# Low Frequency Oscillations in a Hydroelectric Generating System to the Variability of Wind and Solar Power

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## Abstract

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## 1. Introduction

- A promising hybrid system of the hydropower generation integrating with wind farm and solar photovoltaic system is established using MATLAB/Simulink, in order to enable the stability analysis. This contributes to the current international pool of the integration modelling knowledge.
- The sensitivity of hydropower low frequency oscillations to its governor regulation capacity is quantified under the volatility influence of wind and solar energies. The main adopted methods include Nyquist response and root-locus analysis.
- To understand the stability conditions of the hybrid system, the influence of different wind/solar/hydropower quotas (i.e., W: S: H) and the various transmission line distance ratios on the low frequency oscillation mode of hydropower system are also quantified. The assessment indicators in this part include the three-phase parallel RLC load, the grounding transformer, the three-phase PI section line and the wind-farm transmission line, and the assessment criteria are oscillation frequency and damping ratio.

## 2. Mathematical Model of the Hybrid Power System

#### 2.1. Hydropower System

#### 2.1.1. Penstock

_{W}is the water flow inertia time constant, ${T}_{W}=\frac{{L}_{r}{Q}_{r}}{{gAH}_{r}}$. T

_{r}is the water hammer pressure time constant, ${T}_{r}=\frac{2{L}_{r}}{a}$. The variable f

_{1}is represented by ${f}_{1}=\frac{f{Q}_{r}^{2}{L}_{r}}{2D{A}^{2}g{H}_{r}}$, wherein parameters L

_{r}, Q

_{r}and H

_{r}are the total pipe length, design flow and design head.

_{q}, Z

_{01}and T

_{01}are transient head of hydraulic turbine, standardized value of hydraulic surge impedance of pipeline and elastic time, respectively.

#### 2.1.2. Governor

_{P}, K

_{I}and K

_{D}are the proportional adjustment coefficient, integral adjustment coefficient and differential adjustment coefficient, respectively. △F, Y

_{c}, b

_{p}and T

_{1v}are the relative deviation of the input frequency, relative predefined guide vane opening, permanent interpolation coefficient and differential loop time constant, respectively. If the permanent interpolation coefficient (b

_{p}) equals to zero, thus Equation (8) is rewritten as:

#### 2.1.3. Hydro-Turbine

_{t}, q, h, f

_{p}, p

_{m}, q

_{nl}, △ω, D

_{t}, and y denote the hydro-turbine gain, relative deviation of hydro-turbine flow, relative deviation of hydro-turbine head, relative head loss coefficient in penstock, mechanical output power, relative no-load flow, relative deviation of hydro-turbine rotational speed, damping factor, and relative deviation of guide vane opening, respectively. Thus, the block diagram of the hydro-turbine is performed in Figure 4.

#### 2.1.4. Synchronous Generator

_{d}, L

_{q}, L

^{’}

_{fd}, L

^{’}

_{kd}, L

^{’}

_{kq1}and L

^{’}

_{kq2}are self-inductance of winding; and L

_{md}and L

_{mq}are mutual inductance of winding.

#### 2.1.5. Excitation Sector

#### 2.2. Wind Farm

#### 2.2.1. Wind Turbine Model

_{p}) and the swept area (A

_{r}) of the blade. The operational phenomenon of wind turbines can be described by the aerodynamics theory, and based on this theory, the power captured by blades of the wind turbine can be expressed as:

_{W}, ρ, C

_{p}, A

_{r}, and v

_{ω}are the mechanical power output of the wind turbine, air density, wind energy efficiency, swept area of the blade, and wind velocity, respectively. Here, the wind energy efficiency C

_{p}can expressed as the ratio between the power captured by the wind turbine (P

_{w}) and the total power through the wind turbine (P

_{a}), i.e.,

_{a}is the tip velocity ratio and β is the pitch angle of the blade. Using all these equations, the wind characteristic curves can be obtained and described as shown in Figure 7.

#### 2.2.2. Mechanical Drive Shaft Model

_{ls}, T

_{t}, T

_{m}, ω

_{t}, ω

_{g}, J

_{r}, J

_{g}, B

_{r}, B

_{g}, B

_{ls}and n

_{g}are the low speed shaft torque, aerodynamic torque, generator electromagnetic torque, rotor speed, generator speed, rotor inertia, generator inertia, rotor external damping, generator external damping, low speed shaft damping, and ratio constant, respectively.

#### 2.2.3. DFIG Model in dq Frame

_{sd,q}, v

_{rd,q}, i

_{sd,q}, i

_{rd,q}, Φ

_{sd,q}, Φ

_{rd,q}, R

_{s}, R

_{r}, ω

_{s}, and ω

_{r}are the stator voltage in dq frame, rotor voltage in dq frame, stator current in dq frame, rotor current in dq frame, stator flux in dq frame, rotor flux in dq frame, stator resistance, rotor resistance, stator speed, and rotor speed, respectively.

_{s}, L

_{r}, and M are the stator inductance, rotor inductance, and mutual inductance, respectively.

_{p}is the pole pairs of the DFIG.

#### 2.2.4. PWM Converter Model

_{a}T and d

^{’}

_{a}T are the conduction times of switch S1 and switch S2, respectively; with T and d

_{a}as the switching cycle of the PWM switches and the duty ratio of phase d

_{a}; i

_{abc}represents the currents flowing through three phases; e

_{abc}represents the voltages at the grid connection point for three phases; R

_{f}and L

_{f}are the filter resistance and inductance, respectively; v

_{dc}is the DC-link voltage at the input of the PWM converter; and v

_{N0}is the neutral voltage at the grid connection point. The subscripts a, b, and c with different variables represent the phase a, phase b, and phase c, respectively.

_{dc}is the Dc current at the input of the PWM converter.

#### 2.3. Solar Photovoltaic System

## 3. Low Frequency Oscillation Response to Hybrid Regulation

_{p}), the integral adjustment coefficient (k

_{i}) and the differential adjustment coefficient (k

_{d}), are selected to analyze their regulation performance on low frequency oscillation of hydropower generation based on the Nyquist response and root-locus analysis.

#### 3.1. Nyquist Response to PID Regulation

#### 3.1.1. Nyquist Profile

#### 3.1.2. Influences of Governor Parameters on Nyquist and Step Responses

_{p}), the integral adjustment coefficient (k

_{i}) and the differential adjustment coefficient (k

_{d}) are determined. In this case study, the wind, solar and hydropower ratio keeps 40 vs. 1 vs. 150 to benefits the identification of leading variate. The Nyquist and step response results are shown in Figure 13, Figure 14 and Figure 15.

_{p}) increases from 0.8 to 2.4. The Nyquist trajectory is a transversely zygomorphic closed curve starting at the origin, and it surrounds twice in the clockwise direction and then returns back to the origin. The curvature of Nyquist trajectory increases after it first go through the real-axis, but the curvature gradually decreases after the second real-axis traverse. All Nyquist trajectories intersect at the origin or at a point near the origin, whereas the peak response is continuously enhanced with the decrease of the proportional adjustment coefficient (k

_{p}). Moreover, it is clearly observed in Figure 13b, there are some similarities for the step response of the hydropower system under a different proportional adjustment coefficient (k

_{p}). For example, the step response within k

_{p}= 2.4 first increases to the maximum overshoot (i.e., 0.014) and then gradually stabilized at the value of 0.002. However, notr that there is the smaller proportional adjustment coefficient, the smaller maximum overshoot. This reveals that the hydropower block has an excellent regulation capacity to the low frequency oscillation of the hybrid system within k

_{p}= 0.8 compared with the situation within k

_{p}= 2.4

_{i}= [0.25, 1.25]. These curves only intersect at the origin, although the curvature of the first circle is greater than that of the second circle for a specific value of k

_{i}. Additionally, regarding all different values of k

_{i}, their second circles move right along with the real-axis and the corresponding peak response improved from k

_{i}= 0.25 to k

_{i}= 1.25. As performed in Figure 14b, all step responses have a decaying trend under different k

_{i}conditions, differently, the peak value and final value of the step response increase gradually with the increase of integral adjustment coefficient (k

_{i}). For instance, the peak value and final value for k

_{i}= 1.25 are 0.0115 and 0.0081 in comparison with the relatively smaller values of 0.0098 and 0.001 for k

_{i}= 0.25. However, fortunately, all overshoots for various parameter settings of k

_{i}are less than the value of 10

^{−3}, thus the hydropower system can keep its stability during the selected variation domain of k

_{i}.

_{d}= 0.5 and k

_{d}= 1.5. For a certain value of k

_{d}, its Nyquist trajectory starts and also finally converges at the origin by turning clockwise twice; meanwhile, the curvature of the first circle is greater than that of the second circle. The peak response of various Nyquist trajectories increase as the differential adjustment coefficient rises from k

_{d}= 0.5 to k

_{d}= 1.5. Additionally, the step responses of the hydropower system perform decaying trends under various k

_{d}conditions, and these response curves finally converge to a certain value, around 0.0017. The peak values for different values of k

_{d}are extremely close to the mean value of 0.0098, although the peak value increases slightly with the change of the differential adjustment coefficient from k

_{d}= 0.5 to k

_{d}= 1.5. This implies that the studied value domain of the differential adjustment coefficient between (k

_{d}) has a lower sensitivity to the low frequency oscillation of hydropower system.

#### 3.1.3. Root-Locus Profile

_{0}and A

_{i}denote the Laplace transformation of the step response, and residues corresponding to zero and pole, respectively. Hence, the general stability rule influenced by the pole and zero point in the root-locus trajectory is summarized two aspects: (1) The stability condition needs all closed-loop poles be located in the left half plane, meaning that all root-locus trajectories are in the left half plane, and (2) the rapidness of system response depends on the distance between the closed-loop pole (s

_{i}= σ + jω) and the imaginary-axis. If such a distance is shorter, there exists a rapider decaying speed of the e

^{sit}and thus results in an excellent rapidness of system response.

#### 3.1.4. Influences of Governor Parameters on Root-Locus Response

_{p}= 1.6, k

_{i}= 0.75 and k

_{d}= 0.5, and the zero point coordinates are (−0.0925, 0) and (−1.69, 0) as well as the pole coordinates are (−1, 0), (−1.05, 0) and (−15, 0). There are three root-locus responses: (i) The first root-locus trajectory begins at the pole (−1, 0) and ends at the zero point (−0.0925, 0). (ii) The second root-locus trajectory starts from the pole (−1.05, 0) and is extended towards the zero point (−1.69, 0), and (iii) the third root-locus trajectory stems from the pole (−15, 0) and goes to the negative infinity. All root-locus trajectories are located in the left half plane, and thus the hydropower system is stable in this situation. With the increase of the differential adjustment coefficient from k

_{d}= 0.8 to k

_{d}= 2.4, the pole coordinate (Figure 16b) remains unchanged, but the zero point is gradually closer to the imaginary-axis. This indicates that there is an excellent rapidness response of the hydropower system to the disturbance of low frequency oscillations within a smaller k

_{p}.

_{p}= 1.6, k

_{i}= 0.75 and k

_{d}= 1. The zero point coordinates include (−0.255, 0) and (−1.84, 0), and the pole coordinates involve in (−1, 0), (−1.1, 0) and (−15, 0). There are three root-locus trajectories: the first trajectory is between the pole (−1, 0) and zero point (−0.255, 0), the second trajectory is from the pole (−1.1, 0) to the zero point (−1.84, 0), and the last trajectory starts from the pole (−15, 0) and links to the negative infinity. The hydropower system operates safely and stably since these root-locus trajectories are in the left halt plane. Figure 17b illustrates the regulation capacity of the integral adjustment coefficient from k

_{i}= 0.25 to k

_{i}= 1.25. With the decrease of parameter k

_{i}, the pole coordinate keeps constant while the zero point moves closely to the imaginary-axis. This indicates that the hydropower system has a better rapidness as the parameter k

_{i}increases to 1.25.

_{p}= 1.6, k

_{i}= 0.75 and k

_{d}= 1 in Figure 18a. The zero points coordinates are (−0.255, 0) and (−1.84, 0), while the pole coordinates are (−1, 0), (−1.1, 0) and (−15, 0). The first root-locus trajectory is from the pole (−1, 0) to the zero point (−0.255, 0), the second root-locus trajectory is from the pole (−1.1, 0) to the zero point (−1.84, 0), and the last root-locus trajectory experiences from the pole (−15, 0) to negative infinity. All the three trajectories lie in the left half plane, resulting in a steady operational condition of the hydropower system responding to the low frequency oscillations. Moreover, as shown in Figure 18b, the zero point has an opposite trend with the pole with the increase of differential adjustment coefficient (k

_{d}), where the zero point moves closely to the imaginary-axis, but the pole is far away from the imaginary-axis. This reveals that there is a rapid response of the hydropower system to the low frequency oscillation within k

_{d}= 1 in comparison to the condition of k

_{d}= 1.6.

## 4. Low Frequency Oscillation Response to Renewable-Quota

^{2}vs. 7.48 × 10

^{2}vs. 7.71 × 10

^{2}. Besides, the damping ratios for these four stability indicators are constant. This suggests that the appropriate increase of the wind farm installed capacity is beneficial for the electricity production without system stability defects. As regards the condition of SL:WL = 2:1, the oscillation frequency of the three-phase parallel RLC load and the wind-farm transmission line is unchanged while the oscillation frequency of the grounding transformer and the three-phase PI section line obviously increases with the capacity-increasing improvement of wind farm. The oscillation damping ratio of the three-phase parallel RLC load and the grounding transformer has no variation in contrast to the decline damping ratio of the three-phase PI section line and the wind-farm transmission line. It is indicated that the wind farm installed capacity is unable to design too large, otherwise the stability problem is likely to occur under the condition of SL:WL = 2:1. Regarding the condition of SL:WL = 3:1, there is no variation for the oscillation frequency of the three-phase parallel RLC load, the three-phase PI section line and the wind-farm transmission line, but the oscillation frequency of the grounding transformer rises. Differently, the oscillation damping ratio of the three-phase parallel RLC load, the grounding transformer and the wind-farm transmission line totally remains unchanged, whereas the oscillation damping ratio of the grounding transformer first declines and then rises. Thus, it is possible to increase the installed capacity of the wind farm since the hydropower generation has a low sensitivity to the low frequency oscillations in this case.

## 5. Conclusions

_{p}, the integral adjustment coefficient k

_{i}, and the differential adjustment coefficient k

_{d}) to the system stability profile. Furthermore, the influence of different renewable-quota and transmission line distance ratios on the low frequency oscillation mode of the hydropower system is quantified. The major conclusions can be drawn as follows.

- Under the case where the wind, solar and hydropower ratio is 40:1:150, it is interesting that the smaller the governor parameters (k
_{p}, k_{i}, and k_{d}), the smaller the Nyquist overshoot and step fluctuation. Herein, the studied domains for k_{p}, k_{i}, and k_{d}are [0.8, 2.4], [0.25, 1.25] and [0.5, 1.5], and thus the optimal values for maximally reducing hydropower low frequency oscillation are finally determined as k_{p}= 0.8, k_{i}= 0.25 and k_{d}= 0.5. - The wind/solar/hydropower hybrid system keeps global stability in the studied governor parameter domains since the Nyquist and root-locus low frequency oscillation responses meet the relevant stability criteria, i.e., the clockwise number around the point of −1/K equals to the negative pole numbers in the G(s)H(s) right half plane, as well as all root-locus trajectories are in the left half plane. Despite this merit, the overshoot problem is expected to arouse great attention and discussion to reduce the fatigue damage of hydropower components.
- Aiming at different wind/solar/hydropower quotas (i.e., 20: 1: 150, 30: 1: 150, and 40: 1: 150), the four quantified indicators (i.e., the three-phase parallel RLC load, the grounding transformer, the three-phase PI section line, and the wind-farm transmission line) show that it is beneficial to increase the wind farm installed capacity to maximize the electricity production without system stability defects under the solar-load and wind-load line ratios for 1:1 and 3:1 excepting for 2:1. This is contributed by the smaller quantified values of oscillation frequency and damping ratio.
- Regarding a certain wind/solar/hydropower quota, it is a promising strategy to increase the solar-load transmission line in order to achieve the safe and stable operation of the hybrid system and a relatively excellent dynamic regulation capacity of the hydropower governor.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic diagram of wind/solar/hydropower integrated system. Images of wind turbine, PV generator, and electric load from the ref. [20].

**Figure 2.**Block diagram of the PID controller in hydropower system. Parameters k

_{p}, k

_{i}and k

_{d}are the proportional adjustment coefficient, integral adjustment coefficient and differential adjustment coefficient, respectively.

**Figure 4.**Block diagram of the hydro-turbine. Parameters A

_{t}, q, h, f

_{p}, p

_{m}, q

_{nl}, △ω, D

_{t}, y h

_{fc}and h

_{q}denote the hydro-turbine gain, relative deviation of hydro-turbine flow, relative deviation of hydro-turbine head, relative head loss coefficient in penstock, mechanical output power, relative no-load flow, relative deviation of hydro-turbine rotational speed, damping factor, relative deviation of guide vane opening, relative head friction loss and hydro-turbine head change caused by flow, respectively.

**Figure 8.**Diagram of a two-mass drive shaft model of a mechanical drive shaft model. In this figure, the parameters T

_{t}, T

_{hs}, T

_{ls}, T

_{m}, ω

_{t}, ω

_{g}, ω

_{ls}, J

_{r}, J

_{g}, B

_{r}, B

_{g}, B

_{ls}, K

_{ls}, and n

_{g}represent the aerodynamic torque, high speed shaft torque, low speed shaft torque, generator electromagnetic torque, rotor speed, generator speed, low speed shaft speed, rotor inertia, generator inertia, rotor external damping, generator external damping, low speed shaft damping, low speed shaft stiffness, and ratio constant, respectively.

**Figure 9.**Block diagram of the wind turbine and mechanical drive shaft model. The symbols in this figure, W_wt, Pitch_deg, WS, Tt, T_wt, ωg, T_hs and Tm denote the wind turbine speed, blade pitch angle, wind speed, torque of the mechanical drive shaft, wind turbine torque, generator speed, torque transmitted through the shaft, and torque at the input of the generator, respectively.

**Figure 10.**Block diagram of a state-average PWM converter model. In this figure, the parameters Uctrl_grid_conv, Uctrl_rotor_conv, Uavg_grid_conv, Uavg_rotor_conv, Iabc_grid_conv_pu, Iabc_rotor_pu, Vab_gc, Vbc_gc, Vab_rc, Vbc_rc, Vdc, and Asyncmac_sig represent the grid-side converter voltage control signal, rotor-side converter voltage control signal, grid-side average converter voltage signal, rotor-side average converter voltage signal, grid converter current signal, rotor converter current signal, grid-side voltage between phases a and b, grid-side voltage between phases b and c, rotor-side voltage between phases a and b, rotor-side voltage between phases b and c, DC-link voltage, and the asynchronous machine signal, respectively.

**Figure 13.**Nyquist and step responses to the variation of the proportional adjustment coefficient, k

_{p}= [0.8, 2.4]. (

**a**) Nyquist response, and (

**b**) step response.

**Figure 14.**Nyquist and step responses to the variation of the integral adjustment coefficient, k

_{i}= [0.25, 1.25]. (

**a**) Nyquist response, and (

**b**) step response.

**Figure 15.**Nyquist and step responses to the variation of the differential adjustment coefficient, k

_{d}= [0.5, 1.5]. (

**a**) Nyquist response, and (

**b**) step response.

**Figure 16.**Root-locus response to the variation of the proportional adjustment coefficient, k

_{p}= [0.8, 2.4]. (

**a**) Root-locus response at k

_{p}= 1.6, k

_{i}= 0.75 and k

_{d}= 0.5, and (

**b**) root-locus response with the variation of k

_{p}= 0.8, 1.6 and 2.4.

**Figure 17.**Root-locus response to the variation of the integral adjustment coefficient, k

_{i}= [0.25, 1.25]. (

**a**) Root-locus response at k

_{p}= 1.6, k

_{i}= 0.75 and k

_{d}= 1, and (

**b**) root-locus response with the variation of k

_{i}= 0.25, 0.75 and 1.25.

**Figure 18.**Root-locus response to the variation of the differential adjustment coefficient, k

_{d}= [0.5, 1.5]. (

**a**) Root-locus response at k

_{p}= 1.6, k

_{i}= 0.75 and k

_{d}= 1, and (

**b**) Root-locus response with the variation of k

_{d}= 0.5, 1 and 1.5.

**Table 1.**Low frequency oscillation response of hydropower system to wind/solar/hydropower quota and line distance ratio.

Line Length Ratio (S:W) | W:S:H | Eigenvalue | Frequency/HZ | Damping Ratio/% | Remark |
---|---|---|---|---|---|

1:1 | 20:1:150 | −1.14 × 10^{−1} + j0 | 0.0182 | 100 | Three-Phase Parallel RLC Load |

−1.26 × 10^{1} + j0 | 2.01 | 100 | Grounding Transformer | ||

−9.18 × 10^{2} − j4.47 × 10^{3} | 7.26 × 10^{2} | 20.1 | Three-Phase PI Section Line | ||

−3.15 × 10^{3} − j9.43 × 10^{3} | 1.58 × 10^{3} | 31.7 | Windformsystem/30 km line | ||

30:1:150 | −1.13 × 10^{−1} + j0 | 0.018 | 100 | Three-Phase Parallel RLC Load | |

−1.65 × 10^{1} + j0 | 2.63 | 100 | Grounding Transformer | ||

−9.47 × 10^{2} − j4.61× 10^{3} | 7.48 × 10^{2} | 20.1 | Three-Phase PI Section Line | ||

−3.15× 10^{3} − j9.43× 10^{3} | 1.58 × 10^{3} | 31.7 | Windformsystem/30 km line | ||

40:1:150 | −1.13 × 10^{−1} + j0 | 0.018 | 100 | Three-Phase Parallel RLC Load | |

−1.99 × 10^{1} + j0 | 3.17 | 100 | Grounding Transformer | ||

−9.77 × 10^{2} − j4.74 × 10^{3} | 7.71 × 10^{2} | 20.2 | Three-Phase PI Section Line | ||

−3.15 × 10^{2} − j9.43 × 10^{3} | 1.58 × 10^{3} | 31.7 | Windformsystem/30 km line | ||

2:1 | 20:1:150 | −1.14 × 10^{−1} + j0 | 0.0182 | 100 | Three-Phase Parallel RLC Load |

−1.26 × 10^{1} − j0 | 2.01 | 100 | Grounding Transformer | ||

−4.62 × 10^{3} − j0 | 7.36 × 10^{2} | 100 | Three-Phase PI Section Line | ||

−3.23 × 10^{2} − j7.11 × 10^{3} | 1.13 × 10^{3} | 4.54 | Windformsystem/30 km line | ||

30:1:150 | −1.13 × 10^{−1} + j0 | 0.018 | 100 | Three-Phase Parallel RLC Load | |

−1.65 × 10^{1} + j2.37 × 10^{−10} | 2.63 | 100 | Grounding Transformer | ||

−9.67 × 10^{2} − j4.58 × 10^{3} | 7.45 × 10^{2} | 20.7 | Three-Phase PI Section Line | ||

−3.22 × 10^{2} − j7.11 × 10^{3} | 1.13 × 10^{3} | 4.53 | Windformsystem/30 km line | ||

40:1:150 | −1.13 × 10^{−1} + j0 | 0.018 | 100 | Three-Phase Parallel RLC Load | |

−1.99 × 10^{1} + j0 | 3.17 | 100 | Grounding Transformer | ||

−9.98 × 10^{2} − j4.71 × 10^{3} | 7.66 × 10^{2} | 20.7 | Three-Phase PI Section Line | ||

−3.22 × 10^{2} − j7.11 × 10^{3} | 1.13 × 10^{3} | 4.52 | Windformsystem/30 km line | ||

3:1 | 20:1:150 | −1.14 × 10^{−1} + j0 | 0.0182 | 100 | Three-Phase Parallel RLC Load |

−1.26 × 10^{1} + j0 | 2.01 | 100 | |||

−8.82 × 10^{2} − j4.39 × 10^{3} | 7.13 × 10^{2} | 19.7 | Three-Phase PI Section Line | ||

−5.85 × 10^{2} + j627 × 10^{3} | 1.00 × 10^{3} | 9.28 | Windformsystem/30 km line | ||

30:1:150 | −1.13 × 10^{−1} + j0 | 0.018 | 100 | Three-Phase Parallel RLC Load | |

−1.65 × 10^{1} + j0 | 2.63 | 100 | Grounding Transformer | ||

−1.72 × 10^{2} − j4.48 × 10^{3} | 7.13 × 10^{2} | 3.85 | Three-Phase PI Section Line | ||

−5.85 × 10^{2} + j6.27 × 10^{3} | 1.00 × 10^{3} | 9.28 | Windformsystem/30 km line | ||

40:1:150 | −1.13 × 10^{−1} + j0 | 0.018 | 100 | Three-Phase Parallel RLC Load | |

−1.99 × 10^{1} − j0 | 3.17 | 100 | Grounding Transformer | ||

−1.76 × 10^{2} − j4.47 × 10^{3} | 7.12 × 10^{2} | 3.95 | Three-Phase PI Section Line | ||

−5.85 × 10^{2} + j6.27 × 10^{3} | 1.00 × 10^{3} | 9.28 | Windformsystem/30 km line |

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## Share and Cite

**MDPI and ACS Style**

Xu, B.; Lei, L.; Zhao, Z.; Jiang, W.; Xiao, S.; Li, H.; Zhang, J.; Chen, D.
Low Frequency Oscillations in a Hydroelectric Generating System to the Variability of Wind and Solar Power. *Water* **2021**, *13*, 1978.
https://doi.org/10.3390/w13141978

**AMA Style**

Xu B, Lei L, Zhao Z, Jiang W, Xiao S, Li H, Zhang J, Chen D.
Low Frequency Oscillations in a Hydroelectric Generating System to the Variability of Wind and Solar Power. *Water*. 2021; 13(14):1978.
https://doi.org/10.3390/w13141978

**Chicago/Turabian Style**

Xu, Beibei, Liuwei Lei, Ziwen Zhao, Wei Jiang, Shu Xiao, Huanhuan Li, Junzhi Zhang, and Diyi Chen.
2021. "Low Frequency Oscillations in a Hydroelectric Generating System to the Variability of Wind and Solar Power" *Water* 13, no. 14: 1978.
https://doi.org/10.3390/w13141978