Valuing hydrological forecasts for a pumped storage assisted hydro facility

Journal of Hydrology 373 (2009) 453–462

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Journal of Hydrology

journal homepage: www.elsevier .com/ locate / jhydrol

Valuing hydrological forecasts for a pumped storage assisted hydro facility

Guangzhi Zhao a, Matt Davison a,b,*

a Dept. of Applied Mathematics, The University of Western Ontario, London, ON, Canada N6A 5B7b Dept. of Statistical and Actuarial Sciences, The University of Western Ontario, London, ON, Canada N6A 5B7

a r t i c l e i n f o

Article history:Received 23 December 2008Received in revised form 7 May 2009Accepted 10 May 2009

This manuscript was handled by G. Syme,Editor-in-Chief

Keywords:Hydroelectric facilityInflow rateTurbine efficiencyHeadOptimizationOptimal control

0022-1694/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.jhydrol.2009.05.009

* Corresponding author. Address: Dept. of Appliedof Western Ontario, London, ON, Canada N6A 5B7. Te519 661 3523.

E-mail address: [email protected] (M. Davison).

s u m m a r y

This paper estimates the value of a perfectly accurate short-term hydrological forecast to the operator of ahydro electricity generating facility which can sell its power at time varying but predictable prices. Theexpected value of a less accurate forecast will be smaller.

We assume a simple random model for water inflows and that the costs of operating the facility,including water charges, will be the same whether or not its operator has inflow forecasts. Thus, theimprovement in value from better hydrological prediction results from the increased ability of the fore-cast using facility to sell its power at high prices. The value of the forecast is therefore the differencebetween the sales of a facility operated over some time horizon with a perfect forecast, and the salesof a similar facility operated over the same time horizon with similar water inflows which, though gov-erned by the same random model, cannot be forecast.

This paper shows that the value of the forecast is an increasing function of the inflow process varianceand quantifies how much the value of this perfect forecast increases with the variance of the water inflowprocess. Because the lifetime of hydroelectric facilities is long, the small increase observed here can leadto an increase in the profitability of hydropower investments.

� 2009 Elsevier B.V. All rights reserved.

Introduction

What is the value of the forecast of an environmental variable,such as temperature, wind speed, or water inflow rate, for somepoint? This is a question which has intrigued people for manyyears. Such forecasts have value to companies active in many dif-ferent industry sectors, including transportation, leisure, and en-ergy. In the electrical power industry, temperature is a crucialvariable, the value of which to providers of electrical ‘‘spinning re-serve” has been quantified in Hobbs et al. (1999). For producers ofhydroelectricity, water inflow and water level in various points isof great interest. The value of long range forecasts to operators ofhydro dams on the large Columbia River watershed was estimatedin Hamlet et al. (2002).

This paper provides a theoretical framework for investigatingthe value of much shorter lead time hydrological forecasts to hydroproducers, operating both with and without pumped storage facil-ities, on a small watershed. The operator of such a facility faceshydrological inflows that are somewhat uncertain even on the48 h time scale. Improved hydrological forecasts can add value tohydro power producers in two ways: by improving their abilityto forecast prices and to improve their ability to manage variable

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Mathematics, The Universityl.: +1 519 661 3621; fax: +1

water inflows. This paper focuses on the latter, volumetric uncer-tainty, aspect and assumes that the operator is certain of the pricefor electrical power over the same 48 h time scale, either becausethe plant operates in a regulated electricity market or becauseaccurate short-term price forecasts are available.

To compute Vd, the sales of a facility operated over some timehorizon with a perfect forecast, we use Monte Carlo techniquesto find the average best practices sales with forecast information.To do this we first simulate a water inflow time series from ourhydrological model. We then use dynamic programming to findthe optimal way for a model hydro plant, operating with water le-vel constraints, to use this water for power generation and theoptimal sales level which results. The value of Vd is the averageover many realizations of this process.

Vr, the sales of a similar facility operated over the same timehorizon with similar water inflows which cannot be forecast, iscomputed using stochastic dynamic programming to find the opti-mal operation protocol for the same power plant with the sameconstraints exposed to the same random inflow process. Vr is theexpected sales of the plant operated in this optimal way.

The structure of the paper is as follows: ‘‘Difficulties inherent inhydrological modelling” briefly summarizes the many difficultiesinherent in hydrological modelling and describe our simple modelfor random water inflows. ‘‘Modelling of hydroelectric facility”introduces modelling of hydroelectric facility and also describethe pumped storage facility model. ‘‘Optimization algorithm”focuses on the programming algorithm that is used to determine

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454 G. Zhao, M. Davison / Journal of Hydrology 373 (2009) 453–462

the optimal operation. ‘‘Modelling of hydroelectric facility” and‘‘Optimization algorithm” draw heavily on the earlier work ofThompson et al. (2004) and Zhao and Davison (2009). ‘‘Analysisof results” describes the numerical experiments conducted andthe results of these calculations. ‘‘Value of water inflow forecast”analyzes the value of hydrological forecast for a pumped storagefacility. The last section presents conclusions and suggestions forfuture work.

Difficulties inherent in hydrological modelling

A hydrological forecast for a given region can be undertakenwith a model for forecasting precipitation over a given watershed.Precipitation is delivered to streams both as overland flow to trib-utary channels and by subsurface flow rates as groundwater(Freeze and Cherry, 1979). Determining where and how the watertravels, as overland flow, on the terrain requires an understandingof hydraulics. A detailed model also requires accurate topographicinformation (which can be obtained from various data sources,such as digital elevation models, and then entered into Geograph-ical Information Systems (GIS)).

In principle, once this hydrologic information is integrated, acomputationally intensive process can be used to make predictionsof water levels (referenced to some datum such as sea level) andflow rates. Water level information can then be used to calculatethe volume stored in each surface mass of water within a GISframework. Given the inherent difficulty of making accurate pointforecasts of rainfall, this approach is very challenging toimplement.

The other missing piece of the puzzle is measuring the amountof time it takes for a raindrop to travel from where it fell to anotherpoint on Earth’s surface. This travel time is called the ‘‘characteris-tic time” and, with this measurement in hand, the flow and level atthis point can be obtained by ‘‘looking backwards” and determin-ing the rainfall at that time (either by forecasting or by lookingat observed data of past rainfalls). It should be noted that even this‘‘backcasting” step, if applicable, is not trivial as detailed finegrained rain measurements are rarely made throughout a wa-tershed, let alone recorded.

The forecasting step is even more challenging, because the grid-or mesh-size on a General Circulation Model (GCM) is much largerthan the area over which rainfall forecasts are needed. However,the prediction of rainfall is very difficult. Rainfall is extremely var-iable on spatial and temporal scales, with scaling reported to befractal (Breslin and Belward, 1999; Rajagopalan and Tarboton,1993). It is created by a variety of interacting physical processeswhich occur on scales varying from micrometers to hundreds ofkilometers (Rajagopalan and Tarboton, 1993). As weather modelshave a grid size of just a few kilometers, physical processes, includ-ing the important ones relating to cloud formation occurring onsmaller length scales must be approximated, for instance by the‘‘dynamical downsizing” of these grids (Cluckie et al., 2006).

Even though there are many challenges in precipitation fore-casting and the difficulty of modelling watersheds (Krzysztofowiczand Herr, 2001; Krzysztofowicz et al., 1993), there are some avail-able streamflow-based models, which capture the random charac-teristics of the adjusted historical streamflows data, able toperform sufficiently accurate inflow predictions (Wurbs, 1993).For a case study of how hydrological inflow forecasts are used byhydroelectric operators, see Druce (1990) and Druce (1994).

In this paper, we create a simple random model for a hourlywater flow into the upper reservoir of a pump-assisted storagefacility. We optimize the value of the facility, run for the next48 h, in the face of this random inflow. Then, for 100 different real-izations of this random inflow process, we optimize the value of

the same facility with the inflows known for the next 48 h. Theaverage of these values, which is equal to the uncertainty opti-mized value, is less than the deterministic value—we quantifythe exact difference in ‘‘Value of water inflow forecast”.

Modelling of hydroelectric facility

There are two main categories of hydroelectric power genera-tion: conventional methods (dams and run-of-the-river), whichproduce electricity via water flow in one direction; and pumpedstorage methods, which are both producers and consumers ofelectricity.

The optimal operation of hydroelectric generation facilities de-pends on the price of power, p, inflow rate, f, the total amount ofwater, w, and the power function, ~E, as discussed in Thompsonet al. (2004) and Zhao and Davison (2009). We seek the controllingflow rate, c, to maximize the cash value, ~Vðt; T; p;wÞ, which is de-fined as

~Vðt; T;p;wÞ ¼ maxc

ER T

t e�rðs�tÞpðsÞ~Eðc;wÞdsþ e�rðT�tÞ~RðwðTÞÞh i

;

dw ¼ lðf ; cÞdsþ rðf ÞdBs:

8<:

ð1Þ

Here, l(f,c) is the mean and r(f) is the variance, of water amount, tand T are the beginning and ending time, r is a discount factor forthe time value of money, ~RðwTÞ is the residual value of the waterremaining in the reservoir at the end of the time horizon, and B isBrownian motion.

Because the reservoir cannot be drained below some minimumwater level nor filled above some maximum level, the amount ofwater stored is constrained to lie within w 2 [wmin,wmax], and therelease flow c is also constrained (depending on w) to lie in therange c 2 [cmin(w),cmax(w)].

The theoretical power P available from a given head of water isin direct proportion to the head h and the release rate c (Hydro-power basics, 2007). The output power of hydro–turbo generationsis a function of both the net hydraulic head and the water dis-charge (El-Hawary and Christensen, 1979). If P is measured inWatts, c in m3/s and h in meters, q = 1000 kg/m3, g is the acceler-ation due to gravity m/s2, the gross power of the flow of wateris: P = qgch. When we release or pump water at c m3/s (i.e.3600c m3/h) at water head h, the water head changes at the ratedh, dh = c/S, where S is the bottom area of the reservoir. To simplifythe numerical details of the computation we henceforth assumethat the reservoir is cuboid and S = 3600p m2. In this case if we re-lease water at rate c m3/s, dh = c/p m/h, and our computations aresimplified by using the water head h rather than water volume w.Suppose dh = (f � c)/Sdt + rf/S dBt and rewrite the objective func-tion (1), so,

Vðt; T;p;hÞ ¼maxc

ER T

t e�rðs�tÞpðsÞEðc;hÞdsþ e�rðT�tÞRðhðTÞÞh i

;

dh ¼ ðf � cÞ=Sdsþ rf=SdBs:

8<:

ð2Þ

Discretize objective (2) as

Vðt; T;p;hÞ ¼maxc

EPTs¼t

e�rðs�tÞpEDsþ e�rðT�tÞRðhðTÞÞ� �

;

hsþDs ¼ hs þ ðfs � csÞ=SDsþ rfs=SDBs:

8<: ð3Þ

In particular, we assume that the turbine efficiency g is given byfunction (4) as described in Thompson et al. (2004),

gðc; hÞ ¼ �gmax10�6qgch

w� 1

!2

þ gmax: ð4Þ

G. Zhao, M. Davison / Journal of Hydrology 373 (2009) 453–462 455

Suppose the pump operates at a fixed input power level of ap

megawatts with a constant efficiency of c < 1, and the cross-sec-tional area of the pipe from the reservoir to the turbine is Sp = pm2,Sp � S, correspondingly the diameter is d = 2 m. We also need toinclude the head loss hf given by Darcy–Weisbach equation (5)(Brown, 2002), which is directly proportional to the length of pipeL, square of water velocity v, and a term accounting for the frictionfactor ff, and is inversely proportional to the diameter of the pipe d:

hf ¼ ffLv2

2gd: ð5Þ

We suppose that the downstream level does not change whenwater is pumped into the upper reservoir. The head loss in thepump must, however, be considered. From Thompson et al.(2004) and Zhao and Davison (2009), the power function E(c,h),giving the total power generated measured in megawatts, is

Eðc;hÞ ¼10�6gqgcðh� hf Þ; 0 6 c 6

ffiffiffiffiffiffiffiffi2gh

pSp; release or wait;

�ap; c ¼ �apc10�6qgðhþhf Þ

; pump;

hmin 6 h 6 hmax:

8>><>>:

ð6Þ

Without pump storage capability, (6) must be simplified to removethe ‘‘pump” contingency.

We consider a market participant too small to alter the supply–demand balance of the electricity market through its productiondecisions; the ‘‘price-taker” assumption. Of course, for a large util-ity operating numerous dams in concert this price-taker assump-tion will fail and a more difficult stochastic unit commitmentproblem must be solved (Ethier, 1999; Skantze et al., 2000).

Even within this price-centred framework, this deterministic,time varying market price model is open to a number of criticisms.A cursory glance at an electricity price versus time graph displaysextremely variable and apparently unpredictable prices (Davisonet al., 2002). However, on a short time horizon of a few days, theseprices are actually somewhat predictable, set as they are in largepart by the interplay between weather driven demand, known onthose time horizons because of reliable short term temperatureforecasts and well understood temperature-demand relationships,and plant reliability-driven demand, again fairly predictable onshort time scales.

In future work it would be interesting to examine the value ofinformation in a more realistic stochastic price model. This wouldadd another source of randomness to the model considered here,with the concomitant computational difficulties. We believe thatthe results presented in this paper point towards the results whichmight be expected by this larger, more complicated, model.

We analyze a pumped storage facility with the same engineer-ing parameters as presented in Thompson et al. (2004) and Zhaoand Davison (2009). Let gmax = 0.85, w = 60, ap = 15p, c = 75%,ff = 0.01, and L = 120. We discuss the model encoded by (6) inwhich the control constraint depends on head h. Also supposeh 2 [120,180] and use the average hourly price function (7),

pðtÞ ¼ 27þ 15 sin2pt � 15:4p

24; ð7Þ

obtained from actual price data Thompson et al. (2004), to discussthe control strategy for the generated value.

Optimization algorithm

This section presents a numerical method for solving the opti-mization problem of Eq. (3) in face of the water head and flow rateconstraints. As the time horizon is very small, it can be assumedr = 0 without much affecting the control strategy. Suppose Ds = 1,

the residual value R(hT) = 0, and the volatility of the inflow rate isDf, from (3), we have

Vt ¼ maxcs

EPTs¼t

psEðcs;hsÞ� �

;

hsþ1 ¼ hs þ fs � cs þ Dfs:

8<: ð8Þ

E½Dfs� ¼ 0; E½DfsDfs� ¼ 0 ðs–sÞ: ð9Þ

This objective can be achieved backwards in time using dy-namic programming (Kamien and Schwartz, 1991), i.e., the optimalchoices will be made not only at s = t but also at s = t + 1, t + 2, . . .,T,therefore,

Vtþ1 ¼maxcs

EXT

s¼tþ1

psEðhs; csÞ" #

;

Vt ¼maxct

E ptEðht ; ctÞ þ Vtþ1½ �:

While V(t,T,p,h) is the maximum, we use these values moving for-ward to determine the optimal control path,

c� ¼ arg maxct

E ptEðht; ctÞ þ Vtþ1½ �:

The computation steps are:

(1) At the last time s = T, VT = 0 for hT 2 H.(2) Starting at s = T � 1, find the control c that

Vs ¼maxcs

E psEðhs; csÞ þ Vsþ1½ �;

c�s ¼ arg maxcs

E psEðhs; csÞ þ Vsþ1½ �:

(3) Continue the process of step (2) decreasing time by one unitper step, until s = t.

Refer to Zhao and Davison (2009) for the detailed computation.

Analysis of results

To analyze the value of a forecast, we isolate the value of a waterinflow forecast by creating a model for the optimal control of a facil-ity with random head. If the random change of the head only de-pends on the water inflow, then a random water inflow implies arandom head. To simplify, suppose the average inflow ft is constantor deterministic, and the inflow rate is ft + �tft, where �t is a randomvariable taking I possible values, �t,i, with the corresponding proba-bility qi, i ¼ 1;2; . . . ; I, and �t is independent identically distributed.In the rest of this paper, we use this distribution instead of Brownianmotion as the model for stochastic inflow rate to study the controlstrategy. If the head is ht at time t, it reaches hi

tþ1 ¼ ht þ ft�ct þ �t;ift with probability qi at the next time t + 1. If the maximumvalue is Vtþ1;htþ1 at time t + 1, the maximum value, Vt, is

Vt ¼maxc

ptEðct ;htÞ þ EVtþ1;htþ1

� �¼max

cptEðct;htÞ þ

Xi

qiVtþ1;hitþ1

( ):

ð10Þ

Actually, any real-world data can be input into Eq. (10) to findthe optimal control strategy. Let �t,i = {�0.4, �0.2, 0, 0.2, 0.4} withcorresponding probability qi = {0.1, 0.2, 0.4, 0.2, 0.1}, and use thenumerical method to find the optimal control at the initial timefor any given initial head. This optimal control gives the best strat-egies ‘‘on average” but will not necessarily be best, in hindsight, fora given realization of random head value. We cannot find the posthoc optimal control path, because there is more than one possiblehead at the next time step. We determine the optimal control pathbased on the expected value of the random future inflows. Thealgorithm works by maximizing the expected value of the facility.


In this paper, for the sake of illustration and to quantify the va-lue of short-term hydrological forecasts, we will adopt a time hori-zon of 48 h and discuss how the random inflow rate affects thecontrol strategy. ‘‘Constant expected inflow rate” discusses thecase of constant expected inflow rate, while ‘‘Variable expected in-flow rate” presents results in which the expected inflow rate is afunction of time.

Constant expected inflow rate

We can compare the control strategy to some results in Zhaoand Davison (2009). As is reasonable, the expected value for therandom head is less than that for deterministic head, since the in-flow water cannot be used as efficiently as in the deterministic casebecause of the random head. Fig. 1 clearly depicts the total valuedifference for the different initial head between the random inflowand deterministic inflow.

For the control strategy with respect to head and time, the con-trol surface is smoother for the random head (Fig. 2) than the

120 130 140 13.5

3.55

3.6

3.65

3.7

3.75

3.8

3.85

3.9

3.95

4 x 104

He

Valu

e $

DeterministicRandom

Fig. 1. Cash value

120010

2030

40

−10

−5

0

5

10

15

20

Time: hour

Cont

rol:×

πm3 /s

Fig. 2. Expected c

deterministic case (Fig. 3). This makes sense because the determin-istic control can perfectly anticipate known future inflows whilethe stochastic control must average, or smooth, over possible fu-ture inflows.

Fig. 4 depicts the control path with initial head h0 = 120 m(empty) from t = 0 to t = T. When the inflow rate is small and res-ervoir is empty, there is no big difference between the expectedand deterministic control, because the reservoir is far from full,therefore, the variance has little effect on the control strategy.

Fig. 5 depicts the control strategy for initial heads ranging fromh0 = 120 m (empty) to h0 = 180 m (full). The control for uncertainhead is smoother and results in more waiting and less pumpingthan for the deterministic case. This is what we would anticipatebecause of ‘‘uncertainty” introduced by requiring the constraintsto be met. We need to maintain the water head to reduce thechance of a fluctuation resulting in an overly low head. On theother hand for the deterministic control, we know the exact headfor the next step and so can be more ‘‘aggressive” in using thewater most efficiently.

50 160 170 180ad: m

: t = 0, f = 5p.

130140

150160

170180

Head: m

ontrol: f = 5p.

120130

140150

160170

180

010

2030

40

−10

−5

0

5

10

15

20

Head: mTime: hour

Cont

rol:×

πm3 /s

Fig. 3. Deterministic control: f = 5p.

0 5 10 15 20 25 30 35 40 45

0

10

20

30

40

50

60

Time: hour

Head

(m) a

nd C

ontro

l(πm

3 /s)

Expected headExpected controlDeterministic headDeterministic controlInflowprice

Fig. 4. Control path: f = 5p.

120 130 140 150 160 170 180−10

−5

0

5

10

15

Time : hour

Cont

rol:×

πm3 /s

Deterministic ControlExpected Control

Fig. 5. Control at t0 = 0 as a function of initial head: f = 5p.


0 5 10 15 20 25 30 35 40 45

0

10

20

30

40

50

60

Time: hour

Head

(m) a

nd C

ontro

l(πm

3 /s)




0

10

20

30

40

50

60

Head

(m) a

nd C

ontro

l(πm

3 /s)

0 5 10 15 20 25 30 35 40 45Time: hour



0 5 10 15 20 25 30 35 40 45Time: hour

0

10

20

30

40

50

60

Head

(m) a

nd C

ontro

l(πm

3 /s)

Fig. 8. Control path: �f ¼ 5.



As the inflow rate,f = 8p, increases, there is less chance to pump,because the variance �f also increases making the future waterhead more random. It is usually better to wait than to pump andhere the optimal controls greatly differ, as shown in Fig. 6. Theoptimal random control is to keep the head lower than the maxi-mum head in order to make best use of random inflows. As indi-cated by power function (6), the power generated is proportionalto both release rate and head. A higher release rate diminishes fu-ture head forcing a tradeoff between the quantities. Fig. 6 showsthat the expected control tends to emphasize release rate overmaintaining a high head. On the other hand, for the deterministiccase, a control which simultaneously keeps the head high and usesa lot of water is possible.

When f = 12p, the random control continues to be less aggres-sive with more waiting time (Fig. 7). The price does not have a sub-stantial effect on either strategy but does affect the random case alittle more. As the inflow rate is much greater than the release rate,which maximizes the power function, the control strategy is tokeep the reservoir full to optimize energy and simply to run thefacility as if it were run-of-the-river by using all inflows as they ar-rive. This is the same strategy used in the deterministic case.


0 5 10 15 20Tim

0

10

20

30

40

50

60

Head

(m) a

nd C

ontro

l(πm

3 /s)

Fig. 9. Control p


0 5 10 15 20Tim

0

10

20

30

40

50

60

Head

(m) a

nd C

ontro

l(πm

3 /s)

Fig. 10. Control path: �f

Variable expected inflow rate

If the inflow rate is a function of time or any forecast data, theexpected optimal control also can be found. Suppose there is a 48-hcycle inflow rate, f ðtÞ ¼ ð�f þ 3 cosðpt=24þ 5ÞÞp m3=s, where �f isthe mean inflow rate. We discuss a 4-day control strategy in orderto understand how the inflow rate affects the control in thissection.

Fig. 8 depicts the control for �f ¼ 5, and the control strategy isthe same for the constant inflow rate—release at higher price andat the highest head in order to maximize the value of water. As be-fore the random case leaves some room to store water against ran-dom inflow water, while the deterministic case maximizes head atall times.

Fig. 10 depicts the control strategy with inflow uncertainty dou-ble than that used in the Fig. 9 case. The resulting control leavesmore room for large random inflows, in order to use as much in-flow water as possible.

For inflow rates of �f ¼ 15 (as depicted in Fig. 11) or larger, thecontrols for the deterministic and random inflows are quite simi-lar. As the inflow rate increases or the variance decreases, the head

25 30 35 40 45e: hour

ath: �f ¼ 10.

25 30 35 40 45e: hour

¼ 10, big variance.


0 5 10 15 20 25 30 35 40 45Time: hour

0

10

20

30

40

50

60

Head

(m) a

nd C

ontro

l(πm

3 /s)

Fig. 11. Control path: �f ¼ 15.

120 130 140 150 160 170 1800

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Head: m

Adde

d va

lue:

%

Fig. 12. Added value if forecast is accurate: k� = 1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Variance change rate: kε

Adde

d va

lue:

%

Fig. 13. Average percent increased with perfect forecast: h0 = 120.



will become close to or even equal to the highest head. As the in-flow rate becomes large enough to maximize the power function E,the control strategy is to keep outflow and head conditions so as tomaintain this maximum power. In this case, inflow rate, electricityprice, and random water head variance will not affect the control.This can be proved by

Vðt; T; p;wÞ ¼maxc

E

Z T

te�rðs�tÞpðsÞEðc;hÞds

� �

6maxc

E

Z T

te�rðs�tÞpðsÞEmaxds

� �¼Z T

te�rðs�tÞpðsÞEmaxds

¼ Emax

Z T

te�rðs�tÞpðsÞds:

Here, Emax is constant.

Value of water inflow forecast

From the above analysis, we know that the value of running afacility is higher for deterministic than for variable inflows. This

120 130 140 153.9

4

4.1

4.2

4.3

4.4

4.5 x 10 4

Initial

Valu

e $

Fig. 14. Compare the generated v

120 130 140 151.2

1.4

1.6

1.8

2

2.2

2.4 x 10 4

Initial

Valu

e $

Fig. 15. Compare the generated

is true even when the statistical properties of the inflow processare identical. This insight suggests that better knowledge of waterinflows allows the capture of value. Although in this paper we fo-cus on random water inflows without considering random pricefluctuations, in a deregulated market a large hydro producer mightalso be able to obtain value from the correlation between futurewater inflows and future power prices. With this in mind, howmuch is an accurate water inflow forecast worth? If we can predictthe exact water inflow for the next N hours, the ‘‘deterministic” va-lue of the facility is denoted by Vd(N). If, on the other hand, we can-not make exact predictions, the expected value of the facility isdenoted by Vr(N), Vr(N) < Vd(N). Thus we should be willing to pur-chase a water inflow forecast if its cost is less than Vd(N) � Vr(N).

Now we give a simple explanatory example. Suppose the ex-pected inflow rate is the constant f = 5p, the inflow variability isk��f, where k� is variance change rate, k� 2 [0, 5], and the otherparameters are as before. As the variance decreases, the value in-creases, and, by comparing the value difference between differentvariance change rate k�, we can determine the forecast value.

For the 2-day water inflow forecast, two curves in Fig. 1, respec-tively, depict the generated value for the deterministic and the ran-

0 160 170 180 head: m.

Deteministic ValueExpected Value of Random InflowSimulated valueDeterministic Value(no pump)Expected Value(no pump) Simulated value(no pump)

alue: 2 days, f = 6p, Df = 2p.

0 160 170 180head: m.

Deteministic ValueExpected Value of Random InflowSimulated valueDeterministic Value(no pump)Expected Value(no pump) Simulated value(no pump)

value: 2 days, f = 2p, Df = p.


dom case; if the forecast cost is less than the difference betweenthe two curves, it is valuable to purchase a forecast predictingthe water inflow. Fig. 12 indicates the added value (in percent) offorecast as described in Fig. 1, where k� = 1. Fig. 13 depicts the aver-age percent of increased value as a function of the variance changerate k�; the bigger the water inflow variance, the more generatedvalue (with or without pumping) can be added for an accurate in-flow forecast. The forecast value is an increasing function of inflowprocess variance.

Furthermore, we also compute the simulated value, which isequal to the expected value, and compare the deterministic, ex-pected, and simulated values with pumping and without pumping.We use six curves to describe these values (three curves are for thepumping case and the other three are for the no pumping case), seeFigs. 14 and 15. As a matter of convenience, here we suppose boththe inflow rate, f, and the variance, Df, are constant, and there areonly two cases for the inflow rate, f ± Df. When f = 6p, Df = 2p, thevalue with pumping is equal to that with no pumping for deter-ministic case and random case respectively, and this means thatpumping is not optimal (Fig. 14). If the inflow rate is small, as forinstance in Fig. 15 where f = 2p, a facility that pumps always gen-erates more value than one that does not, regardless of the initialhead.

Conclusions and future work

This paper presents, for the first time, a framework for deter-mining the value of a perfect short-term hydrological forecast tothe operator of a hydroelectric facility. This technique dependson the solution of one stochastic optimal control problem andthe average solution of a large number of deterministic optimalcontrol problems. With the help of the valuation framework wewere able to determine the value of this perfect forecast increaseswith the variance of the water inflow process.

Accurate hydrological forecasts allow a power output increaseof several percent. Because the lifetime of hydroelectric facilitiesis long, this small increase observed here can lead to an increasein the profitability of hydropower investments.

Of course no forecast is perfect, despite what we assumed inthis paper. The value of a perfect forecast must represent an upperbound to the value of an imperfect forecast. In future work we planto extend these results to value imperfect water forecasts.

Acknowledgements

The authors would like to thank the anonymous reviewers fortheir very insightful and practically relevant comments.

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Valuing hydrological forecasts for a pumped storage assisted hydro facility