# Adaptively Learned Modeling for a Digital Twin of Hydropower Turbines with Application to a Pilot Testing System

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## Abstract

**:**

## 1. Introduction

## 2. Turbine System Model

#### 2.1. Turbine Speed (Frequency) Control System

#### 2.2. Torque and Water Flow Module

#### 2.3. State Space Model for Non-Elasticity Water System Dynamics

^{2}·s. By integrating the linearized Equations (3)–(9) into a state space format, the following state space model can be readily obtained.

#### 2.4. State Space Model for Elastic Water Flows

#### 2.5. PID Controller for Shaft Speed

#### 2.6. Discretization

#### 2.7. Hydraulic Servo for the Guide Vane Opening

## 3. Discretized Input and Output Models

#### 3.1. Input and Output Model for Nonelastic Water Flow

#### 3.2. Input and Output Model for Elastic Water Flow

## 4. Least Squares Adaptive Learning Scheme Using Real-Time Data for Elastic Water Flows Case

## 5. Experimental and Data Processing

^{3}/s during testing and data collection.

#### 5.1. Learning of Input and Output Models for Elastic Case

#### 5.2. Direct Estimate of the Six Coefficients

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- ARUP. Digital Twin: Towards a Meaningful Framework; ARUP: London, UK, 2019. [Google Scholar]
- Parrott, A.; Warshaw, L. Industry 4.0 and the Digital Twin: Manufacturing Meets Its Match; Deloitte: Olathe, KS, USA, 2017. [Google Scholar]
- Tao, F.; Zhang, H.; Liu, A.; Nee, A.Y.C. Digital Twin in Industry: State-of-the-Art. IEEE Trans. Ind. Inform.
**2019**, 15, 2405–2415. [Google Scholar] [CrossRef] - Vachálek, J.; Bartalský, L.; Rovný, O.; Šišmišová, D.; Morháč, M.; Lokšík, M. The digital twin of an industrial production line within the industry 4.0 concept. In Proceedings of the 2017 21st International Conference on Process Control (PC), Štrbské Pleso, Slovakia, 6–9 June 2017; pp. 258–262. [Google Scholar]
- Knapp, G.L.; Mukherjee, T.; Zuback, J.S.; Wei, H.L.; Palmer, T.A.; De, A.; DebRoy, T. Building blocks for a digital twin of additive manufacturing. Acta Mater.
**2017**, 135, 390–399. [Google Scholar] [CrossRef] - Costello, K.; Omale, G. Gartner Survey Reveals Digital Twins Are Entering Mainstream Use. Available online: https://www.gartner.com/en/newsroom/press-releases/2019-02-20-gartner-survey-reveals-digital-twins-are-entering-mai (accessed on 13 October 2022).
- Goasduff, L. Confront Key Challenges to Boost Digital Twin Success. Available online: https://www.gartner.com/smarterwithgartner/confront-key-challenges-to-boost-digital-twin-success (accessed on 13 October 2022).
- Kosan, L. Digital Twin Technology: Where Are We Now? Available online: https://www.iotworldtoday.com/2019/05/08/digital-twin-technology-where-are-we-now/ (accessed on 13 October 2022).
- Martynova, O. Digital Twin Technology: A Guide for Innovative Technology. Available online: https://intellias.com/digital-twin-technology-guide/ (accessed on 13 October 2022).
- Lund, A.M.; Mochel, K.; Lin, J.-W.; Onetto, R.; Srinivasan, J.; Gregg, P.; Bergman, J.E.; Hartling, K.D.; Ahmed, J.A.; Chotai, S. Digital Wind Farm System. US Patent No. US20160333855A1, 17 November 2016. [Google Scholar]
- Water Power Technologies Office. Water Power Technologies Office Releases First Multi-Year Program Plan; Water Power Technologies Office: Washington, DC, USA, 2022.
- Wang, H.; Ahmed, O.; Smith, B.T.; Bellgraph, B. Developing a digital twin for hydropower systems—An open platform framework. Int. Water Power Dam Constr. Mag.
**2021**, 81, 24–25. [Google Scholar] - Wang, H.; Liu, Y.Q.; You, D.H. Application of a Nonlinear Self-tuning Controller for Regulating the Speed of a Hydraulic Turbine. J. Dyn. Syst. Meas. Control
**1991**, 113, 541–544. [Google Scholar] [CrossRef] - Giosio, D.R.; Henderson, A.D.; Walker, J.M.; Brandner, P.A. Physics-Based Hydraulic Turbine Model for System Dynamic Studies. IEEE Trans. Power Syst.
**2017**, 32, 1161–1168. [Google Scholar] [CrossRef] - Pennacchi, P.; Chatterton, S.; Vania, A. Modeling of the dynamic response of a Francis turbine. Mech. Syst. Signal Process.
**2012**, 29, 107–119. [Google Scholar] [CrossRef] - Gracino, R.; Hansen, V.; Goia, L.; Campo, A.; Campos, B. System Identification of a Small Hydropower Plant. In Proceedings of the 2021 14th IEEE International Conference on Industry Applications (INDUSCON), São Paulo, Brazil, 15–18 August 2021; pp. 1430–1434. [Google Scholar]
- Jakobsen, S.H.; Bombois, X.; Uhlen, K. Non-intrusive identification of hydro power plants’ dynamics using control system measurements. Int. J. Electr. Power Energy Syst.
**2020**, 122, 106180. [Google Scholar] [CrossRef] - Adedayo, O.O.; Gbadamosi, S.L.; Ale, D.T. Neural Network Predictive Controller for Improved Operational Efficiency of Shiroro Hydropower Plant. Int. J. Sci. Eng. Res.
**2015**, 6, 1454–1459. [Google Scholar] - Liu, D.; Xiao, Z.; Li, H.; Liu, D.; Hu, X.; Malik, O.P. Accurate Parameter Estimation of a Hydro-Turbine Regulation System Using Adaptive Fuzzy Particle Swarm Optimization. Energies
**2019**, 12, 3903. [Google Scholar] [CrossRef] - Zeng, Y.; Zhang, L.X.; Qian, J.; Guo, Y.K.; Xu, T.M. Additional Mechanical Torque Coefficients of Hydro Turbine and Governor System. Adv. Mater. Res.
**2012**, 443–444, 954–961. [Google Scholar] [CrossRef] - Li, H.; Chen, D.; Zhang, H.; Wang, F.; Ba, D. Nonlinear modeling and dynamic analysis of a hydro-turbine governing system in the process of sudden load increase transient. Mech. Syst. Signal Process.
**2016**, 80, 414–428. [Google Scholar] [CrossRef] - Fang, H.; Shen, Z. Modeling and Simulation of Hydraulic Transients for Hydropower Plants. In Proceedings of the 2005 IEEE/PES Transmission & Distribution Conference & Exposition: Asia and Pacific, Dalian, China, 18 August 2005; pp. 1–4. [Google Scholar]
- Yang, W. Hydropower Plants and Power Systems: Dynamic Processes and Control for Stable and Efficient Operation. Doctoral Dissertation, Electricity, Department of Engineering Sciences, Technology, Disciplinary Domain of Science and Technology, Uppsala University, Uppsala, Sweden, 2017. [Google Scholar]
- Mu, C.; Wang, K.; Ni, Z. Adaptive learning and sampled-control for nonlinear game system using dynamic event-triggering strategy. IEEE Trans. Neural Netw. Learn. Syst.
**2022**, 33, 4437–4450. [Google Scholar] [CrossRef] [PubMed] - Wang, Q.; Psillakis, H.E.; Sun, C.; Lewis, F.L. Adaptive NN distributed control for time-varying networks of nonlinear agents with antagonistic interactions. IEEE Trans. Neural Netw. Learn. Syst.
**2021**, 32, 2573–2583. [Google Scholar] [CrossRef] [PubMed]

**Figure 6.**Data of normalized incremental responses of shaft speed, water flow rate, water head, torque, and guide vane opening collected for Tests 1 and 2.

**Figure 7.**Shaft speed $\omega $ in rmp, its estimated value $\widehat{\omega}=(1+\widehat{x}){\omega}_{0}$, and the guide vane opening $\u2206u$ in its normalized incremental value.

**Figure 8.**Response of the shaft speed modeling error ${e}_{speed}\left(k\right)$ when the adaptive learning Equations (57)–(59) is progressing for Tests 1 and 2.

**Figure 11.**Actual system responses of shaft speed, water flow rate, pressure, guide vane opening, and torque all in the normalized incremental sense as in Equation (3).

**Figure 12.**

**Top**: actual and estimated shaft speeds in rpm.

**Bottom**: actual guide vane opening in its normalized incremental values in the closed loop shaft speed control mode.

**Figure 13.**Actual and estimated water flow rate in the normalized incremental values. The blue line stands for real data and the red line for the model output.

Symbol | Description |
---|---|

$a$ | grid interface parameter |

$b$ | turbine interface parameter |

${e}_{g}$ | damping ratio contributed by the power grid |

${e}_{speed}$ | tracking error of the incremental speed control for the incremental shaft speed |

${e}_{qx}$, ${e}_{qu}$, ${e}_{qh}$ | linearized coefficients for the water flow rate ($q$) calculated at a fixed operating point O |

${e}_{x}$, ${e}_{u}$, ${e}_{h}$ | linearized coefficients for the turbine torque ($m$) calculated at a fixed operating point O |

$h$ | normalized incremental water head |

$H$ | water head (time-variant) |

${H}_{0}$ | average water head |

$J$ | equivalent inertia |

${K}_{I}$ | integral gain in the speed controller |

${K}_{I,v}$ | integral gain in the voltage controller |

${K}_{p}$ | proportional gain in the speed controller |

$l$ | length of the water pipeline |

$L$ | equivalent load when connected to the grid |

$m$ | normalized turbine torque |

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

$M$ | turbine torque (time-variant) |

${M}_{0}$ | average turbine torque |

$O$ | a selected and fixed operating point for the hydropower generation unit connected to the grid |

$q$ | normalized incremental water flow rate |

$Q$ | water flow rate (time-variant) |

${Q}_{0}$ | average water flow rate |

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

$u$ | guide vane opening (time-variant) |

${u}_{0}$ | average guide vane opening |

${u}_{p}$ | set point of active power, which reflects the power demand from the grid |

${u}_{s}$ | output of the governor (speed controller) |

$x$ | normalized incremental turbine shaft speed |

$\tau $ | time interval, 0.2 s |

$\omega $ | turbine shaft speed (time-variant) |

${\omega}_{0}$ | average turbine shaft speed |

$\mathsf{\Delta}u$ | normalized incremental guide vane opening angle |

$u$ | guide vane opening angle |

Variable | Unit | Average Sampling Frequency |
---|---|---|

Water head (H) | Pa | 5060/s |

Water flow rate (Q) | m^{3}/s | 5060/s |

Turbine torque (M) | N·m | 5060/s |

Load torque (L) | N·m | 5060/s |

Guide vane opening angle (u) | ° | 10/s |

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## Share and Cite

**MDPI and ACS Style**

Wang, H.; Ou, S.; Dahlhaug, O.G.; Storli, P.-T.; Skjelbred, H.I.; Vilberg, I.
Adaptively Learned Modeling for a Digital Twin of Hydropower Turbines with Application to a Pilot Testing System. *Mathematics* **2023**, *11*, 4012.
https://doi.org/10.3390/math11184012

**AMA Style**

Wang H, Ou S, Dahlhaug OG, Storli P-T, Skjelbred HI, Vilberg I.
Adaptively Learned Modeling for a Digital Twin of Hydropower Turbines with Application to a Pilot Testing System. *Mathematics*. 2023; 11(18):4012.
https://doi.org/10.3390/math11184012

**Chicago/Turabian Style**

Wang, Hong, Shiqi (Shawn) Ou, Ole Gunnar Dahlhaug, Pål-Tore Storli, Hans Ivar Skjelbred, and Ingrid Vilberg.
2023. "Adaptively Learned Modeling for a Digital Twin of Hydropower Turbines with Application to a Pilot Testing System" *Mathematics* 11, no. 18: 4012.
https://doi.org/10.3390/math11184012