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This paper presents dynamic modelling of a Francis turbine with a surge tank and the control of a hydro power plant (HPP). Non-linear and linear models include technical parameters and show high similarity to measurement data. Turbine power control with an internal model control (IMC) is proposed, based on a turbine fuzzy model. Considering appropriate control responses in the entire area of turbine power, the model parameters of the process are determined from a fuzzy model, which are further included in the internal model controller. The results are compared to a proportional-integral (PI) controller tuned with an integral absolute error (IAE) objective function, and show an improved response of internal model control.

In HPPs, a digital turbine governor is an indispensable part of the control system. The stability of frequency, active power control, water flow control, turbine start-up procedure and emergency shut-down implemented with algorithms are major functions of digital turbine governor. Connected to Supervisory Control and Data Acquisition (SCADA), the unit controller, the excitation system and the digital voltage controller, it enables the operator to change the operating states of generator,

With reference to achieving suitable control results and exploring the dynamic responses of a hydro power plant unit, it is necessary to obtain a mathematical hydraulic model of the hydro turbine and other parts of water system. Different types of hydraulic non-linear models were proposed in [

Many recommendations of modelling, design and testing control systems for hydraulic turbines are described in international standards [

In [

In this paper, the focus is on mathematical modelling and the fuzzy power control approach of a Francis turbine of Aggregate 1 at hydro power plant (HPP) Moste. In the second section, the paper includes a comparison of non-linear and linearised models, with and without the surge tank of the Francis Turbine 1 in HPP Moste. In the third section, a fuzzy model of the Francis turbine has been applied based on a first order linearised model. Section 4 presents a fuzzy IMC controller with tuning parameter

Slovenian hydro power production consists of three HPP chains placed on the Sava, Soca and Drava Rivers. As the first HPP on the Sava River chain with a unique accumulative operation type, HPP Moste plays an important role in the Slovenian hydro power plant generation portfolio.

HPP Moste is part of the HPP Moste and HPP Zavrsnica water system depicted in ^{3} of useful volume of water storage, a −6.25 m water level deviation is allowed. After the water intake object, there is a surge tank connected by a tunnel. The pressure penstock connects the surge tank with a powerhouse, comprised of two Francis turbines (HPP Moste) and one Francis turbine (HPP Zavrsnica) with a joint tail-water conduit.

Turbine dynamics with the penstock are derived from second Newton's law [^{3}/s, with the penstock area section ^{2} and length _{0} is the static head of the water column; _{f}_{sp}

With reference to water inertia time [

In _{w}_{i}_{i}_{r}_{r}_{f}_{sp}

Turbine power _{m}_{nl}_{t}_{m}_{,}_{r}_{g}_{,}_{r}_{r}_{r}

With purpose of analysing dynamic responses, linearisation in the surrounding area of operating point

We obtain linearised

The linearisation process for all relations between _{m}

In connection to negative numerator coefficient _{0} in _{m}_{f}_{sp}

In high-head HPPs, the surge tank is often included in the water system for providing the limits of water pressure fluctuation and water speed decreases in water tunnels and penstocks. There are some other important functions of surge tanks that influence the water system [

With reference to model improvement, the surge tank with _{s}_{w}_{T}_{s}_{w}

The surge tank implemented on HPP Moste has two extended horizontal chambers with enlarged upper and bottom surface areas, _{1_kom} and _{3_kom}. Therefore, three different areas _{w}_{1_kom}, _{2_kom} and _{3_kom}) were added to the model, depending on absolute surge tank water level calculated with relation:
_{abs}_{b}_{,max} is the maximum available head and _{abs}

The friction loss of head _{s}_{fr}_{p}_{2} and _{p}_{3} in the water tunnel are presented in _{0} is the intake loss coefficient. The net head of surge tank is expressed in p.u. with _{0} is static head of water column; _{w}_{1} is the water inertia time of the water passages before the surge tank intake and _{w}_{2} is the water inertia time between the surge tank and the Francis turbine. A dynamic system with differential _{m}_{m}_{m}

Coefficients of the numerator, denominator and all other parameters in simulation are related to the real system (Francis turbine 1 HPP Moste with penstock and surge tank) and are described in

With the real ramp reference (slower opening rate) obtained from the HPP measurement (from 5 to 6 MW), the transient response is practically without undershot (

Surge tank level _{vod}_{w}

In connection with the high similarity of transient responses (models with/without surge tank) depicted in _{1}, _{2} and gain

Due to the nonlinearity of the process, a fuzzy model was realised to determine the parameters in the middle sections of the working points. Membership functions, describing degrees of membership for input parameter _{m}_{1} and _{2} are presented in

Membership functions with triangular and trapezoidal shapes were determined based on the knowledge acquired from operating experiences and intuitiveness. Fuzzy rules added to the fuzzy model are presented with rule list interpretation:
_{m}_{1}_{2}_{m}_{1}_{2}_{m}_{1}_{2}

Determination of output fuzzy set was made with an aggregation method (maximum) and with a defuzzification process, where the centroid method (also called centre of area, centre of gravity) was used to obtain crisp values of parameters [

In connection to IMC calculation, the fuzzy model of the Francis turbine proposed in Section 3 is used. With IMC, a model is parallel-added to the process (

In an ideal case, when disturbance

In the first step, the minimal phase and all-pass part of the process transfer function are expressed with

The ideal IMC controller is an inversed minimal phase part that results in mirror-transformed zeros over the ordinate axis:

Due to easier realisation and increased robustness, a PT1 (first order) filter with tuning parameter

Considering _{R}

Referring to different parameters in working points from the fuzzy model, the IMC controller is presented with a matrix of transfer functions _{R}

The results of IMC with a nonlinear model and surge tank are presented in _{m}_{m}_{m}

The fuzzy IMC controller algorithm was additionally formed with a fuzzy system to determine parameter _{m}_{1} and _{2} are observed from a turbine fuzzy model. Results after the defuzzification process (centroid method) of tuning parameter

If a hydraulic model of turbine and water passages are known, a PID or PI controller could be used for turbine power control, tuned with IAE expressed with

In _{r}_{m}_{p}_{I}

_{f}_{sp}_{m}

The fuzzy IMC controller in middle section _{m}_{p}

The smallest under-control power effect with 0.200 p.u. undershoot is obtained with IMC in _{m}_{m}_{m}_{m}

In comparison to the PI controller, fuzzy IMC presents better results, with lower undershoots _{p}_{m}

Fuzzy IMC also provides smaller variations of head Δ

In this paper, the dynamic modelling of a Francis turbine 1 (HPP Moste) with and without a surge tank with linearisation and IMC power control was proposed. With the actual HPP technical data (water tunnels and conduits), the presented models show a good match to the real data measurement.

With linearisation, a first-order transfer function with non-minimal phase was used for fuzzy modelling. Concerning appropriately defined input and output membership functions, a fuzzy rule list and the defuzzification process (centroid method), the crisp parameters of the Francis turbine model were calculated.

Based on the fuzzy model, a fuzzy IMC controller was proposed with calculation of tuning parameter _{p}

The applied methodology with the extension of working points with different static heads can be used as algorithm in digital turbine governor in order to ensure better results in water turbine power control.

The authors declare no conflict of interest.

Model parameters.

Generator rated power | _{g, r} |
9 MVA |

Maximum turbine rated power | _{m, r} |
7.5 MW |

Rated turbine power | _{r} |
6.692 MW |

Maximum turbine flow | _{max} |
14 m^{3}/s |

Rated turbine flow | _{r} |
13 m^{3}/s |

No-load flow | _{nl} |
0.19 p.u. |

Rated turbine head p.u. | _{r} |
0.824 p.u. |

Static head p.u. | _{0} |
0.9740 p.u. |

Static head | _{b} |
68.62 m |

Gravity constant | 9.80665 m/s^{2} | |

Water inertia time | _{w}_{1} |
2.2361 s |

Water inertia time | _{w}_{2} |
0.7267 s |

Water inertia time | _{w}_{3} |
2.9628 s |

Surge tank intake loss | _{0} |
0.0005 |

Friction loss penstock | _{p}_{1} |
0.0136 |

Friction loss water tunnel | _{p}_{2} |
0.0136 |

Additional friction loss of the water tunnel | _{p}_{3} |
0.01 |

Additional friction loss of the penstock | _{p}_{4} |
0.003 |

Surge tank parameter | _{s} |
225.5150 s |

Turbine damping | 0.5 p.u./p.u. | |

Surge tank upper chamber area | 316.6 m^{2} | |

Surge tank main chamber area | 44.2 m^{2} | |

Surge tank bottom chamber area | 276.4 m^{2} |

Hydro power plant (HPP) Moste and HPP Zavrsnica water system.

Turbine power of the non-linear model (red solid line), linearised model (blue dashed line) and the real system measurement (green dashed line).

Dynamic response on the ramp reference of realised models at turbine power working point 0.6657 p.u.

Simulated water level and water flow of the surge tank.

Membership functions of Francis turbine model.

Internal model control.

IMC control of nonlinear turbine model with conduit and surge tank for 3 different working points.

Wicket gate opening, turbine flow, head at turbine admission, friction head loss, and tail water in p.u. with fuzzy internal model control (IMC) and proportional-integral (PI) control in the middle section at turbine power 0.484 p.u.

Step response of the turbine power with fuzzy IMC and PI controller in the middle sections of working points.

Francis turbine model parameters in working points and middle sections, calculated with fuzzy model.

_{1} |
_{2} | ||
---|---|---|---|

_{m} |
1.3583 | 0.7610 | 0.6614 |

_{m} |
1.3565 | 0.8922 | 0.7150 |

_{m} |
1.3935 | 1.2863 | 0.9139 |

_{m} |
1.4470 | 1.3918 | 0.9506 |

_{m} |
1.3537 | 1.6694 | 1.0515 |

_{m} |
1.2929 | 2.0928 | 1.2593 |

_{m} |
1.2671 | 2.3073 | 1.3348 |

Tuning parameter _{p}_{I}

_{P} |
_{I} | ||
---|---|---|---|

_{m} |
0.3000 | 0.2983 | 0.3154 |

_{m} |
0.2786 | 0.2912 | 0.3016 |

_{m} |
0.2178 | 0.2709 | 0.2594 |

_{m} |
0.2000 | 0.2764 | 0.2437 |

_{m} |
0.1671 | 0.2520 | 0.2318 |

_{m} |
0.1064 | 0.2329 | 0.1932 |

_{m} |
0.0800 | 0.2225 | 0.1759 |

Turbine maximum undershoot _{p}

_{m} |
_{m} |
_{m} |
_{m} | |
---|---|---|---|---|

_{p} |
0.200 | 0.202 | 0.340 | 0.410 |

_{p} |
0.915 | 0.990 | 1.088 | 1.076 |

Δ |
0.027 | 0.022 | 0.022 | 0.020 |

Δ |
0.086 | 0.083 | 0.061 | 0.050 |