the inclusion of PVPs in frequency and voltage regulation. The study's findings demonstrate that PVPs equipped with SPCs can effectively reduce frequency deviations and oscillations. In [14], the study focuses on examining various loading scenarios for high penetration level PVPs using the WECC converter model, which represents a large-scale dynamic model of PVPs. Within these loading scenarios, different PVPs are substituted with various synchronous generators, with the active power value being kept constant. The study finds that critical synchronous generators can maintain system stability in these loading scenarios. Additionally, the study tests different voltage controls provided by the WECC model for PVPs. It is concluded that PVPs can have a positive impact on voltage stability. [15] investigates the influence of a high penetration level of PV systems on the Texas 2000 grid. The study reveals that the voltage and frequency stability of the grid can be negatively impacted by significant PV penetration. This is attributed to the reduction in the overall moment of inertia and reactive power support of the grid resulting from the substitution of conventional power plants with PV systems. Therefore, a strategy to replace PVPs with conventional power plants with low moments of inertia has been proposed. In [16], the study focuses on the replacement of certain conventional generation sources in the power system with large-scale and rooftop PV systems. The research utilizes a test system that represents a section of the western US interconnection. It is determined that rooftop PV systems have a lesser impact on stability during failure conditions compared to large-scale PV power plants. Furthermore, the study thoroughly examines the positive and negative effects of both large-scale and rooftop PV systems on the power system, providing a detailed analysis of their implications. In [17], three distinct scenarios for PV systems with different dynamic models are examined on the Ontario grid. These scenarios primarily involve distributed PV systems (as rooftop) and centralized PV systems (as large-scale). Furthermore, the study compares large-scale PV power plants with and without voltage regulation capability. According to the findings, distributed PV systems demonstrate significant advantages in terms of stability compared to large-scale PV power plants. The study also indicates that the impact on stability is similar for PV power plants with and without voltage regulation capabilities. In [18], the impact of high penetration PVPs on the US Eastern Interconnection is analyzed. The study focuses on analyzing these impacts in terms of the oscillation modes within the grid. The findings indicate that as the penetration level of PVPs increases, the grid's ability to withstand and dampen oscillations decreases. Additionally, the study reveals that changing the control method of PVPs can result in a change in the mode shape. Furthermore, it has been discovered that improper adjustment of certain parameters in the control method can give rise to new oscillation modes within the system. In [9], high PV penetration scenarios are examined in three major US grids: the Western Electricity Coordinating Council, the Eastern Interconnection, and the Electric Reliability Council of Texas. The study focuses on preserving the frequency response of the grid under high PV penetration levels. The research findings are presented as recommendations, which include: keeping a portion of synchronous generation units active to provide stability and support to the grid; adjusting the governor settings of synchronous generators to enhance their response to frequency deviations; implementing under-frequency relaying mechanisms to detect and mitigate frequency drops; and allowing converter-based resources, such as PV systems, to participate in providing frequency support and regulation. These recommendations aim to maintain grid stability and ensure reliable frequency response in the presence of high PV penetration. In [19], the study focuses on the integration of large-scale and rooftop PV systems as replacements for conventional generation sources in the power system. The research analyzes the impact of PV penetration levels on critical modes and transient response. It is observed that as the PV penetration level increases, the damping ratio decreases in critical modes. Additionally, an eigenvalue sensitivity analysis is conducted for disabled synchronous generators, leading to the conclusion that critical generators should provide support to the grid. This emphasizes the importance of ensuring sufficient support from synchronous generators for maintaining grid stability in the presence of high PV penetration. In [20], the study explores the influence of large-scale PVPs equipped with a mandatory frequency-sensitive mode and a high inertial response on the grid frequency. The research utilizes the IEEE12 bus power system (IEEE12-BPS) and models changes in generation and consumption based on the Danish winter load profile. This modeling approach enables an investigation of both short-term and long-term effects on the grid frequency. When examining the short-term impact, the study considers the combined and separate effects of the frequency-sensitive mode and inertial response in PVPs. The findings demonstrate the positive contribution of PVPs with power reserves and frequency support in enhancing frequency stability, particularly at high penetration levels. [21] presents a master–slave-based power control method for PVPs to participate in frequency regulation by providing power reserve control. The presented control method focuses on the transition between MPPT and power reserve. A damping and inertia control scheme is also added to PVP. The proposed control method is compared with the MPPT method for different cases. These cases (three-phase fault and load disturbance) are different power system models (three-area power system and IEEE11-BPS) and different irradiance values for PVP. In [22], a two-level active power control method is proposed for PVP. In this method, which is proposed to reduce electromechanical oscillations in the power system, PVP has a power reserve. In the method, according to the frequency deviation signal, the fuzzy logic controller determines a power set point for PVP. In this method, an inner control mechanism is used (in PVP) that tracks the voltage so that PVP follows the set power set point. The proposed method is tested in IEEE11-BPS by modeling a three-phase short-circuit fault. [23] considers the impact of PVPs at high penetration levels on low-frequency oscillations in power systems. In [23], a time-delay feedback-control-based wide-area damping controller design is proposed to dampen low-frequency oscillations for PVP. The performance of the proposed controller is tested on IEEE11-BPS and IEEE68-BPS. The tests considered in these systems include load change, PVP generation change, three-phase short-circuit fault, and permanent line outage. The performance of the proposed controller is verified by testing it through real-time simulation using OPAL-RT.

1.3 Motivation and Contributions

The widespread use of PVPs in power systems is the motivation for this study. As the penetration level of PVPs in power systems increases, their impact on frequency stability increases significantly. Investigating these effects for PVPs is among the main research topics in the literature. In the literature, the impact of PVPs at different penetration levels without frequency support is usually analyzed [9, 14,15,16,17,18,19, 23]. Furthermore, the majority of these analyses are limited to a single event type in the power system [9, 16,17,18,19]. A single event does not provide a complete picture of the effect of PVPs on frequency.

Because PVPs have a significant impact on frequency at high penetration, it is necessary to investigate the provision of frequency support by PVPs. Therefore, there have been studies on frequency-supporting PVPs published in the literature [11,12,13, 20,21,22]. The PVPs used in [11,12,13, 20,21,22] have a reserve that allows them to increase their generation when the frequency decreases. This prevents maximum efficiency from the PVP. For a power plant with no generation costs, this is economically very harmful. Also, two shortcomings stand out in these studies. The first of these is generally not investigating the frequency support of PVPs for different events in power systems [11, 20, 22]. The second of these is that the depth and impact of this support have not been investigated in detail in studies of PVPs with frequency support.

The effect of voltage on frequency stability is not considered in the literature on PVPs mentioned above. The integration of the PVP into the power system at different penetration levels has an effect on the voltage levels at the buses in both steady-state and transient events. Because loads in power systems have constant impedance or constant current load models, they are affected (except for constant power models) by voltage. Because the PVP's penetration level affects the voltage in the power system, it also affects the load. This is important because frequency stability is the balance between generation and consumption.

Another aspect that is rarely discussed in the literature is the use of time-domain parameters in the analysis. The use of time analysis parameters allows the study's findings to be more meaningful. It also allows the results of studies in the literature to be compared with each other.

In this study, IEEE12 bus power system is modeled, and PVP is integrated into this model. PVP, which provides frequency support (only when the frequency increases) at different depths, is used at different penetration levels. This results in nine different scenarios. In the study, load change, load outage, generator outage, and three-phase short-circuit events are realized. These events are considered separately. The rotor speed of generator G2, which is located at the topographic center of the power system, is considered to examine frequency stability. The results (rotor speed transient responses) are compared using time-domain parameters. The time-domain parameters used are settling time (ST), overshoot (OS), and undershoot (US). The analyses and modeling in the study are carried out using the DigSilent Powerfactory simulation program.

This paper addresses several analyses to address the shortcomings of the previous studies in the literature. Table 1 compares the distinguishing points of this study with the existing literature. In addition, the main contributions of this study can be summarized as follows.

  • In studies in the literature, PVPs usually have a reserve for frequency support. This is not economically profitable. Therefore, this paper suggests a PV power plant strategy that will provide frequency support only when the frequency increases. The suggested strategy is based on maximizing the efficiency of PVPs while at the same time improving the frequency stability. For this strategy, many scenarios and events are considered, and a detailed analysis is presented.

  • In the literature, the effect of the PVP model on frequency is usually considered for a single event, so the response of the PVP model to other events is not known. In this paper, all main events (generator outage, load outage, and short-circuit fault) in power systems are considered, and the transient response of the PVP model to these events is presented. PVP transient responses are compared using time-domain parameters.

  • The effect of the frequency support depth of PVP has not been considered in the literature. This study investigates the frequency support depth of a PVP at different penetration levels. In this way, the effect of frequency support depth at low and high penetration levels is presented.

  • The load model is usually not considered in frequency stability studies in the literature. In this study, the relationship between voltage changes and frequency is analyzed in detail based on load models.

Table 1 Comparison of this paper with the literature
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The second section of the study focuses on the swing equations, which serve as the foundation for frequency stability. In the third chapter, the power systems used for the analysis are discussed. In the fourth section, a detailed description is provided regarding the modeling of large-scale PVP systems. The fifth section sheds light on the various scenarios that have been taken into consideration. The sixth section presents the results obtained from the frequency stability analysis. Finally, in the seventh section, the study concludes by emphasizing the key findings and conclusions.

2 Swing Equation

The swing equation is very important to understand the change in rotor speed (frequency). For this reason, the swing equation is discussed. The relationship between steady state and transient state can be expressed for rotor speed by the swing equation. The relative position of the rotor axis is fixed to the position of the magnetic field axis at a steady state [24]. When a disturbance occurs (transient state), there is a difference between these two positions [24]. The rotor can either accelerate or decelerate in this case. In other words, the synchronous generator will affect the grid frequency in both situations.

In steady-state operation where losses are neglected, the following equation is obtained at synchronous speed (\({\omega }_{{\text{sm}}}\)).

$$ T_{{\text{m}}} = T_{{\text{e}}} $$
(1)

where \({T}_{{\text{m}}}\) is mechanical torque, and \({T}_{{\text{e}}}\) is electromagnetic torque. Equation 2 is obtained if there is a difference between mechanical and electromagnetic torque.

$${T}_{{\text{a}}}={T}_{{\text{m}}}-{T}_{{\text{e}}}$$
(2)

where \({T}_{{\text{a}}}\) is the acceleration torque. Equation 3 can be written if frictional and damping torques are neglected.

$$J\frac{{d}^{2}{\theta }_{{\text{m}}}}{{\text{d}}{t}^{2}}={T}_{{\text{m}}}-{T}_{{\text{e}}}$$
(3)

where J is the combined moment of inertia of the generator and prime mover.

$${\theta }_{{\text{m}}}={\omega }_{{\text{sm}}}t+{\delta }_{{\text{m}}}$$
(4)

where \({\theta }_{{\text{m}}}\) is the angular displacement of the rotor with respect to the stator axis. The rotor angular velocity can be obtained by taking the derivative of the angular displacement.

$${\omega }_{{\text{m}}}=\frac{{\text{d}}{\theta }_{{\text{m}}}}{{\text{d}}t}={\omega }_{{\text{sm}}}+\frac{d{\delta }_{{\text{m}}}}{{\text{d}}t}$$
(5)

where \({\delta }_{{\text{m}}}\) is the rotor position before the disturbance.

The acceleration of the rotor is given below.

$$\frac{{d}^{2}{\theta }_{{\text{m}}}}{{\text{d}}{t}^{2}}=\frac{{d}^{2}{\delta }_{{\text{m}}}}{{\text{d}}{t}^{2}}$$
(6)

Using Eqs. 3 and 6, Eq. 7 is obtained.

$$J\frac{{d}^{2}{\delta }_{{\text{m}}}}{{\text{d}}{t}^{2}}={T}_{{\text{m}}}-{T}_{{\text{e}}}$$
(7)

Multiplying both sides of Eq. 7 by \({\omega }_{{\text{m}}}\) gives Eq. 8.

$$J{\omega }_{{\text{m}}}\frac{{d}^{2}{\delta }_{{\text{m}}}}{{\text{d}}{t}^{2}}={\omega }_{{\text{m}}}{T}_{{\text{m}}}-{\omega }_{{\text{m}}}{T}_{{\text{e}}}$$
(8)

The angular velocity multiplied by the torque gives the power. The difference in power (mechanical and electromagnetic) can be presented in Eq. 9. This equation is called the swing equation.

$$J{\omega }_{{\text{m}}}\frac{{d}^{2}{\delta }_{{\text{m}}}}{{\text{d}}{t}^{2}}={P}_{{\text{m}}}-{P}_{{\text{e}}}$$
(9)

where \({P}_{{\text{m}}}\) and \({P}_{{\text{e}}}\) are mechanical and electromagnetic power, respectively. As shown in Eq. 9, the rotor speed decreases as the active power demand increases (\({P}_{{\text{e}}}>{P}_{{\text{m}}}\)). When the active power demand decreases (\({P}_{{\text{e}}}<{P}_{{\text{m}}}\)), the rotor speed increases.

3 Modified IEEE12 Bus Power System Modeling

In this study, IEEE12-BPS is used [25]. In the power system, G1, G2, G3, and G4 represent synchronous generators; L1, L2, L3, L4, L5, and L6 represent loads; L7 represents shunt reactance; and C1 and C2 represent shunt capacitances. In addition, IEEE12-BPS is modified by connecting PVP to the power system (at Bus 4). Governor, automatic voltage regulator (AVR), and power system stabilizer (PSS) are used in all generators. The modified IEEE12-BPS is given in Fig. 2.

Fig. 2
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The modified IEEE12-BPS

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The initial power and voltage values for the power system shown in Fig. 2 are given in Table 2.

Table 2 Bus data of the modified IEEE12-BPS
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In Table 2, Bus 9 is defined as a slack bus, and Buses 10, 11, 12, and X are defined as generator buses. Buses 1, 2, 3, 4, 5, 6, and 7 are defined as load buses. Also, shunt capacitors have a negative sign, and shunt reactance has a positive sign in Table 2.

In this study, the polynomial load model is used. This model is also known as the ZIP load model. The mathematical expression of this model is given below [26].

$$P={P}_{0}[{Z}_{p}{\overline{V} }^{2}+{I}_{p}{\overline{V} }^{1}+{P}_{p}{\overline{V} }^{0}]$$
(10)
$$Q={Q}_{0}[{Z}_{q}{\overline{V} }^{2}+{I}_{q}{\overline{V} }^{1}+{P}_{q}{\overline{V} }^{0}]$$
(11)

where \({P}_{0}\) and \({Q}_{0}\) are the initial values for active and reactive components of the load, respectively. V is the voltage, and the exponent (0, 1, 2) of the voltage determines the relationship of the load (P/Q) to the voltage. \({Z}_{{\text{p}}}\), \({I}_{{\text{p}}}\), and \({P}_{{\text{p}}}\) for active component (P) and \({Z}_{{\text{q}}}\), \({I}_{{\text{q}}}\), and \({P}_{{\text{q}}}\) for reactive component (Q) are the constants of the ZIP load model. The ZIP load model consists of a combination of three load types. These load types are constant impedance (\({Z}_{{\text{p}}}/{Z}_{{\text{q}}}\)), constant current (\({I}_{{\text{p}}}/{I}_{{\text{q}}}\)), and constant power (\({P}_{{\text{p}}}/{P}_{{\text{q}}}\)). The active/reactive power value of “\({Z}_{{\text{p}}}/{Z}_{{\text{q}}}\)” and “\({I}_{{\text{p}}}/{I}_{{\text{q}}}\)” load types depends on voltage. However, “\({P}_{{\text{p}}}/{P}_{{\text{q}}}\)” load type does not depend on voltage. The sum of the voltage coefficients (“\({Z}_{{\text{p}}}\), \({I}_{{\text{p}}}\), \({P}_{{\text{p}}}\)”, “\({Z}_{{\text{q}}}\), \({I}_{{\text{q}}}\), \({P}_{{\text{q}}}\)”) must be equal to one \(({Z}_{{\text{p}}}+{I}_{{\text{p}}}+{P}_{{\text{p}}}={Z}_{{\text{q}}}+{I}_{{\text{q}}}+{P}_{{\text{q}}}=1)\). In this way, a load can be expressed proportionally to different types of loads. To clearly understand the effect of voltage on frequency stability, the load type is chosen as constant impedance (\({Z}_{{\text{p}}}/{Z}_{{\text{q}}}=1, {I}_{{\text{p}}}/{I}_{{\text{q}}}=0, {P}_{{\text{p}}}/{P}_{{\text{q}}}=0)\).

In this study, the penetration levels of PVP are calculated based on the total active power (1850 MW) of the loads, neglecting the losses. The penetration levels of PVP for 300, 500, and 700 MW are 16.22% (300 MW/1850 MW), 27.03%, and 37.84%, respectively. As the penetration level of PVP increases (16.22%, 27.03%, and 37.84%), the active power generated by the G1 synchronous generator (at the slack bus) is 516.5, 307.3, and 110 MW, respectively. The scheduled powers of the rest of the generators remain constant.

4 Large-Scale PVP Modeling

PVPs used in the grids can be classified according to their installed power. PVPs with an installed power higher than 20 MW and connected to a voltage higher than 66 kV can be considered large-scale PVPs [27]. For this reason, PVP used in this study is considered large-scale. Control modules designed by Western Electricity Coordinating Council (WECC) are used to model the large-scale PVP. WECC control modules used in PVPs are renewable energy generator convertor (REGC_A), renewable energy electrical control (REEC_B), and renewable energy plant control (REPC_A) [28]. REGC_A control module generates current according to current commands from REEC_B module [29]. The REEC_B module is used for local control, while REPC_A module is used for plant control [30]. The parameter values of the WECC modules in this study are the same as those in the model validation guideline for PVPs [30]. Detailed information about these modules can be found in [28,29,30].

In this study, PVPs with or without frequency support are discussed. To provide frequency support, REPC_A module is used. The active power block used for frequency support of REPC_A module is given in Fig. 3 [30].

Fig. 3
figure 3

Active power block of REPC_A module in WECC [30]

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The active power generation of PVP can be changed according to the frequency by using the “Freq_flag” switch [30, 31]. In other words, the synthetic governor response can be enabled (1) or disabled (0) with this switch. “Freg” and “Freg-ref” represent frequency deviation and initial frequency deviation (0), respectively. REPC_A active power block operates when there is a deviation (“Freq”- “Freg-ref”) in frequency. The over-frequency and under-frequency dead bands are represented by fdbd1 and fdbd2, respectively. The “ddn (down-regulation droop)” and “dup (up-regulation droop)” parameters are used for the depth of the active power response of PVP to the frequency increase and decrease in the power system, respectively [30, 31]. “Pbranch” and “Plant-pref” represent branch active power flow and initial branch active power flow, respectively. “femax” and “femin” represent the maximum and minimum power error in droop regulators, respectively. "Pmax" and "Pmin" are the maximum and minimum plant active power commands, respectively. “Kpg” and “Kig” are proportional and integral gains for the droop regulator. In other words, they are controller parameters for the droop regulator. “Tlag” is the lag time constant for the active power command (“Pref”).

Due to PVP generating active power when the frequency decreases, it must allocate some of its generation power as a reserve. Since solar energy is free as a source of generation, it is preferred to use PVPs at maximum power generation. Therefore, PVP is used for only responding to the frequency increase. Parameters are set as in Table 3 to control PVP as described.

Table 3 Active power control methods for PVPs with WECC structure [30]
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In the case of no governor response (NGR), PVP does not respond to frequency [30]. In the case of governor response to down-regulation (GRDR), PVP responds to the frequency increase by reducing its active power [30].

5 Scenarios

In this study, four different events are analyzed. These are load change, load outage, generator outage, and three-phase short-circuit fault. Nine different scenarios are considered for these events. These scenarios are defined based on different frequency support depths and penetration levels. The scenarios are given in Table 4.

Table 4 Scenarios for different frequency support depths and penetration levels of PVP
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For Scenarios I, II, and III, the depth of PV frequency support is 0 pu, 15 pu, and 30 pu, respectively. The PVP's output power for these three scenarios is 300 MW. The other scenarios in the table are designed similarly to these first three scenarios.

The events considered for scenarios include outages of L4 (350 MW) and G4 (350 MW) units, a load change of L6 (50% decrease and increase), and a three-phase short-circuit fault at Bus 6. Outage and fault events are started at \(t=1\) second and ended at \(t=1.2\) second. The load change in the scenario involves a 50% decrease in \(t=1\) second, followed by a 50% increase (returns to initial load value) in \(t=30\) second. The rotor speed of G2 generator is analyzed since it is at the topographic center of the power system. To evaluate the frequency stability, the load imbalance, which is dependent on voltage changes, in the system is considered. Voltage analysis provides a better understanding of frequency stability analysis. This proves the importance of voltage analysis for frequency stability analysis.

6 Frequency Stability Analysis

6.1 Load Change

In the proposed strategy, the entire output power of the PVP is utilized. In other words, the output power of the PVP is not constrained by a reserve. In this strategy, the PVP provides frequency support when the consumption (load) increases in the imbalance between generation and consumption in the system. This sub-heading discusses load change and presents the suggested strategy. The L6 load (250 MW) is increased by 50% in \(t=1\) second and then decreased by 50% in \(t=30\) second to return it to its initial value. The transient response of the G2 rotor speed for load change is given in Fig. 4.

Fig. 4
figure 4

G2 rotor speed response at L6

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Figure 4 shows that PVP frequency support is active when the load decreases in accordance with the suggested strategy. This figure also shows that as the penetration level increases, deviations increase in the lack of frequency support. In order to examine the results of the load change in more detail, Fig. 5 is obtained by zooming into Fig. 4.

Fig. 5
figure 5

G2 rotor speed response for load change a + 50% and b − 50%

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Figure 5a shows that as the penetration level increases, the deviations in G2 rotor speed in the lack of PVP frequency support increase similarly for the same penetration level. Figure 5b shows the importance of the frequency support depth of the PVP at a high penetration level. Comparing all scenarios, it is observed that the highest overshoot (1.0069 pu) occurs (Scenario VII) at the highest penetration level in the lack of frequency support. Overshoot is best dampened (1.0031 pu) in the scenario (Scenario IX) with the highest frequency support and penetration level. G2 rotor speed before load change and maximum deviation (US/OS) values and rates after load change are given in Table 5.

Table 5 Time-domain analysis results for rotor speed of G2 generator during L6 load change
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PVP active power outputs during load change are given in Fig. 6. Figure 6 shows that PVP outputs are inactive when the load change is positive (+ 50%) and active when the load change is negative (− 50%).

Fig. 6
figure 6

PVP active power response at L6 load change

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6.2 Load Outage

Figure 7 shows the change in rotor speed of G2 generator when L4 load is experiencing a momentary outage. The outage creates an imbalance between generation and consumption. Due to the reduced consumption, the rotor speed of G2 generator increased in all scenarios during the disturbance.

Fig. 7
figure 7

G2 rotor speed response at L4 load outage

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Figure 7 shows that OS of G2 generator rotor speed increases (both during and after the disturbance) when the penetration level of PVP in the power grid increases and the frequency support decreases. After the disturbance, it is clear that the frequency support of PVP at the high penetration level has a very high effect on the rotor speed of G2 generator. To explain the increase in rotor speed of G2 generator, the effect of total load and load changes needs to be looked at. The change in voltage will also change the load demands. When the voltage increases, the load demand will increase, and when the voltage decreases, the load demand will decrease. Table 6 shows the active power demand during the disturbance (measured at 1.1 s) and the normal condition for each scenario.

Table 6 The active powers (MW) demanded by the loads during L4 load outage
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Table 6 shows that in Scenarios I, II, III, IV, V, and VI, where the voltage is higher than the nominal voltage, the active power of the loads exceeds the nominal power. In Scenarios VII, VIII, and IX, in which the voltage is lower than the nominal voltage, the active power of the loads is below the nominal power. Scenarios VII, VIII, and IX have high penetration levels. Table 7 shows the voltage values at the load buses during the disturbance (measured at 1.1 s) and the normal condition for each scenario.

Table 7 Voltage values (pu) at load buses during L4 load outage
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In Fig. 7, the frequency support and penetration level of PVP after the disturbance is quite effective in the OS value. This is most clearly seen when looking at Scenario VII. In this scenario, PVP provides a high active power (700 MW) to the grid and does not reduce its active power during the disturbance, causing a high OS in G2 generator rotor speed. In addition, the frequency support of PVP plays an effective role in the US and ST. In scenarios (V, VI, VIII, and IX) where PVP provides frequency support at a high penetration level, it is seen that G2 generator is experiencing reduced deviations in rotor speed. As the penetration level of PVP decreases, the results of scenarios (at the same penetration levels) come close to each other as shown in Fig. 7. It is seen clearly in Scenarios I, II, and III. Table 8 clearly shows the effect of PVP penetration level and frequency support using the time-domain analysis for all scenarios.

Table 8 Time-domain analysis results for rotor speed of G2 generator during L4 load outage
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When all scenarios are compared, Scenario VII has reached the maximum OS (1.0087 pu) and minimum US (0.9978 pu) points, while Scenarios VI and IX with the shortest ST (6.55 s) have reached as shown in Table 8. Figure 8 shows the active power outputs of PVP during the disturbance.

Fig. 8
figure 8

PVP active power response at L4 load outage

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6.3 Generator Outage

Figure 9 shows the change in rotor speed of G2 generator when G4 generator is experiencing a momentary outage. Like a load outage, it creates an imbalance between generation and consumption. As a result of the outage, the rotor speed of G2 generator decreased in all scenarios until the end of the disturbance.

Fig. 9
figure 9

G2 rotor speed response at G4 generator outage

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Figure 9 shows that the rotor speed of G2 generator shows a very similar response during the outage in all scenarios. This is because the changes in load demands are close to each other during the outage. Table 9 shows the active power demand during the disturbance (measured at 1.1 s) and the normal condition for each scenario.

Table 9 The active powers (MW) demanded by the loads during G4 generator outage
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Table 9 shows that there is a decrease in load demands during the fault in all scenarios. This means that there is a voltage drop on the load buses in all scenarios. Since the voltage drops at the load buses are close to each other, the total loads also are similarly close to each other in all scenarios. Table 10 shows the voltage values at the load buses during the disturbance (measured at 1.1 s) and the normal condition for each scenario.

Table 10 Voltage values (pu) at load buses during G4 generator outage
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In scenarios where the frequency support of PVP is high and the penetration level is low, less oscillation occurs in the rotor speed of G2 generator after the disturbance. With the increase in the penetration level of PVP, the deviations in the rotor speed of G2 generator increase. This is evident at different penetration levels without frequency support, as in Scenarios I, IV, and VII. The frequency deviation decreases as the frequency support increases. This is easily seen if the results of Scenarios VII, VIII, and IX are compared. Table 11 demonstrates the impact of PVP penetration level and frequency support through the time-domain analysis for each scenario.

Table 11 Time-domain analysis results for rotor speed of G2 generator during G4 generator outage
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As shown in Fig. 10, in the scenario, where PVP does not react during the disturbance (like in Scenario VII), the rotor speed of G2 generator reaches its maximum OS which is 1.0134 pu due to the high penetration level. The minimum US (0.9928 pu) and the shortest ST (11.63 s) are reached in Scenario IX.

Fig. 10
figure 10

PVP active power response at G4 generator outage

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6.4 Three-Phase Short-Circuit Fault

Figure 11 shows the change in the rotor speed of generator G2 during a three-phase short-circuit fault at Bus 6.

Fig. 11
figure 11

G2 rotor speed response for three-phase short-circuit fault at Bus 6

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At the location (Bus 6) where a short-circuit fault occurs without fault impedance, the voltage drops to zero. Voltage drops are also very high around this location. As a result, the active power demand of the L6 load at the fault location becomes zero, while the active power demand of the loads in close proximity to this location experiences a significant decrease. Table 12 shows the active power demand during the fault (measured at 1.1 s) and the normal condition for each scenario.

Table 12 The active powers (MW) demanded by the loads during the three-phase short-circuit fault
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Table 12 shows that in all scenarios, the total load demand during a fault is close to each other. This result is also evident in the zoomed image in Fig. 11. This indicates that the voltage changes at the bus according to the nominal scenarios are close to each other for the same bus. Table 13 shows the voltage values at the load buses during the fault (measured at 1.1 s) and the normal condition for each scenario.

Table 13 Voltage values (pu) at load buses during the three-phase short-circuit fault
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At high penetration levels, this event clearly highlights the significance of PVP frequency support depth in effectively damping oscillations. Table 14 shows the results of the time-domain parameters for each scenario.

Table 14 Time-domain analysis results for rotor speed of G2 generator during the three-phase short-circuit fault
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When comparing all the scenarios, it is observed that Scenario VII reaches the maximum OS at 1.0203 pu. Scenario IX reaches the minimum US at 0.9909 pu. Scenario IX exhibits the shortest settling time (ST) at 11.57 s. PVP active power outputs that are effective in obtaining these results are given in Fig. 12.

Fig. 12
figure 12

PVP active power response for three-phase short-circuit fault at Bus 6

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7 Conclusions

The penetration levels of PVPs in electricity grids are increasing very rapidly. This makes it necessary for PVPs to provide frequency support for the grids. For this reason, PVP is discussed at different penetration levels and different frequency support rates in this study. In this study, the strategy is based on maximizing the output power from PVPs and providing frequency support during consumption (load) decreases. This strategy is considered for load change (L6), load outage (L4), generator outage (G4), and three-phase short-circuit fault (at Bus 6). As a result, the events are examined in detail, and the effect of PVP on the grid frequency is analyzed. The results of this study show that the suggested strategy for PVP effectively reduces the frequency oscillations and deviations (while maximizing the output power from PVP) at high penetration levels. The results can be summarized as follows.

  • L6 Load Change In this event, in accordance with the suggested strategy, the PVP does not provide frequency support when the load change is positive and provides frequency support when the load change is negative. Therefore, while the load change is positive, it is observed that the frequency deviation increases (US) as the PVP penetration level increases. If the load change is negative, the frequency support depth of the PVP has a significant impact on the deviations and oscillations. This proves the positive effect of frequency support depth on deviation and oscillations at high PVP penetration levels.

  • L4 Load Outage In this event, the rotor speed of G2 generator increases during the disturbance at all penetration levels. The deviations in the rotor speed increase as the penetration level of PVP increases and the frequency support decreases. When the penetration level of PVP is low, the voltage rises in the event of the outage. When the penetration level of PVP increases too much, there is a voltage drop in the event of the outage. Both load outage and voltage drop reduce the total load demand considerably. This increases the deviations in frequency. In addition, the frequency support of PVP plays an important role in reducing the oscillations in rotor speed.

  • G4 Generator Outage In this event, the rotor speed of G2 generator decreases during the disturbance at all penetration levels. During the disturbance, the rotor speeds in all scenarios are very close to each other due to the similar imbalance between generation and consumption in all scenarios. The small change in voltages causes small changes in load demands. After the disturbance, the importance of the frequency support of PVP becomes prominent as the penetration level of PVP increases. High-frequency support may create a large US in the rotor speed. On the other hand, a lack of frequency support may create a large OS in the rotor speed.

  • Three-Phase Short-Circuit Fault at Bus 6 This event causes results close to the generator outage. Deviations increase more in this event as it has more impact on the power system than a generator outage. A different aspect of the generator outage is that the total active power demand during the event is very close in all scenarios. This scenario also proves the significance of frequency support for PVP with high penetration levels.

The results obtained from this study show that the penetration level and frequency support depth of PVP should be adjusted very carefully for the frequency stability of the grid. High-frequency support can increase the minimum US, while low-frequency support can increase the maximum OS. Extremely high PVP penetration levels can increase the voltage drop during events. These problems can be avoided by analyzing different events, such as load and generator outages, short circuits, and load changes. Also, the type of load should be considered in the analysis because of the effect of voltage changes on the load.