ARTICLE IN PRESS

0960-1481/$ - se

doi:10.1016/j.re

�Correspondfax: +643 364 2

E-mail addre

peter.giddens@

Renewable Energy 33 (2008) 507–519

www.elsevier.com/locate/renene

Technical Note

Optimum penstocks for low head microhydro schemes

K.V. Alexandera,�, E.P. Giddensb

aDepartment of Mechanical Engineering, University of Canterbury, Christchurch, P.O. Box 4800, Christchurch, New ZealandbFormally of Department of Civil Engineering, University of Canterbury, 81 Grange Street, Opawa, Christchurch, New Zealand

Received 8 March 2006; accepted 16 January 2007

Available online 19 March 2007

Abstract

This paper presents an analysis for penstock optimization, for low head microhydro schemes. The intent of the optimization is to

minimize capital cost per kilowatt rather than maximize the energy from the site. It is shown that site slope is an important consideration

that affects the economics.

While this work stands alone it has been generated as part of a research program that is in the final stages of developing a modular set

of cost-effective low-head microhydro schemes for site heads below those currently serviced by Pelton Wheels.

The rationale for the work has been that there is a multitude of viable low-head sites in isolated areas where microhydro is a realistic

energy option, and where conventional economics are not appropriate. This is especially the case in third world countries. The goals of

this paper have been to illustrate the issues and show how to decide on the most cost-effective penstock solutions that systematically

cover the 0.2–20 kW supply. The paper presents the results as a matrix of the most cost-effective penstocks, and in the larger project it

matches them to a modular set of turbines. It shows how to find the relative cost-effectiveness of alternative penstocks, and concludes

with examples illustrating the results.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Microhydro; Penstock; Hydropower; Renewable energy; Home energy; Low head

1. Introduction

The microhydro region of interest in this paper isapproximately defined in Fig. 1.

The goal of this paper is the optimization of microhydropenstocks, by which is meant achieving the maximumpower for the minimum capital investment. While thispaper describes penstocks in relation to a larger projectdescribed in [1], the outline on penstocks provided heregives standalone results that can be applied independentlyof the larger project.

The objective of the larger research program is toprovide a set of economical, small hydropower solutionsfor medium- and low-head sites to supply small isolated

e front matter r 2007 Elsevier Ltd. All rights reserved.

nene.2007.01.009

ing author. Tel.: +643 3667 001x7385;

078.

sses: keith.alexander@canterbury.ac.nz (K.V. Alexander),

xtra.co.nz (E.P. Giddens).

communities of various sizes. The plan, as described in [1]is to achieve economy by optimizing each component, andby running the system continuously at the maximumefficiency point. The strategy for each site is to pick theappropriate components from the optimized selection andassemble them in a modular fashion to produce a cost-efficient whole installation. The required components areshown in Fig. 2, and this includes the penstock. It is thetask of this paper to detail the optimization process for thepenstock piping in the microhydro range.

2. The importance of minimizing capital cost

In conventional large-scale hydropower, attention isgiven to maximizing the energy that will be produced fromthe site over the life of the scheme, which may be a centuryor more. The efficiency of energy recovery tends todominate over considerations of initial capital cost. Onthe other hand, microhydro installations typically servicesmall communities with limited resources, and the initialcapital cost becomes the overriding issue, so it is more

ARTICLE IN PRESS

Notation

C cost per unit length of penstock material ($/m)Cost cost of the full length of the penstockD penstock piping inside diameter (m) (fre-

quently referred to in mm, but must beconverted to m for calculations)

e penstock wall roughness (m)f friction factorg acceleration due to gravity, 9.81m/s2

H turbine head (m)h penstock friction head loss (m)Hg site gross head (m)K1 constantK2 constant, K2 ¼ rwg ¼ 9810K3 constant that defines costsL penstock length (m)M power per unit cost (kWm/$)Ns specific speed as defined in [1]OD/ID typical outside diameter to inside diameter

ratio for penstock material

P hydraulic power delivered by the penstock tothe turbine (W)

Q penstock flow rate; or discharge (m3/s) (fre-quently referred to in l/s, but must beconverted to m3/s for calculations)

Re Reynolds numberS site slopeV flow velocity (m/s)$/kW cost per unit power for the penstock in a given

installation$/m3 cost per cubic meter of penstock materialrw density of fresh water, 1000 kg/m3

d trial penstock diameter/reference penstockdiameter, D2/D1

n kinematic viscosity of fresh water.

Subscripts

1 reference case2 trial case

K.V. Alexander, E.P. Giddens / Renewable Energy 33 (2008) 507–519508

important to maximize power per unit cost than tomaximize power alone.

Since the penstock cost is typically 1/3 of the overallinstallation costs, it is one of the most expensiveitems and has to be carefully chosen. There are twocomponents to achieving the best power per unit cost,first finding the maximum power per unit length ofpenstock and second, making a realistic choice of penstockslope.

Pelton W

heel Ns = 3

6

0.2 kW Electical O

utput.

1

10

100

Gross Head, Discharge, Specific Speed at 1

011

Discha

Gro

ss S

ite H

ead

(m

) .

Pelton Wheel Region

Microhyd

Power too low to consider

Fig. 1. The site head and discharge ran

3. Optimum friction loss ratio

Clearly larger diameter piping costs more, so the task inthis section is to determine the maximum power that can beobtained for a given diameter of penstock. The classicsolution for power per unit discharge (from [2] forexample), is as follows:Consider a site with a fixed gross head Hg, a penstock of

fixed length L, and a fixed diameter D; but consider a

25 kW Electical O

utput.

Highest Pra

ctical N

s = 600

550 RPM and Electrical Power Delivered

0001100

rge (l/s)

Minihydro Region

ro Region

Specific Speed too high to achieve

ges defining the microhydro region.

ARTICLE IN PRESS

Turbine

Flywheel & Brake*

Generator

Electrical Controller

Penstock

Intake

PowerhouseOutflow; Draft tube

Consumers

Transmission Line

Electrical Dump Load

* Depending on generator

Fig. 2. Components of a typical medium-head microhydro installation.

Variation of Power and Head with Flow Rate

0

5

10

15

20

25

0 20 40 60 80 100

Flow Rate Q L/s

Po

wer

(kW

), H

ead

(m

)

Turbine Power (kW) Turbine Head (m) Gross Head (m)

Flow at max Power OptimumTurbine Head

Fig. 3. A sample case to illustrate that for a given diameter, maximum turbine power occurs when the turbine head is 2/3 of gross head (conditions:

150mm uPVC pipe, 100m long.).

K.V. Alexander, E.P. Giddens / Renewable Energy 33 (2008) 507–519 509

variable discharge Q. If the sum of the friction losses is h,then the turbine head H, will be given by

H ¼ Hg � h. (1)

The head loss, h, is related to the discharge, Q, by thefollowing square-law relationship:

h ¼ K1LQ2, (2)

where K1 is to all intents, a constant (ignoring the smallvariation in Reynolds number). The input water power P

available at the turbine is then given by

P ¼ K2HQ, (3)

where K2 is another constant. Hence

P ¼ K2ðHg � K1LQ2ÞQ. (4)

Differentiating P with respect to Q, equating to zero andsubstituting Eq. (2) into the result identifies that themaximum power P, occurs when

h ¼ Hg=3. (5)

From Eq. (1) this also means that the turbine head H isgiven by

H ¼ 2=3�Hg. (6)

Fig. 3 illustrates this analysis. For the purposes of micro-hydro this result can be stated as follows:

For a given penstock diameter D, the power in the flowis a maximum when the flow rate Q causes the penstockhead loss h to be one-third of the gross head Hg.

ARTICLE IN PRESSK.V. Alexander, E.P. Giddens / Renewable Energy 33 (2008) 507–519510

It should be pointed out that this is not the maximumpower that can be obtained from the given discharge Q

with the given site gross head Hg. More power can beobtained if the penstock diameter is increased. This willreduce the head loss h and more power will be available atthe turbine. But the purpose here is to maximize the poweragainst capital cost. And an increase in diameter willincrease the cost; typically the next available piping size willcost around 30% more. So the result above gives themaximum power that can be obtained for a given diameterof penstock.

While this analysis gives maximum power for flow andtherefore diameter, the actual concern is maximum powerper unit cost. To address this it is assumed that since wallthickness will increase in proportion to diameter, a realisticcost function may be given as

C ¼ K3D2, (7)

where K3 is a constant and C is a cost per unit length ofpenstock. The power per unit cost is defined as

M ¼ P=C. (8)

Substituting for P from Eq. (3), C from Eq. (7), H fromEq. (1) and h from Eq. (2), the equation becomes

M ¼K2Q

K3D2ðHg � K1LQ2Þ. (9)

Differentiating M with respect to Q, equating to zero andsubstituting Eq. (2) into the result shows that the maximumpower per unit cost M, occurs when h ¼ Hg/3, which is thesame result as for Eq. (5) above. This confirms that themaximum power per unit cost does in fact correspond tothe maximum power for a given diameter, at least for theassumed cost function of Eq. (7) and the assumption thatK1 is constant. Fig. 4 illustrates the relationship. (It may benoted in Fig. 4 that the peak of the curve of kW/unit cost,does not quite align with the point corresponding to

0.333

0.0

0.2

0.4

0.6

0.8

1.0

1.2

100 120 140 160Diame

Hea

d lo

ss a

s a

frac

tion

of

Gro

ss H

ead

(h/H

g)

Fig. 4. Maximum power against penstock diameter (and cost) occurs when he

20m gross head.).

h ¼ Hg/3. This is because in Fig. 4 K1 was allowed to varyrather than being kept constant. In this kW/unit cost case,analysis shows that h ¼ 2Hg/7 is in fact a closer figure. Forthe purposes of the exercise however, h ¼ Hg/3 is quiteclose enough to select appropriately from the availablechoices of diameter which are widely spaced for commer-cial convenience.)

4. Other penstock losses

Any bends, the tailrace, the inlet grill, valving, and so onwill reduce the head delivered by a given penstock. Thismust be compensated for by providing a higher gross headin the first place. Typically these losses add up to about20% of the penstock gross head. In any given scheme thisshould be assessed carefully and taken into account indetermining the available turbine head H, to ensure thefinal scheme produces the power predicted.

5. Penstock slope

Penstock slope and features of the penstock installationare sketched in Fig. 5. As shown, a tail tank is part of thedesign, and this is where the turbine outlet pipe discharges.The spillway of the tail tank should be set clear ofmaximum flood level to allow the water to discharge backinto the river. The purpose of the tail tank is to fix theturbine tail water level and make it independent ofvariations in river level during flood conditions. If theturbine outlet pipe is discharged directly into the river,variations in the water level could vary the turbine headand complicate the operation of the turbine, apart from therisk of endangering the installation.The intake arrangements should be such that the forebay

tank water level is constant and it follows that, for aconstant operating flow rate, the turbine head will then beconstant. The design of the powerhouse would normally set

180 200 220 240ter (mm)

Pow

er p

er U

nit

Cos

t (A

rbit

rary

sca

le)

Head Loss

kW/UnitCost

ad loss is approximately 1/3 of gross head (149mm ID uPVC, 100m long,

ARTICLE IN PRESS

Hg

Forebay tank water level

Tail tank water level

Total energyline

Penstock

Turbine

Tail tank Variable

river level

LH

h

Slope: S = Hg/L

Fig. 5. Diagram of penstock arrangement and the definition of penstock slope, S.

1On slopes steeper than 0.25 there are likely to be rock faces and keeping

components in place becomes difficult; slopes shallower than 0.125 and the

penstock becomes very long and the cost is too high.

K.V. Alexander, E.P. Giddens / Renewable Energy 33 (2008) 507–519 511

the tail tank water level about 1m below the turbine axis.To a first approximation this extra meter will be ignored,and the turbine head H will be considered to be developedin the penstock at its entry to the turbine.

The slope S of the penstock is defined as the change inelevation between the forebay tank and the turbine entry,divided by the length of the penstock, as shown in thefigure. This is equivalent to the gross head Hg divided bythe length of the penstock L, i.e.,

S ¼ Hg=L. (10)

This definition makes it consistent with the hydraulicgradient which is the penstock pipe friction loss, divided bythe length of the penstock; i.e.,

hydraulic gradient ¼ h=L. (11)

6. The influence of penstock slope

The slope of the penstock route turns out to be asignificant parameter in evaluating proposed sites, and isexamined below.

Dividing through Eq. (2) by Hg gives

h

Hg¼

K1LQ2

Hg. (12)

But from Eq. (10) Hg/L ¼ S, the slope of the penstock,therefore

h

Hg¼

K1Q2

S. (13)

But from Eqs. (5) and (9) the optimum is when h/Hg isfixed at 1/3. Thus for a fixed K1 (which contains a fixeddiameter D), the flow rate Q is dependent only on the slope,S. There is however, a limited range of likely slopes forviable sites. These are most likely to range between 0.25

(1 in 4) and 0.125 (1 in 8).1 Because of this limited range ofslopes it is possible to set just one flow rate to cover therange. Thus it is decided that h/Hg ¼ 1/3 at the flatter slopeof 0.125 (1 in 8) for reasons of cost given below. Steeperslopes will then have lesser values of h/Hg and will vary alittle from the optimum in terms of cost.

7. Penstock slope and diameter

K1 in Eq. (2) above can be defined as

K1 ¼f

2gðp=4Þ2D5, (14)

developed from, for example [3], where f is the frictionfactor determined by the surface roughness of the pen-stock, (Eq. (17) below) and g is the acceleration due togravity.To determine the optimum penstock diameter, taking the

slope into account, K1 from Eq. (14), is substituted intoEq. (13), and then Eq. (13) is rearranged as follows:

D ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif

2gðp=4Þ2ðh=HgÞ

Q2

S

5

s. (15)

In this equation g is constant and f is virtually constant (seeEq. (17) below). From the analysis above, the aim is to seth/Hg ¼ 1/3 to achieve maximum power. This leaves theremaining parameters, site slope S and discharge Q whichare now all that is required to find the optimum penstockdiameter D, for a particular site. Thus the site dischargeand slope, found by site survey, can be used to define theoptimum penstock diameter using Eq. (15).An alternative derivation enables the optimum discharge

Q to be found for any given diameter D. Again Eq. (14) is

ARTICLE IN PRESSK.V. Alexander, E.P. Giddens / Renewable Energy 33 (2008) 507–519512

substituted into Eq. (13) and rearranged:

Q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSðh=HgÞ

2gðp=4Þ2D5

f

s. (16)

As before f and g are essentially constant and the aim is tohave h/Hg ¼ 1/3. For a chosen limited slope S, the dischargeQ is now defined by the diameter D. This equation can beused to find the optimum discharge for any diameter.

8. Penstock slope and cost

Penstock slope, S as defined in Fig. 5 has two effects:

�

For a given site power, a lower slope increases the lengthof the penstock, which increases the price (by Eq. (10)). � This increase in length further increases the pipe losses

(by Eq. (2)), which requires the site to have a still greatergross head to overcome the losses in order to provide thesame power.

Given that any scheme will typically have one-third of itscost in the penstock, Fig. 6 illustrates how the cost can getout of hand if the slope is too low. A typical site might havea slope of 0.25 (1 in 4), while a recommended minimumslope is 0.125 (1 in 8). At this minimum recommendedslope the penstock cost rises from 33% to over 50% of thewhole installation cost. Below this slope (that is 1in 9 to 1in 12 say) the penstock cost becomes increasingly prohi-bitive, and is generally considered uneconomic (though thisis a choice for the customer).

The increase in gross head required to deliver the samehydraulic power to the turbine as slope decreases is alsoshown in Fig. 6, completing the point that penstocksbecome less viable as the slope decreases.

0%

10%

20%

30%

40%

50%

60%

70%

80%

0 1 2 3 4 5 6

Site Sl

Pen

sto

ck C

ost

& G

ross H

ead

.

Penstock Cost as a % of Total Cos

Additional Gross Head Requireme(as a % of Turbine Head)

Typical Installation

Recommended Minimum Slope

Fig. 6. Typical effects of site slope on (a

9. Choice of a reasonable minimum slope

The approach taken for the modular design project wasto decide on a reasonable minimum value for the slope anduse that with h/Hg ¼ 1/3 in Eq. (16) to fix the dischargevalues Q for each standard pipe size. (These fixed dischargevalues are then used to determine turbine sizes for themodular design project. In other words, because availablepenstock piping is a large component of overall cost, thepenstock is optimized first, and then the penstock dischargeis allowed to dictate the size of the most cost-effectiveturbines.)This philosophy means that on a low-slope site

(S ¼ 0.125 or 1 in 8) the relatively long and expensivepenstock will be optimized. But this choice of discharge isslightly less than optimum on a site with a steeperslope. For example, on a site with a slope of 1 in 4 orS ¼ 0.25 the chosen discharge results in h/Hg ¼ 0.167and the penstock is potentially 5% or so more expen-sive than the optimum. But on this steeper slope thepenstock is shorter. This results in it being less thanhalf the cost of the penstock in the 1 in 8 slope site. So theslight increase in cost for the 1 in 4 slope is accepted as oneof the costs of developing a modular system where thedischarge has been chosen to be fixed by penstockdiameter.

10. Design discharge for standard pipe sizes

Based on the analysis above it has been decided that theoptimum discharge for any penstock will be determinedwhen the slope S is 0.125 (1 in 8), with the head loss fromEq. (5) as h/Hg ¼ 1/3. Taking an average surface rough-ness, (to allow for ageing) of e ¼ 0.1mm, and a watertemperature of 20 1C, the friction factor f may be calculatedto adequate accuracy from the US adaptation of Moody’s

7 8 9 10 11 12 13

ope: 1 in...

t

nt

) penstock costs and (b) gross head.

ARTICLE IN PRESSK.V. Alexander, E.P. Giddens / Renewable Energy 33 (2008) 507–519 513

formula (from [4]):

f ¼ 0:0055 1þ 20000e

Dþ

106

Re

� �1=3" #

, (17)

where the Reynolds number Re is given as

Re ¼VD

n, (18)

in which the kinematic viscosity, n at 20 1C is 1� 10�6

, andthe flow velocity V is approximately 2m/s but may berefined by iteration. With this information the optimumdischarge Q may be calculated from Eq. (16) for anyproposed penstock diameter.

The most economic penstock material is typically thecommercially available uPVC (for example uPVC in AS/NZS 1477 1999, or ISO 4422-2 [5]). There are severalgrades but the most suitable is 600 kPa uPVC pressure pipewhich is adequate for up to 87m gross head and istherefore suitable for any installation in the microhydrorange in Fig. 1. While it is possible to use lighter gradeuPVC piping for low-head sites, this is not recommended,largely for the reason that the 600 kPa material is betterable to cope with rough handling, transport and installa-tion in rugged environments. Given this decision, thefollowing analysis has been based on the available metricsizes of this 600 kPa material. Using Eqs. (13) and (16)–(18)the diameters and flow properties have been calculated,and the results listed in Table 1.

It will be noted in the table that for the four largepipe sizes the flow velocity has been limited to a maximumof 3m/s. This is in order to comply with the WorldBank requirements [6], which need to be met for fundingassistance for small hydro schemes in some countries. Theeffect is to reduce the head loss ratio below the optimum 1/3,making the four largest penstocks in Table 1 slightly moreexpensive than the ideal determined by Eq. (5).

11. Penstock performance chart

Fig. 7 gives an overview of the modular scheme andincludes the flows for commercially available metric uPVC

Table 1

Calculated conditions for commercially available uPVC pipe

Pipe nominal

diameter (mm)

Pipe inside diameter D

(mm)

Velocity V

(m/s)

Reynolds

number Re

Relati

roughn

80 83.7 1.73 1.45E+05 0.0011

100 107.8 2.04 2.20E+05 0.0009

125 132.2 2.32 3.07E+05 0.0007

150 151.3 2.53 3.83E+05 0.0006

175 190.2 2.92 5.56E+05 0.0005

200 213.8 3.00 6.41E+05 0.0004

225 237.7 3.00 7.13E+05 0.0004

250 266.2 3.00 7.99E+05 0.0003

300 299.5 3.00 8.99E+05 0.0003

penstocks from Table 1. This figure is taken from Fig. 9in [1].The following points are illustrated in Fig. 7:

�

ve

ess

9

3

6

6

3

7

2

8

3

The vertical axis represents both the site gross head andturbine head.

� The horizontal axis represents the flow through the

penstocks and consequently the flow through theturbines.

� The heavy vertical lines on the chart represent the

commercially available metric uPVC pipe sizes placedaccording to their optimum discharge as shown inTable 1.

� The lines sloping from the lower left to the upper right

are lines of constant specific speed, Ns in Eq. (1) in [1].Each line represents one form of turbine at 70% turbineefficiency and 1500 rpm. The six lines cover the sixspecific speeds and therefore the six scalable turbineconfigurations chosen for the modular design project,covering the microhydro range.

� On each specific speed line there are 4–8 intersections

with the vertical penstock lines. These intersections aremarked with circles. These circles represent the turbinesthat could be built to take advantage of the optimumpenstock flows.

� The head on the vertical axis, corresponding to each

circle, represents the turbine head H for each turbine(circle).

� Each turbine (circle) on the penstock line is accompa-

nied directly above by a triangle representing the slopeof 0.25 or 1 in 4, and above that again by a square,representing the slope of 0.125 or 1 in 8. The triangleand square correspond to the required gross head Hg

(read off the vertical axis) to achieve the turbine headand electrical power of their associated turbine (nearestcircle beneath). Because of the close proximity of thetriangle to the square, it is relatively easy to interpolateby eye to find the appropriate point for any other slope.

� The lines sloping from the upper left to the lower right

are lines of constant power; in this case the estimatedelectrical power output from the generator(s) with allefficiencies taken into account. (Note: The electrical

e/D

Friction factor

f

Flow rate Q

(l/s)

Penstock slope

S ¼ Hg/L

Head loss

ratio h/Hg

0.023 9.5 0.125 0.333

0.021 18.6 0.125 0.333

0.020 31.9 0.125 0.333

0.019 45.5 0.125 0.333

0.018 83.0 0.125 0.333

0.018 108 0.125 0.304

0.017 133 0.125 0.267

0.017 167 0.125 0.232

0.016 211 0.125 0.201

ARTICLE IN PRESS

80 m

m

10

0 m

m

1

25

mm

150

mm

1

75

mm

20

0 m

m

225

mm

25

0 m

m

30

0 m

m

Pelton W

heel, Ns = 36

70% Efic

iency Ns = 60

Ns = 104

Ns = 175

Ns = 260

Ns = 400

Ns = 600

0.3 kW Electric

15 kW Electric

35kW

Electric:M

ini Hydro R

egion

1.3 kW Electric

3 kW Electric

5 kW Electric

1

10

100

1000100101

Discharge, Q (l/s)

Tu

rbin

e H

ead

& G

ross H

ead

(m

)

Turbine Head

Gross Head for 1 in 4 Slope

Gross Head for 1 in 8 Slope

Example

Penstock Nominal Diameters

Fig. 7. Preliminary site assessment chart including effects of site slope on gross head. [1].

K.V. Alexander, E.P. Giddens / Renewable Energy 33 (2008) 507–519514

power lines can, of course, only be read in relation to theturbines (circles)).

For example: there is a turbine in Fig. 7 just above theintersection of the Ns ¼ 260 line and the 3 kW electric line,indicated by an enlarged circle. For this turbine theelectrical power is clearly just above 3 kW. The turbinedischarge is 83 l/s and the turbine head is 7.5m. If thisturbine was on a site with a slope of 0.25 (1 in 4) the grosshead would need to be (the triangle) 9m, with the samedischarge of 83 l/s. And if it was on a slope of 0.125 (1 in 8),the gross head would need to be (the square) 11m, againwith the same discharge.

In the figure there are a total of 38 possible turbines of 6different forms. One of each form has already been builtand tested, with several now in service. The intention is toeventually build a useful selection of sizes from the possible38, by scaling from the already-proven models.

12. Questions of stability

The prospect of operation at the point of maximumpenstock power, illustrated in Fig. 3, raises questions ofstability. The reason is that at this particular point thepower is, for a small range of flow at the top of the powercurve, independent of head. This suggests that instabilitymight arise resulting in hunting, periodic changes in rpmand flow, or steps in flow between two points of equalhead. For such instability to be avoided, the turbinehead–discharge curve needs to be compared with thepenstock head–discharge curve to demonstrate that theyare not anywhere near parallel in this region.

The head–discharge characteristic of the turbine, atconstant speed, in the vicinity of the duty point approx-imates to a parabola passing through the origin. If this issuperimposed on the penstock head–discharge characteristicof Fig. 3, the result fortunately, is an almost perpendicularintersection. The lines are nowhere near parallel and so thereis no danger of instability, as is shown in Fig. 8.Similarly, the power–discharge characteristic of the turbine,

at constant speed, in the vicinity of the duty pointapproximates to a cubic through the origin and again, whenit is superimposed on the penstock power discharge character-istic, a good stable intersection is the result, as shown in Fig. 9.These analyses signify that the matching of the turbine to

the penstock, at the point of maximum power, which isnecessary for cost-effectiveness, is stable, and will not causehunting or other instabilities.

13. Practical issues

13.1. Other components

Up to this point consideration has been given to thepenstock piping alone. Of course there will also need to beconnection components, anchoring components, possiblyseveral bends (though these are to be avoided if possible),an inlet system that excludes vegetation and debris (such asreported in [7]) a shutoff and venting arrangement, not tomention earthworks, labor and transport of the piping tothe site. So the costs derived from the piping alone are onlya part of the picture. For a long penstock, the piping costsdominate these other items, but for medium and lowerheads these other items assume greater significance.

ARTICLE IN PRESS

0

2

4

6

8

10

0 20 40 60 80 100

Flow rate (L/s)

Po

wer

(kW

)

Penstock Power-Flow characteristic Turbine Power-Flow characteristic

Fig. 9. Power-flow characteristics of penstock and turbines showing little potential for instability.

0

5

10

15

20

25

0 20 40 60 80 100

Flow rate (L/s)

He

ad

(m

)

Penstock Head-Flow Characteristic TurbineHead-FlowCharacteristic

Fig. 8. Head-flow characteristics of penstock and turbines showing little potential for instability.

K.V. Alexander, E.P. Giddens / Renewable Energy 33 (2008) 507–519 515

13.2. Shutoff system

It should be noted that penstocks full of flowing water canpresent a considerable hazard if they are shutoff suddenly. Itis recommended that any valving or shutoff mechanismshould be at the intake or penstock inlet end, not at theturbine end. If the penstock is to be closed when it is shutdown (rather than draining the forebay tank), it shouldincorporate an air vent that allows it to empty of water in amanaged way, whenever it needs to be shut down.

13.3. Penstock fouling

Over time the penstock may deliver less head if it coatswith slime or other deposits. Experience has shown thatdeposits are hard to predict and are frequently site

dependent. Customers should be warned that they mayneed to accept a small drop in peak power over the firstyears of operation if there are deposits of some sort.Customers tend to take all the power available and this is atrap as some appliances may have to be laid off as powerdrops.

14. Another look at penstock cost

It is now possible to produce a generic cost function forany penstock. In Eq. (7) above a little thought will showthat the constant K3 is given by

K3 ¼p4

$

m3

� �OD

ID

� �2

� 1

!, (19)

ARTICLE IN PRESSK.V. Alexander, E.P. Giddens / Renewable Energy 33 (2008) 507–519516

where ($/m3) represents the cost per cubic meter ofpenstock material, and OD/ID is a representative insideto outside diameter ratio for the penstock material in use(OD/ID is typically 1.07 for 600 kPa uPVC pressurepiping).

In Eq. (7) the cost per unit length was found, so the costfor the full length of the penstock piping at a given site is

Cost ¼ CL ¼ CHg

S. (20)

When combined with Eq. (7) this gives

Cost ¼ K3D2 Hg

S. (21)

In this equation all factors are either known from above orfrom a site survey. This equation may be now used forcomparative analyses and budgeting, with the proviso thatactual costs will need to be determined locally.

15. Alternatives to the optimum penstock

It is frequently the case that the designers will want toexamine alternative penstock options rather than acceptthose given, for example, in Fig. 7. One reason may be thatthe turbine head H, at a particular site is just insufficient.Similarly the modular scheme has based the pipingselection on slopes of 0.125 (1 in 8), while at significantlyhigher slopes there are some piping selections that can giveslightly better economy. Under such circumstances theremight be a case for using the next larger or next smaller sizeof penstock piping, even though it could cost a little moreper kW delivered, or may deliver less power.

The following equations, derived from those above,enable a comparison with any particular penstock choice.In the following, the factor d is the ratio of the trialpenstock diameter (subscript 2), to any previously chosenreference diameter (subscript 1). For example the optimumpenstock already derived from Fig. 7 may be designatedD1, while any alternative is D2:

d ¼D2

D1. (22)

Of particular interest are the power P, cost and turbinehead H, and how an alternative penstock fairs incomparison with the first choice. In deriving the equationsit has been assumed that on a particular site the slope S,gross head Hg, and flow Q are site specific, and remainunchanged during the comparison. On the other hand, thediameter D and consequently the head loss h will change,resulting in a change to the turbine head H and thehydraulic power P, delivered to the turbine. Looking firstat the equations leading to the power comparison, fromEqs. (17) and (18)

f 2

f 1

¼1þ 20; 000ðe=D2Þ þ ðd� 106=Re1Þ

� �1=3� �1þ 20; 000ðe=D1Þ þ ð10

6=Re1Þ� �1=3� � . (23)

From Eq. (14)

ðK1Þ2

ðK1Þ1¼

1

d5f 2

f 1

. (24)

From Eq. (13)

h2

h1¼ðK1Þ2

ðK1Þ1. (25)

From Eq. (1) the turbine heads can be compared:

H2

H1¼ðHg � h2Þ

ðHg � h1Þ. (26)

Then from Eq. (3) the comparison in power between thetwo cases is given as

P2

P1¼

H2

H1. (27)

Now examining the cost comparison, from Eq. (21):

Cost2

Cost1¼ d2. (28)

And finally the value comparison from Eqs. (27) and (28).(Note particularly that H1 is in the denominator on theRHS in this case):

ð$=kWÞ2ð$=kWÞ1

¼ d2H1

H2. (29)

Two figures are used to illustrate these comparisons.Fig. 10 shows a case where 150mm nominal diameter isthe optimum choice for a slope of 0.125 (1 in 8) with aconsequent head loss ratio of h/Hg ¼ 1/3. Alternatediameters are spread along the horizontal axis, andneighboring commercial pipe sizes are shown. The verticalaxis compares the trial cases with the reference for cost,power P, and $/kW. It can be seen, for example, that the125mm pipe which is the next available size smaller thanthe reference:

�

Costs less, at 77% of the value of the 150mm pipe � Delivers only half the power (because of a greater head

loss)

� And has a $/kW value of 1.5 times that of the 150mm

pipe

It is clear that the 150mm penstock gives better valuethat the 125mm pipe. It is also apparent from the shape ofthe $/kW curve that the 150mm reference case gives thebest $/kW value, as would be expected since h/Hg ¼ 1/3 forthis pipe in this example.It can also be seen that the 175mm pipe which is the next

available size on the larger side:

�

Costs more at 156% of the value of the 150mm pipe � Delivers 34% more power (because of a lesser head loss) � And has a $/kW value of 1.15 times that of the 150mm

pipeWhile it is apparent that the 150mm penstock still gives

better value for money, as expected, it illustrates what the

ARTICLE IN PRESS

No

m 1

25

mm

No

m 1

50

mm

(D

1,R

efe

ren

ce

)

No

m 1

75

mm

No

m 1

00

mm

0%

50%

100%

150%

200%

250%

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

δ

($/k

W2)

/ ($

/kW

1),

P2/P

1, C

ost2

/Co

st1

($/kW2) / ($/kW1)

P2/P1, H2/H1

Cost2/Cost1

Penstocks

= D2/D1

Fig. 10. Comparison of adjacent penstock sizes to the 150mm nominal size, for the case where the 150mm piping is optimized for a slope of 0.125

(1 in 8) and with h/Hg ¼ 1/3.

0%

50%

100%

150%

200%

250%

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

δ =D2/D1

($/k

W2)

/ ($

/kW

1),

P

2/P

1, C

ost2

/Co

st1

($/kW2)/($/kW1)

P2/P1,H2/H1

Cost2/Cost1

Penstocks

No

m 1

00

mm

No

m 1

25

mm

No

m 1

50

mm

(D

1,

Re

fere

nce

)

No

m 1

75

mm

Fig. 11. Comparison of adjacent penstock sizes to the 150mm nominal size, for the case where the 150mm piping is not optimized, has a slope of 0.25

(1 in 4) so has h/Hg ¼ 1/6. In this instance the 125mm piping is marginally better economically.

K.V. Alexander, E.P. Giddens / Renewable Energy 33 (2008) 507–519 517

customer might get in terms of power if they wanted to paythe extra for the larger penstock, and if there was a turbineavailable to match that penstock head and discharge.

Fig. 11 illustrates what happens in a minority of cases inthe modular scheme, where penstocks are chosen to havethe optimum head loss ratio at slopes of 0.125 (1 in 8), butwhere all slopes are assumed to have the same flow rate Q

(as illustrated in Fig. 7). In Fig. 7 the 150mm penstock isassumed to discharge 46 l/s, which, at a slope of 0.25 (1 in4) results in a head loss ratio h/Hg ¼ 1/6 rather than theoptimum of h/Hg ¼ 1/3 (which it does have at a slope of0.125 (1 in 8)).

In Fig. 11 with the 150mm reference pipe having a headloss ratio of h/Hg ¼ 1/6, the 125mm pipe:

�

Again costs less at 77% of the value of the 150mm pipe � Delivers only 80% of the power (because of a greater

head loss)

� But has a $/kW value 5% less than that of the reference

150mm pipe.

So in this instance the comparison has shown that thesmaller penstock is slightly more economic. Conceivably thecustomer might decide that the slightly lower cost is moreimportant than the reduction in available power; but of courseif the modular scheme is to be used, then there is no turbine tomatch the 80% power delivered by the 125mm penstock atthis flow, while there is one to match the 150mm penstock. Soa change to the 125mm penstock would not be practical.

ARTICLE IN PRESSK.V. Alexander, E.P. Giddens / Renewable Energy 33 (2008) 507–519518

(Note that this slightly better economy for a step down insize only happens with a few combinations of commercialpipe sizes).

16. Sample calculation

At a particular site the available flow Q is determined tobe 90 l/s, the gross head Hg is reckoned to be 12m, and theslope S ¼ 0.167 (1 in 6). The penstock material cost isfound to be $5000 per cubic meter and the OD/ID ratio ofthe piping material is typically 1.05. The task is to analyzepenstock options.

The most cost-effective penstock may be found byreferring directly to Fig. 7, this being the purpose of thefigure. Alternatively it may be found from the equationsused to derive the figure, namely Eq. (15): assumingh/Hg ¼ 1/3 from Eq. (5), with some refinement iterationsfor f in Eqs. (17) and (18), the numbers in Eq. (15) are

D ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif

2gðp=4Þ2ðh=HgÞ

Q2

S

5

s,

D ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:018

2ð9:81Þ ðp=4Þ2 ð1=3Þ

ð90=1000Þ2

0:167

5

s

¼ 185:3mm;

which is closest to 190.2mm ID or 175mm nominal diameterfrom Table 1. This corresponds with the result from Fig. 7.

The approximate cost of 175mm nominal penstockpiping for this site is calculated from Eqs. (19) and (21) as

K3 ¼p4ð15; 500Þ ð1:07Þ2 � 1

� �¼ 1764

Cost ¼ 1764190:2

1000

� �212

0:167¼ $4595.

(Note that this is the cost of the piping alone, and anyfittings will be extra).

The length of piping required is given by Eq. (10):

L ¼ Hg=S ¼ 12=0:1667 ¼ 72m:

Now since a slightly different diameter is to be used thanthe ideal of 185.3mm calculated, (because the nearestavailable size is used), the actual h/Hg ratio will be differentthan 1/3. From Eq. (14)

K1 ¼f

2gðp=4Þ2D5

¼0:018

2ð9:81Þ ðp=4Þ2ð190:2=1000Þ5

¼ 6:03,

and so in Eq. (13)

h

Hg¼

K1Q2

S

¼6:03ð90=1000Þ2

0:167¼ 0:293.

The approximate power available at the turbine inlet maybe found from Eq. (3) assuming h/Hg ¼ 0.293 andknowing K2 ¼ rwg ¼ 9810:

P ¼ 9810ð1� 0:293Þ1290

1000

¼ 7492W ¼ 7:49 kW:

(Note that after the turbine and generator efficiencies havebeen taken into account something less than half of thispower will be available electrically).If the modular microhydro scheme of Fig. 7 is to

be used then a point on the 175mm penstock line must befound that corresponds to a slope of about 0.167 (1 in 6).This can be seen to occur at a head of about 10m which isbetween the triangle (slope 0.25 or 1 in 4) and the square(slope 0.125 or 1 in 8), so only about 10m of theavailable12m gross head will be used (though it is prudentto have this 20% in reserve for additional losses).The corresponding turbine (the ringed circle below 10mhead on the 175 penstock line) will be at a specificspeed of Ns ¼ 260 with a turbine head of 7.5m anddelivered electrical power, (after all efficiencies have beenaccounted for), of close to 3 kW, as indicated by readingoff the electrical power line in the figure. (And as notedabove, this is less than half of the hydraulic power into theturbine).

17. Sample calculation of off-optimum conditions

In the sample calculation above, the flow and head of thesite were a little above what was matched by the modularscheme of Fig. 7. Suppose the next size up penstock were tobe used. What would be the advantage if any? Would extrapower be available from the site, or would a bettereconomy achieved?First, if the modular system of Fig. 7 is to be used, the

penstocks are matched to turbines and in the first instanceit is easiest to simply use the available systems and foregoany small advantages that might theoretically be achievedwith an exact match with the site. There is however, somelimited scope to run these turbines at different flow ratesand heads than those given in Fig. 7, to better match siteconditions; this is outlined in [1], and can be beneficial in alimited number of cases.For a check on alternative penstock sizes, Eqs. (22)–(29)

may be used to generate data like that plotted in Figs. 10and 11 (except that in this case the different nominaldiameter of 175mm is the reference diameter, D1). Suchcalculations for the adjacent pipe sizes have beenperformed using these equations, and the results are givenin Table 2. In reference to this table it can be seen that theturbine head for the smaller pipe size (150mm nominal) isso low that it is barely able to deliver the flow; this is clearlynot an option. The larger size on the other hand, while it isonly slightly worse in $/kW terms, can provide another1.6m of turbine head resulting in the capacity to deliveranother 1.4 kW of hydraulic power to the turbine. This

ARTICLE IN PRESS

Table 2

Comparison of neighboring pipe sizes for the sample calculation

Downsize Reference Penstock Upsize

Pipe size 150 nom 175 nom 200 nom

D (mm) 151.3 190.2 213.8

f 0.0189 0.0182 0.0178

K1 19.72 6.03 3.29

Hg (m) 12 12 12

h (m) 11.50 3.51 1.92

H (m) 0.499 8.49 10.08

h/Hg 0.96 0.29 0.16

P (W) 440 7492 8898

Cost $2907 $4595 $5805

$/kW $6605 $613 $652

K.V. Alexander, E.P. Giddens / Renewable Energy 33 (2008) 507–519 519

could appear to be an advantage, but would only beso if a matching turbine were available in the scheme.Unfortunately this is not the case; the flow of 90 l/s isbetween two turbines. The alternative to finding a matchingturbine is to belt-drive an existing turbine, a limited optiondiscussed in [1] that sometimes allows a belt-driven matchbetween turbines. The added complexity, cost and main-tenance however, would reduce the attraction of such achoice, and decrease the available kW/dollar. These issuesmust be weighed against using the extra power from the175mm penstock, and they suggest that it is really of noadvantage.

There are sometimes cases where the turbine head withthe optimum penstock, is just insufficient for a match to bemade with the corresponding turbine of the modularscheme. In such instances, going to the next size up ofpenstock material may give sufficient increase in turbinehead for a match to be made with the turbine thatcorresponded with the smaller penstock. In such cases,unlike that above, it may well be an advantage to invest inthe more expensive penstock.

18. Conclusions

While this paper has focused on penstocks for themicrohydro region at heads lower than Pelton Wheels, theprinciples and equations will apply equally well for pen-stocks of higher heads into the Pelton Wheel range. At theother end of the scale, at very low heads of say 2m, thepenstock will be relatively short, and while the equations stillapply, the penstock cost becomes insignificant whencompared to that of the rest of the installation, and theoptimum penstock cost is not such an important issue.A particular goal of the project has been to find the most

economic penstock solutions for microhydro schemes. Toachieve this, equations have been developed, a fittingsummary given in Fig. 7, and an example worked through.Equations have also been given to enable a sensitivityanalysis of any penstock choice. It is hoped this willprovide all that is necessary for a competent practitioner toarrive at the most economic penstock solution for low headmicrohydro sites.

References

[1] Alexander KV, Giddens EP. Microhydro: cost-effective, modular

systems for low heads. Renew Energy, 2007.

[2] Vennard JK, Street RL. Elementary fluid mechanics, 6th ed. Wiley;

1982. p. 400.

[3] Streeter VL. Fluid mechanics, 5th ed. McGraw-Hill; 1971. p. 283.

[4] Massey BS. Mechanics of fluids, 5th ed. Van Nostrand Reinhold;

1984. p. 213.

[5] ISO 4422-2. 1996. Pipes and fittings made of unplasticized poly(vinyl

chloride) (PVC-U) for water supply—specifications—Part 2: pipes

(with or without integral sockets).

[6] Renewable energy for rural economic development, Sri Lanka. Off

grid—village hydro—technical specifications—mechanical compo-

nents. /www.energyservices.lk/index.htmS.

[7] Giddens EP. A self-cleaning intake for mountain streams. In:

Proceedings of the hydraulics in civil engineering conference. National

Conference Publication no 84/7. Adelaide, Australia: Institution of

Engineers Australia; 1984. p. 44–8.