# Mathematic Modelling of a Reversible Hydropower System: Dynamic Effects in Turbine Mode

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## Abstract

**:**

_{R}) and an overpressure of 40–65% of the rated head (H

_{R}), showing significant impacts on the pressure wave propagation along the entire hydraulic circuit. Sensitivity analyses based on systematic numerical simulations of PATs (radial and axial types) and reaction turbines (Francis and Kaplan types) and comparisons with experiments are discussed. These evaluations demonstrate that the full-load rejection scenario can be dangerous for turbomachinery with low specific-speed (n

_{s}) values, in particular when associated with long penstocks and fast guide vane (or control valve) closing maneuver.

## 1. Introduction

## 2. Methods and Materials

#### 2.1. Water Hammer Modelling

#### 2.2. Boundary Conditions for MOC

#### 2.3. Turbine Modelling

_{RW}/Q

_{R}), ${\beta}_{R}$ = relative runaway rotating speed (N

_{RW}/N

_{R}), $n$ = relative runner speed, and $h$ = relative turbine net head.

#### 2.4. CFD Model Description and Mesh

## 3. Dynamic Effects in Hydropower Systems

#### 3.1. Overspeed Effect Due to Full-Load Rejection

^{3}/s) or on (m, kW) as follows:

_{m}(s), is defined based on Equation (22):

^{2}).

_{W}is the hydraulic inertia time constant, defined by the following equation:

_{s}value, the runaway causes a considerable reduction in flow with a high upsurge, even greater than that caused by the closing of the guide vane, resulting in the first peak in head, visible in Figure 7a at about t = 2 s. At this point, the discharge reduction is no longer as strongly influenced by the overspeed effect and begins to be controlled mainly by the flow control (Figure 7b).

#### 3.2. Influence of Other Characteristic Parameters

## 4. Comparisons between Modelling and Laboratory Tests for Different Type of Turbines and Specific Speeds

- -
- For Francis and Kaplan with fixed blades (propeller)

- -
- For radial PAT

- -
- For axial PAT

#### 4.1. PAT Radial Runners—Low ns

^{3}/s) for increasing rotational speed induced a typical wall effect, where the flow discharge reduced significantly. For example, for a constant head of 6 m and by varying the rotational speed between the rated conditions (i.e., $N$ = 1020 rpm) until we attained runaway conditions around the double of rated value, the flow reduced from 4.3 to 2.7 l/s, corresponding to a reduction of around 63%, which is very significant in terms of a water hammer event for a full-load rejection (Figure 11). As the rotational speed increases, the flow rate decreases and the head increases, induced by the centrifugal influence of a wall-type effect of cutting the flow discharge. This can be a dangerous event, inducing an upsurge peak that can be higher than the one induced by a valve or a guide-vane closure.

_{R}= 3.36 l/s; H

_{R}= 4 m; N

_{R}= 1020 rpm.

#### 4.2. PAT Axial Runners—High n_{s}

^{3}/s) and it was found that at a constant head, e.g., $H$ = 2 m, the flow increases from 8 to 14 l/s with the increase of the rotational speed from 500 to 2750 rpm (Figure 14), because of the suction effect that occurs in these types of runners.

_{R}= 4.4 l/s; H

_{R}= 0.34 m; N

_{R}= 750 rpm (Figure 15).

#### 4.3. Francis Turbine—Low ${n}_{s}$

#### 4.4. Kaplan Turbine—High ${n}_{s}$

## 5. Conclusions

_{R}and an overpressure of up to 65% H

_{R,}with significant interaction with the penstock fluid propagation along the all-hydraulic circuit.

_{s}values, both with or without the closure of the turbine’s valve/guide vane. For low specific-speed and small inertia turbines, particularly used in micro or small hydropower plants, the transient overpressure can become significant by the overspeed effect that induces a type of wall effect, cutting the flow in few seconds and thus originating substantial upsurges that propagate along the hydraulic circuit. Conversely, in turbomachines with high n

_{s}values, the overspeed effect can induce an increase in the flow discharge as a suction effect with a pressure drop, which is not dangerous for the hydropower system.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

$a$ | Wave celerity (m/s) |

${A}_{1}$, ${B}_{1}$, ${C}_{1}$ | Constants of a pump head curve |

$A$ | Cross-sectional area of a pipe (m^{2}) |

${A}_{o}$ | Cross-sectional area of an orifice (m^{2}) |

${A}_{c}$ | Cross section of an air vessel (m^{2}) |

$C$ | Constant computed in the initial condition of an air vessel |

${C}_{a}$ | Relationship between $g$, $A$, and $a$ |

${C}^{-}$ | Negative characteristic equation (-) |

${C}^{+}$ | Positive characteristic equation (-) |

${C}_{c}$ | Contraction coefficient (-) |

${C}_{g}$ | Opening gate coefficient (-) |

${C}_{n}$ | Negative characteristic constant (-) |

${C}_{p}$ | Positive characteristic constant (-) |

${C}_{s}$ | Runner’s rotational speed (-) |

${C}_{u}$ | Constant of the turbulent eddy viscosity coefficient |

$D$ | Internal pipe diameter (m) |

$f$ | Friction factor (-) |

${f}_{u}$ | Turbulent viscosity factor |

${F}_{x}$ | Forces in x direction (N) |

$g$ | Gravitational acceleration (m/s^{2}) |

$j$ | Iteration number (-) |

$i$ | Position in a pipe (-) |

${h}_{T}$ | Relative turbine head (-) |

$H$ | Hydraulic grade line (m) |

${H}_{A}$ | Piezometric head on section A (m) |

${H}_{B}$ | Piezometric head on section B (m) |

${H}_{b}$ | Barometric pressure (m) |

${H}_{P}$ | Piezometric head at section P (m) |

${H}_{P,air}$ | Pressure head at the end of an analyzed time step in an air vessel (m) |

${H}_{Pi}$ | Piezometric head on an analyzed side of section P (m) |

${H}_{P,i+1}$ | Piezometric head on the upstream side of section P (m) |

${H}_{P,i-1}$ | Piezometric head on the downstream side of section P (m) |

${H}_{R}$ | Turbine/pump rated head (m) |

${H}_{res}$ | Water level of a reservoir (m) |

$I$ | Rotating mass inertia (kgm^{2}) |

$k$ | Turbulent kinetic energy (m^{2}/s^{2}) |

${K}_{V}$ | Valve head loss coefficient (-) |

$L$ | Pipe length (m) |

$m$ | Water mass (kg) |

$n$ | Relative rotating speed (-) |

$N$ | Rotational speed |

${N}_{t}$ | Number of segments (-) |

${N}_{R}$ | Turbine rated speed (-) |

${N}_{RW}$ | Runaway turbine rotating speed (r.p.m. |

${n}_{R}$ | Nominal runner speed (r.p.m.) |

${n}_{s}$ | Specific velocity (r.p.m.). |

$P$ | Pump power (kW) |

${P}_{R}$ | Reference power (kW) |

${P}_{w}$ | Pipe perimeter (m) |

$p$ | Pressure force (N) |

${p}_{c}$ | Polytropic coefficient (-) |

$Q$ | Discharge (m^{3}/s) |

${q}_{T}$ | Relative flow through the turbine (-) |

$q$ | Relative discharge (-) |

${Q}_{A}$ | Discharge at section A (m^{3}/s) |

${Q}_{B}$ | Discharge at section B (m^{3}/s) |

${Q}_{P}$ | Discharge at section P (m^{3}/s) |

${Q}_{P,orifice}$ | Air vessel orifice discharge (m^{3}/s) |

${Q}_{R}$ | Turbine/pump rated discharge (m^{3}/s) |

${Q}_{RW}$ | Turbine discharge at runaway speed (m^{3}/s) |

$t$ | Time (s) |

${T}_{T}$ | Turbine torque (N·m) |

${T}_{G}$ | Electromagnetic resistance torque (N·m) |

${T}_{H,R}$ | Actuating hydraulic torque (N·m |

${t}_{C}$ | Guide-vane or valve closing time (s) |

$TE$ | Elastic time constant (s) |

${T}_{H}$ | Hydraulic turbine torque (N·m) |

${T}_{m}$ | Start-up time of rotating masses (s) |

${T}_{w}$ | Hydraulic inertia time (s) |

$v$ | Water velocity in a pipe (m/s) |

${V}_{P,air}$ | Volume of the air enclosed in the vessel of an analyzed time step (m^{3}) |

$x$ | Distance along the main direction of a pipe system (m) |

$y$ | Distance from the wall (m) |

$z$ | Initial elevation of the free surface inside an air vessel (m) |

${z}_{P}$ | Free surface elevation at the end of the time step (m) |

$W{D}^{2}$ | Inertial weight of rotating mass (N·m^{2}) |

${\alpha}_{R}$ | Relative runaway discharge (-) |

${\beta}_{R}$ | Relative runaway rotating speed (-) |

$\epsilon $ | Turbulent dissipation |

$\theta $ | Pipe slope (m/m) |

$\u2206t$ | Time step (s) |

$\u2206H$ | Overpressure (m) |

$\rho $ | Water density (kg/m^{3}) |

$\gamma $ | Water unit weight (N/m^{3}) |

$\mu $ | Dynamic viscosity coefficient (-) |

${\mu}_{t}$ | Turbulent eddy viscosity coefficient (-) |

${\tau}_{0}$ | Shear stress (N/m^{2}) |

${\tau}_{ij}$ | Viscous shear stress tensor |

${\tau}_{ij}^{R}$ | Reynolds-stress tensor |

${\delta}_{ij}$ | Kronecker delta function (-) |

$\omega $ | Angular speed (rad/s) |

${\eta}_{R}$ | Unit rated efficiency (-) |

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**Figure 6.**Relationship between maximum relative overpressure head, inertia of the runner, and specific speed: (

**a**) plan view and (

**b**) 3D view, better showing the curvature significance effect for both inertia and specific-speed variation.

**Figure 7.**Full-load rejection for a hydropower case study: (

**a**) Relative pressure head; (

**b**) Relative discharge; (

**c**) Relative rotational speed and (

**d**) Relative torque.

**Figure 8.**Maximum relative overpressure for different combinations of hydraulic circuit length and flow control device closure time.

**Figure 9.**Maximum relative upsurges for a typical hydropower with different characteristics: (

**a**) 2D representation for wave speed, $a$, and guidevane closure time, ${t}_{C}$, variation; and (

**b**) 3D dependency on different combinations of these parameters.

**Figure 10.**Three experimental set-ups for tests performed and calibration of developed mathematical models: (

**a**) for classical turbines – Francis and Kaplan with fixed blades, (

**b**) for the radial PAT, (

**c**) for axial PAT.

**Figure 11.**Different operating conditions for a radial runner by increasing rotational speed until to attain the runaway: (

**a**) characteristic curves H-Q for different N; (

**b**) pressure values and velocity fields for N = 810 and 1500 rpm.

**Figure 13.**The influence of rotational speed on the fast variation of flow (Q/QR—color variation) for constant head values in a radial runner type.

**Figure 14.**Different operating conditions for an axial runner by increasing rotational speed until to attain the runaway: (

**a**) characteristic curves H-Q for different N; (

**b**) stream lines and velocity fields (

**top**), pressure contours values and shear stress (

**bottom**) for N = 750 and 1500 rpm.

**Figure 16.**The influence of rotational speed on the fast variation of flow (Q/QR—color variation) for certain constant head values in an axial runner type.

**Figure 17.**Runaway effect in a Francis turbine with ns = 130 rpm (m, kW): (

**a**) upstream head, flow and rotational speed variation, (

**b**) flow velocity fields and pressure contours variation in 2D simulation for rated and runaway conditions.

**Figure 18.**Francis turbine behavior (with low ${n}_{s}$) during a simultaneously runaway and guide-vane closure events.

**Figure 19.**Runaway effect in a Kaplan turbine with ${n}_{s}$ = 400 rpm (m, kW): (

**a**) upstream head, flow and rotational speed variation, (

**b**) flow velocity fields and pressure contours variation in 2D simulation for rated and runaway conditions.

**Figure 20.**Kaplan turbine behavior (with high ${n}_{s}$) during a simultaneously runaway and guide-vane closure events.

Element | Scheme | Considerations and Additional Equations | Equations System | Notation |
---|---|---|---|---|

Constant-head reservoir | The boundary condition for a constant-head reservoir is computed neglecting entrance head losses as: ${H}_{P}={H}_{res}$ | (10) | ${H}_{res}$ is the water level of a reservoir. | |

Pump | Based on pump characteristic curve for each time step as:${Q}_{P}=\frac{2-{B}_{1}N{C}_{a}-\sqrt{{\left({B}_{1}N{C}_{a}-2\right)}^{2}-4{C}_{1}{C}_{a}\left({A}_{1}{C}_{a}{N}^{2}+{C}_{P}+{C}_{n}\right)}}{2{C}_{1}{C}_{a}}$ | (11) | ${A}_{1}$, ${B}_{1}$, and ${C}_{1}$ are constants of a pump curve, and $N$ is the rotational speed. | |

Valve | The boundary condition in a regulating valve is obtained based on steady-state head loss equation as: ${Q}_{P}=\frac{-2+\sqrt{4+2\frac{{K}_{V}{C}_{a}}{g{A}^{2}}\left({C}_{P}+{C}_{n}\right)}}{\frac{{K}_{V}{C}_{a}}{g{A}^{2}}}$ | (12) | ${K}_{V}$ is the head loss coefficent obtained experiment20ally. | |

Air vessel | The equation of an air vessel is obtained considering the polytropic law $({H}_{P,air}{V}_{P,air}^{{p}_{c}}=C)$, which needs to be solved simultaneously with the following five equations: ${Q}_{P,orifice}=\left({C}_{P}-{C}_{n}\right)-\left({C}_{a,i}+{C}_{a,i+1}\right){H}_{P}$${Q}_{P,orifice}=C{A}_{o}\sqrt{2g\Delta {H}_{P}}$${H}_{P,air}={H}_{P}+{H}_{b}-{z}_{P}-\Delta {H}_{P,orifice}$${V}_{P,air}={V}_{air}-{A}_{C}\left({z}_{P}-z\right)$${z}_{P}=z+0.5\left({Q}_{P,orifice}+{Q}_{orifice}\right)\frac{\Delta t}{{A}_{c}}$ | (13) | ${H}_{P,air}$ is the absolute pressure head at the end of an analysed time step, ${V}_{P,air}$ is the air volumne at the end of an analysed time step, ${V}_{air}$ is the air volume, ${Q}_{P,orifice}$ is the flow through the orifice,${A}_{o}$ is the cross-section of the orifice, $C$ is the discharge coefficient of the orifice, ${p}_{c}$ is the polytropic coefficient (usually takes a value of 1.2), $z$ is the initial elevation of the free surface, ${H}_{b}$ is the barometric pressure, $C$ is the constant computed in the initial condition of the air vessel, ${z}_{p}$ is the free surface elevation at the end of the time step, and ${A}_{c}$ is the cross-section of the air vessel. |

Mesh | Number of Fluid Mesh Cells | Number of Solid Mesh Cells | H | Error (%) | Duration |
---|---|---|---|---|---|

Mesh 1 | 35,888 | 25,378 | 9.8 | 0.15 h | |

Mesh 2 | 60,381 | 30,229 | 6.4 | 34% | 0.49 h |

Mesh 3 | 110,691 | 60,088 | 5.8 | 9% | 2.23 h |

Mesh 4 | 121,936 | 71,162 | 5.4 | 6% | 2.52 h |

Mesh 5 | 134,152 | 81,291 | 5.2 | 3% | 3.28 h |

Mesh 6 | 135,472 | 88,364 | 5.1 | 1% | 3.33 h |

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## Share and Cite

**MDPI and ACS Style**

Ramos, H.M.; Coronado-Hernández, O.E.; Morgado, P.A.; Simão, M.
Mathematic Modelling of a Reversible Hydropower System: Dynamic Effects in Turbine Mode. *Water* **2023**, *15*, 2034.
https://doi.org/10.3390/w15112034

**AMA Style**

Ramos HM, Coronado-Hernández OE, Morgado PA, Simão M.
Mathematic Modelling of a Reversible Hydropower System: Dynamic Effects in Turbine Mode. *Water*. 2023; 15(11):2034.
https://doi.org/10.3390/w15112034

**Chicago/Turabian Style**

Ramos, Helena M., Oscar E. Coronado-Hernández, Pedro A. Morgado, and Mariana Simão.
2023. "Mathematic Modelling of a Reversible Hydropower System: Dynamic Effects in Turbine Mode" *Water* 15, no. 11: 2034.
https://doi.org/10.3390/w15112034