# Exploring the Regulation Reliability of a Pumped Storage Power Plant in a Wind–Solar Hybrid Power Generation System

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## Abstract

**:**

## 1. Introduction

## 2. Model and Method

#### 2.1. Model of the Pumped Storage Power Plant

#### 2.1.1. Penstock

_{0}= L/α and Z

_{0}= αQ

_{r}/A

_{i}gH

_{r}. h

_{q}is the relative value of head change caused by flow change. T

_{0}is the elastic time of the equivalent penstock. α is the water hammer wave speed. L is the length of penstock. Z

_{0}is the surge impedance in per unit of the equivalent penstock. Q

_{r}and H

_{r}are the rated flow and head, respectively. A

_{i}is the section dimension of the penstock. g is the acceleration of gravity. q represents the relative value of flow. s is the Laplace operator.

_{0}s) and omit the higher order term, then Equation (1) can be rewritten as [21]

#### 2.1.2. Hydraulic Speed Regulation System

_{y}is the engager relay time constant.

_{p}, k

_{i}and k

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

#### 2.1.3. Turbine

_{m}stands for the power output of the hydro turbine per unit. A

_{t}and q

_{n}

_{1}denote the gain coefficient of the turbine and the no-loading flow per unit. D

_{t}and Δω represent the mechanical damping coefficient of the hydro turbine and the difference of angular velocity, respectively.

#### 2.1.4. Excitation System

#### 2.1.5. Generator

_{B}are the rotor angle of the generator, the deviation of the relative angular speed and the nominal generator rotor speed, respectively. P

_{m}and P

_{G}represent the hydro turbine output power and the generator magnetic power, respectively. T

_{j}, D

_{t}and ${{T}^{\prime}}_{d0}$ denote the inertia time constant of the generator, the generator damping coefficient and the generator time constant, respectively. X

_{d∑}, X′

_{d∑}, U

_{s}and E

_{f}stand for the d-axis synchronous reactance, the d-axis transient reactance, the bus voltage and the controller output, respectively.

#### 2.1.6. Pumped Storage Power Plant Model

#### 2.2. Wind Power Generation System (WPGS)

_{WT}and P

_{rated}are the power output of a wind turbine and the rated electrical power, respectively. v

_{ci}represents the cut−in wind speed. v

_{r}is the rated wind speed. v

_{co}stand for the cut−off wind speed. A, B and C are the intermediate variables, i.e.,

#### 2.3. Photovoltaic Power Generation System (PPGS)

_{ph}is the photo current. I

_{0}is the diode saturation current. R′

_{s}is the series resistance. R′

_{p}is the shunt/parallel resistance. V

_{t}is the diode thermal voltage.

_{S}and N

_{p}are the number of PV cells in a series for the studied array and the number of PV module in parallel, respectively. P

_{A}, I

_{A}and V

_{A}are the power output, current and voltage of a PV array, respectively.

#### 2.4. Model of the Wind−Solar−Hydro Hybrid System

#### 2.5. Uncertainty Analysis

#### 2.6. Sensitivity Analysis

#### 2.7. Reliability Analysis

#### 2.7.1. First-Order Reliability Method

_{i}. μ, C and F are the vector of mean values, the covariance matrix and the failure domain, respectively.

_{i}.

_{f}is the probability of failure.

#### 2.7.2. Second-Order Reliability Method

**u*** in a standard normal U-space as [43]

**α**is the directional vector at the design point in U-space.

**B**is the scaled second-order derivatives of $\tilde{g}(u)$ at

**u***, known as the scaled Hessian matrix.

## 3. Numerical Experiments

#### 3.1. Dynamic Characteristics of WSH System in Steady and Fault States

#### 3.2. Dynamic Performance Indexes (DPIs)

_{r}), settling time (t

_{s}), peak value (p), peak time (t

_{p}) and overshoot (Os), which are used to characterize the response rapidity and stability of the system. Please refer to Appendix A for more details about DPIs. The corresponding results of the DPIs with Ke and Ki changing are shown in Table 4 and Table 5.

_{i}have little influence on the rise time, settling time, peak value and peak time of the guide vane opening since the difference in each DPI result between different Ke and Ki settings is relatively small. However, the overshoot values of the guide vane opening decrease with the increases of Ke and Ki. The maximum and minimum of the overshoot of the guide vane opening are 189.362 and 181.57, occurring in simulation No. 16 and No. 12, respectively. That is to say that a smaller setting of Ke and Ki causes a slower governor movement, leading to a larger overshoot of the guide vane opening. The above results show that the different Ke and Ki settings have a significant influence on the regulation quality of the guide vane opening. As for the DPIs of the angular velocity, it can be seen that both Ke and Ki have almost no effect on rise time, peak value and peak time. The maximum of settling time occurs in simulation No. 16 where Ke = 6 and Ki = 0.1. In addition, the values of overshoot at Ke = 6 are larger than those at Ke = 7 and Ke = 8. This means that the greater Ke value is, the better the dynamic performance of the system is.

#### 3.3. Uncertainty Analysis

^{−4}and 1.599 × 10

^{−4}. These phenomena mean that most of the rise time values are less than 1.599 × 10

^{−4}. From Figure 13b, the cumulative probability curve is relatively steep, indicating that the rise time value of the generator terminal voltage is comparatively centralized. The cumulative probability of rise time values less than 0.1406 is 99.34%, while the cumulative probability is almost equal to 0 with rise time value less than 0.08581. That is to say that the rise time value is in the range of 0.08581 and 0.1406. From Figure 13c, the cumulative probability is 99.62% when t

_{r}< 0.05674, and the cumulative probability is 0.4066% when t

_{r}< 0.01501. From Figure 13d, most of the values of rise time are less than 1.468 × 10

^{−4}, where the cumulative probability is 99.41%. Meanwhile, the slope of rise time cumulative probability of angular velocity curve changes smoothly compared with that of reactive power, generator terminal voltage and guide vane opening. The above phenomena show that there are great differences in the rise time of different output variables, especially reactive power and angular velocity. The cumulative probability curve in Figure 13b changes faster than those in other subgraphs.

_{s}< 1, the cumulative probability of angular velocity is larger than that of reactive power, generator terminal voltage and guide vane opening. When t

_{s}< 1, the cumulative probabilities of reactive power, generator terminal voltage, guide vane opening and angular velocity are 0, 0, 0.3797% and 66.58%, respectively. When t

_{s}< 2, the cumulative probabilities of reactive power, generator terminal voltage, guide vane opening and angular velocity are 0, 0, 99.83% and 100%, respectively. When t

_{s}< 4, the cumulative probabilities of reactive power, generator terminal voltage, guide vane opening and angular velocity are 54.91%, 82.48%, 100% and 100%, respectively. From the comparative results, the possible value of the settling time of the guide vane opening and angular velocity are larger than that of reactive power and generator terminal voltage in the case of large probability. The cumulative probability distribution of reactive power, generator terminal voltage, guide vane opening and angular velocity is significantly different from each other.

#### 3.4. Sensitivity Analysis

_{1}(9.298%), I

_{r}(2.859%), D

_{t}(2.719%), T

_{q00}(2.144%), Ncellm12 (1.878%), b

_{p}(1.51%), K

_{d}(1.507%), T (1.478%) and q

_{nl}(1.388%). The total contribution rate of the top 10 sensitive parameters is 81.77%, meaning that these parameters have a direct effect on the rise time of angular velocity and the most significant factors affecting the rise time are identified through sensitivity analysis. The contribution rate of other parameters is less than 1.3%, indicating that the sensitivity of interaction among these parameters is small and the parameters are independent. In addition, it is worth noting that the second and the third sensitivity parameters are H

_{1}and I

_{r}coming from WPGS and PPGS, respectively. These phenomena mean that these parameters have the ability to indirectly influence the angular velocity of PSPP by interacting with other parameters.

_{q00}(22.29%), A

_{t}(4.851%), F (3.619%), K

_{p}(1.829%), H

_{1}(1.674%), R

_{r}(1.671%), F

_{1}(1.645%), K

_{a}(1.596%) and R

_{s}(1.57%). The total contribution rate of the top 10 sensitive parameters is 74.45%, meaning that the most significant factors affecting the output are studied and identified through sensitivity. Therefore, the influence of the top 10 sensitive parameters on the settling time should be fully considered in the numerical simulation of the WSH hybrid system. The sensitivity index of other parameters is less than 1.6%, indicating that the sensitivity of interaction among these parameters is small and the parameters are independent.

_{t}with a sensitivity index 5.44%, followed by F, T

_{q00}, q

_{nl}, K

_{a}, R

_{s}, T

_{d00}, H

_{1}and K

_{p}with sensitivity index 2.693%, 2.663%, 2.073%, 2.053%, 1.779%, 1.581%, 1.333% and 1.286%, respectively. The total contribution rate of the top 10 sensitive parameters to the peak value is 93.45%, which indicates that these parameters have a significant influence on the peak value of the angular velocity.

_{0}, Ncellm12, T

_{q00}, L

_{ls}, K

_{a}, T, R

_{s}, I

_{r}and A

_{t}. The corresponding sensitivity indexes are 71.59%, 2.355%, 2.231%, 1.988%, 1.844%, 1.681%, 1.444%, 1.422%, 1.318% and 1.28%, respectively. The total contribution rate of the top 10 sensitive parameters is 87.15%, that is, the top 10 sensitive parameters have a significant influence on the peak time of the angular velocity. In other words, the influence of different parameters on peak time varies greatly. The Ncellm12 is the third sensitivity parameter coming from PPGS, indicating that the parameter of PPGS has the ability to indirectly affect the angular velocity by interacting with other parameters. In addition, H has the greatest impact on the peak time consistent with that of peak value and settling time, indicating that the most sensitive parameters of these DPIs are consistent.

_{0}, q

_{nl}, R

_{s}, L

_{m}, Ncellm12, H

_{1}, K

_{d}, f

_{p}, I

_{r}and H, respectively. The sensitivity results are shown in Table 6. From Table 6, the maximum sensitivity index is 3.177% coming from T

_{0}, and the minimum sensitivity index is 1.328% coming from H. It also can be seen that the total contribution rate of the top 10 sensitive parameters is 17.764%. It is worth noting that the sensitivity index value is relatively small compared with that of rise time, settling time, peak value and peak time. This phenomenon means that although many factors affect the overshoot of angular velocity, the difference of influence degree is small.

## 4. Reliability Analysis

## 5. Conclusions

- (1)
- The influence rules of the model parameters on the WSH hybrid system are obtained from the uncertainty analysis. Parameters of the wind, solar and hydro subsystem show the different influence on DPIs of the PSPP output due to parameters uncertainty. Both PSPP and WPGS parameters have a deterministic effect on the DPIs of reactive power, while the influence of PPGS has no regularity. The uncertain parameters of WPGS, PSPP and PPGS have regularity influence on the DPIs of the generator terminal voltage. Only PSPP parameters show certainty influence on the DPIs of the guide vane opening and angular velocity. The results also mean that the coupling effect of subsystems has the ability to affect the DPIs of PSPP in a certain case.
- (2)
- For the same DPI, the cumulative probability distributions of different output variables are significantly different from each other. Regarding different DPIs, the cumulative probability distributions of the same output variable are also different. In general, the settling time is larger than rising time.
- (3)
- The sensitivity degree of different DPIs to system parameters is obviously different, and even the same parameter has a different effect on the response speed and response stability of the angular velocity. The total contribution rate of the top 10 sensitive parameters on the rise time, settling time, peak value, peak time and overshoot of the angular velocity is 81.77%, 74.45%, 72.55%, 87.15% and 17.764%, respectively. Meanwhile, parameters of WPGS and PPGS have the ability to indirectly affect the angular velocity of PSPP by interacting with other parameters.
- (4)
- The peak value of angular velocity is distributed between 0.017 and 0.034. Most of the peak value of the angular velocity is in the range of 0.022 to 0.024, and the values on both sides are relatively small. There is a 2.5% probability that the system cannot meet the requirements of operation reliability, which may have a bad impact on the corresponding equipment or even threaten the normal operation of the system.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Dynamic Performance Indexes of System under Unit Step Response | ||
---|---|---|

DPIs | Equations | Symbol and physical meaning |

t_{r} | ${t}_{r}=\frac{\pi -\mathrm{arctan}\frac{\sqrt{1-{\xi}^{2}}}{\xi}}{{\omega}_{d}}$ | ξ: the damping ratio ω _{d}: the damped oscillation frequency, ω_{d} = ω_{n}(1 − ξ^{2})^{1/2} |

t_{s} | ${t}_{s}=\left\{\begin{array}{c}\frac{4}{\xi {\omega}_{n}},\Delta =2\%\\ \frac{3}{\xi {\omega}_{n}},\Delta =5\%\end{array}\right.$ | ω_{n}: the underdamped oscillation frequencyΔ: the error band |

t_{p} | ${t}_{p}=\frac{\pi}{{\omega}_{d}}$ | ω_{d}: the damped oscillation frequency, ω_{d}= ω_{n}(1−ξ^{2})^{1/2} |

os | $os={e}^{\frac{-\xi \pi}{\sqrt{1-{\xi}^{2}}}}\times 100\%$ | ξ: the damping ratio |

p | ------ | ------ |

## Appendix B

No. | Parameter | Physical Meaning | Unit | Mean | Variance | Distribution |
---|---|---|---|---|---|---|

1 | T | transfer function parameter | p.u. | 10 | 1 | Normal |

2 | Kp | proportional adjustment coefficient | p.u. | 1.6 | 0.16 | Normal |

3 | bp | adjustment coefficient | p.u. | 0.01 | 0.001 | Normal |

4 | Kd | differential adjustment coefficient | s | 2 | 0.2 | Normal |

5 | At | turbine gain | p.u. | 1.1534 | 0.11534 | Normal |

6 | Dt | damping factor | p.u. | 5 | 0.5 | Normal |

7 | fp | head loss coefficients | p.u. | 0.0028 | 0.00028 | Normal |

8 | qnl | no-load flow deviation | p.u. | 0.15 | 0.015 | Normal |

9 | T0 | transfer function parameter | p.u. | 0.47 | 0.047 | Normal |

10 | Td0 | transient time constant of d-axis in short circuit | p.u. | 1.01 | 0.101 | Normal |

11 | Td00 | super transient time constant of d-axis in short circuit | p.u. | 0.045 | 0.0045 | Normal |

12 | Tq00 | super transient time constant of q-axis in short circuit | p.u. | 0.045 | 0.0045 | Normal |

13 | H | inertia coefficient | p.u. | 1.5 | 0.15 | Normal |

14 | F | friction factor | p.u. | 0.28 | 0.028 | Normal |

15 | Ka | regulator gain | p.u. | 6.5 | 0.65 | Normal |

16 | Rs | stator resistance | p.u. | 0.023 | 0.0023 | Normal |

17 | Lls | stator inductance | p.u. | 0.18 | 0.018 | Normal |

18 | Rr | rotor resistance | p.u. | 0.016 | 0.0016 | Normal |

19 | Llr | rotor inductance | p.u. | 0.16 | 0.016 | Normal |

20 | Lm | magnetizing inductance | p.u. | 2.9 | 0.29 | Normal |

21 | H1 | wind inertia constant | p.u. | 0.685 | 0.0685 | Normal |

22 | F1 | wind friction factor | p.u. | 0.21 | 0.021 | Normal |

23 | WS | wind speed | m/s | 20 | 2 | Normal |

24 | Ncellm12 | number of photorefractive array units | p.u. | 96 | 9.6 | Normal |

25 | Ir | intensity of illumination | w/m^{2} | 1500 | 150 | Normal |

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**Figure 2.**The block diagram of the hydro turbine. A

_{t}is the hydro-turbine gain. h and q are the deviation of the water head and flow of the hydro turbine, respectively. fp is the head loss coefficient p

_{m}is the relative value of the output mechanical power. q

_{n1}is the no-load flow deviation. Δω is the deviation value of the angular velocity of the unit. D

_{t}is the damping factor. hfc is the relative value of the pipe friction head loss. y is the guide vane opening. h

_{q}denotes the variation of the water head of the hydro turbine caused by the flow change of the penstock.

**Figure 3.**A typical excitation system configuration. U

_{t}is the generator terminal voltage. U

_{ref}is the reference voltage. U

_{R}is the output of the voltage regulator. E

_{f}is the excitation voltage. r

_{f}is the excitation winding resistance of generator. x

_{ad}is the inductance coefficient of the d-axis armature reaction. U

_{f}is the output of the excitation system stabilizer. U

_{s}is the output of the power system stabilizer. PSS stands for the power system stabilizer.

**Figure 4.**Model of the pumped storage power plant. V

_{d}and V

_{q}are the stator voltage of the d-axis and q-axis, respectively. Pe, Pm and P

_{ref}represent the electrical power, the power output of the hydro turbine per unit and the reference output, respectively. V

_{ref}, V

_{stab}and V

_{f}are the reference value of the stator terminal voltage, the voltage connected to a power system stabilizer and the field voltage, respectively. A, B and C stand for the stator voltage input/output terminal. a, b and c denote the winding rotor output voltage terminal. dw is the rotor speed deviation.

**Figure 8.**Dynamic characteristics of voltage and current with three−phase short circuit fault of point H occurring as 1.0 s and cleared at 1.04 s. (

**a**) Dynamic characteristics of voltage and current of point W with three−phase short circuit fault of point H occurring at 1.0 s and cleared at 1.04 s. (

**b**) Dynamic characteristics of voltage and current of point S with three−phase short circuit fault of point H occurring at 1.0 s and cleared at 1.04 s. (

**c**) Dynamic characteristics of voltage and current of point W with three−phase short circuit fault of point S occurring at 1.0 s and cleared at 1.04 s. (

**d**) Dynamic characteristics of voltage and current of point S with three−phase short circuit fault of point S occurring at 1.0 s and cleared at 1.04 s.

**Figure 9.**The influence of the wind−solar−hydro system parameters on the DPIs of the reactive power of PSPP. (

**a**) The influence of Ka on the rise time. (

**b**) The influence of Ka on the overshoot. (

**c**) The influence of Td0 on the settling time. (

**d**) The influence of Tq00 and L1s on the peak value and peak time.

**Figure 10.**The influence of the wind−solar−hydro system parameters on the DPIs of the generator terminal voltage of PSPP. (

**a**) The influence of Td0 on the settling time. (

**b**) The influence of Ka on the peak value. (

**c**) The influence of Lm and H1 on the peak value and peak time. (

**d**) The influence of Ncellm12 and Ir on the overshoot.

**Figure 11.**The influence of the wind−solar−hydro system parameters on the DPIs of the guide vane opening of PSPP. (

**a**) The influence of At, F and H on the peak value and peak time. (

**b**) The influence of F on the rise time and settling time. (

**c**) The influence of F on the overshoot.

**Figure 12.**The influence of the wind−solar−hydro system parameters on the DPIs of the angular velocity of PSPP. (

**a**) The influence of H on the settling time. (

**b**) The influence of H on the overshoot.

**Figure 13.**The cumulative probability of the rise time. (

**a**) The cumulative probability of the rise time of the reactive power; (

**b**) The cumulative probability of the rise time of the generator terminal voltage; (

**c**) The cumulative probability of the rise time of the guide vane opening; (

**d**) The cumulative probability of the rise time of the angular velocity.

**Figure 14.**The cumulative probability of the settling time of the reactive power, generator terminal voltage, guide vane opening and angular velocity, respectively.

**Figure 15.**The sensitivity index of parameters of the rise time of the angular velocity. (

**a**) The sensitivity index of 25 parameters; (

**b**) The sensitivity index of the top 10 parameters. The numbers 1 to 25 represent symbols T, Kp, b

_{p}, K

_{d}, A

_{t}, D

_{t}, f

_{p}, q

_{nl}, T

_{0}, T

_{d0}, T

_{d00}, T

_{q00}, H, F, K

_{a}, R

_{s}, L

_{ls}, R

_{r}, L

_{lr}, L

_{m}, H

_{1}, F

_{1}, WS, Ncellm12 and I

_{r}, respectively. For the physical meaning and definitions of these parameters, see Appendix B.

**Figure 16.**The sensitivity index of parameters of the settling time of the angular velocity. (

**a**) The sensitivity index of 25 parameters; (

**b**) The sensitivity index of the top 10 parameters. The numbers 1 to 25 represent symbols T, Kp, b

_{p}, K

_{d}, A

_{t}, D

_{t}, f

_{p}, q

_{nl}, T

_{0}, T

_{d0}, T

_{d00}, T

_{q00}, H, F, K

_{a}, R

_{s}, L

_{ls}, R

_{r}, L

_{lr}, L

_{m}, H

_{1}, F

_{1}, WS, Ncellm12 and I

_{r}, respectively. For the physical meaning and definitions of these parameters, see Appendix B.

**Figure 17.**The sensitivity index of parameters of the peak value of the angular velocity. (

**a**) The sensitivity index of 25 parameters; (

**b**) The sensitivity index of the top 10 parameters. The numbers 1 to 25 represent symbols T, Kp, b

_{p}, K

_{d}, A

_{t}, D

_{t}, f

_{p}, q

_{nl}, T

_{0}, T

_{d0}, T

_{d00}, T

_{q00}, H, F, K

_{a}, R

_{s}, L

_{ls}, R

_{r}, L

_{lr}, L

_{m}, H

_{1}, F

_{1}, WS, Ncellm12 and I

_{r}, respectively. For the physical meaning and definitions of these parameters, see Appendix B.

**Figure 18.**The sensitivity index of parameters of the peak time of the angular velocity. (

**a**) The sensitivity index of 25 parameters; (

**b**) The sensitivity index of the top 10 parameters. The numbers 1 to 25 represent symbols T, Kp, b

_{p}, K

_{d}, A

_{t}, D

_{t}, f

_{p}, q

_{nl}, T

_{0}, T

_{d0}, T

_{d00}, T

_{q00}, H, F, K

_{a}, R

_{s}, L

_{ls}, R

_{r}, L

_{lr}, L

_{m}, H

_{1}, F

_{1}, WS, Ncellm12 and I

_{r}, respectively. For the physical meaning and definitions of these parameters, see Appendix B.

**Figure 19.**The sensitivity index of parameters of the overshoot of the angular velocity. (

**a**) The sensitivity index of 25 parameters; (

**b**) The sensitivity index of the top 10 parameters. The numbers 1 to 25 represent symbols T, Kp, b

_{p}, K

_{d}, A

_{t}, D

_{t}, f

_{p}, q

_{nl}, T

_{0}, T

_{d0}, T

_{d00}, T

_{q00}, H, F, K

_{a}, R

_{s}, L

_{ls}, R

_{r}, L

_{lr}, L

_{m}, H

_{1}, F

_{1}, WS, Ncellm12 and I

_{r}, respectively. For the physical meaning and definitions of these parameters, see Appendix B.

Unit | Equation | Parameter |
---|---|---|

Measure unit | ${G}_{M}(s)=\frac{1}{{T}_{r}s+1}$ | T_{r}: the time constant of the measure units: the Laplace operator |

Voltage regulator | T_{b}, T_{c}: the time constants used to model equivalent time constants inherentKa: the regulator gain T _{a}: the regulator time constantU _{R}: the output of the voltage regulatorU _{t}: the generator terminal voltageU _{R},_{max}, U_{R,min}: the limitation of the voltage | |

Exciter | ${G}_{E}(s)=\frac{1}{{T}_{e}s+{K}_{e}}$ | T_{e}: the exciter time constantK _{e}: the exciter gain |

Excitation system stabilizer | ${G}_{ESS}(s)=\frac{{K}_{f}}{1+{T}_{f}s}$ | K_{f}: the gain of the excitation system stabilizerT _{f}: the time constant of the excitation system stabilizer |

Symbol | Characteristics | Value |
---|---|---|

Voc | the open circuit voltage | 64.2 V |

Vmp | the optimum operating voltage | 54.7 V |

Isc | the short circuit current | 5.96 A |

Imp | the optimum operating current | 5.58 A |

NCellm | the number of photorefractive array units | 96 |

beta | the temperature coefficient of Voc | −0.27269 mV/°C |

alpha | the temperature coefficient of Isc | 0.061745 mA/°C |

Symbol | Physical Meaning | Symbol | Physical Meaning |
---|---|---|---|

h_{q} | the relative value of head caused by flow | E_{fd} | the exciter output voltage |

H | the inertia coefficient | E_{f} | the regulator output |

q | the relative value of flow | Te | the exciter time constant |

T_{0} | the elastic time of the equivalent penstock | Ke | the exciter gain |

α | the water hammer wave speed | Ka | the regulator gain |

L | the length of penstock | T_{a} | the time constant |

Q_{r} | the rated flow | K_{f} | the gain of the excitation system stabilizer |

H_{r} | the rated head | T_{f} | the time constant of the excitation system stabilizer |

A_{i} | the section dimension of penstock | T_{b}, T_{c} | the time constants used to model equivalent time constants inherent |

g | the acceleration of gravity | V_{t0} | the initial values of the terminal voltage |

s | the Laplace operator | V_{f0} | the initial values of the field voltage |

T_{y} | the engager relay time constant | tr | the low-pass filter time constant |

K_{p} | the proportional adjustment coefficient | Pe | the electrical power |

K_{i} | the integral adjustment coefficient | P_{ref} | the reference output |

K_{d} | the differential adjustment coefficient | A, B, C | the stator voltage input/output terminal |

δ | the relative value of the rotor angle | a, b, c | the winding rotor output voltage terminal |

ω | the relative value of the generator rotor speed | dw | the rotor speed deviation |

y | the relative value of the guide vane opening | Q | the output reactive power |

P_{m} | the power output of the hydro turbine per unit | δ | the power angle |

A_{t} | the gain coefficient of the turbine | ifd | the field current |

q_{n}_{1} | the no-loading flow per unit | t_{r} | the rise time |

D_{t} | the mechanical damping coefficient of the turbine | t_{s} | the settling time |

Δω | the difference of the angular velocity | p | the peak value |

hfc | the relative value of the pipe friction head loss | t_{p} | the peak time |

Ka | the regulator gain | os | the overshoot |

V_{ref} | the reference value of the stator terminal voltage | T | the transfer function parameter |

V_{d} | the stator voltage of the d-axis | V_{q} | the stator voltage of q-axis |

V_{tf} | the stator terminal voltage | F1 | the wind friction factor |

Rs | the stator resistance | H1 | the wind inertia constant |

Llr | the rotor inductance | Lm | the magnetizing inductance |

WS | the wind speed | Ncellm12 | the number of photorefractive array units |

Ir | the intensity of illumination | PL | the load power |

Xl | the positive sequence reactance | Xd | the d-axis synchronous reactance |

Xd0 | the d-axis transient reactance | Xd00 | the d-axis super-transient reactance |

Xq00 | the q-axis super-transient reactance | Xq | the q-axis synchronous reactance |

Rs1 | the stator resistance | x | the possible value of the uncertain component |

V_{f} | the field voltage | V_{stab} | the voltage connected to the power system stabilizer |

Z_{0} | the surge impedance per unit of the equivalent penstock | Td0 | the transient time constant of the straight axis in short circuit |

Tq00 | the super transient time constant of the quadrature axis in short circuit | Td00 | the super transient time constant of the straight axis in short circuit |

S | the state domain | F’ | the failure domain |

μ | the vector of mean values | μ_{i}^{N} | the equivalent normal mean |

F | the friction factor | C | the covariance matrix |

[R] | the correlation matrix | β | the Hasofer–Lind index |

α | the directional vector at the design point in U-space | B | the scaled second-order derivatives of $\tilde{g}(u)$ at u* |

φ(β) | the cumulative distribution function of the standard normal variable | P_{f} | the probability of failure |

X | the vector representing the set of random variables x_{i} | σ_{i}^{N} | the equivalent normal standard deviation of random variable x_{i} |

U_{t} | the generator terminal voltage | U_{R} | the output of the voltage regulator |

U_{ref} | the reference voltage | E_{f} | the excitation voltage |

x_{ad} | the inductance coefficient of d-axis armature reaction | r_{f} | the excitation winding resistance of the generator |

U_{s} | the output of the power system stabilizer | U_{f} | the output of the excitation system stabilizer |

T_{r} | the time constant of the measure unit | L | the inductance |

ψ | the magnetic flux | L_{m} | the mutual inductance |

T_{L} | the resistance torque of load | J | the rotational inertia |

p_{n} | the pole pairs | u_{s}, i_{s}, R_{s} | the voltage, current, resistance of stator |

P_{WT} | the power output of the wind turbine | P_{rated} | the rated electrical power of the wind turbine |

v_{ci}, v_{co} | the cut-in and cut-off wind speed | v_{r} | the rated wind speed |

I_{ph} | the photo current | I_{0} | the diode saturation current |

R^{’}_{s} | the series resistance | R^{’}_{p} | the shunt/parallel resistance |

V_{t} | the diode thermal voltage | P_{A}, I_{A}, V_{A} | the power output, current, and voltage of the PV array |

**Table 4.**The statistics of dynamic performance indexes of the reactive power and generator terminal voltage.

Simulation No. | K_{e} (p.u.) | K_{i} (s^{−1}) | Reactive Power | Generator Terminal Voltage | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

t_{r} (s) | t_{s} (s) | p (p.u.) | t_{p} (s) | t_{r} (s) | t_{s} (s) | p (p.u.) | t_{p} (s) | Os (p.u.) | |||

1 | 6 | 0.55 | 0.00029 | 0.70835 | 7.50472 | 0.0063 | 0.05893 | 0.59551 | 226.588 | 0.24475 | 1.405 |

2 | 7 | 0.55 | 0.00017 | 0.83774 | 7.50482 | 0.0063 | 0.04869 | 1.15137 | 226.26 | 0.24475 | 1.99228 |

3 | 7 | 0.55 | 0.00017 | 0.83774 | 7.50482 | 0.0063 | 0.04869 | 1.15137 | 226.26 | 0.24475 | 1.99228 |

4 | 7 | 0.55 | 0.00017 | 0.83774 | 7.50482 | 0.0063 | 0.04869 | 1.15137 | 226.26 | 0.24475 | 1.99228 |

5 | 7 | 0.1 | 0.00016 | 0.83847 | 7.50482 | 0.0063 | 0.04869 | 1.15186 | 226.261 | 0.24475 | 1.99339 |

6 | 6 | 1 | 0.00029 | 0.70833 | 7.50472 | 0.0063 | 0.05892 | 0.59554 | 226.588 | 0.24475 | 1.40557 |

7 | 7 | 0.55 | 0.00017 | 0.83774 | 7.50482 | 0.0063 | 0.04869 | 1.15137 | 226.26 | 0.24475 | 1.99228 |

8 | 7 | 0.55 | 0.00017 | 0.83774 | 7.50482 | 0.0063 | 0.04869 | 1.15137 | 226.26 | 0.24475 | 1.99228 |

9 | 7 | 1 | 0.00016 | 0.83862 | 7.50482 | 0.0063 | 0.04869 | 1.1528 | 226.26 | 0.24475 | 1.99509 |

10 | 7 | 0.55 | 0.00017 | 0.83774 | 7.50482 | 0.0063 | 0.04869 | 1.15137 | 226.26 | 0.24475 | 1.99228 |

11 | 7 | 0.55 | 0.00017 | 0.83774 | 7.50482 | 0.0063 | 0.04869 | 1.15137 | 226.26 | 0.24475 | 1.99228 |

12 | 8 | 1 | 0.00007 | 1.39728 | 7.50489 | 0.0063 | 0.04716 | 1.9153 | 226.013 | 0.24475 | 2.47662 |

13 | 8 | 0.55 | 0.00007 | 1.39712 | 7.50489 | 0.0063 | 0.04716 | 1.91419 | 226.014 | 0.24475 | 2.47599 |

14 | 8 | 0.1 | 0.00007 | 1.37754 | 7.50489 | 0.0063 | 0.04717 | 1.9127 | 226.014 | 0.24475 | 2.47537 |

15 | 7 | 0.55 | 0.00017 | 0.83774 | 7.50482 | 0.0063 | 0.04869 | 1.15137 | 226.26 | 0.24475 | 1.99228 |

16 | 6 | 0.1 | 0.00029 | 0.70835 | 7.50472 | 0.0063 | 0.05893 | 0.5955 | 226.588 | 0.24475 | 1.40489 |

_{r}, t

_{s}, p, t

_{p}and Os are the rise time, settling time, peak value, peak time and overshoot, respectively.

Simulation No. | K_{e} (p.u.) | K_{i} (s^{−1}) | Guide Vane Opening | Angular Velocity | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

t_{r} (s) | t_{s} (s) | p (p.u.) | t_{p} (s) | Os (p.u.) | t_{r} (s) ×10^{−5} | t_{s} (s) | p (p.u.) | t_{p} (s) | Os (p.u.) | |||

1 | 6 | 0.55 | 0.01837 | 1.15621 | 0.21895 | 0.22785 | 188.507 | 3.99 | 1.10165 | 1.05118 | 0.248 | 5.1204 |

2 | 7 | 0.55 | 0.01853 | 1.16495 | 0.21793 | 0.22795 | 185.165 | 3.32 | 0.98561 | 1.05078 | 0.2485 | 5.0801 |

3 | 7 | 0.55 | 0.01853 | 1.16495 | 0.21793 | 0.22795 | 185.165 | 3.32 | 0.98561 | 1.05078 | 0.2485 | 5.0801 |

4 | 7 | 0.55 | 0.01853 | 1.16495 | 0.21793 | 0.22795 | 185.165 | 3.32 | 0.98561 | 1.05078 | 0.2485 | 5.0801 |

5 | 7 | 0.1 | 0.01843 | 1.16483 | 0.2175 | 0.22795 | 187.636 | 3.41 | 0.98563 | 1.05077 | 0.2485 | 5.07975 |

6 | 6 | 1 | 0.01855 | 1.15657 | 0.21967 | 0.2279 | 184.326 | 3.95 | 1.10156 | 1.05118 | 0.248 | 5.12074 |

7 | 7 | 0.55 | 0.01853 | 1.16495 | 0.21793 | 0.22795 | 185.165 | 3.32 | 0.98561 | 1.05078 | 0.2485 | 5.0801 |

8 | 7 | 0.55 | 0.01853 | 1.16495 | 0.21793 | 0.22795 | 185.165 | 3.32 | 0.98561 | 1.05078 | 0.2485 | 5.0801 |

9 | 7 | 1 | 0.01863 | 1.16506 | 0.21836 | 0.24795 | 182.75 | 3.28 | 0.98559 | 1.05078 | 0.2485 | 5.08045 |

10 | 7 | 0.55 | 0.01853 | 1.16495 | 0.21793 | 0.22795 | 185.165 | 3.32 | 0.98561 | 1.05078 | 0.2485 | 5.0801 |

11 | 7 | 0.55 | 0.01853 | 1.16495 | 0.21793 | 0.22795 | 185.165 | 3.32 | 0.98561 | 1.05078 | 0.2485 | 5.0801 |

12 | 8 | 1 | 0.0187 | 1.17919 | 0.21736 | 0.228 | 181.57 | 3.28 | 0.98559 | 1.05047 | 0.2485 | 5.0493 |

13 | 8 | 0.55 | 0.01859 | 1.17913 | 0.21693 | 0.228 | 183.92 | 2.91 | 0.99159 | 1.05047 | 0.2485 | 5.0493 |

14 | 8 | 0.1 | 0.01849 | 1.17906 | 0.2165 | 0.228 | 186.321 | 2.93 | 0.99159 | 1.05047 | 0.2485 | 5.04902 |

15 | 7 | 0.55 | 0.01853 | 1.16495 | 0.21793 | 0.22795 | 185.165 | 3.32 | 0.98561 | 1.05078 | 0.2485 | 5.0801 |

16 | 6 | 0.1 | 0.01834 | 1.15613 | 0.21881 | 0.22785 | 189.362 | 4.00 | 1.10166 | 1.05118 | 0.248 | 5.12006 |

_{r}, t

_{s}, p, t

_{p}and Os are the rise time, settling time, peak value, peak time and overshoot, respectively.

Rise Time (t_{r}) | Settling Time (t_{s}) | Peak Value (p) | |||||||||

No. | Parameter | Sensitivity Index | Ranking | No. | Parameter | Sensitivity Index | Ranking | No. | Parameter | Sensitivity Index | Ranking |

1 | T | 1.48% | 9 | 2 | K_{p} | 1.83% | 5 | 2 | K_{p} | 1.29% | 10 |

3 | b_{p} | 1.51% | 7 | 5 | A_{t} | 4.85% | 3 | 5 | A_{t} | 5.44% | 2 |

4 | K_{d} | 1.51% | 8 | 12 | T_{q}_{00} | 22.29% | 2 | 8 | q_{nl} | 2.07% | 5 |

6 | Dt | 2.72% | 4 | 13 | H | 33.70% | 1 | 11 | T_{d}_{00} | 1.58% | 8 |

8 | q_{nl} | 1.39% | 10 | 14 | F | 3.62% | 4 | 12 | T_{q}_{00} | 2.66% | 4 |

9 | T_{0} | 56.99% | 1 | 15 | K_{a} | 1.60% | 9 | 13 | H | 72.55% | 1 |

12 | T_{q}_{00} | 2.14% | 5 | 16 | R_{s} | 1.58% | 10 | 14 | F | 2.69% | 3 |

21 | H_{1} | 9.30% | 2 | 18 | R_{r} | 1.67% | 7 | 15 | K_{a} | 2.05% | 6 |

24 | Ncellm12 | 1.88% | 6 | 21 | H_{1} | 1.67% | 6 | 16 | R_{s} | 1.78% | 7 |

25 | I_{r} | 2.86% | 3 | 22 | F_{1} | 1.65% | 8 | 21 | H_{1} | 1.33% | 9 |

Total | -- | 81.77% | -- | Total | -- | 74.45% | -- | Total | -- | 93.45% | -- |

Peak time (p_{t}) | Overshoot (O_{s}) | Note | |||||||||

No. | Parameter | Sensitivity index | Ranking | No. | Parameter | Sensitivity index | Ranking | Colour in cells: gradient change from green through yellow to red represents sensitivity from good to bad. | |||

1 | T | 1.44% | 7 | 4 | K_{d} | 1.49% | 7 | ||||

5 | A_{t} | 1.28% | 10 | 7 | f_{p} | 1.44% | 8 | ||||

9 | T_{0} | 2.36% | 2 | 8 | q_{nl} | 2.30% | 2 | Theses sensitivity indexes values of dynamic performance indexes are based on angular velocity. | |||

12 | T_{q}_{00} | 1.99% | 4 | 9 | T_{0} | 3.18% | 1 | ||||

13 | H | 71.59% | 1 | 13 | H | 1.33% | 10 | ||||

15 | K_{a} | 1.68% | 6 | 16 | R_{s} | 1.78% | 3 | ||||

16 | Rs | 1.42% | 8 | 20 | L_{m} | 1.66% | 4 | Physical meaning and definitions of these parameters see Table 3. | |||

17 | L_{ls} | 1.84% | 5 | 21 | H_{1} | 1.55% | 6 | ||||

24 | Ncellm12 | 2.23% | 3 | 24 | Ncellm12 | 1.62% | 5 | The longer the blue data bar, the weaker the sensitivity of the parameter. | |||

25 | I_{r} | 1.32% | 9 | 25 | I_{r} | 1.42% | 9 | ||||

Total | -- | 87.15% | -- | Total | -- | 17.76% | -- |

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## Share and Cite

**MDPI and ACS Style**

Xu, B.; Zhang, J.; Egusquiza, M.; Zhang, J.; Chen, D.; Egusquiza, E. Exploring the Regulation Reliability of a Pumped Storage Power Plant in a Wind–Solar Hybrid Power Generation System. *Water* **2021**, *13*, 2548.
https://doi.org/10.3390/w13182548

**AMA Style**

Xu B, Zhang J, Egusquiza M, Zhang J, Chen D, Egusquiza E. Exploring the Regulation Reliability of a Pumped Storage Power Plant in a Wind–Solar Hybrid Power Generation System. *Water*. 2021; 13(18):2548.
https://doi.org/10.3390/w13182548

**Chicago/Turabian Style**

Xu, Beibei, Jingjing Zhang, Mònica Egusquiza, Junzhi Zhang, Diyi Chen, and Eduard Egusquiza. 2021. "Exploring the Regulation Reliability of a Pumped Storage Power Plant in a Wind–Solar Hybrid Power Generation System" *Water* 13, no. 18: 2548.
https://doi.org/10.3390/w13182548