# Fractional-PID and Its Parameter Optimization for Pumped Storage Units Considering the Complicated Conduit System

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Hydraulic Systems’ Modeling

#### 2.1.1. Overall Pipeline Modeling

- Upstream diversion tunnel:

- 2.
- Penstock:

- 3.
- Downstream tailwater tunnel

#### 2.1.2. Surge Tank Modeling

#### 2.2. Pump-Turbine Modeling

#### 2.3. Generator Modeling

#### 2.4. Modeling of Electro-Hydraulic Servomechanism

#### 2.5. PSGS Model

## 3. Fractional PID Algorithm of the Speed Governor

#### 3.1. Definition of Fractional Calculus

#### 3.2. FOPID Controller Structure

## 4. Improved Chaotic Gravitational Search Algorithm

#### 4.1. Standard Gravitational Search Algorithm

#### 4.2. Improved Gravitational Search Algorithm (IGSA)

#### 4.3. Objective Function

## 5. Simulation Results Analysis

#### 5.1. Simulation Parameter Setting

#### 5.2. Load Disturbance Scenario

#### 5.2.1. Comparison of SA, GA, PSO and IGSA

#### 5.2.2. Comparison of PID and FOPID

#### 5.3. Frequency Disturbance Scenario

#### 5.3.1. Comparison of SA, GA, PSO and IGSA

#### 5.3.2. Comparison of PID and FOPID

#### 5.4. Results Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

PSGS: | |

${T}_{w}$ | water inertia time constant |

${h}_{f}$ | head loss of the tunnel |

${T}_{w1},{T}_{w2},{T}_{w3},{T}_{w4}$ | water inertia time constant of diversion tunnel, penstock, tailrace tunnel and downstream tailrace tunnel |

${h}_{s}$ | relative water pressure deviation |

${h}_{s1},{h}_{s2}$ | relative water pressure deviation of upstream surge tank and downstream tailrace tunnel |

${h}_{1},{h}_{2},{h}_{3}$ | relative water pressure deviation of the diversion tunnel, penstock, tailrace tunnel |

${h}_{f1},{h}_{f2},{h}_{f3},{h}_{f4}$ | relative head loss of diversion tunnel, penstock, tailrace tunnel and downstream tailrace tunnel |

${q}_{s}$ | relative flow deviation |

${q}_{1}{,q}_{2},{q}_{3},{q}_{4}{,q}_{s1}{,q}_{s2}$ | relative flow deviation of diversion tunnel, penstock, tailrace tunnel, downstream tailrace tunnel, upstream surge tank and downstream surge tank |

$\u2206\mathrm{H}$ | water level in the surge tank changes |

$\u2206\mathrm{Q}$ | flow in and out of surge tank |

${A}_{s}$ | cross-sectional area of surge tank |

${T}_{j}$ | time constant of surge tank |

${T}_{j1},\text{}{T}_{j2}$ | time constant of upstream surge tank and downstream surge tank |

${m}_{t}$ | relative value of hydraulic turbine torque deviation |

${h}_{t}$ | relative value of hydraulic turbine head deviation |

${{q}_{t},q}_{t-1}$ | relative flow deviation of the pump-turbine at moments $t$ and $t-1$ |

$x,\text{}y$ | relative value of rotate speed deviation and guide vane opening deviation |

$y\text{}(\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{v}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{s}\mathrm{m})$ | stroke of the servo motor |

${e}_{x},{e}_{y},{e}_{h}$ | partial derivatives of the torque with respect to head, guide vane and turbine speed |

${e}_{qx},{e}_{qy},{e}_{qh}$ | partial derivatives of the flow with respect to head, guide vane and turbine speed |

${T}_{a}$ | time constant of mechanical inertia of the generator |

${m}_{g0}$ | load torque |

${e}_{g}$ | load self-regulation factor |

${e}_{n}$ | synthetic self-regulation coefficient |

$G{D}^{2}$ | flywheel torque of rotating part of the unit |

${n}_{r}$ | rated speed of the unit |

${P}_{r}$ | rated output of the unit |

$u$ | control signal output by the controller |

${T}_{y}$ | servomotor response time |

FOPID: | |

${}_{a}D_{t}^{\alpha}$ | operation basic operation operator of fractional calculus |

$a$$,\text{}t$ | upper and lower bounds of the operation operator |

$\alpha $ | order of the calculus |

$h$ | calculation step |

$\left[\frac{t-\alpha}{h}\right]$ | the largest integer smaller than the real number $\frac{t-\alpha}{h}$ |

${K}_{p}$ | proportional adjustment coefficient |

${K}_{i}$ | integral adjustment coefficient |

${K}_{d}$ | differential adjustment coefficient |

$e\left(t\right)$ | error signal |

$u\left(t\right)$ | controller output signal |

$\lambda $$,\text{}\mu $ | integral order and differential order |

s | the Laplace operator |

IGSA: | |

${x}_{i}^{d},{v}_{i}^{d}$ | the position and velocity components of individual $i$ in the d-dimensions |

${F}_{ij}^{d}\left(t\right)$ | gravitational force of individual j on $i$ |

$G\left(t\right)$$,\text{}{R}_{ij}\left(t\right)$ | the universal gravitational constants at the tth iteration and the Euclidean distances of individuals $i$ and j |

${G}_{0}$$,\text{}\alpha $$,\text{}\epsilon $ | gravitational constant |

$t$, $T$ | current number of iterations and the maximum number of iterations |

${rand}_{j}$ | a random variable obeying a uniform distribution between $\left[\mathrm{0,1}\right]$ |

$Kbest$ | the collection of the top $k$ individuals with the optimal fitness value and maximum quality |

${a}_{i}^{d}\left(t\right)$ | the acceleration of individual $i$ in the d-dimensions |

${r}_{t}$ | the number of chaos generated in the tth iteration |

$u$ | chaos control parameters |

${c}_{1}^{d}$ | a d-dimensional random vector |

$\xi $ | factor that controls the range of chaos |

${\alpha}_{0}$ | initial value of the gravitational decay factor |

$\overline{w}$, $\delta $ | scaling factors |

$\theta $ | shift factor |

$Ub\left(d\right)$, $Lb\left(d\right)$ | the upper and lower boundaries of the positions of the individuals in d-dimension |

${c}_{1}$, ${c}_{2}$ | adaptive learning factors |

${F}_{best}^{d}$, ${X}_{best}^{d}$ | individual optimal position and the global optimal position in the d-dimensions |

$f$ | fitness value |

${w}_{1}$,${w}_{2}$ | weighting factors |

$N$, $T$ | the number of samples and time sequence |

${h}_{tmin}$ | the minimum value of water pressure deviation |

## Appendix A

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

Rated rotation speed | 500 r/min |

Power frequency | 50 Hz |

Maximum head | 565 m |

Rated head | 540 m |

Minimum head | 526 m |

Rated discharge | 62.09 m^{3}/s |

100% GVO | 20.47° |

Rated Power output | 300 MW |

## Appendix B

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

Rated capacity (generator) | 334 MVA |

Rated power (generator) | 300 MW |

Rated capacity (electric motor) | 338 MVA |

Rated power (electric motor) | 325 MW |

Rated voltage | 15.75 kV |

Power factor (generator) | 0.9 |

Power factor (electric motor) | 0.975 |

Rated rotation speed | 500 rpm |

Number of rotor magnetic poles | 12 (Pole logarithm 6) |

Rated frequency | 50 Hz |

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**Figure 2.**Block diagram of the transfer function of the pumped storage power plant governing system.

**Table 1.**HTGS simulation parameters [44].

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

$\mathrm{Water}\text{}\mathrm{inertia}\text{}\mathrm{time}\text{}\mathrm{constant}\text{}{T}_{w1},{T}_{w2},{T}_{w3},{T}_{w4}$ | 0.6, 0.5, 0.5, 0.6 |

$\mathrm{Head}\text{}\mathrm{loss}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{tunnel}\text{}{h}_{f1},{h}_{f2},{h}_{f3},{h}_{f4}$ | 0.0026, 0.003, 0.003, 0.0015 |

$\mathrm{Time}\text{}\mathrm{constant}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{surge}\text{}\mathrm{tan}\mathrm{k}\text{}{T}_{j1},{T}_{j2}$ | 10, 10 |

$\mathrm{Time}\text{}\mathrm{constant}\text{}\mathrm{of}\text{}\mathrm{mechanical}\text{}\mathrm{inertia}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{generator}\text{}{T}_{a}$ | 3.4 |

$\mathrm{Load}\text{}\mathrm{self}-\mathrm{regulation}\text{}\mathrm{factor}\text{}{e}_{g}$ | 0.05 |

$\mathrm{Servomotor}\text{}\mathrm{response}\text{}\mathrm{time}\text{}{T}_{y}$ | 0.2 |

${\mathit{e}}_{\mathit{x}}$ | ${\mathit{e}}_{\mathit{y}}$ | ${\mathit{e}}_{\mathit{h}}$ | ${\mathit{e}}_{\mathit{q}\mathit{x}}$ | ${\mathit{e}}_{\mathit{q}\mathit{y}}$ | ${\mathit{e}}_{\mathit{q}\mathit{h}}$ |
---|---|---|---|---|---|

−0.45 | 1.0 | 1.5 | 0.0 | 1.0 | 0.5 |

${\mathit{K}}_{\mathit{p}}$ | ${\mathit{K}}_{\mathit{i}}$ | ${\mathit{K}}_{\mathit{d}}$ | $\mathit{\lambda}$ | $\mathit{\mu}$ | |
---|---|---|---|---|---|

FOPID | [0, 5] | [0, 5] | [0, 5] | [0, 2] | [0, 2] |

PID | [0, 5] | [0, 5] | [0, 5] | - | - |

**Table 4.**Dynamic indicators of SA, GA, PSO and IGSA optimal control parameters under load disturbance.

$\mathit{f}$ | Overshoot | Stabilization Time | $\left|{\mathit{h}}_{\mathit{m}\mathit{i}\mathit{n}}\right|$ | |
---|---|---|---|---|

SA | 0.0509 | -- | 45.0 | 0.034 |

GA | 0.1230 | -- | 60.6 | 0.036 |

PSO | 0.0482 | -- | 40.1 | 0.033 |

IGSA | 0.0452 | -- | 21.2 | 0.029 |

$\mathit{f}$ | Overshoot | Stabilization Time | $\left|{\mathit{h}}_{\mathit{m}\mathit{i}\mathit{n}}\right|$ | |
---|---|---|---|---|

PID | 0.0505 | -- | 40.2 | 0.0332 |

FOPID | 0.0452 | -- | 21.2 | 0.0290 |

**Table 6.**Dynamic indicators of SA, GA, PSO and IGSA optimal control parameters under frequency disturbance.

$\mathit{f}$ | Overshoot | Stabilization Time | $\left|{\mathit{h}}_{\mathit{m}\mathit{i}\mathit{n}}\right|$ | |
---|---|---|---|---|

SA | 0.0587 | 23.26% | 44.85 | 0.0180 |

GA | 0.1667 | 42.22% | 67.70 | 0.0213 |

PSO | 0.0319 | 10.14% | 32.10 | 0.0140 |

IGSA | 0.0254 | 2.28% | 17.05 | 0.0084 |

**Table 7.**Dynamic indicators of PID and FOPID optimal control parameters under frequency disturbance.

$\mathit{f}$ | Overshoot | Stabilization Time | $\left|{\mathit{h}}_{\mathit{m}\mathit{i}\mathit{n}}\right|$ | |
---|---|---|---|---|

PID | 0.0452 | 45.98% | 30.75 | 0.0339 |

FOPID | 0.0254 | 2.28% | 17.05 | 0.0084 |

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## Share and Cite

**MDPI and ACS Style**

Zhou, X.; Zheng, Y.; Xu, B.; Liu, W.; Zou, Y.; Chen, J.
Fractional-PID and Its Parameter Optimization for Pumped Storage Units Considering the Complicated Conduit System. *Water* **2023**, *15*, 3851.
https://doi.org/10.3390/w15213851

**AMA Style**

Zhou X, Zheng Y, Xu B, Liu W, Zou Y, Chen J.
Fractional-PID and Its Parameter Optimization for Pumped Storage Units Considering the Complicated Conduit System. *Water*. 2023; 15(21):3851.
https://doi.org/10.3390/w15213851

**Chicago/Turabian Style**

Zhou, Xuan, Yang Zheng, Bo Xu, Wushuang Liu, Yidong Zou, and Jinbao Chen.
2023. "Fractional-PID and Its Parameter Optimization for Pumped Storage Units Considering the Complicated Conduit System" *Water* 15, no. 21: 3851.
https://doi.org/10.3390/w15213851