0046351af6b072b4f7000000

Short-cut design of small hydroelectric plants

N.G. Voros*, C.T. Kiranoudis, Z.B. Maroulis

Department of Chemical Engineering, National Technical University of Athens, Polytechnioupoli,

Zografou, Athens, 15780, Greece

Received 20 September 1997; accepted 5 May 1999

Abstract

The problem of designing small hydroelectric plants has been properly analysed andaddressed in terms of maximizing the economic bene®ts of the investment. An appropriateempirical model describing hydroturbine e�ciency was developed. An overall plant model

was introduced by taking into account their construction characteristics and operationalperformance. The hydrogeographical characteristics for a wide range of sites have beenappropriately analyzed and a model that involves signi®cant physical parameters has beendeveloped. The design problem was formulated as a mathematical programming problem,

and solved using appropriate programming techniques. The optimization covered a widerange of site characteristics and three types of commercially available hydroturbines. Themethodology introduced an empirical short-cut design equation for the determination of the

optimum nominal ¯owrate of the hydroturbines and the estimation of the expected unitcost of electricity produced, as well as of the potential amount of annually recoveredenergy. # 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction

The term hydropower refers to generation of shaft power from falling water.The power could then be used for direct mechanical purposes or, more frequently,for generating electricity. Hydropower is the most established renewable resourcefor electricity generation in commercial investments. Although, hydroelectricgeneration is regarded as a mature technology, there are still possibilities for

0960-1481/00/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.

PII: S0960 -1481 (99)00083 -X

Renewable Energy 19 (2000) 545±563

www.elsevier.com/locate/renene

* Corresponding author.

Nomenclature

a, b, c turbine e�ciency model coe�cients de®ned in Eq. (2)c0, c1, c2 capital cost coe�cients de®ned in Eq. (15)cE unit cost of electricity produced ($/kWh)cEL conventional unit cost of electricity ($/kWh)cOP operational cost coe�cient de®ned in Eq. (16) ($/kW)CCP capital cost ($)COP operational cost ($/h)CT total annual cost ($)e percentage of the capital cost on an annual rateE electrical energy annually recovered (W)g gravity constant (m/s2)H available hydraulic head (m)H0 available vertical fall of water (m)Hr nominal vertical fall of water de®ned in Eq. (9) (m)k ¯ow duration curve parameter de®ned in Eq. (11)P electrical power (W)PI investment e�ciencyPr nominal hydroturbine power (W)q �50 ¯owrate duration curve parameter de®ned in Eq. (13)q �min ¯owrate duration curve parameter de®ned in Eq. (12)qmax hydroturbine maximum working ¯owrate fraction de®ned in Eq.

(5)qmin hydroturbine minimum working ¯owrate fraction de®ned in Eq.

(4)Q hydroturbine ¯owrate (m3/s)Q� available water ¯owrate (m3/s)Q �50 mid-year stream ¯owrate (m3/s)Q �max annual highest stream ¯owrate (m3/s)Q �min annual lowest stream ¯owrate (m3/s)S total annual pro®ts expected from the investment ($)t time (s)tOP operating time of the plant (s)tY time of the calendar year (s)

Greek lettersg short-cut model parameter de®ned in Eq. (21)Z turbine e�ciencyZr turbine nominal e�ciency

N.G. Voros et al. / Renewable Energy 19 (2000) 545±563546

improvement. While some elements of hydropower, such as turbine e�ciency andcost, have reached an extreme, the same cannot be said for the system itself. Newturbine designs and transmission technology (which is of great importance tohydropower development) with respect to regional characteristics, continue toadvance. Moreover, environmental concerns are driving changes in the design,construction, operation and optimization of hydroelectric plants [1,2].

Taking into consideration the aforementioned technological advances, as well asthe economic bene®ts of this technology, the generation of electricity derived fromhydroturbines has now a growing capacity of total world-wide installations ofabout 5% per year, doubling about every 15 years. More speci®cally, hydroinstallations and plants are long lasting (due to continuous steady-state operationwithout high temperatures and mechanical stresses), thus producing electricity atlow cost with consequent economic bene®ts [3].

No consensus has been reached on the de®nition of small, mini, and microhydropower plants. Micro hydropower schemes are usually described as thosehaving capacities below 100 kW, mini hydropower plants as those ranging from100 kW to 1 MW while small hydroelectric plants as those that produce electricpower ranging from 1 to 30 MW. Recent international surveys on smallhydropower facilities (with capacities below 10 MW) reveal that small hydropowerplants are under construction or have been already constructed in more than 100countries [4±6].

The design of reliable and cost e�ective small hydropower plants capable oflarge-scale electrical energy production is a prerequisite for the e�ective use ofhydropower as an alternative resource. In this sense, the design of a smallhydroelectric plant or equivalently the determination of type and energy load ofthe particular hydroturbines, should maximize the energy output together with thelife-time of the machines. In all cases, the design objective is closely related to thetotal annual output of the overall hydroturbine operation in power terms.Obviously, given the power curve of the hydroturbine to be used and regional¯ow duration statistical data as well as the topology of the site, we can estimatethe total annual energy output of a small hydroelectric plant to be installed. Inthis case, both the type and size of the hydroturbine, expressed in terms of itsnominal ¯owrate have to be determined under a certain economic environment.The determination of the optimal plant characteristics must be based on speci®cdesign objectives. In this case, the design problem may be formulated as amathematical programming problem, involving an objective function representingthe investment e�ciency which is expressed by the pro®ts expected per unit ofcapital invested.

The construction and operation of various types of hydroturbines to be usedhas been investigated to an extent that several operating experimental data arecurrently available in the literature. Furthermore, hydrogeographical statisticaldata and hydropower potential have been thoroughly investigated for a widerange of regions where hydropower seems promising for exploitation. In addition,suitable mathematical models describing the operation of hydroturbines have beendeveloped and used for the simulation of hydroelectric plants. Fasol and Pohl [7]

N.G. Voros et al. / Renewable Energy 19 (2000) 545±563 547

developed suitable mathematical models for the simulation of the operation of ahydroelectric plant and its advanced control schemes. However, design e�orts inthis ®eld is limited with respect to the general objectives described previously.Charles et al. [8] used advanced computational ¯uid dynamics for the design ofhydroelectric plants and calculated in detail ¯ow quantities, including energy lossand plant e�ciency.

Short-cut design is a technical procedure for expressing in a straightforwardway the optimal results of a detailed design problem through empirical equationsinvolving the corresponding design variables. In this way, all other modelvariables are directly computed through the model equations. The parameters ofthe empirical equations are evaluated by ®tting the short-cut model equations tothe corresponding design problem optimal results computed using the full processmodel. Short-cut design is extremely important for preliminary selection ofalternative design scenaria for diversi®ed policies of investment at a regional level.In this way, short-cut evaluations of optimal designs for certain sites is anindispensable tool for assessing regional planning strategies at a national orinternational level. Moreover, it is extremely important for determining the waythat alternative energy sources could possibly penetrate the energy market by anappropriate subsidy policy.

This work addresses the problem of small hydroelectric plant short-cut design interms of maximizing the economic bene®ts of the investment. The mathematicalmodel of hydroturbines was developed taking into consideration theirperformance with respect to construction and operation. An empirical model wasused for estimating the overall turbine e�ciency. The objective function to bemaximized was the investment e�ciency. The hydrogeographical characteristics ofa site have been analyzed in terms of signi®cant physical parameters and modeledappropriately. Optimization was carried out for a wide range of site characteristicsexpressed by the corresponding model parameters and for three di�erent types ofcommercial hydroturbines. An empirical short-cut model equation was introducedfor determining the optimum nominal ¯owrate of the hydroturbines. The regionsof applicability for all turbines involved, was determined as a function of modelparameters. From the engineering point of view, such an analysis will directlyserve as an evaluation tool for explicitly determining the pro®tability of exploitinghydropower at a certain region.

2. Mathematical modeling of hydroturbines

The power obtained by a hydroturbine operated in a small hydroelectric plant isproportional to the potential energy lost by the falling ¯uid and is given by thefollowing equation [9]:

P � ZgQH �1�The turbine e�ciency, Z, involved in the calculation of water potential converted


energy through expression (1), represents the actual utilization of the availablepotential energy of the system. Normally, a hydroturbine is designed so that it canbe operated for a wide range of working ¯uid ¯owrates around a nominaloperating point. The performance of the turbine is characterised by its nominal¯owrate, Qr, that is an explicit indication of its size. For a speci®c type of turbine(con®guration of equipment), its size and capacity that are directly analogous toits diameter are proportionally related to its nominal ¯owrate. Therefore, theturbine nominal ¯owrate is a suitable variable for sizing the turbine and allrelevant equipment of the plant. The turbine e�ciency depends on the working¯uid ¯owrate and actual turbine characteristics. Experimental data for the turbinee�ciency, in the case of three commercial hydroturbines studied in this work (i.e.FRANCIS, PELTON and AXIAL), are given in Fig. 1, as a function of the ratioof working ¯owrate to its corresponding nominal ¯owrate and the turbinee�ciency at the nominal ¯owrate, Zr [10]. All curves exhibit a maximum at a¯owrate representing the nominal performance of the hydroturbine. An empiricalexpression for representing the turbine e�ciency characteristic curve is proposed:

ZZr

� a

�Q

Qr

�2

�b�Q

Qr

�� c �2�

In this expression, the turbine characteristics are the nominal turbine e�ciency, Zr,the nominal turbine ¯owrate, Qr, and three parameters expressing the shape of thecurves. It must be noted that the nominal power of the turbine is given by thefollowing expression:

Pr � ZrgQrH �3�

Excellent ®ts to actual experimental data were detected when expression (2) wasused as real turbine data. The predictions of the empirical equation proposed forthe case of turbine e�ciency experimental values are also given in Fig. 1,indicating the excellence of the ®t and suggesting the practical signi®cance of Eq.(2). The estimated turbine parameters of Eq. (2) for all commercial hydroturbinesstudied, are given in Table 1. The nominal turbine e�ciency is independent of thenominal ¯owrate for all turbines examined, and its corresponding values are alsogiven in Table 1. Each hydroturbine is constructed to operate between twoextremes, a minimum and a maximum working ¯owrate. We introduce twoturbine characteristic parameters qmin and qmax, representing the fraction of itsnominal ¯owrate corresponding the lower and upper extreme working ¯owrates,respectively. These values are also included in Table 1 for each one of thecommercial turbines studied. The minimum and maximum working ¯owrates forthe turbine are accordingly given by the following equations:

Qmin � qminQr �4�

Qmax � qmax Qr �5�


Fig. 1. Hydroturbine e�ciency curves (points are experimental data and lines are model predictions).

(a) FRANCIS, (b) PELTON, (c) AXIAL.


The water ¯owrate through the turbine, Q, is determined by the followingrelationship as a function of the available water ¯owrate, Q� [9±11]:

Q �8<: 0, Q� < Qmin

Q�, Qmin < Q� < Qmax

Qmax , Qmax < Q��6�

The available hydraulic head involved in Eq. (1) can be estimated by subtractingthe friction losses through the penstock from the available vertical fall of water[11]:

H � H0

"1ÿ l

�Q

Qr

�2#

�7�

The friction coe�cient, l, involved in Eq. (7) depends on the penstock size,con®guration and topology of the region in the sense that the e�ect of pipingnetwork and the dam layout is embodied in this parameter. Eqs. (1), (2) and (7)can now be combined to express the actual power converted as a function ofturbine working ¯owrate and available vertical fall of water.

The nominal power of the plant is given by:

Pr � ZrgQrHr �8�where:

Hr � H0�1ÿ l� �9�The potential of a stream is characterized by the available vertical fall (H0) and its¯owrate that is usually expressed by the ¯owrate duration curve. The ¯owduration curve provides the time period during which the ¯ow rate of the streamis greater than a speci®c value (cumulative distribution of stream ¯owrate on anannual basis). Natural river ¯ow is highly variable, a characteristic that hasimportant implications for the design of hydroelectric plants and theirincorporation into the electrical generation system. Most rivers exhibitpronounced seasonal variation in their ¯ow. In some cases, the average three-

Table 1

Hydroturbine e�ciency and operational parameters

Turbine FRANCIS PELTON AXIAL

a ÿ0.537 ÿ0.224 ÿ0.219b 1.047 0.483 0.476

c 0.490 0.741 0.743

Zr 0.9 0.9 0.9

qmin 0.55 0.35 0.35

qmax 1.1 1.5 1.6


month high ¯ow may be more than ten times greater than the average three-month low ¯ow, while in others, it is less than double. In order to provide ageneralized model for predicting a stream ¯owrate duration curve the followingexpression is suggested:

Q� � Q�max � �Q�min kÿQ�max �t1� �kÿ 1�t �10�

where:

k � Q�max ÿQ�50Q�50 ÿQ�min

�11�

Eq. (10) describes the ¯owrate duration curve of a stream by utilizing only threeparameters; the annual highest stream ¯owrate, Q �max, the annual lowest stream¯owrate, Q �min, and the stream ¯owrate corresponding to the mid-year point ofthe ¯ow duration curve, Q �50. All these characteristic parameters of the streamduration curve along with the stream duration curve itself are presented in Fig. 2.The ®rst two parameters indicate the poles of the ¯ow duration curve, while thethird one, its shape (curvature). Obviously this parameter describes the sharpness

Fig. 2. Placement of a hydroturbine on a site stream duration curve.


of seasonal variations in stream ¯owrate. To illustrate the way that ahydroturbine could be placed on the ¯ow duration curve, Fig. 2 also includes thecharacteristic variables of a hydroturbine placed randomly. In order to describemore e�ciently the site characteristic parameters, we introduce the followingvariables, expressing the corresponding ¯owrate duration curve parameters as afraction of the maximum annual ¯owrate value of the stream:

q�min �Q�min

Q�max

�12�

q�50 �Q�50Q�max

�13�

The annual energy obtained by the operation of the hydroelectric plant iscalculated by integrating Eq. (1) for the entire year:

E ��tOP

0

P dt �14�

The installation cost of the plant is given by the following equation as a functionof the turbine nominal power and the vertical free fall of the site [10]:

CCP � c0Pc1r H

c20 �15�

The operational cost of the plant is proportional to the installed plant capacityand is given by the following equation [10]:

COP � cOPPr �16�

The total annual cost of the plant is therefore calculated by means of thefollowing equation:

CT � eCCP � tOPCOP �17�

As a result the unit cost of energy produced is the ratio of total annual cost andannual energy recovered:

cE � CT

E�18�

The expected pro®ts from the operation of the hydroelectric plant is thereforegiven below:

S � E�cEL ÿ cE� �19�

The investment e�ciency is expressed as the ratio of expected pro®ts per investedcapital:


PI � S

CCP

�20�

Eqs. (1)±(20) constitute the mathematical model of the entire hydroelectric plant.In this case, the design objective is to maximize the investment e�ciency from theoperation of such a plant. Given the type of hydroturbine and sitehydrogeographical characteristics and topology, there is only one design variableto be computed by means of maximizing the objective function; the nominal¯owrate of the hydroturbine. The optimization procedure throughout this paperwas carried out by means of the successive quadratic programming algorithmimplemented in the form of the subroutine E04UCF/NAG. All runs wereperformed on a SG Indy workstation under Unix.

3. Short-cut design of small hydroelectric plants

On the basis of the above, the design strategy for small hydroelectric plants cannow be clearly stated. Given the type of hydroturbine and site hydrogeographicalcharacteristics (i.e. stream duration curve parameters) and topology (i.e. availablevertical fall of water) the nominal ¯owrate at which the hydroturbine shouldoperate must be determined by means of optimizing appropriatetechnoeconomical criteria under speci®c operational and environmentalconstraints.

As a consequence of the above, the determination of the optimum plantcon®guration must be based on speci®c design objectives. In practice, therepresentation of the design problem for small hydroelectric plants should focus tocorresponding mathematical ®gures obtained through an adequate mathematicalmodel as previously formulated. In all cases, the design procedure should involvean objective function representing the economic bene®ts from the operation ofsuch a plant or its e�ciency in terms of energy availability towards regionaldemand. Certain alternative objective function types may be taken intoconsideration regarding the bene®ts expected:

1. Maximization of the investment e�ciency. This case suits to design problemsconfronted by individual power producing industries (either private ormunicipal) that have invested or plan to invest in this ®eld, in countries wherelegislation permits so. In other words, this objective refers to the directeconomical bene®ts expected from such an investment under a speci®ccompetitive economic environment, thus determining the feasibility ofexploiting this type of renewable energy source.

2. Maximization of the amount of energy annually produced from the availablehydro resources. This case suits to design problems usually confronted forregions where no other sources of energy are technically exploitable, and theobjective is to exploit the highest possible energy potential of a region in orderto cover the local demand, assuming that the use of hydropower is still


pro®table compared to the unit cost of electricity available in remote nationalregions due to increased transportation cost.

Throughout this paper we choose the ®rst possibility for our objective function.Characteristic economical ®gures concerning capital and operational costcomponents for a typical economic environment were taken into considerationand are listed in Table 2. Between these two cases, the former evaluates anoptimum plant size that is completely di�erent (smaller) than the latter. However,it can be shown that the optimal results of the ®rst objective that is aneconomically driven function coincide with the ones of the second objective that isa purely technical function (independent of economics) when the unit cost ofconventional electricity approaches in®nity. In this case the plant operates at thepoint of maximum energy recovery.

In order to illustrate the above mentioned situation we examine the case of areal site (Kourtaliotis, Crete) involving a ¯ow duration curve described by amaximum annual stream ¯owrate of 17,640 m3/h, minimum annual stream¯owrate of 17% of the maximum, mid-year stream ¯owrate of 25% and availablevertical water fall of 65 m. The hydroturbine type used for this example wasFRANCIS. The e�ect of hydroturbine nominal ¯owrate on investment e�ciencyand total energy recovered is presented in Fig. 3. Obviously, each case results incompletely di�erent plant size and economic ®gures. For increasing unit cost ofconventional electricity, optimum plant size determined by optimizing theinvestment e�ciency increases to the one determined by maximizing the energyrecovery.

The power performance curve of the optimal plant size using FRANCIShydroturbine, determined by the maximization of the annual pro®ts for the atypical site involving a linear ¯ow duration curve described by maximum annualstream ¯owrate of 10,000 m3/h, zero level of minimum annual stream ¯owrate andavailable vertical water fall of 100 m, is given in Fig. 4. The power curve ischaracterised by two di�erent regions: a constant power region that lasts for 257days and a decreasing power region that lasts for the remaining 108 days of theyear. The ®rst region corresponds to days where stream ¯owrate is greater orequal to the maximum hydroturbine ¯owrate. The second region corresponds to

Table 2

Design and cost data

c0 (k$) 40

c1 0.7

c2 ÿ0.35cOP 0.01

e 0.1

l 0.1

tY (h/y) 8760

cEL (c/kWh) 6.7


days where the stream ¯owrate is less than the maximum hydroturbine ¯owrate.

In this case, the turbine operates for only 55 days where the stream ¯owrate isgreater or equal to the minimum hydroturbine ¯owrate. Obviously, for theremaining 53 days the plant is out of operation since the stream ¯owrate is lessthan the minimum hydroturbine ¯owrate. Therefore, the total annual energy

Fig. 3. E�ect of hydroturbine nominal ¯owrate on certain process variables. (1) Investment e�ciency,

(2) energy recovered. (a) cEL=4 c/kWh, (b) cEL=8 c/kWh, (c) cEL=20 c/kWh.


recovered corresponds to the area below the hydroturbine power curve thatextends up to the 312th day.

The optimization procedure for the determination of the optimal turbine sizedescribed earlier, was concentrated on the solution of a speci®c design probleminvolving a prede®ned turbine type and site hydrogeographical characteristics.This procedure can be extended to include a wide range of turbine types andstream particularities described by di�erent values for the parameters of its ¯owduration curve as well as for di�erent vertical water fall values. When the resultsof the optimization for each turbine and site combination are systematicallycompiled and presented, an empirical short-cut design equation can be evaluatedso that the design engineer can automatically determine the optimal size of eachplant and subsequently evaluate its performance in terms of the recovered amountof energy and the unit cost of power produced. A short-cut design empiricalequation of the following form is proposed:

Qr �"

gq�501� �gÿ 1�q�50

�1ÿ q�min

qmax

�� q�min

qmax

#Q�max �21�

It involves only one parameter, g, and expresses the optimal hydroturbine nominal

Fig. 4. Power performance curve.


¯owrate in terms of investment e�ciency maximization. The determination of thisshort cut model parameter was carried out by ®tting Eq. (21) to the optimalresults of the full design problem for all turbines studied. The values of g for allturbines are given in Table 3. The ®tting of the short-cut empirical model to theoptimal plant nominal ¯owrate for all turbines studied are given in Figs. 5±7.Obviously, all ®ts were extremely satisfactory and Eq. (21) can be safely used forshort-cut design purposes in the case of small hydroelectric plant design. Eq. (21)

Table 3

Short-cut model parameters

Turbine g

FRANCIS 0.422

PELTON 0.369

AXIAL 0.364

Fig. 5. Fitting of the short-cut empirical equation (lines) on optimal plant nominal ¯owrate for

FRANCIS hydroturbine (points) (1) q50=20%, (2) q50=40%, (3) q50=60%, (4) q50=80%. (a)

qmin=0%, (b) qmin=30%, (c) qmin=60%.


does not include any dependence of the optimum plant size on the availablevertical fall. For the entire range of sites studied, PELTON and AXIALhydroturbines involve smaller optimum installation than the one evaluated forFRANCIS. All hydrogeographical model parameters (Q �max, Q

�min and Q �50) have

positive e�ect on the optimum value of the plant nominal ¯owrate (Qr). Betweenthe site hydrogeographical parameters, the maximum annual stream ¯owrate hasthe greatest impact on the plant nominal ¯owrate, while the mid-year annualstream ¯owrate has the smallest one. Clearly, for sites with high hydropotential,larger hydroturbines should be utilised to fully exploit hydropower in this case.

Figs. 5±7 expressing Eq. (21) are the essence of design of small hydroelectricplants with capacities up to 100 MW. Given the type of hydroturbine used andthe regional hydrogeographical and topological characteristics (expressed byappropriate model parameters), the engineer can automatically evaluate the plantoptimum nominal size, the corresponding total amount of energy recovered and areasonable estimation of the total plant cost and pro®ts expected at a preliminarydesign level. At this stage of design, the short-cut design equation for small

Fig. 6. Fitting of the short-cut empirical equation (lines) on optimal plant nominal ¯owrate for

PELTON hydroturbine (points) (symbols as in Fig. 5).


hydroelectric plants produced is a tool of great signi®cance for feasibility studieson such investments.

The e�ect of variation of selected model parameters on optimal processvariables for the case of FRANCIS hydroturbine and the typical site of Fig. 4 isgiven in Fig. 8. Optimal nominal ¯owrate is greatly a�ected by the shape of thestream ¯owrate duration curve, while as indicated above the available vertical fallof water, has no e�ect at all. The unit cost of electricity produced is greatlya�ected by the available vertical fall of water, while the shape of the stream¯owrate duration curve has the smallest e�ect. Moreover, the available verticalfall of water has the greatest impact on the total energy recovered while theminimum annual stream ¯owrate has the lowest one.

The short-cut design model equation parameter, g, is generally a�ected by theeconomic environment assumed. More speci®cally, no impact of capital turbinecost was observed on this parameter. On the contrary, the unit cost ofconventional electricity had a strong positive e�ect on it, indicating that largerhydroturbines are favored as conventional electricity cost increases. This e�ect is

Fig. 7. Fitting of the short-cut empirical equation (lines) on optimal plant nominal ¯owrate for AXIAL

hydroturbine (points) (symbols as in Fig. 5).


given in Fig. 9 for the FRANCIS hydroturbine. The results are directly related tothe ones obtained from the case study of Fig. 3.

In order to determine the range of applicability of each hydroturbine, allhydroturbines were directly compared for a wide range of model parametersexpressing topology and hydrogeography. Only two parameters could discernbetween the various types of hydroturbines. The range of application for allhydroturbines are given in Fig. 10. Clearly FRANCIS hydroturbine is inferior toboth others while PELTON is chie¯y preferred for literally most commonpractical cases.

4. Conclusions

The design of small hydroelectric plants can be properly analysed and addressedby means of optimizing the expected bene®ts from such an investment in the ®eld

Fig. 8. E�ect of variations of selected model parameters on optimal process variables for the

FRANCIS hydroturbine. (1) QR, (2) cE (3) E (a) H, (b) q �50, (c) q �min. Squares: +25%; triangles:

ÿ25%.


Fig. 9. E�ect of conventional unit cost of electricity on short-cut model parameter g.

Fig. 10. Regions of applicability of all hydroturbines studied.


of renewable energy exploitation. Optimization can be carried out by developingthe mathematical model of the hydroturbines, taking into account theirconstruction characteristics and operational performance. The turbine e�ciencycan be successfully modeled by means of an empirical equation. The plant modelmust also involve the regional characteristics in terms of hydrogeographical andtopological model parameters. The design problem can be formulated as amathematical programming one, and can be solved using appropriateprogramming techniques. An empirical short-cut design equation describesoptimal size of the plant for a wide range of site characteristics and allcommercially available hydroturbines studied. In this case, the optimum nominal¯owrate, the amount of energy recovered and a reasonable estimation of the plantcost can be automatically determined.

References

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[7] Fasol KH, Pohl GM. Simulation, controller design and ®eld tests for a hydropower plantÐa case

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[10] Prefeasibility Study for the Renewable Energy Sources Exploitation in Crete, ALTENER

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