Small-Signal Stability Analysis of Photovoltaic-Hydro Integrated Systems on Ultra-Low Frequency Oscillation

: In recent years, ultralow-frequency oscillation has repeatedly occurred in asynchronously connected regional power systems and brought serious threats to the operation of power grids. This phenomenon is mainly caused by hydropower units because of the water hammer e ﬀ ect of turbines and the inappropriate Proportional-Integral-Derivative (PID) parameters of governors. In practice, hydropower and solar power are often combined to form an integrated photovoltaic (PV)-hydro system to realize complementary renewable power generation. This paper studies ultralow-frequency oscillations in integrated PV-hydro systems and analyzes the impacts of PV generation on ultralow-frequency oscillation modes. Firstly, the negative damping problem of hydro turbines and governors in the ultralow-frequency band was analyzed through the damping torque analysis. Subsequently, in order to analyze the impact of PV generation, a small-signal dynamic model of the integrated PV-hydro system was established, considering a detailed dynamic model of PV generation. Based on the small-signal dynamic model, a two-zone and four-machine system and an actual integrated PV-hydro system were selected to analyze the inﬂuence of PV generation on ultralow-frequency oscillation modes under di ﬀ erent scenarios of PV output powers and locations. The analysis results showed that PV dynamics do not participate in ultralow-frequency oscillation modes and the changes of PV generation to power ﬂows do not cause obvious changes in ultralow-frequency oscillation mode. Ultra-low frequency oscillations are mainly a ﬀ ected by sources participating in the frequency adjustment of systems.


Introduction
There are differences in mechanism and characteristics between ultralow-frequency oscillation and traditional low-frequency oscillation. The frequency range of low-frequency oscillation is 0.1-2.5 Hz, and frequencies of ultralow-frequency oscillation is below 0.1 Hz. At present, researchers in this field generally believe that ultralow-frequency oscillation is caused by hydropower units. In recent years, ultralow-frequency oscillation has occurred frequently. As early as 1964, a frequency oscillation with a period of about 20 s was observed in the Southwestern United States [1]. Ultralow-frequency oscillations with frequencies below 0.05 Hz have also been observed in Turkey and Bulgaria [2], but due to their small impacts, they have not attracted widespread attention from researchers. In 2016, in an asynchronous networking test conducted by Yunnan Power Grid in China, a relatively severe ultralow-frequency oscillation event occurred, which lasted about half an hour [3]. After tripping the governor of some hydropower units, the oscillation decayed. According to researchers' studies, similar possible troubles of ultralow-frequency oscillations exist in Sichuan Power Grid in China, which also

Damping Torque Analysis
Negative damping problems of hydropower units in the ultralow-frequency band are mainly caused by the water hammer effect of a turbine and improper governor parameters. The damping torque analysis of the hydraulic turbine and the governor can obtain the damping characteristics of the ultralow-frequency band. In the following, we provide a damping torque analysis for a single hydropower unit, which reveals the basic mechanism and impact factors of ultralow-frequency oscillations [19].
The open-loop system model of a governor and a turbine is shown in Figure 1. The turbine transfer function was written as: where Tw is the water hammer time constant. The governor transfer function was described as: where KP, KI, and KD are the proportional, integral, and differential parameters, respectively, bp is the adjustment coefficient, and TG is the time constant of the servo system. The open-loop transfer function of the governor and turbine system was expressed as: Decomposing Equation (3) in the Δδ-Δω coordinate system, Equation (4) can be obtained as: where DT is the damping torque and ST is the synchronous torque. The torque position is shown in Figure 2. For DT > 0, it provides positive damping to the system.  The turbine transfer function was written as: where T w is the water hammer time constant. The governor transfer function was described as: G gov (s) = K D s 2 + K P s + K I b p K I + s where K P , K I , and K D are the proportional, integral, and differential parameters, respectively, b p is the adjustment coefficient, and T G is the time constant of the servo system. The open-loop transfer function of the governor and turbine system was expressed as: Decomposing Equation (3) in the ∆δ−∆ω coordinate system, Equation (4) can be obtained as: where D T is the damping torque and S T is the synchronous torque. The torque position is shown in Figure 2. For D T > 0, it provides positive damping to the system. The rest of this paper is organized as follows. Section 2 analyzes the damping characteristics of governors and turbines. Section 3 builds a detailed small-signal model of an integrated PV-hydro system. Section 4 analyzes the influence of PV generation on ultralow-frequency oscillation. Conclusions derived from these analyses are presented in Section 5.

Damping Torque Analysis
Negative damping problems of hydropower units in the ultralow-frequency band are mainly caused by the water hammer effect of a turbine and improper governor parameters. The damping torque analysis of the hydraulic turbine and the governor can obtain the damping characteristics of the ultralow-frequency band. In the following, we provide a damping torque analysis for a single hydropower unit, which reveals the basic mechanism and impact factors of ultralow-frequency oscillations [19].
The open-loop system model of a governor and a turbine is shown in Figure 1. The turbine transfer function was written as: where Tw is the water hammer time constant. The governor transfer function was described as: where KP, KI, and KD are the proportional, integral, and differential parameters, respectively, bp is the adjustment coefficient, and TG is the time constant of the servo system. The open-loop transfer function of the governor and turbine system was expressed as: Decomposing Equation (3) in the Δδ-Δω coordinate system, Equation (4) can be obtained as: where DT is the damping torque and ST is the synchronous torque. The torque position is shown in Figure 2. For DT > 0, it provides positive damping to the system.  The damping characteristics of a system composed of a governor and a turbine in a frequency range of 0-2.5 Hz are shown in Figure 3. For the water hammer time constant T w , a larger T w had a more negative damping torque in the ultralow-frequency band. For K P and K I in the PID governor, a larger value had more negative damping in the ultralow-frequency band. K D is generally set to 0. However, the water hammer effect is an inherent characteristic of hydro turbines and cannot be changed. The primary frequency regulation ability of a governor generally requires larger K P and K I , which contradicts the suppression of ultralow-frequency oscillation.
Energies 2020, 13, 1012 4 of 18 The damping characteristics of a system composed of a governor and a turbine in a frequency range of 0-2.5 Hz are shown in Figure 3. For the water hammer time constant Tw, a larger Tw had a more negative damping torque in the ultralow-frequency band. For KP and KI in the PID governor, a larger value had more negative damping in the ultralow-frequency band. KD is generally set to 0. However, the water hammer effect is an inherent characteristic of hydro turbines and cannot be changed. The primary frequency regulation ability of a governor generally requires larger KP and KI, which contradicts the suppression of ultralow-frequency oscillation.
(a) Different Tw Although the damped torque analysis method can analyze the damping characteristics of the governor and the turbine at different frequencies, it is difficult to analyze multimachine systems and the impact of PV generation.

Small-Signal Dynamic Model of an Integrated PV-Hydro System
In order to analyze the impact of PV generation on the ultra-low-frequency oscillation mode of multimachine systems, a detailed small-signal model of an intergrated PV-hydro system needed to be established for small-signal stability analysis.

Modeling of PV Generation
A PV generation model mainly included a PV array, an inverter, and controllers. Figure 4 shows the structure of a PV generation model connected to a power system. Although the damped torque analysis method can analyze the damping characteristics of the governor and the turbine at different frequencies, it is difficult to analyze multimachine systems and the impact of PV generation.

Small-Signal Dynamic Model of an Integrated PV-Hydro System
In order to analyze the impact of PV generation on the ultra-low-frequency oscillation mode of multimachine systems, a detailed small-signal model of an intergrated PV-hydro system needed to be established for small-signal stability analysis.

Modeling of PV Generation
A PV generation model mainly included a PV array, an inverter, and controllers. Figure 4 shows the structure of a PV generation model connected to a power system.

PV Array
The accurate model of a PV cell is very complicated, and some parameters are difficult to measure directly [20]. Thus, it is not convenient for research and application. By simplifying calculation equations, a practical engineering model was used in this paper [21]. The standard conditions for PV cells are Sref = 1000 W/m 2 and Tref = 25 °C. In addition, the voltage-current equation under nonstandard conditions can be descried as: where Isc is the short-circuit current, Uoc is the open-circuit voltage, Im and Um are the current and the voltage at the maximum power, respectively. The parameters under nonstandard conditions can be obtained as: where T and Tair are the temperatures of the PV cell and air, S is the light intensity, Uocref is the opencircuit voltage, Iscref is the short-circuit current, Umref is the voltage of the maximum power point, Imref is the current of the maximum power point in standard conditions, and k, α, β, and γ are compensation coefficients.
If the number of PV cells in series is n and the number of parallel connections is m, the voltage and the current of PV array were written as: Figure 4. The structure of photovoltaic (PV) generation. Symbols: C dc , direct current (DC) capacitor; U dc , DC-side output voltage; i C , the current of a DC capacitor; V k , alternating current (AC)-side output voltage; L f , AC inductor; i g , AC-side output current; V g , the voltage of the point connected with a power system.

PV Array
The accurate model of a PV cell is very complicated, and some parameters are difficult to measure directly [20]. Thus, it is not convenient for research and application. By simplifying calculation equations, a practical engineering model was used in this paper [21]. The standard conditions for PV cells are S ref = 1000 W/m 2 and T ref = 25 • C. In addition, the voltage-current equation under nonstandard conditions can be descried as: where I sc is the short-circuit current, U oc is the open-circuit voltage, I m and U m are the current and the voltage at the maximum power, respectively. The parameters under nonstandard conditions can be obtained as: where T and T air are the temperatures of the PV cell and air, S is the light intensity, U ocref is the open-circuit voltage, I scref is the short-circuit current, U mref is the voltage of the maximum power point, I mref is the current of the maximum power point in standard conditions, and k, α, β, and γ are compensation coefficients. If the number of PV cells in series is n and the number of parallel connections is m, the voltage and the current of PV array were written as: According to Equations (5) and (13), Equation (14) can be obtained as: Energies 2020, 13, 1012 6 of 17

DC Capacitor
Assume that the loss of the inverter can be ignored. Then, the output power of a PV array is equal to the sum of the power of a DC capacitor and the output power of an inverter, which can be described as: The voltage of the capacitor was selected as a state variable, which can be written as: According to Equations (15) and (16), Equation (17) can be obtained as:

Inverter and Controller
The PV controller consisted of a voltage controller and a current controller, which can achieve main functions [22]. The voltage controller regulated the DC voltage to control or maximize the power extracted from the PV array. The current controller realized the control of an actual current to the current reference value. Figure 5 shows the structures of voltage and current controllers. i * gq was assigned as 0. The voltage and current control equations were given as Equations (18) and (19), respectively: Energies 2020, 13 According to Equations (5) and (13), Equation (14) can be obtained as: 3.

DC Capacitor
Assume that the loss of the inverter can be ignored. Then, the output power of a PV array is equal to the sum of the power of a DC capacitor and the output power of an inverter, which can be described as: The voltage of the capacitor was selected as a state variable, which can be written as: According to Equations (15) and (16), Equation (17) can be obtained as: Voltage controller Current controller Figure 5. The structure of controllers. Symbols: U * dc, the reference value of a DC-side voltage; i * gd, the reference value of a d-axis current; i*gq, the reference value of a q-axis current; igd, the d-axis current; igq, the q-axis current; vgd, the d-axis voltage; vgq, the q-axis voltage; ω, the angular frequency of the system.
The PV controller consisted of a voltage controller and a current controller, which can achieve main functions [22]. The voltage controller regulated the DC voltage to control or maximize the power extracted from the PV array. The current controller realized the control of an actual current to the current reference value. Figure 5 shows the structures of voltage and current controllers. i * gq was Figure 5. The structure of controllers. Symbols: U * dc , the reference value of a DC-side voltage; i * gd , the reference value of a d-axis current; i* gq , the reference value of a q-axis current; i gd , the d-axis current; i gq , the q-axis current; v gd , the d-axis voltage; v gq , the q-axis voltage; ω, the angular frequency of the system. X v , Y d , and Y q were introduced as the state variables of the controllers [23]. The dynamic equations were described as: Considering the structure of the filter L f , the dynamic equations of filterwere written as:

PV Generation
According to Equations (14), (17), (20), and (21), a small-signal model of a PV generation model can be obtained by linearization as following: where ∆i gq ] T , and the coefficient matrices are shown in Equations (23) and (24):

Synchronous Generator
All generators were synchronous generators with a fourth-order model. The model was shown as: where ω 0 is the base angular frequency, H is the inertia constant, P m is the mechanical power, P e is the electromagnetic power, D is the damping coefficient, E ' d and E ' q are the d-axis and q-axis transient voltages, respectively, X d and X q are the unsaturated reactances, X ' d and X ' q are the unsaturated transient reactances, I d and I q are the d-axis and q-axis currents, respectively, E fd is the excitation voltage, and T ' d0 and T ' q0 are the unsaturated subtransient times. The detailed meanings of the symbols is given in [19].

Exciter
An excitation system is the main cause of low-frequency oscillations, and it is unclear whether it has an effect on ultralow-frequency oscillations. Therefore, a detailed typical fourth-order excitation system was selected [24]. The block diagram of the excitation system is shown in Figure 6.
where ω0 is the base angular frequency, H is the inertia constant, Pm is the mechanical power, Pe is the electromagnetic power, D is the damping coefficient, E ' d and E ' q are the d-axis and q-axis transient voltages, respectively, Xd and Xq are the unsaturated reactances, X ' d and X ' q are the unsaturated transient reactances, Id and Iq are the d-axis and q-axis currents, respectively, Efd is the excitation voltage, and T ' d0 and T ' q0 are the unsaturated subtransient times. The detailed meanings of the symbols is given in [19].

Exciter
An excitation system is the main cause of low-frequency oscillations, and it is unclear whether it has an effect on ultralow-frequency oscillations. Therefore, a detailed typical fourth-order excitation system was selected [24]. The block diagram of the excitation system is shown in Figure 6. Uex1, Uex2, and Uex3 were selected as the state variables. The mathematical model was shown as: , where Um, Uref, and Efd are the terminal voltage, reference input excitation voltage, and generator excitation potential, respectively, and Ka, Kf, Ta, Tf, Tr, and Te are the amplifier gain, stabilizer gain, amplifier time constant, stabilizer time constant, measurement time constant, and excitation circuit time constant, respectively. The expressions of Se and Uex were shown as: U ex1 , U ex2 , and U ex3 were selected as the state variables. The mathematical model was shown as: , where U m , U ref , and E fd are the terminal voltage, reference input excitation voltage, and generator excitation potential, respectively, and K a , K f , T a , T f , T r , and T e are the amplifier gain, stabilizer gain, amplifier time constant, stabilizer time constant, measurement time constant, and excitation circuit time constant, respectively. The expressions of S e and U ex were shown as:

Governor and Turbine
In order to study ultralow-frequency oscillation, a detailed model of a governor and a turbine was selected [19]. It consisted of a regulating system, an electro-hydraulic servo system, and a turbine model. In an actual running system, K D is generally set to 0. The hydraulic turbine and the PID governor are shown in Figure 7.

Governor and Turbine
In order to study ultralow-frequency oscillation, a detailed model of a governor and a turbine was selected [19]. It consisted of a regulating system, an electro-hydraulic servo system, and a turbine model. In an actual running system, KD is generally set to 0. The hydraulic turbine and the PID governor are shown in Figure 7. X1, X2, PGV, and Pm were seleted as the state variables of the model composed of a governor and a turbine.

Small-Signal Model of the Integrated PV-Hydro System
Suppose the system has n generator nodes, one PV generation, and l connected nodes. The lines and loads of the system can be expressed by algebraic Equation (29) By eliminating the connected nodes, the nodal admittance matrix can be simplified as: By integrating the PV small-signal model into the hydropower system, a small-signal dynamic model of the integrated system can be obtained as: Figure 7. Block diagram of a water turbine and a PID governor. Symbols: K W , the gain of frequency deviation; b p , permanent difference coefficient; K P1 , the gain of the governor; K I1 , the integral gain of the governor; K P2 , the gain of the servo system; T F , the time constant of stroke feedback; T W , the time constant of the water hammer. X 1 , X 2 , P GV , and P m were seleted as the state variables of the model composed of a governor and a turbine.

Small-Signal Model of the Integrated PV-Hydro System
Suppose the system has n generator nodes, one PV generation, and l connected nodes. The lines and loads of the system can be expressed by algebraic Equation (29): By eliminating the connected nodes, the nodal admittance matrix can be simplified as: By integrating the PV small-signal model into the hydropower system, a small-signal dynamic model of the integrated system can be obtained as: where ∆X sys = [∆X w1 , . . . , ∆X wn , ∆X PV ] T , A sys is the complete system state matrix, and ∆X w1 , . . . , ∆X wn are the state variables of n hydropower units, and ∆X PV is the state variables of the PV generation. By analyzing the eigenvalues and the eigenstructures of A sys , the system small-signal stability can be evaluated.

Small-Signal Stability Analysis
According to the small-signal model above, the effect of grid-connected PV generation on ultralow-frequency oscillation was studied based on two test systems, i.e., a modified two-zone and four-machine system and an actual system.

•
The two-zone and four-machine system is a typical benchmark system with standard parameters to study power system oscillations [19]. This paper selected it as a case study system and added PV generation into this system. The steam turbines of the two-zone and four-machine system were replaced by water turbines for hydropower studies.

•
In order to study the effect of PV generation in an actual system, an actual integrated PV-hydro system in Sichuan Province, China was selected, so that the research has practical significance.

Modified Two-Area and Four-Machine System
Based on the two-zone and four-machine system, an integrated PV-hydro system was constructed. The structure of the integrated PV-hydro system is shown in Figure 8. The parameters of the two-zone and four-machine system can be found in Reference [19]. The characteristic matrix of the system can be obtained by Equation (31). evaluated.

Small-Signal Stability Analysis
According to the small-signal model above, the effect of grid-connected PV generation on ultralow-frequency oscillation was studied based on two test systems, i.e., a modified two-zone and four-machine system and an actual system.  The two-zone and four-machine system is a typical benchmark system with standard parameters to study power system oscillations [19]. This paper selected it as a case study system and added PV generation into this system. The steam turbines of the two-zone and four-machine system were replaced by water turbines for hydropower studies.  In order to study the effect of PV generation in an actual system, an actual integrated PV-hydro system in Sichuan Province, China was selected, so that the research has practical significance.

Modified Two-Area and Four-Machine System
Based on the two-zone and four-machine system, an integrated PV-hydro system was constructed. The structure of the integrated PV-hydro system is shown in Figure 8. The parameters of the two-zone and four-machine system can be found in Reference [19]. The characteristic matrix of the system can be obtained by Equation (31).

Gen1
Bus1 T1 Bus5 In order to make the damping characteristics of each hydropower unit different, different water hammer time constants were set for each hydroelectric unit. The detailed parameters of governors and turbines are shown in Table 1.
The ultralow-frequency oscillation of the system calculated by the small-signal model is shown in Table 2. The oscillation frequency was less than 0.1 Hz, which belongs to the ultralow-frequency range. In order to make the damping characteristics of each hydropower unit different, different water hammer time constants were set for each hydroelectric unit. The detailed parameters of governors and turbines are shown in Table 1. The ultralow-frequency oscillation of the system calculated by the small-signal model is shown in Table 2. The oscillation frequency was less than 0.1 Hz, which belongs to the ultralow-frequency range. Participation factors are the multiplication of the corresponding elements in the right and left eigenvectors of a state matrix. It can be used for evaluating the association degree between state variables and modes. In this paper, we performed the participation factor analysis based on the state matrix A sys in Equation (31). The participation factors of state variables for the ultralow-frequency oscillation mode are shown in Figure 9. As can be seen from Figure 9, the dynamics of synchronous machines, governors, and turbines were mainly involved in the ultralow-frequency oscillation mode, and the generators with a larger T w were more involved. The dynamics of PV hardly participate in the ultralow-frequency oscillation mode. This is mainly because PV generation uses power control modes and does not participate in the frequency regulation.

Participation Factor Analysis
Participation factors are the multiplication of the corresponding elements in the right and left eigenvectors of a state matrix. It can be used for evaluating the association degree between state variables and modes. In this paper, we performed the participation factor analysis based on the state matrix Asys in Equation (31).
The participation factors of state variables for the ultralow-frequency oscillation mode are shown in Figure 9. As can be seen from Figure 9, the dynamics of synchronous machines, governors, and turbines were mainly involved in the ultralow-frequency oscillation mode, and the generators with a larger Tw were more involved. The dynamics of PV hardly participate in the ultralowfrequency oscillation mode. This is mainly because PV generation uses power control modes and does not participate in the frequency regulation.

Different Output Powers
When the output power of PV generation increased from 100 to 600 MW, the root locus of the ultralow-frequency oscillation mode changed, as shown in Figure 10, and the corresponding damping ratio and frequency are shown in Table 3. In Figure 10, the abscissa axis correspond to the real parts of eigenvalues, and the vertical axis corresponds to the imaginary parts of eigenvalues. It Figure 9. Participation factors. The state variables of synchronous machines contain δ, ω, E ' d , and E ' q . The state variables of excitation systems contain U ex1 , U ex2 , and U ex3 . The state variables of governors and turbines contain X 1 , X 2 , P GV , and P m . The state variables of PV generation contain U dc , X V , Y d , Y q , i gd , and i gq .

Different Output Powers
When the output power of PV generation increased from 100 to 600 MW, the root locus of the ultralow-frequency oscillation mode changed, as shown in Figure 10, and the corresponding damping ratio and frequency are shown in Table 3. In Figure 10, the abscissa axis correspond to the real parts of eigenvalues, and the vertical axis corresponds to the imaginary parts of eigenvalues. It can be seen from the results that the changes in PV output power had little effect on the ultralow-frequency oscillation mode.

Different Locations
The ultralow-frequency oscillation modes for PV generation connected to different locations are shown in Table 4. It can be seen that the connections of PV generation with different buses had little effect on the ultralow-frequency oscillation mode.  Table 5 shows the ultralow-frequency oscillation modes when a hydropower unit was replaced by PV generation. According to the results of the damping torque analysis, a larger T w of a hydropower unit provided more negative damping. Because T w values of Gen1 and Gen2 were small, they provided less negative damping to the system. When they were replaced by PV generation, the system damping ratio reduced. Since the T w of Gens 3 and 4 were large, they provided more negative damping to the system. When they were replaced by PV generation, the system damping ratio was improved.

Actual System of a County in Sichuan Province, China
An integrated PV-hydro system in a county of Sichuan Province in China was selected as the second test system with its structure shown in Figure 11. The system was connected to an external grid through a double feeder, which could be disconnected from an outside grid and then achieve an islanded operation.

Actual System of a County in Sichuan Province, China
An integrated PV-hydro system in a county of Sichuan Province in China was selected as the second test system with its structure shown in Figure 11. The system was connected to an external grid through a double feeder, which could be disconnected from an outside grid and then achieve an islanded operation. The output powers of the sources are shown in Table 6. The ultralow-frequency oscillation modes of the system under different operating modes are shown in Table 7. When connected to the network, the overall damping of the system was relatively strong, since the external power grid can help stabilize the frequency. During island operation, the damping of the ultralow-frequency oscillation mode became smaller, and it was easier to excite the ultralow-frequency oscillation.  The output powers of the sources are shown in Table 6. The ultralow-frequency oscillation modes of the system under different operating modes are shown in Table 7. When connected to the network, the overall damping of the system was relatively strong, since the external power grid can help stabilize the frequency. During island operation, the damping of the ultralow-frequency oscillation mode became smaller, and it was easier to excite the ultralow-frequency oscillation.  The participation factors are shown in Figure 12. Figure 12 indicates that the dynamics of PV hardly participated in the ultralow-frequency oscillation mode. The root locus of the PV output power increasing from 20 to 70 MW is shown in Figure 13. In Figure 13, the abscissa axis corresponds to the real parts of eigenvalues, and the vertical axis corresponds to the imaginary parts of eigenvalues. Figure 13 indicates that the root positions of the ultralow-frequency oscillation mode changed very little. The conclusion is the same as that obtained by studying the two-zone and four-machine system. The participation factors are shown in Figure 12. Figure 12 indicates that the dynamics of PV hardly participated in the ultralow-frequency oscillation mode. The root locus of the PV output power increasing from 20 to 70 MW is shown in Figure 13. In Figure 13, the abscissa axis corresponds to the real parts of eigenvalues, and the vertical axis corresponds to the imaginary parts of eigenvalues. Figure 13 indicates that the root positions of the ultralow-frequency oscillation mode changed very little. The conclusion is the same as that obtained by studying the two-zone and fourmachine system. Remark: In order to diminish the negative influences of the ultralow-frequency oscillation, some methods have been proposed. First, by quitting the frequency regulation function of hydropwer generators with negative damping, the oscillation could be eliminated [25]. Second, some optimization methods for the PID parameters of hydropower governors were proposed, which take The participation factors are shown in Figure 12. Figure 12 indicates that the dynamics of PV hardly participated in the ultralow-frequency oscillation mode. The root locus of the PV output power increasing from 20 to 70 MW is shown in Figure 13. In Figure 13, the abscissa axis corresponds to the real parts of eigenvalues, and the vertical axis corresponds to the imaginary parts of eigenvalues. Figure 13 indicates that the root positions of the ultralow-frequency oscillation mode changed very little. The conclusion is the same as that obtained by studying the two-zone and fourmachine system. Remark: In order to diminish the negative influences of the ultralow-frequency oscillation, some methods have been proposed. First, by quitting the frequency regulation function of hydropwer generators with negative damping, the oscillation could be eliminated [25]. Second, some optimization methods for the PID parameters of hydropower governors were proposed, which take Remark: In order to diminish the negative influences of the ultralow-frequency oscillation, some methods have been proposed. First, by quitting the frequency regulation function of hydropwer generators with negative damping, the oscillation could be eliminated [25]. Second, some optimization methods for the PID parameters of hydropower governors were proposed, which take into account the tradeoff between the performance of primary frequency regulation and the suppression of ultralow-frequency oscillations [4]. In addition, some researchers have added a governor's power system stabilizer on the speed control side of a hydropower generator to increase its damping in the ultralow-frequency band, thereby suppressing ultralow-frequency oscillations [9]. In this paper, we mainly focused on analyzing the impact of PV generation on ultralow-frequency oscillations. The methods to suppress ultralow-frequency oscillations will be included in our future work.