# Multi-Objective Optimization of Back-to-Back Starting for Pumped Storage Plants under Low Water Head Conditions Based on the Refined Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- It is the first to establish a nonlinear PSP model combining the electrical subsystem with the fine hydraulic and mechanical subsystems for BTBS;
- (2)
- The effects of the hydraulic-mechanical-electrical parameters on the BTBS are comprehensively investigated on the basis of the model mentioned above;
- (3)
- An innovative multi-objective optimization scheme is proposed for the control strategy of BTBS at low water head conditions for the first time, which is proven to be suitable for a variety of working conditions.

## 2. Refined Modelling of a Pumped Storage Plant for Back-to-Back Starting

#### 2.1. The Hydraulic-Mechanical Subsystem

- (1)
- Conduit System

- (2)
- The Pump turbine Model

_{11}in the way of logarithmic projection so as to overcome the difficulty. The equation is as follows:

_{11}and M

_{11}with abscissa X. It immensely reduces the complexity of the calculation procedure problems whether compared with the Suter or improved Suter methods [26].

- (3)
- The Turbine Governor System

#### 2.2. The Electrical Subsystem

- (1)
- Synchronous Machine

- (2)
- The Excitation System

#### 2.3. Model Validation with On-Site Measurement

## 3. Analysis of Factors Affecting Back-to-Back Starting

#### 3.1. Excitation Current

#### 3.2. Control Way of the Excitation System

#### 3.3. Initial Difference between Rotor Positions

#### 3.4. Water Head

#### 3.5. Control Scheme of the Governor

_{s}. After accelerating to 0.9, the PID controller is put into operation in closed-loop regulation according to the rated speed, as shown in Figure 17. The rotational speed transition curves of BTBS are plotted at different points of starting time T

_{s}in Figure 18. Generally, the faster the guide vanes are opened, the faster the speeds of the units rise, but this is not absolute. As shown in Figure 18, when the starting time T

_{s}=15 s, the rising speed of the driven machine is faster than the scheme with starting time T

_{s}=10 s.

## 4. Optimization of BTBS Strategy Based on Multi-Objective Control

#### 4.1. Objective Function

#### 4.2. Decision Variables

- Scheme 1: The traditional CEC mode with one-stage DGVC+PID control. In one-stage DGVC, the guide vane of the driving machine is first opened at the rate k
_{c}, and remains unchanged when the guide vane opening reaches y_{c}. When the speed reaches 90% of the rated speed, the PID controller will be put into operation. The excitation system of the two units shall first operate with the given excitation current, and switch to the constant terminal voltage mode when the speed reaches 90% of the rated speed. k_{c}generally takes the maximum rate, which is a known parameter. Therefore, the given excitation current ${i}_{g}^{*},{i}_{m}^{*}$, and given opening y_{c}are selected as the decision variables in the open-loop stage. Parameters of PID controller, K_{p}, K_{i}, and K_{d}, are adopted as the decision variables in the closed-loop stage.$${X}_{1}=\left[{i}_{g}^{*},{i}_{m}^{*},{y}_{c},{K}_{p},{K}_{i},{K}_{d}\right]$$ - Scheme 2: The proposed CEV mode with one-stage DGVC+PID control. Similarly, the given excitation voltage${V}_{g}^{*},{V}_{m}^{*}$, given opening y
_{c}, and the three parameters of PID controller, K_{p}, K_{i}, and K_{d}, are chosen as the decision variables.$${X}_{2}=\left[{V}_{g}^{*},{V}_{m}^{*},{y}_{c},{K}_{p},{K}_{i},{K}_{d}\right]$$

#### 4.3. Constraint Conditions

- (1)
- Operation Time Constraint$$0<{t}_{s2}<{t}_{\mathrm{max}}$$
- (2)
- The Boundary of Decision Variables$${X}_{i}\in \left[L,U\right]$$
- (3)
- Rotor Speed Difference Between Two MachinesAccording to the requirements of the BTBS, the rotational speed difference between the two units shall not exceed a certain limit value; otherwise, it is considered a startup failure:$$\left|{\omega}_{1}-{\omega}_{2}\right|\le {\omega}_{\mathrm{max}}^{d}$$

#### 4.4. Optimization Procedures

## 5. Case Study and Analysis

#### 5.1. Model Parameters Setting

_{1}and X

_{2}are obtained, as shown in Table 3.

#### 5.2. Introduction to Comparative Experiments

#### 5.3. Effectiveness Analysis

_{max}represents the speed overshoot, t

_{r}represents the speed rise time, and t

_{s}represents the stability time.

#### 5.4. Validation of the Proposed Optimization Strategy

## 6. Conclusions

- The given value and control way of excitation current, the control scheme of the governor, and the water head have great influence on the transient process of BTBS. The control scheme of excitation current and guide vane should be selected as the decision variables in the BTBS optimization; the worst BTBS condition can be identified by the lowest water head.
- The overshoot and stable time of the speed are contradictory. The traditional single-objective optimization scheme merely considers the single objective, which can very easily cause the unit to fall into the S-shaped area, resulting in severe fluctuations in speed and power.
- Compared with the single-objective, the optimization strategy proposed can considerably improve the speed overshoot and the speed stable time by at least 68.27% and by 3.22% under the worst working condition. The optimization results show that the multi-objective scheme is a better choice than the single-objective scheme.
- Compared with the MOCEC scheme, when the MOCEV scheme is adopted, the overshoot, rise time, and stable time are improved by 68.35%, 3.7%, and 3.2% in PSU-1, and 45.4%, 3.7%, and 3.2% in PSU-2. Thus, the MOCEV scheme is superior.
- The proposed MOCEV optimal control scheme can effectively keep away from the S-shaped area and is verified by a real PSU.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Abbreviations | |

PSP | pumped storage plant |

BTBS | back-to-back starting |

PSU | pump storage unit |

CEV | constant excitation voltage |

CEC | constant excitation current |

LCP | logarithmic curve projection |

MOC | method of characteristics |

GVO | gate valve opening |

MOCEC | multi-objective CEC |

MOCEV | multi-objective CEV |

SOCEC | single-objective CEC |

SOCEV | single-objective CEV |

ITAE | integrated time and absolute error |

pu | per-unit value |

Symbols | |

Parameters | |

A | cross sectional area of pipeline (m^{2}) |

a | wave velocity (m/s) |

b_{p} | permanent difference coefficient (pu) |

D | the diameter of the turbine runner (m) |

d | pipeline diameter (m) |

f | friction coefficient (pu) |

g | acceleration of gravity (m/s^{2}) |

K_{d} | differential gain (pu) |

K_{e} | self-excitation coefficient of exciter (pu) |

K_{f} | damping coefficient (pu) |

k_{0} | forward amplification factor (pu) |

K_{i} | integral gain (pu) |

K_{p} | proportional gain (pu) |

R_{a} | resistance of stator winding (pu) |

S_{e} | exciter saturation factor (pu) |

T_{a} | amplifier time constant (s) |

T_{b} | lead lag time constant (s) |

T_{c} | lead lag time constant (s) |

x | distance calculated from upstream (m) |

T_{d} | differential time constant (s) |

T_{e} | exciter time constant (s) |

T_{f} | damping time constant (s) |

T_{j} | mechanical time constant (s) |

T_{y} | main servomotor response time (s) |

T_{y}_{1} | assistant servomotor response time (s) |

${T}_{d0}^{\prime},{T}_{d0}^{\prime \prime}$ | transient and sub-transient time constants of open-circuit d-axis (s) |

${T}_{q0}^{\prime},{T}_{q0}^{\prime \prime}$ | transient and sub-transient time constants of open-circuit q-axis (s) |

${X}_{q},{X}_{q}^{\prime},{X}_{q}^{\prime \prime}$ | synchronous, transient and sub-transient reactance of q-axis (pu) |

${X}_{q},{X}_{q}^{\prime},{X}_{q}^{\prime \prime}$ | synchronous, transient and sub-transient reactance of q-axis (pu) |

Variables | |

${E}_{d}^{\prime},{E}_{d}^{\prime \prime}$ | the transient and sub-transient internal EMF of d-axis (pu) |

E_{fd} | excitation EMF (pu) |

${E}_{q}^{\prime},{E}_{q}^{\prime \prime}$ | the transient and sub-transient internal EMF of q-axis (pu) |

H | piezometric head (m) |

H_{t} | the working head of pump turbine (m) |

${i}_{d},{i}_{q}$ | the current of d- and q-axis (pu) |

i_{fd} | excitation current (pu) |

${i}_{g}^{*},{i}_{m}^{*}$ | excitation current setting value of generator and motor (pu) |

K_{a} | amplifier coefficient (pu) |

M_{e} | electromagnetic torque (pu) |

M_{t} | the moment of pump turbine (pu) |

M_{11} | unit torque (N/m^{3}) |

N | the rotational speed of the turbine (r/min) |

N_{11} | unit speed (m^{1/2}/s) |

N_{11r} | rated unit speed (m^{1/2}/s) |

Q | the water flow rate (m^{3}/s) |

Q_{t} | the flow of pump turbine (m^{3}/s) |

Q_{11} | unit flow (m^{1/2}/s) |

Q_{11r} | rated unit flow (m^{1/2}/s) |

u | controller output signal (pu) |

${V}_{g}^{*},{V}_{m}^{*}$ | excitation voltage setting value of generator and motor (pu) |

${v}_{d},{v}_{q}$ | d- and q-axis component of the voltage (pu) |

Y | guide vane opening (deg) |

y | main servomotor output signal (pu) |

ω, ω* | relative and given value angular shaft velocity (m) |

${\phi}_{d},{\phi}_{q}$ | the internal EMF of d- and q-axis (pu) |

$\delta $ | rotor angle (rad) |

## Appendix A

Parameters | Values | Parameters | Values |
---|---|---|---|

Rated speed (r/min) | 250 | Rated capacity (MVA) | 334 |

Rated water-head (m) | 195 | Rated voltage (kV) | 15.75 |

Rated water flow (m^{3}/s) | 176.1 | Rated current (A) | 12,244 |

Power rating (MW) | 306 | Rated frequency (Hz) | 50 |

Turbine runner diameter (m) | 5.26 | Power factor | 0.90 |

100% guide-vane opening (°) | 43.01 | Moment of inertia (ton/m^{2}) | 19,300 |

Number | Length (m) | Diameter (m) | Wave Velocity (m/s) | Roughness |
---|---|---|---|---|

L_{1} | 1113.94 | 9.21 | 1100 | 0.014 |

L_{2} | 206.71 | 8.97 | 1120 | 0.014 |

L_{3} | 250.77 | 4.77 | 1204 | 0.011 |

L_{4} | 173.52 | 6.90 | 1161 | 0.010 |

L_{5} | 260.30 | 4.80 | 1204 | 0.011 |

L_{6} | 173.52 | 6.89 | 1160 | 0.012 |

L_{7} | 295.27 | 10.82 | 1050 | 0.014 |

Sectional Area of the Impedance Hole (m^{2}) | Inflow Loss Coefficient | Outflow Loss Coefficient | Sectional Area (m^{2}) | Altitude (m) |
---|---|---|---|---|

19.63 | 0.0009217 | 0.0006767 | 19.63 | 231.70~268.30 |

380.13 | 268.30~310.00 | |||

530.93 | 310.00~320.00 |

Excitation System | Synchronous Machine | ||||
---|---|---|---|---|---|

T_{a} | 0.001 | R_{a} | 0.00125 | X^{’’}_{d} | 0.2 |

T_{b} | 0 | X_{d} | 1.015 | X^{’’}_{q} | 0.195 |

T_{c} | 0 | X_{q} | 0.627 | T^{’}_{d0} | 12.6 |

T_{e} | 0 | X^{’}_{d} | 0.253 | T^{’’}_{d0} | 0.189 |

T_{f} | 0.1 | T_{j} | 10.8 | T^{’’}_{q0} | 0.519 |

K_{a} | 300 | Speed Regulation System | |||

K_{e} | 1 | b_{p} | 0.01 | k_{0} | 1 |

K_{f} | 0.001 | T_{y}_{1} | 0.02 | T_{y} | 0.2 |

**Table A5.**The detailed parameters corresponding to the Pareto front solutions of constant excitation current.

Solutions | ${\mathit{F}}_{1}$ | ${\mathit{F}}_{2}\left(\mathbf{s}\right)$ | ${\mathit{i}}_{\mathit{g}}^{*}$ | ${\mathit{i}}_{\mathit{m}}^{*}$ | ${\mathit{y}}_{\mathit{c}}$ | K_{p} | K_{i} | K_{d} |
---|---|---|---|---|---|---|---|---|

1 | 0.0000 | 61.1400 | 0.9805 | 0.6135 | 0.1923 | 3.4725 | 0.1000 | 5.1410 |

2 | 0.0000 | 60.9600 | 0.9881 | 0.6272 | 0.1924 | 3.5060 | 0.1000 | 4.9311 |

3 | 0.0001 | 60.8600 | 1.0098 | 0.6266 | 0.1932 | 3.4515 | 0.1000 | 5.0734 |

4 | 0.0002 | 60.3600 | 1.0188 | 0.5930 | 0.1938 | 3.6432 | 0.1000 | 5.2704 |

5 | 0.0003 | 60.2800 | 1.0329 | 0.6127 | 0.1942 | 3.5734 | 0.1000 | 5.1753 |

6 | 0.0003 | 60.2200 | 1.0125 | 0.6208 | 0.1945 | 3.5620 | 0.1000 | 5.3608 |

7 | 0.0003 | 60.0400 | 0.9989 | 0.6175 | 0.1944 | 3.5887 | 0.1000 | 5.1362 |

8 | 0.0004 | 59.8600 | 1.0071 | 0.5973 | 0.1949 | 3.5649 | 0.1000 | 5.1444 |

9 | 0.0005 | 59.7000 | 1.0080 | 0.6061 | 0.1954 | 3.3926 | 0.1000 | 4.9045 |

10 | 0.0006 | 59.3400 | 0.9943 | 0.6156 | 0.1964 | 3.5366 | 0.1000 | 5.3577 |

11 | 0.0007 | 59.2200 | 0.9977 | 0.6136 | 0.1962 | 3.4684 | 0.1000 | 4.9254 |

12 | 0.0007 | 59.1400 | 0.9970 | 0.6167 | 0.1970 | 3.4364 | 0.1000 | 5.1984 |

13 | 0.0008 | 59.0000 | 1.0160 | 0.6366 | 0.1971 | 3.5003 | 0.1000 | 5.0073 |

14 | 0.0009 | 58.8800 | 1.0035 | 0.6084 | 0.1968 | 3.3919 | 0.1000 | 4.6497 |

15 | 0.0010 | 58.6400 | 1.0143 | 0.6353 | 0.1984 | 3.3848 | 0.1000 | 5.1553 |

16 | 0.0010 | 58.5200 | 1.0118 | 0.6244 | 0.1980 | 3.3945 | 0.1000 | 4.7860 |

17 | 0.0011 | 58.4600 | 1.0115 | 0.6289 | 0.1988 | 3.3993 | 0.1000 | 5.1672 |

18 | 0.0012 | 58.1600 | 1.0112 | 0.6271 | 0.1997 | 3.3300 | 0.1000 | 5.1576 |

19 | 0.0012 | 57.8800 | 1.0027 | 0.6149 | 0.1999 | 3.4411 | 0.1000 | 5.2033 |

20 | 0.0015 | 57.7200 | 1.0255 | 0.6064 | 0.2001 | 3.2400 | 0.1000 | 4.5951 |

21 | 0.0016 | 57.2200 | 0.9945 | 0.6315 | 0.2014 | 3.3322 | 0.1000 | 4.9067 |

22 | 0.0018 | 57.0600 | 1.0057 | 0.6203 | 0.2022 | 3.2075 | 0.1000 | 4.8830 |

23 | 0.0018 | 56.8000 | 1.0073 | 0.6276 | 0.2028 | 3.3360 | 0.1000 | 5.0474 |

24 | 0.0020 | 56.7800 | 0.9945 | 0.6364 | 0.2038 | 3.3257 | 0.1000 | 5.1467 |

25 | 0.0020 | 56.7400 | 1.0045 | 0.6287 | 0.2029 | 3.1622 | 0.1000 | 4.6525 |

**Table A6.**The detailed parameters corresponding to the Pareto front solutions of constant excitation voltage.

Solutions | ${\mathit{F}}_{1}$ | ${\mathit{F}}_{2}\left(\mathbf{s}\right)$ | ${\mathit{V}}_{\mathit{g}}^{*}$ | ${\mathit{V}}_{\mathit{m}}^{*}$ | ${\mathit{y}}_{\mathit{c}}$ | K_{p} | K_{i} | K_{d} |
---|---|---|---|---|---|---|---|---|

1 | 0.0000 | 58.9400 | 0.6049 | 0.4437 | 0.1929 | 3.5941 | 0.1000 | 5.0779 |

2 | 0.0001 | 58.8800 | 0.5815 | 0.4172 | 0.1928 | 3.4206 | 0.1000 | 4.6254 |

3 | 0.0001 | 58.7200 | 0.5967 | 0.4302 | 0.1940 | 3.5515 | 0.1000 | 5.3448 |

4 | 0.0002 | 58.3600 | 0.6055 | 0.4150 | 0.1949 | 3.6035 | 0.1000 | 5.4766 |

5 | 0.0003 | 58.2200 | 0.5762 | 0.4380 | 0.1946 | 3.5755 | 0.1000 | 5.1809 |

6 | 0.0004 | 57.9400 | 0.5310 | 0.4862 | 0.1955 | 3.6367 | 0.1000 | 5.3869 |

7 | 0.0004 | 57.9200 | 0.5631 | 0.4490 | 0.1956 | 3.3992 | 0.1000 | 4.9775 |

8 | 0.0005 | 57.8800 | 0.5369 | 0.4775 | 0.1963 | 3.4578 | 0.1000 | 5.4568 |

9 | 0.0005 | 57.6000 | 0.5851 | 0.4409 | 0.1962 | 3.5772 | 0.1000 | 5.2510 |

10 | 0.0006 | 57.4600 | 0.5933 | 0.4596 | 0.1962 | 3.5212 | 0.1000 | 4.9430 |

11 | 0.0007 | 57.2600 | 0.6137 | 0.4421 | 0.1975 | 3.5201 | 0.1000 | 5.4057 |

12 | 0.0007 | 57.1400 | 0.5369 | 0.4708 | 0.1974 | 3.4673 | 0.1000 | 5.1295 |

13 | 0.0008 | 57.0000 | 0.5759 | 0.4408 | 0.1976 | 3.4549 | 0.1000 | 5.0178 |

14 | 0.0008 | 56.9400 | 0.5776 | 0.4511 | 0.1979 | 3.4562 | 0.1000 | 5.1085 |

15 | 0.0009 | 56.8600 | 0.5853 | 0.4415 | 0.1986 | 3.3694 | 0.1000 | 5.2359 |

16 | 0.0010 | 56.6200 | 0.5782 | 0.5092 | 0.1990 | 3.5058 | 0.1000 | 5.3278 |

17 | 0.0010 | 56.5000 | 0.5681 | 0.4310 | 0.1991 | 3.4715 | 0.1000 | 5.2946 |

18 | 0.0010 | 56.4200 | 0.5768 | 0.4593 | 0.1992 | 3.5151 | 0.1000 | 5.2587 |

19 | 0.0012 | 56.0000 | 0.5805 | 0.4346 | 0.2003 | 3.4980 | 0.1000 | 5.2770 |

20 | 0.0014 | 55.7800 | 0.5440 | 0.4646 | 0.2010 | 3.3650 | 0.1000 | 5.1138 |

21 | 0.0015 | 55.5800 | 0.5772 | 0.4403 | 0.2013 | 3.3715 | 0.1000 | 5.0147 |

22 | 0.0016 | 55.3200 | 0.5720 | 0.4422 | 0.2023 | 3.3496 | 0.1000 | 5.1389 |

23 | 0.0016 | 55.2200 | 0.5543 | 0.4567 | 0.2024 | 3.3941 | 0.1000 | 5.1392 |

24 | 0.0018 | 55.0600 | 0.5587 | 0.4434 | 0.2027 | 3.2578 | 0.1000 | 4.8341 |

25 | 0.0020 | 54.8600 | 0.5710 | 0.4508 | 0.2037 | 3.1818 | 0.1000 | 4.8972 |

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**Figure 3.**Characteristic curves of a pump turbine. (

**a**) Flow characteristic curve. (

**b**) Moment characteristic curve.

**Figure 8.**Simulation results of PSU-1 during a single-unit load rejection. (

**a**) Guide vane opening. (

**b**) Rotational speed. (

**c**) Volute water pressure. (

**d**) Draft tube water pressure.

**Figure 9.**Successful BTBS of constant excitation current ${i}_{fd}^{g*}=1.2,{i}_{fd}^{m*}=0.6$. (

**a**) Stator’s current; (

**b**) Rotor speed; (

**c**) Field current; (

**d**) Rotor angle difference.

**Figure 10.**Unsuccessful BTBS of constant excitation current ${i}_{fd}^{g*}=0.8,{i}_{fd}^{m*}=0.5$. (

**a**) Stator’s current; (

**b**) Rotor speed; (

**c**) Field current; (

**d**) Rotor angle difference.

**Figure 12.**Simulation results of BTBS with CEC. (

**a**) Excitation current; (

**b**) Rotor speed; (

**c**) Rotor angle difference.

**Figure 13.**Simulation results of BTBS with CEV. (

**a**) Excitation current; (

**b**) Rotor speed; (

**c**) Rotor angle difference.

**Figure 15.**Dynamic process trajectory of moment and torque under different water heads. (

**a**) Dynamic trajectory of flow; (

**b**) Dynamic trajectory of the moment.

**Figure 16.**The critical curves of the stable domain and S-shaped region. (

**a**) Flow critical curve; (

**b**) Moment critical curve.

**Figure 21.**BTBS process of four schemes. (

**a**) Rotor speed of PSU-1; (

**b**) Rotor speed of PSU-2; (

**c**) Field current of PSU-1; (

**d**) Field current of PSU-2; (

**e**) Guide vane opening; (

**f**) Rotor angle difference; (

**g**) Terminal voltage; (

**h**) Active power.

**Figure 23.**BTBS process of three typical particles. (

**a**) Rotor speed of PSU-1; (

**b**) Rotor speed of PSU-2; (

**c**) Flow trajectory; (

**d**) Moment trajectory.

**Figure 24.**The MOCEV result of the optimal solution under N0–N5. (

**a**) Maximum rotor speed; (

**b**) Speed stable time; (

**c**) Speed rising time; (

**d**) Flow trajectory.

Unit Number | Category | Maximum Pressure at Measuring Point of Volute Inlet | Minimum Water Pressure at MEASURING Point of Draft Tube | Maximum Speed |
---|---|---|---|---|

1# | Measurement | 299.32 m | 26.6 m | 140% |

Refined model | 297.05 m | 27.9 m | 136% | |

Absolute error | −2.27 m | 1.3 m | −4% | |

Relative error | −0.75% | 4.88% | 2.85% |

Initial Rotor Angel Difference (°) | Start Time of Speed Rise (s) | Rotor Angle Difference at Steady State (°) | Description of BTBS |
---|---|---|---|

0 | 3 | 17.66 | Successful start; Slight oscillation |

−90 | 8 | 19.24 | Successful start; Slight oscillation |

−180 | 10 | 20.89 | Successful start; Moderate oscillation |

−270 | 10 | 21.19 | Successful start; Moderate oscillation |

Decision Variables | Boundaries | Values | |||||
---|---|---|---|---|---|---|---|

X_{1} | L_{1} | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0 |

U_{1} | 2.0 | 2.0 | 0.4 | 6 | 1 | 6 | |

X_{2} | L_{2} | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0 |

U_{2} | 2.0 | 2.0 | 0.4 | 6 | 1 | 6 |

Schemes | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ |
---|---|---|---|

MOCEC | (0.90, 0.10) | (0.53, 0.47) | (0.91, 0.09) |

MOCEV | (0.90, 0.10) | (0.63, 0.36) | (0.94, 0.06) |

Schemes | SOCEC | SOCEV | MOCEC | MOCEV | |
---|---|---|---|---|---|

Variables | |||||

${i}_{g}^{*}({V}_{g}^{*})$ | 1.2331 | 0.6599 | 0.9881 | 0.6049 | |

${i}_{m}^{*}({V}_{m}^{*})$ | 0.7422 | 0.5413 | 0.6272 | 0.4437 | |

y_{c} | 0.3985 | 0.3976 | 0.1924 | 0.1929 | |

K_{p} | 3.9083 | 4.6298 | 3.5060 | 3.5941 | |

K_{i} | 0.3630 | 0.3900 | 0.1000 | 0.1000 | |

K_{d} | 0.3190 | 0.7031 | 4.9311 | 5.0779 |

Schemes | SOCEC | SOCEV | MOCEC | MOCEV | |||||
---|---|---|---|---|---|---|---|---|---|

Indexes | PSU-1 | PSU-2 | PSU-1 | PSU-2 | PSU-1 | PSU-2 | PSU-1 | PSU-2 | |

$\Delta {\omega}_{\mathrm{max}}$ | 0.0515 | 0.0518 | 0.0468 | 0.0472 | 2.7796 × 10^{−5} | 4.1466 × 10^{−5} | 8.7973 × 10^{−6} | 2.2633 × 10^{−5} | |

t_{r} (s) | 28.20 | 28.18 | 26.84 | 26.82 | 50.74 | 50.74 | 48.86 | 48.84 | |

t_{s} (s) | 90.68 | 90.68 | 99.30 | 99.30 | 60.98 | 60.94 | 59.00 | 58.98 |

Working Conditions | Water Level | Water Head (m) | |
---|---|---|---|

Upstream (m) | Downstream (m) | ||

N0 | 291 | 106 | 185 |

N1 | 295 | 105 | 190 |

N2 | 298 | 103 | 195 |

N3 | 303 | 103 | 200 |

N4 | 303 | 98 | 205 |

N5 | 308 | 98 | 210 |

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## Share and Cite

**MDPI and ACS Style**

Feng, C.; Li, G.; Zheng, Y.; Zhou, D.; Mai, Z.
Multi-Objective Optimization of Back-to-Back Starting for Pumped Storage Plants under Low Water Head Conditions Based on the Refined Model. *Sustainability* **2022**, *14*, 10289.
https://doi.org/10.3390/su141610289

**AMA Style**

Feng C, Li G, Zheng Y, Zhou D, Mai Z.
Multi-Objective Optimization of Back-to-Back Starting for Pumped Storage Plants under Low Water Head Conditions Based on the Refined Model. *Sustainability*. 2022; 14(16):10289.
https://doi.org/10.3390/su141610289

**Chicago/Turabian Style**

Feng, Chen, Guilin Li, Yuan Zheng, Daqing Zhou, and Zijun Mai.
2022. "Multi-Objective Optimization of Back-to-Back Starting for Pumped Storage Plants under Low Water Head Conditions Based on the Refined Model" *Sustainability* 14, no. 16: 10289.
https://doi.org/10.3390/su141610289