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* Corresponding author: Faculty of Mechanical Engineering, University of Montenegro, Džordža Vašingtona bb, Podgorica, Montenegro, phone: +382 69 381834, fax: +382 20 206131, email: uros.karadzic@ac.me

WATER HAMMER EFFECTS DURING PELTON TURBINE

LOAD REJECTION

Uroš KARADŽI *

Faculty of Mechanical Engineering, University of Montenegro, Montenegro

Anton BERGANT

Litostroj Power d.o.o., Slovenia

Petar VUKOSLAV EVI

Faculty of Mechanical Engineering, University of Montenegro, Montenegro

ABSTRACT This paper deals with water hammer phenomena in a high-head hydropower plant Peru ica, Montenegro. During its first phase of modernisation and refurbishment new distributors (needle valves) have been installed on the first two Pelton turbine units. Water hammer is investigated for the case of load rejection under governor control from different initial powers of the first turbine unit. Closure of the Pelton turbine distributors is modelled by two different closing laws i.e. assumed (numerical) closing law and law that considers actual (measured) needle stroke. Standard quasi-steady friction model for calculating friction losses in the plant’s penstocks is used. Turbine rotational speed change, for all cases, is calculated taking into account all influential torques. Numerical results for different distributors closing laws are compared with results of measurements. All numerical models show satisfactory agreement between numerical and measured results. KEYWORDS Water hammer; Pelton turbine; Load rejection; Turbine speed change. 1. INTRODUCTION

Planning of new or revitalisation and modernisation of existing hydropower plants (HPPs) requires detailed water hammer analysis in order to get maximum and minimum pressures as the most important parameters in the design process of the plant components. Causes of water hammer in hydropower plants may be different, for example: turbine start-up and stop, load acceptance and load reduction, load rejection under governor control, emergency shut-down and unwanted runaway, and closure and opening of safety valves. This paper investigates water hammer effects in Peru ica hydropower plant (HPP), Montenegro during load rejection under governor control from different initial powers of the first Pelton turbine unit. In the first part of the paper mathematical tools for solving water hammer equations are presented. Water hammer is described by two hyperbolic partial differential equations; continuity and momentum equation. These equations are traditionally solved by the method of characteristics (MOC) [1], [2]. Friction losses in the plant’s penstocks are calculated using standard quasi-steady model. Two different Pelton turbine distributor’s closing laws are presented, (1)

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closing law that considers the actual (measured) needle stroke (ANS) and (2) the assumed (numerical) closing law (CNS). Turbine speed change during load rejection under governor control is calculated taking into account dissipative torques including the shaft bearing friction torque and ventilation losses in the turbine housing. In the second part of the paper comparisons of numerical and measured results are made for turbine load rejection cases from different initial powers. It is shown that all the numerical results give good agreement with the results of site measurements.

2. THEORETICAL MODELLING

Water hammer is propagation of plane pressure waves in pipelines. For most engineering applications simplified water hammer equations are used [1], [2]:

0 + 2

= xQ

gAa

tH

∂∂

∂∂

(1)

02

|| +

1 + 2 =

gDAQQf

tQ

gAxH

∂∂

∂∂

(2)

where, H = piezometric head (head), t = time, a = pressure wave speed, g = gravitational acceleration, A = pipe area, Q = discharge, x = distance along the pipe, f = friction factor, and D = pipe diameter. Traditionally a quasi-steady friction factor is used in water hammer analysis. This approach is satisfactory for the cases considered in this paper (slow transients [3]). The method of characteristics (MOC) is a standard numerical method for the solution of water hammer equations. The staggered (diamond) grid [1] in applying the MOC is used in this paper. At a boundary (reservoir, Pelton turbine), a device-specific equation replaces one of the MOC water hammer compatibility equations. Pelton turbine output is regulated by control of discharge that acts on the turbine wheel. Discharge control is performed by means of a needle which closes or opens the throat of the nozzle (see Fig.1) and with appropriate position of the jet deflector. The discharge through the nozzle depends on the position of the needle valve and it is not dependent on the turbine rotational speed [4]. Consequently, water hammer equations and the dynamic equation of the turbine unit rotating parts can be solved separately. In this way the instantaneous head at the nozzle inlet and the instantaneous discharge through the nozzle are calculated by the MOC algorithm and these values are then used as an input in the solution method for the dynamic equation of the unit rotating parts.

Fig.1 Pelton turbine distributor (needle valve)

The instantaneous discharge through the nozzle ((Qu)t) is determined from the following relation: )(2)( , dtumQtu HHgAKQ −= (3)

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where, KQ = nozzle discharge coefficient, Am = nozzle area (Am = πdm2/4), Hu,t = head at the

nozzle inlet, and Hd = const. = head downstream the nozzle. The needle closing law is expressed as follows: maxss ⋅= τ (4) where, s = needle stroke, and τ = dimensionless nozzle opening. The load rejection under governor control of the turbine unit is the most frequent transient regime. The turbine is disconnected from the electrical grid followed by simultaneous gradual closure of the needle(s) to the speed-no-load position and controlled manouvre of the jet deflector(s) i.e. rapid activation at the first instant followed by gradual adjustment of the deflector to the speed-no-load position. The equation that describes dynamic behaviour of the Pelton turbine unit rotating parts during load rejection under governor control is [5]:

airfrha mmmdtd

T −−=ϕ (5)

where, ϕ = n/nr = relative speed change, n = turbine rotational speed (traditionally in rpm), r = rated conditions, Ta = mechanical starting time [2], mh = dimensionless hydraulic torque, mfr = dimensionless shaft bearing friction torque, and mair = dimensionless fluid damping torque (ventilation losses in the turbine housing). The dimensionless hydraulic torque is expressed as follows:

2

1 kh

rh

DF

Mm = (6)

where, Mr = rated torque and Fh = jet hydraulic force defined by the following equation [6], [7]: ( )uVQF mmh −= ρ2 (7) with u = peripheral velocity:

( )160

+= ϕπ rk nDu (8)

where, Dk = wheel diameter, Qm = discharge to the turbine wheel, and Vm = jet velocity (Vm = Qm / Am). The discharge that acts on the turbine wheel can be given from the following equation (t ≤ tdef) [5]:

11.0

1−=def

um tt

QQ (9)

where, tdef = jet deflector operating time. The dimensionless shaft bearing friction torque is [7], [8]:

( )BbAbb

br

fr RRD

Mm +=

21 µ (10)

where, Db = shaft bearing diameter, and µb = shaft bearing friction coefficient. The resultant forces in the shaft bearings RAb, RBb of the horizontal-shaft unit are due to hydraulic force, weight of the wheels, weight of the shaft and weight of the generator. The dimensionless fluid damping torque is expressed by [8]:

( )22

2

160

1 += ϕrair

rair

nK

Mm (11)

where, Kair = fluid (air) damping coefficient. The hydraulic torque affects the turbine wheel until the jet deflector deflects all the water into the tailrace (t=tdef). At t > tdef the hydraulic torque is set to zero. When the needle reaches speed-no-load position (about 3% of the nozzle opening), the jet deflector moves to its open position allowing the jet to reach the turbine

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wheel. At the needle speed-no-load position the distributor provides sufficient discharge to maintain the rated turbine rotational speed. In this case the hydraulic torque is balanced with the disipation torques. The dynamic equation of unit rotating parts is soloved analitically for all cases considered [5]. 3. PERU ICA HPP FLOW-PASSAGE SYSTEM

Peru ica HPP flow-passage system is a complex system comprised of a concrete tunnel (LT = 3335 m, DT = 4.8 m), orifice type surge tank (orifice head loss coefficients: ζin = 1.65 and ζout

= 2.48 during inflow and outflow, respectively) of cylindrical cross-section (DST = 8.0 m) with an expansion at elevation z = 611.0 m (DST = 12.0 m) and overflow (elevation: zov = 628.0 m; width of the overflow weir: bov = 7.98 m with discharge coefficient µov = 0.4) and three parallel steel penstocks with horizontal-shaft Pelton turbines built at their downstream ends (see Fig.2). The maximum water level at the intake is 613 m and the minimum one is 602.5 m. The penstock I feeds two turbine units (A1 and A2) with rated unit power of 39 MW, penstock II feeds three turbine units (A3, A4 and A5) of 39 MW each and penstock III feeds two units (A6 and A7) of 59 MW each. A new turbine unit (A8) with a rated power of 59 MW is to be installed in the near future. The length of each penstock is 1920 m, 1966 m and 2014 m, respectively. The respective equivalent penstock diameter [2] is 1965 mm, 2160 mm and 2570 mm. Calculated pressure wave speeds are as follows, aT = 1354 m/s, aI = 1148 m/s, aII = 1123 m/s and aIII = 1152 m/s.

Fig.2 Layout of Peru ica HPP, Montenegro The Pelton wheel diameter of units A1 to A5 is Dk = 2400 mm and for units A6 and A7 is Dk

= 2100 mm. Basic characteristics of the Pelton turbine units are presented in Tab.1 below.

Turbine unit Rated unit power Pr (MW)

Rated net head Hr (m)

Rated speed nr (min-1)

A1,A2,A3,A4 39 526 375 A5 39 526 375

A6,A7 59 526 428

Turbine unit

Number of runners per turbine unit

The polar moment of inertia of the unit

rotating parts J (tm2)

Number of needles per turbine runner

A1,A2,A3,A4 2 168.8 1 A5 2 168.8 1

A6,A7 2 200 2

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Turbine unit Stroke of the needle smax (mm)

Closing time of the needle tc (s)

Opening time of the needle to (s)

A1,A2,A3,A4 150 85 30 A5 195 80 30

A6,A7 166 80 50

Tab.1 Characteristics of Pelton turbine units

Influential quantities have been continuously measured during transient operating regimes including pressure at the upstream end of the distributor, stroke of the needle, stroke of the jet deflector and turbine rotational speed. These measurements have been carried out on turbine units A1 and A2.

4. COMPARISONS OF NUMERICAL AND FIELD TEST RESULTS

Various transient operating regimes have been investigated in the Peru ica HPP during commissioning of the turbine units A1 and A2, including the unit start-up, load acceptance and reduction, load rejection under governor control and emergency shut-down, and closure of turbine inlet valve against the discharge. In this paper the results of measurements and corresponding numerical simulations for the case of load rejection under governor control of the turbine unit A1 are presented. Load rejection cases of A1 have been done from different initial powers i.e. P0 = {10, 20, 30 and 40} MW (Tests A1P10MW, A1P20MW, A1P30MW, A1P40MW, respectively). Tabs.2, 3 and 4 present the main initial parameters for these four tests. Flow in penstock I is turbulent with a large Reynolds number.

Test QI (m3/s) ReI × 106 QII (m3/s) QIII (m3/s) QT (m3/s) A1P10MW 2.2 1.42 0 16.6 18.8 A1P20MW 4.1 2.7 0 16.6 20.7 A1P30MW 6.4 4.15 0 16.6 23.0 A1P40MW 8.4 5.46 2.1 19.7 30.2

Tab.2 Initial discharges through tunnel and penstocks

Penstock I Penstock II Penstock III Tunnel Test

f0 f0 f0 f0 A1P10MW 0.012 0.012 0.016 0.015 A1P20MW 0.011 0.012 0.016 0.015 A1P30MW 0.011 0.012 0.016 0.015 A1P40MW 0.011 0.013 0.016 0.015

Tab.3 Steady friction factors for all tests

Test zR (m) tc (s) s0 (mm) tdef (s)

A1P10MW 606.3 14.1 25.2 1.6 A1P20MW 605.3 28.2 50.6 1.6 A1P30MW 607.0 45.1 82.3 1.6 A1P40MW 605.8 73.0 128.0 2.0

Tab.4 Closure time, intake level and initial opening of the nozzle

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4.1 Comparison of numerical and measured heads at the turbine inlet Transient head and discharge for the Peru ica HPP flow-passage system have been calculated using a diamond grid MOC code with the basic time step of ∆t = 0.040 s. Calculated and measured results from two distictive load rejection cases (A1P10MW and A1P40MW) are shown in Figs.3 and 4. The maximum heads of all four cases are summarised in Table 5. Calculated and measured heads at the turbine inlet for the load rejection under governor control of the turbine unit A1 from the initial power P0 = 10 MW are shown in Fig.3 (Test A1P10MW). The speed-no-load opening of the nozzle in this case is 3.4 % (Fig.3(a)). Note that the process of attaining the speed-no-load position for all investigated turbine load rejection cases is governed by the turbine control system. The maximum measured head of 558.5 m occurs at time t = 3.7 s when the reflected pressure wave arrives back to the nozzle with the head rise of 18.8 m. The maximum calculated heads obtained by the ANS and CNS models are 558.8 m and 558.3 m, respectively (Figs.3(b) and 3(c)). The calculated maximum heads match the measured one and are lower than the maximum permissible system head of 602.0 m. The ANS and CNS models show very good agreement with the measured data during the nozzle closure period. After this, both models give some smaller head amplitudes but with reasonable shape of pressure traces. At time t = 260 s the nozzle opening is increased (load increase) with the system head drop that is properly simulated by the ANS model (see also Fig.3(b)). The CNS model is not able to simulate this process because the nozzle opening is assumed to be at constant position during this phase of transient regime (Fig.3(c)).

Fig.3 Comparison of needle stroke (s) and head at the turbine inlet (datum level z = 65.8 m; time step ∆t=0.040 s). Load rejection under governor control of A1 from P0 = 10.0 MW (Test A1P10MW).

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Numerical and measured heads at the turbine inlet for the load rejection under governor control of the turbine unit A1 from the maximal initial power of P0 = 40 MW are shown in Fig.4 (Test A1P40MW). The speed-no-load opening of the nozzle in this case is 4.1 % (Fig.4(a)). The maximum measured system head for A1P40MW case occurs at the end of the nozzle closure period and its value is 555.6 m with head rise of 23.5 m. The maximum calculated heads given by the ANS and CNS model are the same and its value is 555.2 m (Figs.4(a) and 4(b)). Numerical models show excellent agreement with measured data during and after the nozzle closure period.

Fig.4 Comparison of needle stroke (s) and head at the turbine inlet (datum level z = 65.8 m; time step ∆t=0.040 s). Load rejection under governor control of A1 from P0 = 40.0 MW (Test A1P40MW).

Tab.5 shows the maximum measured and computed heads for all four load rejection cases, their head rise and time when the maximum head occurs, measured from the beginning of the nozzle closure period. Tests A1P10MW and A1P20MW produce the maximum head t = 3.7 s. This may be explained by smaller effect of system friction losses compared with those in tests A1P30MW and A1P40MW.

Measurement ANS model CNS model Test case Hmax (m) ∆H (m) tocc (s) Hmax (m) ∆H (m) tocc (s) Hmax (m) ∆H (m) tocc (s)

A1P10MW 558.5 18.8 3.7 558.8 19.1 3.7 558.3 18.6 3.7 A1P20MW 553.2 15.9 3.7 553.7 15.4 3.7 552.8 15.5 3.7 A1P30MW 553.9 17.9 42.0 553.9 17.9 42.0 553.7 17.7 42.0 A1P40MW 555.6 23.5 69.0 555.2 23.2 69.0 555.2 23.2 69.0

Tab.5 Maximum measured and computed head, head rise and time when the maximum head occurs

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4.2 Comparison of computed and measured turbine rotational speeds The turbine rotational speed change of the turbine unit A1 during load rejection under governor control from different initial power has been calculated using the appropriate solution method of the dynamic equation of the unit rotating parts (Eq.(5)). The instantaneous head and discharge through the nozzle that is used as an input into Eq.(5) solution formulae have been previously calculated by the MOC. Fig.5 shows comparison between the computed and measured rotational speed changes for all case studies considered.

Fig.5 Rotational speed change (n0 = 375 min-1) during load rejection under governor control of A1 from different initial powers P0 = {10, 20, 30, 40} MW

The maximum measured and calculated turbine speed rise for all tests occurs at time t = tdef. The computed maximum turbine rotational speed rise perfectly matches the maximum measured value for all four cases (Fig.5). As expected, the maximum speed rise of 11.0 % occurs for the Test A1P40MW. After the jet deflector deflects the water into the tailrace, the turbine wheel is not affected by the hydraulic torque and the turbine speed decrease is influenced by the dissipation torques. There are some discrepancies in the phase of speed decrease due to the complex flow behavior in the turbine housing. The ventilation losses are dependent on fluid density inside the turbine housing. At the needle speed-no-load position the turbine reaches the rated rotational speed with some minor numerical oscillations. In all four cases the maximum turbine speed rise is well below the permissible speed rise of 25%.

5. CONCLUSIONS

Numerical results obtained from the two different distributor (needle valve) closing laws are compared with results of measurements given for the turbine load rejection under governor

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control cases from different initial powers. From comparisons between the computed and the measured results of measurements it may be seen that the general patterns of the computed transient pressure histories agree well with measured data. Numerical model that considers the actual (measured) needle stroke (ANS) and the numerical model with the assumed needle stroke (CNS) show similar behavior. It means that CNS model properly simulates physical process in the hydraulic system and can be employed in cases when recorded data of needle stroke are not available. The turbine rotational speed change is calculated separately by the appropriate solution of the dynamic equation of the turbine unit rotating parts. Developed numerical model accurately determines the maximum rotational speed rise for all cases. There are some discrepancies during speed-no-load conditions but with reasonable agreement between computational and measured results.

6. REFERENCES

[1] Wylie, E.B., Streeter, V.L.: Fluid Transients in Systems. Prentice Hall. Englewood Cliffs.

1993.

[2] Chaudhry, M.H.: Applied Hydraulic Transients. Van Nostrand Reinhold Company. New

York. 1987.

[3] Karadži , U., Bergant, A., Vukoslav evi , P.: A Novel Pelton Turbine Model for Water

Hammer Analysis. Strojniški Vestnik – Journal of Mechanical Engineering. 55. 2009.

Accepted for publication.

[4] Benišek, M.: Hydraulic Turbines. Faculty of Mechanical Engineering. Belgrade. 1998

(in Serbian).

[5] Karadži , U.: Modelling of complex boundary conditions for transients in hydraulic

systems. PhD Thesis. University of Montenegro. Podgorica. 2008 (in Serbian).

[6] Nechleba, M.: Hydraulic Turbines – Their Design and Equipment. Artia. Prague. 1957.

[7] Zhang, Zh., Müller, J.: Efficiency and runaway characteristics of a Pelton turbine.

Hydro 2007, 2007. CD-ROM.

[8] Thake, J.: The Micro-Hydro Pelton Turbine Manual. ITDG Publishing. London. 2000.

7. NOMENCLATURE

A (m2) pipe area Q (m3.s-1) discharge Am (m2) nozzle area Qm (m3.s-1) discharge to wheel a (m.s-1) wave speed Qu (m3.s-1) nozzle discharge bov (m) width of weir Rb (N) resultant force D (m) pipe diameter Re (-) Reynolds number Db (m) shaft diameter s (m) needle stroke Dk (m) wheel diameter Ta (s) mechanical starting time DST (m) surge tank diameter t (s) time Dk (m) valve diameter tc (s) needle closing time dm (m) nozzle diameter tdef (s) jet deflector operating time Fh (N) jet hydraulic force to (s) needle opening time

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f (-) friction factor tocc (s) time at maximum head g (m.s-2) gravi. acceleration u (m.s-1) peripheral velocity H (m) piezometric head Vm (m.s-1) jet velocity Hd (m) nozzle down. head x (m) distance along the pipe Hu (m) nozzle upstream head z (m) water level, elevation J (kg.m2) moment of inertia ∆H (m) head rise Kair (N.m.s-2) damping coefficient ∆t (s) time step KQ (-) discharge coefficient ϕ (-) relative speed change L (m) length ρ (kg.m-3) water mass density mair (-) damping torque µb (-) shaft bearing friction coef. mfr (-) friction torque µov (-) overflow discharge coef. Mr (N.m) rated torque τ (-) dimensionless nozzle open. n (rpm) rotational speed ζin (-) surge tank inflow coef. Subscripts: max maximum R reservoir r rated conditions T tunnel 0 initial conditions I, II, III penstock number Abbreviations: ANS actual needle stroke CNS assumed needle stroke HPP hydropower plant MOC method of characteristics