Source Link: https://www.nature.com/articles/s41598-023-43228-1

Quaranta et al.^{14} proposed a set of equations to estimate the weight of hydropower equipment. The equations, with an average absolute error which is typically below 20%, proved to be appropriate for large-scale preliminary estimates. These equations were here applied to the EU hydropower database (already described in^{15} and including turbine type, head (*H,* m), flow rate (*Q*, m^{3}/s), power (*P*, MW) and number of units per plant), to estimate the weight of the electro-mechanical equipment, assuming a specific weight of 78 kN/m^{3}. The cumulative installed capacity included in this database is 154 GW (158 GW including UK), which reduces to 151.6 GW when considering the power plants whose all the abovementioned data are known. In this work, the analysis was applied to the 151.6 GW of power plants with known data representing almost the whole European Union’s hydropower fleet in terms of installed capacity. Finally, results were linearly extrapolated to the current installed capacity of 154 GW of the EU. The geographical distribution of the European hydropower fleet is depicted in Fig. 2.

### Kaplan, Propeller and Francis runners

The weight *G* (kN) of steel runners is a function of the turbine diameter *D* (m):

$$G = { 6}D^{{{2}.{75}}} ;{text{for Francis}}$$

(1)

$$G = { 2}.{3}D^{{3}} ;{text{for Propeller}}$$

(2)

$$G = { 3}0D^{{{1}.{9}0}} ;{text{for Kaplan}}$$

(3)

where the diameter *D* (m) can be calculated as proposed in^{16} as a function of the rotational speed *N* (rpm). The rotational speed can be estimated as a function of the dimensionless flow rate *Q**(=frac{{Q}_{nom}}{sqrt{2gH}{H}^{2}}) (Eq. 4), as detailed in^{17}. Under the category of Francis turbines we also included Pump-as-Turbines, Deriaz and Girard turbines.

$$N= left(alpha {Q}^{*beta }right)frac{sqrt{2gH}}{H}$$

(4)

where α = 20.3 and 26.8 for Francis and Kaplan-Bulb turbines, respectively. β = − 0.36 and − 0.38 for Francis and Kaplan-Bulb turbines, respectively. When applying Eq. (4), the same coefficients used for Kaplan turbines were also used for Bulb turbines, when the specific speed, expressed as *N *(sqrt{P}{H}^{-1.25})*,* was above 700. The specific weight was set to 78 kN/m^{3}.

### Pelton runners

The weight of Pelton runners can be estimated by Eq. (5)

$$Gleft( {{text{kN}}} right) = {439}0;f_{1}^{2.5} ;{text{with R}}^{{2}} = 0.{95}$$

(5)

with *f*_{1} = (sqrt{frac{Q/ {n}_{j}}{sqrt{2 g H}}}), where *Q* is the flow per unit (m^{3}/s), *n*_{j} is the number of jets and *H* is the net head (m). The parameter *f*_{1} is an indicator of the jet diameter, and hence of the runner diameter and width. The average error of Eq. (5), estimated in^{14}, was 19.8%.

### Banki runners

The following equations were used:

$$G = {24}.{7};Q^{{0.{85}}} ,{text{ for the mechanical}}/{text{hydraulic group}}$$

(6)

$$G = , 0.{11};P_{el}^{{0.{98}}} ,{text{ for the generator}}$$

(7)

where *P*_{el} is the electrical power in in kW and *Q* is the design flow in m^{3}/s. The weight calculated by Eq. (6) includes the runner, the external casing, the inlet nozzle and supports, according to data of IREM SpA (Italy). In this study, we assumed that half of the weight is of the runner and the rest is considered as “casing”.

### Electric generator

The weight depends on the generator rotational frequency *N* (rpm) and power *P* (MVA) as per Eq. (8).

$$G ={ alpha }{left(frac{P}{sqrt{N}}right)}^{beta }$$

(8)

The average absolute error estimated in^{14} was 7.4%. For Banki turbines, Eq. (7) was used to estimate the generator weight. The values of the coefficients α and β are listed in Table 1.

### Casing

The casing, normally made of steel, is a static component that encloses the rotating runner and the vanes. Its shape is rather simple for Pelton turbines, while it is spiral for reaction turbines (e.g., Kaplan and Francis turbines).

For Pelton turbines, the casing weight can be estimated by the following equations:

$$G = { 1177}f + {4}.{text{92 for vertical axis }} < { 1}0{text{ MW}}$$

(9)

$$G = { 763}f + { 2}.{24};{text{for horizontal axis }} < { 1}0;{text{MW}}$$

(10)

$$G = { 134}.{8}f + {text{ 99 for vertical axis }} > { 1}0;{text{MW}}$$

(11)

with *f* = *D*^{2} (sqrt{frac{Q/ {n}_{j}}{sqrt{2 g H}}}), *D* is the runner diameter (m), *Q* is the flow per unit (m^{3}/s), *n*_{j} is the number of jets and *H* is the net head (m). R^{2} ranges between 0.71 and 0.94 depending on the configuration, and the average absolute error was estimated to be below 30%^{14}. The vertical axis configuration was chosen when the number of jets was higher than 2. The number of jets can be estimated as:

$${n}_{j}= frac{Q}{0.0039 {P}^{0.5643}}$$

(12)

where *P* (kW) and *Q* (m^{3}/s) are the values per unit. For further details see^{18}.

For reaction turbines, Brekke^{19} showed that the weight *G* of high head large Francis spiral casing has almost reached a stable value with the years of 30 kN/MW of installed capacity, and no further weight reduction is expected. For smaller turbines, the following equations were used^{14}:

$$frac{G}{P}=2.84{left(frac{Q}{sqrt{H}}right)}^{-0.81}text{ for Vertical axis turbines}$$

(13)

$$frac{G}{P}=frac{1032}{H}text{ for Horizontal axis turbines}$$

(14)

where *P* is the installed power capacity expressed in MW and the *G* is the weight in kN, *Q* is the flow in m^{3}/s and *H* the net head in m. R^{2} = 0.8 and average absolute error was estimated to be 20%. In our analysis, we applied these equations to power plants below 10 MW, and assumed 30 kN/MW for larger power plants. According to data of Zeco Hydropower, vertical axis turbines are used below 1.25 m^{3}/s (from a simple statistic, with no particular technical justification).

The weight of guide vanes was calculated considering the traditional engineering practice, as *G* = *z*∙(*l∙s*)∙*m*∙*h*∙*ρg*, where *l* is the vane length, *s* is the vane thickness (20%*l*), *m* = 0.5 is the filling ratio of the cross section (the real cross section—NACA profile—is inscribed in the rectangular area *l*∙*s*), *h* is the vane height and *z* is the number of vanes. The number of vanes can be calculated as 0.25*D*^{0.5} + 5^{20} with *D* the diameter in mm, *l* is calculated to ensure that when all the vanes are closed, the distributor diameter is completely closed (and vanes should overlap by 15%), *h* = (0.134 ln(*N*_{s}) − 0.45)*D* with the diameter *D* in m^{16}.

### Draft tube

The following empirical equations were used to estimate the weight of the draft tube for turbines below 10 MW:

$$frac{G}{{N}_{s}}=0.305, {D}^{3} quad text{for Horizontal axis Francis turbines with elbow draft tube}$$

(15)

$$frac{G}{{D}^{3}}=16.2 ,{D}^{-1.85} quad text{for Vertical axis Kaplan turbines with elbow draft tube}$$

(16)

$$frac{G}{{N}_{s}}=0.0205, {D}^{1.95} quad text{for Propeller turbines}$$

(17)

The elbow draft tube, which is the most used and efficient type, was assumed for Francis and Kaplan turbines, while Propeller turbines use straight draft tubes^{14}. R^{2} > 0.82 and the estimated average absolute error was 20%. *N*_{s} is the specific speed expressed as *N* (sqrt{P}{H}^{-1.25}), with *P* in kW, the net head *H* in m and *N* in rpm. *D* is the draft tube inlet diameter in m, calculated by equations proposed in^{16}. For Kaplan turbine units with elbow and S-type draft tube, the weight estimated by Eq. (16) only includes the first part of the draft tube, because the second (diffuser) part is typically made of concrete built on-site. These equations are considered valid also for power plants above 10 MW.

### Steel in the powerhouse structure

The steel used in the powerhouse can be estimated according to^{21}, in KN, as:

$$G=k (n+R){C}^{0.358 }{D}^{1.074}$$

(18)

where *k* = 32.5 and 79.6 for reaction and impulse turbines, respectively. *C* = Crane capacity in tons, *D* = Runner throat diameter in meters, *n* = Number of units in powerhouse, *R* = Repair bay ratio, that can be assumed as equal to 0.5^{14}. The Crane capacity can be estimated as *G* = 46 ({left(frac{P}{sqrt{N}}right)}^{0.8})^{22}, similarly to Eq. (8). This value does not belong to the electro-mechanical equipment.