Efficiencies and Objectives in Short Term HydroSchedulingEgill Benedikt Hreinsson

School of Engineering and Natural Sciences, University of IcelandHjardarhagi 6, 107 Reykjavik, Iceland

Email: egill@hi.is

Abstract—In short term hydro scheduling and optimization ina pure hydro system, a viable objective function is to minimize thewater drawdown from a long term reservoir, thereby utilizing thecharacteristics and shape of efficiency curves for individual units,waterways and stations. This paper addresses the representationof key concepts related to these efficiencies, as represented by thegeneration and production functions in hydroelectric stations. Toillustrate different aspects, data have been collected and analyzedfor actual stations in the Icelandic hydro dominated powersystem and graphical results are presented for key concepts.These key results of the paper include marginal and averagecost or objective function curves and definitions. The underlyingobjective could be to minimize water usage for all stations, aspart of different possible objectives in short term hydro systemoptimization. The results should be helpful when designingoptimization algorithms and interpreting their outputs, and incasting a light on what are important aspects in solving suchpractical optimization problems.

I. INTRODUCTION

Consider the short term optimization of operating a set ofhydro stations, with the objective of minimizing used water,by efficiency curves, turbine and waterway characteristics. Theobjective could be the released water from upstream reservoirssubject to generating a given amount of MW at each time stepand a specified energy for an extended period [3]. By “shortterm” we assume a time step e.g. of one hour and a horizonof days or weeks.

In such a pure hydro system, the objective is minimizingthe water from reservoirs, since no thermal units are normallyrunning, precluding fuel costs as an objective. Therefore, tominimize the risk of curtailment later, we want to keep asmuch water as possible in upstream long term reservoirs. Thenthe amount of water used per hour, day or week replaces atraditional thermal cost function in hydrothermal scheduling,where this fuel cost is typically minimized.

In this paper we focus on analyzing the characteristicsof cost functions, such as marginal and average costs withhydraulic losses included. We will illustrate various aspects ofthe resulting objective in a production function for a set ofunits/stations. We demonstrate these characteristics in a casestudy with three actual hydro station in Iceland. The datais analyzed in terms of various aspects in optimization [1].Therefore this can form an input into the optimization problemin an actual setting, as for instance in [3].

The paper is organized so that in section II we develop amodel for a production function for the short term problemoutlined above. Section III presents the case study. SectionsIV discusses the results, followed by final sections withacknowledgements and references.

II. PRODUCTION FUNCTIONS FOR RUNNING UNITS

A. The power output for a subset of units

Consider first a single hydro-power station with N tur-bine/generator pairs or units, indexed as i = 1, 2, ..., N . Theset of all these units is called SN and let S be any subsetof units in SN running at a given instant, that is S ⊆ SN .Of course, the number of different subsets, S is 2N − 1, notcounting the empty set. Let Si be the set of a single unit irunning and denote the total power output of all units in Si

and S as Pi and PS respectively, where PS =∑i∈S

Pi. Let the

corresponding flow of water through the running units be Qi

and QS where QS is the total flow in all units in S.Finally, let QN = [Q1, Q2, . . . , QN ]T be a (column) vector

defining the configuration of flows through all units and QS

be this N dimensional vector with non-zero elements only forthe running units in S and other elements in S, correspondingto units that are shut down, equal to zero. Therefore wedistinguish between the flow configuration vector QS and thesum of its components QS .

Note that QS 6=∑i∈S

Qi since the flow through individual

units is generally not additive due to possible losses incommon waterways, if these are present.

If the single unit in set Si is running, its power output is Pi

according to the well known conversion formula for convertingwater to energy in hydro stations, as shown in (1).

Pi = ri(Qi, Hi)hgQiHi(Qi) (1)

Qi,min 6 Qi 6 Qi,max (2)

where Pi is the real power output (W) of unit i in Si

and ri(Qi, Hi) is the unit efficiency, which depends on theflow Qi (in m3/s) through this unit [2] and its net head Hi.Furthermore, h is the density of water (∼ 1000 kg/m3), g isthe gravitational constant (9.81 m/s2) and Hi(Qi) = Hi(QSi)is the net head for unit i when only this unit is running. The“box constraints” (2) on Qi correspond to similar constraintsxxx-1-xxxx-xxxx-2/xx/$31.00 c©IEEE

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Station power output (MW)

Hra

uney

jars

tatio

nef

ficie

ncy

(%)

Figure 1. Efficiency curves for Hrauneyjar hydro project with 3 identicalunits of about 70 MW each. The curve on the left shows the total stationefficiency with any one unit running. The middle curve corresponds to anyof the identical two units running and the curve to the right corresponds toall three units running. The tick marks at the end of each curve show themaximum and minimum power output in each case. Efficiency in all cases isshown in %.

on Pi, which will be omitted here. The concept of net headHi will be discussed further below.

Inserting the appropriate measurement units and quantitiesinto (1) we get (3) and (4):

Pi [W] ∼= ri ·1000[kg/m3]·Qi

[m3/s

]·9.81

[m/s2]·Hi [m] (3)

Pi [MW] ∼= ri · 0.00981 ·Qi

[m3/s

]·Hi [m] (4)

Now, on the left hand side in (4), use [GWh/h] and at rightuse [Gl/h] (Gigaliters) instead of [m3/s]. Then we get (5):

Pi [GWh/h]1

1000∼= ri · 0.00981 ·Qi [Gl/h]

3600

106·Hi [m] (5)

leading to (6) for these engineering measurement units (Andomitting h):

Pi [GWh] ∼= ri · 0.0353 ·Qi [Gl] ·Hi [m] (6)

Now consider the set of running units S ⊆ SN . From (1)we get (7)

PS =∑i∈S

ri(Qi, Hi)hgQiHi(QS) (7)

The net head Hi(QS) at unit i ∈ S, assumes the flow QS

is distributed optimally among units in S. For instance, if theunits are identical, the flow should be equal. This is similarto the economic dispatch problem [5], with identical units,marginal cost (“lambda”) and loading.

However, for any distribution of flows in the configurationvector QS , we have for the net head Hi(QS) according to (8):

Hi(QS) = Y (QS)− U(QS)−HL,i(QS) (8)

where HL,i are the head losses, Y is the headwater leveland U is the tailwater level. Both Y and U depend generallyonly on the total flow QS in the set S. Both also may dependon a basic level at zero flow and a deviation or increase. For

instance, in the case of U , the deviation would be ∆U(QS).Therefore we can write (9):

U(QS) = U(0) + ∆U(QS) (9)

and similarly for Y (QS). Both levels, although dependingon the total flow QS , generally do not depend on the flowconfiguration or distribution QS among tunnels or penstocks.

However, the head losses HL,i(QS) at unit i may dependon the entire flow configuration QS through running units, in-cluding the unit i itself (Qi). This could include a dependencyon the total flow through all units, and may also depend onhow the units are connected to common penstocks.

Finally, we define the concepts of gross head and idealhead. The gross head Hg,S is defined as follows: Hg,S(QS) =Y (QS) − U(QS), that is here we ignore the head lossesrepresented by HL,i(QS). The ideal head HI,S does not takeinto account any deviation of either headwater or tailwaterlevel, i.e. is defined by (9):

HI,S = Y (0)− U(0) (10)

where 0 may indicate a configuration vector of zero ele-ments corresponding to zero flow.

B. A production function for a subset of units running

A configuration vector QS of flows through units in S, canlead to different power outputs for each S in terms of PS from(7), but assuming suboptimal flow distribution. However, withspent water as a measure of the objective to be minimized,as discussed above, we want to define a production functionQ = fS(P ) for each subset S of units running that minimizeswater usage for a given power output P and assuming anoptimal distribution of flows. Therefore define a productionfunction fS for the set S as follows:

Q = fS(P ) = minQS

{QS} = minQS

{∑i∈S

Qi

}subject to

P =∑i∈S

Pi =∑i∈S

ri(Qi, Hi)hgQiHi(QS)

(11)

where we also have the box constraints in (2), repeated in(12)

Pi,min 6 Pi 6 Pi,max (12)

Ignoring (2) and (12), the necessary conditions for solving(11) can be stated by differentiating the Lagrangian L in (13),

L(QS) = QS − λ(PS −∑i∈S

Pi) (13)

and as λ is the Lagrange multiplier, we get:

∂L

∂Qi=∂QS

∂Qi− λ ∂Pi

∂Qi= 0 (14)

which leads to1

λ=∂Pi

∂Qi

∼=∆Pi

∆Qii ∈ S (15)

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Station power output (MW)

Siga

lda

stat

ion

effic

ienc

y(%

)

Figure 2. Efficiency curves for Sigalda hydro project with 3 identical unitsof 50 MW each. Efficiency is shown in %. See Figure 1 for explanation ofindividual curves and markings.

Therefore (14) and (15) state that the marginal “cost” ofwater ∆Qi/∆Pi in (15) should be equal for all units running(In set S) and equal to the Lagrange multiplier λ. Therefore,the importance of studying this marginal cost λ is clear, as isthe study the efficiency characteristics for subsets S of units,including the average cost. Therefore, this ratio of incrementalvalues for water (Q) and power (P ) is our topic in he casestudy in Section III, below.

As seen in (11) - (15) the problem of dispatching individualunits within a subset of units running is very similar to theeconomic dispatch problem ([5],[6]) for thermal systems, anddetails are skipped here for brevity. This definition is basic informing an optimization problem and will be discussed furtherin Section II-C.

C. The station production function and its convex approxima-tion

The production or cost function for subsets S of unitsrunning is defined above in (11). Notice that there maybe overlapping intervals between subsets, corresponding topower ranges which will dictate when to switch units onand off, considering also a possible start-up cost. Thereforethe following station production function fG(P ) is defined asfollows, by combining all possible subsets of units running,as discussed by [4]:

fG(P ) = minS⊆SN

{fS(P )} (16)

As demonstrated in Section III the function may be definedover discreet intervals and have a gap between power ranges[2], [3].

Finally, we will define the convex approximation to (16) asfollows.

fC(P ) = convex hull {fG(P)} (17)

The application of the convex approximation in (17), asdiscussed in more detail formally, for instance in [4] and [3],will not be discussed further here.

III. A CASE STUDY WITH 3 HYDRO STATIONS

Here we apply these concepts to data from the three Icelandpower station. Their basic characteristics are shown in TableI and utilized to illustrate the cost concepts.

Table IMAJOR CHARACTERISTICS OF THREE STATIONS IN THIS STUDY

Hydropower station name Búrfell Sig+) HRF ∗)Station number j 1 2 3Number of turbines N in the station 6 3 3Number of penstocks 2 3 3Minimum turbine power Pi,min (MW) 27 30 40Maximum turbine power Pi,max (MW) 35 50.5 72∗∗) Headwater level Y (m.a.s.) 243 495 424∗∗) Minimum tailwater U (m.a.s.) 123 424 333Unit loss factor kj (m/m6/s2·10−4) 6.88 1.25 2.29Station loss factor ktot,j (m/m6/s2·10−4) 1.15 0.085 0Energy factor (GWh/Gl) 0.263 0.169 0.213Station maximum power (MW) 210 150 210Legend: ∗) HRF = Hrauneyjar station∗∗) m.a.s. = meters above sea level+) Sig=Sigalda

A. Unit and station properties in the case study

For simplicity we can safely assume for all three stationsthat the head losses HL,i are expressed as in (18), that isdepending on the flow Qi through unit i and the total flowQtot = QS of units running in the station whether there are 1,2 or 3 units running (Or 4,5 or 6 in the case of Burfell).

HL,i = ktotQ2tot + kiQ

2i (18)

For each station, the values for ki and ktot are shown inTable I. For Hrauneyjar and Sigalda with N = 3 units each,we assume that all unit are identical and each has its dedicatedpenstock. Then we get an output function for each number ofunits running, that is, for one unit running, the cases S = {1},S = {2} and S = {3} are assumed identical. Also the casesS = {1, 2}, S = {1, 3} and S = {2, 3} for two units runningare assumed identical. Finally we have for these two stationthe case of S = {1, 2, 3}.

However, for Búrfell station, with 6 units, three units areconnected to each of the penstocks/tunnels with the corre-sponding head losses in individual penstocks affecting thetotal head losses, as represented by the first term in (18).Therefore, as each of the two head race tunnels split into 3and connect at each turbine, we can assume a certain start-uporder to minimize the head losses for each given number ofunits running in order to minimize the total water drawdown.It is easy to see that starting alternatively a turbine connectedto each penstock, and thereby distributing the flow betweenpenstocks as much as possible, will minimize this objective.This is indicated by the presence of the square terms in (18).

B. Total efficiencies of sets of units running

Next, consider the important concept of total efficiency,considering the real power output vs. ideal output. The totalefficiency ηS for each set S of units running, is therefore

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92

94

Station power output (MW)

Bur

fell

stat

ion

effic

ienc

y(%

)

Figure 3. Efficiency curves for Búrfell with 6 units of 35 MW each. Eachcurve corresponds to a given number of units, one at left and 6 at right.Efficiency is in %. Notice the dramatic drop in efficiency from the case ofone unit to all six units running, or from 93% to about 86%.

defined as the actual power output PS divided by the idealpower output PI,S without any losses in generators, turbinesor waterways, which is:

PI,S =∑i∈S

hgQiHI,S = hgQSHI,S (19)

where HI,S is the ideal head from (10). Notice the absenceof turbine efficiency ri from (2) in (19).

Therefore, from (7), the efficiency ηS for a subset S of unitsrunning is defined:

ηS =

∑i∈S

ri(Qi, Hi)hgQiHi(QS)

hgQSHI,S(20)

Now we turn to representing graphically the key conceptsdiscussed in the paper. Figure 1 shows efficiency curves forHrauneyjar hydro station, while Figure 2 shows the same forSigalda station. Finally, Figure 3 shows the efficiency curvesfor Búrfell station.

These curves are plotted against power output PS ratherthan flow QS which is more meaningful for the function in(11). Each power range is between the circled points. Notethe overlapping ranges and the impact of common waterways,leading to reduced efficiency with increased number of units.

Also in Figure 3, we can see the consequence of the start-up order, previously mentioned. When one or two units arerunning, the efficiency is similar, while it drops in the case of3 or 4 units.

C. Turbine Efficiencies of Units

Turbine efficiencies ri(Qi, Hi) for unit i basically dependon turbine flow Qi and head Hi. This is shown, for instance,in Figure 4 for Búrfell and Hrauneyjar power stations. In theseFigures, turbine efficiencies are shown as one curve for eachof the different heads.

Furthermore, Figure 5 (at right) shows turbine efficiencyri for each of the identical units in the Sigalda station. Inaddition, at left, Figure 5 shows the increased tailwater level∆U(QS) and how it depends on the flow QS .

20 25 30 35 40

89

90

91

92

93

Individual turbine flow (m3/s)

Turb

ine

effic

ienc

y(%

)

40 60 80 100 12084

86

88

90

92

94

96

Individual turbine flow (m3/s)

Turb

ine

effic

ienc

y(%

)

Figure 4. Turbine efficiencies at different heads in Búrfell station at left forheads of 109, 115, and 122 m.a.s respectively. At the right for Hrauneyjarstation for heads of 63, 72, 81, 86 and 89 m.a.s. Both curves are based onthe original manufacturer’s data.

0 100 200 300 4000

0.5

1

1.5

2

2.5

Station flow (m3/s)

Tailw

ater

leve

linc

reas

e(m

)

40 50 60 70 80 90 10082

84

86

88

90

92

94

Individual turbine flow (m3/s)

Turb

ine

effic

ienc

y(%

)

Figure 5. Tailwater level ∆U increase at Sigalda stain (Left) and efficiencycurves at Sigalda station (Right). The curves at right, counted from top tobottom, correspond to 71, 68, 65, 63 and 59 m gross head.

D. Separate plots of marginal and average costs

Since the relation between station power P and flow Qis close to being linear, plotting it does relatively little toillustrate the nonlinearities graphically. However, the ratio orthe derivative, i.e. the average or marginal costs as discussedpreviously, are more useful for system operation.

The average cost ca is simply the ratio between output and

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1.6

1.62

1.64

1.66

1.68

1.7

1.72

1.74

Station Power Output (MW)

Siga

lda

Ave

rage

Cos

t(m

3 /s/M

W)

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1.4

1.6

1.8

2

2.2

2.4

2.6

Station power output (MW)

Siga

lda

Mar

gina

lCos

t(m

3 /s/M

W)

Figure 6. Cost functions for Sigalda station. At left is the average cost. Atthe right is the marginal cost function. Both curves are based on the originalmanufacturer’s data.

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1.24

1.26

1.28

1.3

Station Power Output (MW)

Hra

uney

jarA

vera

geC

ost(

m3 /s

/MW

)

0 50 100 150 200 2501

1.2

1.4

1.6

1.8

Station power output (MW)H

raun

eyja

rMar

gina

lCos

t(m

3 /s/M

W)

Figure 7. Cost functions for Hrauneyjar station. At left is the average cost.At the right is the marginal cost function. Both curves are based on theoriginal turbine manufacturer’s data.

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0.95

1

1.05

Station power output (MW)

Bur

fell

Ave

rage

Cos

t(m

3 /s/M

W)

Figure 8. Average cost curves for Búrfell station with 6 units of approx-imately 35 MW each. Each curve corresponds to a given number of unitsrunning, one unit at left and 6 units at right.

input, that is defined as follows:

ca(P ) =Q

P=fS(P )

P(21)

while the marginal cost is defined as:

cm(P ) =∆Q

∆P=

∆fS(P )

∆P(22)

Figure 6 shows these cost concepts and results for Sigaldapower station, while Figure 7 shows these cost concepts resultsfor Hrauneyjar power station. Again the three curves from leftto right in each diagram show the cases where 1, 2 or 3 unitsare running.

Finally, with a different configuration in Búrfell, with 6 unitsand 3 units per penstock, Figures 8 and 9 show similar resultsfor average and marginal costs. Therefore these results differfrom the cases of Hrauneyjar and Sigalda.

IV. DISCUSSIONS AND CONCLUSIONS

Finally we make the following comments:1) We have discussed, calculated and plotted key cost con-

cepts in optimization objectives for real hydro stationsapplicable in practical settings.

2) An extension would be to combine water at stations byweighting factors, such as Lagrange multipliers.

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1.2

1.4

1.6

1.8

2

2.2

Station power output (MW)

Bur

fell

Mar

gina

lCos

t(m

3 /s/M

W)

Figure 9. Marginal cost curves for Búrfell station with 6 units. Again, eachcurve corresponds to a given number of units running, one unit at left and 6units at right. See Figure 1 for explanation of individual curves and markings.

Figure 10. Blanda hydropower station, Iceland. This penstock arrangements,(Not a part of the case study) shows a case with three non-identical units,where hydraulic losses may depend on the vector of all flows.

3) The function fG(P ) from (16) and the convex versionfC(P ) could be an extension.

4) One extension would be to simultaneously minimizeturbine, waterway and transmission losses.

5) The results assume identical units. However, stationsmay have different waterways and losses, as in Figure10.

ACKNOWLEDGMENTS

The author would like to thank Landsvirkjun, the NationalPower Co., for providing the data used in the paper.

REFERENCES[1] Edwin Kah Pin Chong and Stanislaw H Zak. An introduction to

optimization. John Wiley & Sons, Hoboken, N.J., 4th edition, 2013.[2] Egill Benedikt Hreinsson. Framleiðslustýring vatnsaflsvirkjana. [Gener-

ation control in water power stations]. Technical report, Landsvirkjun,Reykjavík, April 1984.

[3] Egill Benedikt Hreinsson. Optimal short term operation of a purelyhydroelectric system. IEEE Transactions on Power Systems, 3(3):1072–1077, Aug 1988.

[4] A. B. Philpott, M. Craddock, and H. Waterer. Hydro-electric unit com-mitment subject to uncertain demand. European Journal of OperationalResearch, 125(2):410–424, 2000.

[5] Hadi Saadat. Power system analysis. PSA Pub., 3rd edition, 2010.[6] Allen J. Wood, Bruce F. Wollenberg, and Gerald B. Sheblé. Power

generation, operation, and control. Wiley, third edition, 2013.

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