Uncertainty in reservoir operation

Optimal Allocation of Water Resources (Proceedings of the Fxeter Symposium, Julv 1982). IAHSPubl. no. 135.

Uncertainty in reservoir operation

J, W. DELLEUR & M, KARAMOUZ

School of Civil Engineering, Purdue University, West Lafayette, Indiana 47907, USA

ABSTRACT This paper is concerned with the determination of the annual and monthly reliabilities of a single reservoir with releases regulated in accordance with a rule which minimizes losses. The emphasis is placed on the generation of the synthetic flows and on the

estimation of the reliability curves while the development of the release rule is summarized very briefly. Annual flows are investigated first to account for possible long term dependence and to analyse the annual reliability of the reservoir. These flows are then disaggregated into monthly flows to evaluate the monthly reliability of the reservoir.

SIMULATION OF ANNUAL FLOWS

The following procedures are general. For the purpose of the exposition the annual series of the Gunpowder River inflows to Loch Raven Reservoir is examined (data: 1883-1963 from John Eastman (1972) and 1963-1977 from the Department of Public Works, Baltimore,

Maryland, USA). The Gunpowder River has been one of three principal sources of water supply for the city of Baltimore, Maryland, since 1881. The mean of the annual flows is 11 mJs " , their standard deviation is 3.6 m s , and the lag-one serial correlation coefficient is 0.302.

The probability distribution of the annual flows is examined first. If the flows are not normal, a power, a logarithmic or a Box

& Cox (1964) transformation may be used to normalize them. The latter is given by:

X+ (Qt + C) " - 1 /A A

Ï

(l) Xt = log (Qt + C) A = O

A chi-square test can be performed to determine if the hypothesis of normality of the flows is accepted or rejected. The annual flows of the Gunpowder River pass the test at the 10% significance level and no transformation is necessary.

The long range dependence is examined next by means of the Hurst coefficient. Consider the sequence of annual flows or normalized annual flows {x} as inflows into an ideal reservoir subject to a draft equal to the mean annual flow <X >, construct the adjusted partial sums (storages)

Si = Si-1 + Xi - < Xn > <2 )

7

and consider the maximum and minimum

max (O, Sj_, . . . , S ) ; min (O, S±, . . . , Sn) then the rescaled adjusted range is defined by

V S = <Mn - mn) / S

(3)

(4) where S is the standard deviation of the series. The accumulated

sums of deviations from the mean are shown in Fig.l for the Gunpowder

* H \

\ \ \%

H, i 81 I # tt 1

u

\\ I

SO.O 90.0 ;OD.O

FIC.l Accumulated sums of deviations from the mean for the annual flows of the Gunpowder River.

River. For the first 25 years the flows are seen to be mostly larger than the mean whereas for the last 24 years they are mostly smaller than the mean. This results in a relatively large value of the rescaled range of 15.10 and a value of the Hurst coefficient of 0.70 estimated from

K = log (R /S)/log (n/2) (5)

This value of K is well above the value of 0.5 predicted by Hurst (1951) and Feller (1951) for a purely random normal process. This indicates the need to search for a model capable of reproducing the rescaled range or equivalently the Hurst coefficient.

The adjusted rescaled range or equivalently the Hurst coefficient

Uncertainty in reservoir operation

have been used as measures of long range d e p e n d e n c e , for example by Mandelbrot & Wallis (1969). Klemes (1974) and Potter (1979) have shown that high Hurst coefficients may also be due to changes in the m e a n . Hashino & Delleur (1981) analysed the 140 annual series given by Yevjevich (1963) and found that 51 of them had significant

changes in the mean level continuing longer than 10 y e a r s . Removal of these changes in the mean almost always greatly reduced the Hurst coefficient. Of course, removal of nonhomogeneities due to man-made interventions is necessary before the analysis of the time series.

Hipel & McLeod (1980) have shown that the autoregressive-moving average (ARMA) models with carefully selected p a r a m e t e r s can account for the Hurst phenomenon. With this purpose in m i n d , optimal or near optimal ARMA models are obtained for the standardized series. The ARMA (p,q) model may be written (Box & J e n k i n s , 1976) as

z

= tf

*. z

_. - l ]

9. e ^ 6

= -l

6)

where Zt = (Qt - <Q>) / S where Qt is the flow or transformed flow in year t and <Q> and S are the mean and standard deviation of the Q ' s , the (J*-A and 8^ are the autoregressive and moving average p a r a m e t e r s . For model stationarity and invertibility the parameters m u s t be chosen such that the characteristics equations 1 - (f)]_B - §2^ - (f>2B-^ - O and 1 - 0]^B - 6 2 8 ^ - ... - 9qB<^ = 0, respectively, have their roots outside the u n i t circle. In these equations B is the backward shift operator defined by Br Zt = Zt_r.

For the Gunpowder River the autocorrelation and the partial autocorrelation functions each exhibit a significant value at lag o n e , then oscillate at higher lags without any significant v a l u e . This could possibly indicate an ARMA (1,1) model which has been considered by O'Connell (1974) as a possible model to preserve long range d e p e n d e n c e . However, Hashino & Delleur (1981) have shown that this model i s , in g e n e r a l , not the optimum. They concluded that higher orders of the autoregressive component, up to p = 1 3 , and moving average of order q up to 3 may be necessary to reproduce the Hurst coefficient or the rescaled range.

A slightly modified version of the Kitagawa (1977) procedure for the search of the optimal ARMA model is used h e r e . The procedure m a k e s use of the minimum Akaike (1974) information criterion

AIC = n log a" + 2 (p + q) (7) for the selection of the optimal model and of the determination

coefficient R = 1 - a / 0 „2, where CJ is the variance of the residuals £•£ and 0Z xs the variance of Z. .

The heuristic search algorithm of Kitagawa is outlined w i t h the following steps with reference to Table 1 for the G u n p o w d e r River

(a) For r = 1, 2, ..., pm fit r-th order autoregressive model and compute the A I C (r,0) and the corresponding R v a l u e s . For example, for the AR (9,0) the A I C (9,0) = -21.52 and R2 = 0.34.

(b) Find the local m i n i m a (rj_,0) i = 1, 2, ..., k of the AIC (r,0) along the column q = O. Minimal points are found for p = 1, 4, 7 and 9.

(c) Fit ARMA models along the diagonal in the p-q p l a n e passing through the local minima found in ( b ) . Along the first diagonal fit

TABLE 1 AIC and (R ) values of the ARMA (p,q) models fitted to the annual flow series of the Gunpowder River

p

0 1 2 3 4 5 6 7 8 9 10 11 12 13

q:

0

-8.81 (.108) -7.96 (.118) -6.88 (.127)

-7. 74 (.153)

-6.44 (.159) -4.97 (.164) -16.99 (.278) -15.09 (.279) -21.52

( .340) -21.30

(.353) -20.30 (.359) -19.23 (.366) -17.48 (•367)

1 -9.93 (•118) -8.01 (•119) -7.07 (.129) -4.88 (.127) -6.20 (.157) -4.50 (.159) -8.83 (.214) -16.06 (.286) -24.81*

-19.53 (.340) -19.50 (.353) -27.45*

-17.22 (.366) -27.22*

2 -8.62 (.124) -10.46 (.159) -8.40 (.159) -6.11 (.156) -8.28 (.192) -27.41*

-23.66*

-10.14*

-16.95 (.322) -30.47*

-18.05 (.358) -43.33*

-40.85*

-14.09 (.371)

3 -6.81 (.126) -6.67 (.143) -7.02*

-4.69 (.161) -7.51 (.203) -19.40*

-17.40*

-15.40*

-16.29 (.332) -14.29 (.332) -16.66 (.362) -14.98 (.364) -18.48*

-24.37*

* Non-stationary or non-invertible or did not converge.

the model ARMA model (0,1)» along the second diagonal fit the models (3,1), (2,2) and (1,3), along the third diagonal, models (6,1), (5,2), (4,3) and along the fourth diagonal, models (8,1), (7,2), (6,3) .

(d) Find the local m i n i m a (p.,q.) of the A I C along the d i a g o n a l s . They are ( 0 , 1 ) , ( 2 , 2 ) , (5,2) a n d1( 8 ^ 1 ) .

(e) Fit an ARMA model to each of the points Pj_,q-j around each minimum found in ( d ) . Going around the first m i n i m u m fit the m o d e l s

( 1 , 1 ) , (0,2) and ( 1 , 2 ) , for the second minimum fit the m o d e l s ( 1 , 3 ) , ( 2 , 3 ) , ( 3 , 3 ) , (3,2) and (2,1) and likewise for the third and fourth m i n i m a .

(f) Discard the m o d e l s w h i c h are not stationary or not invertible; the m o d e l s (5,2) and (8,1) are discarded.

(g) Find the global minimum which is achieved for ( 9 , 0 ) .

(h) If the global minimum lies on the boundary of pm and qm, the

Uncertainty in reservoir operation

The ARMA models are fitted by the procedures given in Salas et al.

(1980). The optimum ARMA model for the Gunpowder River annual flows is

Zt = 0.3291 Zt_x - 0.1770 Z^t_2 + 0.0677 Zt_3 + 0.0317 Zfc_4 - 0.0737 Z 5 + 0.0431 Zt_6 - 0.0508 Zt_7 - 0.0324 Zt_8 + 0.1138 Zt_9 + £t

The Waterloo Simulation Procedure 1 of McLeod & Hipel (1978) has been used for the generation of the annual series because random realizations of the underlying process are used as initial values, thus avoiding bias in the simulations.

Three thousand replicate series of 95 years were generated. The averages of the means, standard deviations and lag-one serial

3 — 1 3 — 1

correlation coefficients are 11.01 m s , 3.9 m s and 0.219, respectively, which are close to the respective historical statistics. The mean and the standard deviation of the Hurst coefficients are 0.6511 and O.05906, respectively. The probability of the Hurst coefficient to be larger than that of the historical series is 19.4%. The generating scheme is thus seen to conserve the first and second order statistics of the annual flows as well as the Hurst coefficient and hence the rescaled range.

OPERATING RULE

The general inflow series are routed through a single reservoir of known capacity operated in accordance with a release rule designed to minimize the total losses from the operation. The loss function is defined as a piecewise exponential function. Within a specified safe release range (RLOW £ release £ RUP) there is no loss as the release is large enough to satisfy the demand and yet is small enough to prevent flooding. Outside this range, an increase in the release above RUP may cause losses due to flooding, whereas a decrease in release below RLOW may cause losses due to water deficiencies. The loss function is thus defined as

Loss(Rt) = A[exp(Rt/RUP) - exp(l)] if Rt > RUP (8a) Loss(Rt) = 0 if RLOW £ Rt £ RUP (8b)

Loss(Rt) = B[exp(-Rt/RLOW) - exp(-l)] if Rt $ RLOW (8c) where A and B are known constants that depend on the price of the

water and on how extensive the property damage is, and Rt is the release during year t. For annual flows the values of the constants are taken as follows A = 3.88 x 1 05, B = 1.58 x 1 06, RUP = 1 . 2 MAF

(mean annual flow), RLOW = 0.8 MAF. The safe range is thus within 20% of the mean annual flow, and the values of A and B result in losses of 10 units when the release is zero or twice the mean annual

flow. This loss function responds to the release only. In some applications, as in hydroelectric power generation or recreation, a more elaborate loss function may be required as the losses may also be a function of storage and possibly other system characteristics.

The objective function is to minimize the total losses for the T years of expected economic life of the reservoir:

r T 1

minimize [E Loss(R,)J (9) subject to the following constraints:

(i) the mass balance of the reservoir (continuity)

St+1 ~ St + Rt = It t = 1, 2, ..., T (9a)

where It = inflow during year t, St = storage at the beginning of year t,

(ii) Rt :> R ™l n (9b)

(iii) Rt < R^a X (9c)

(iv) S. > s"11" (9d)

, „ , „ni3x , _ .

v St < S^ 9e

u v t

where the superscripts min and max indicate the minimum or maximum allowable value.

Karamouz & Houck (1981) solved this problem as an iterative discrete dynamic problem and regression analysis, using 20 discrete storage volumes uniformly distributed between zero and full reservoir capacity. They regressed the optimal storage, optimal release and concurrent inflow by means of the equation

Rt = a It + b St + c (lO)

for different bounds on R and R as follows t t

Rmax = (]_ + B O U N D) (a It + b St + c) (11a)

m i n r ~i R = maximum [0; (1 - BOUND) (a It + b St + c)J (lib) In (11a,b) the quantity (a I^ + b St + c) represents the r e l e a s e rule

obtained in the p r e v i o u s i t e r a t i o n . T h e introduction of these b o u n d s generalizes the p r o c e d u r e of Young (1967).

Combining the continuity equation (9a) w i t h t h e e q u a t i o n for the release rule (10) g i v e s

st + l = t1 - a ) I t + (1 - b ) St - c (12)

If St +] _ > 0 then R. is given by t h e release rule (10) , w i t h t replaced by t + 1 , if from (12) St + 1 < O, then St + 1 is set equal to zero and Rt = It + S^. T h e storage S t + 1 cannot exceed the r e s e r v o i r capacity, C A P , and the excess h a s to b e released. T h e r e f o r e if Rt > It + St - C A P t h e n Rt = It + St - C A P and St + 1 = C A P .

Uncertainty in reservoir operation 13

The occurrence based annual reliability, Ra, is defined as the number of non-failure years expressed as a percentage of the total number of years in the given period, it is thus equivalent to the probability that the reservoir will deliver the expected demand throughout its lifetime without incurring a deficiency. Usually, the economic life of a reservoir is taken between 50 and lOO years. However, to obtain reliability characteristics that are not influenced by initial

conditions of storage the reliability characteristics have been computed for a 500-year operation. This is done through the generation of 750 replicate series of 500 years by means of the previously described ARMA model. These inflow series are routed through reservoirs of various combinations of sizes and drafts making use of the previously developed release rules and the reliability characteristics are then calculated. As an example, the release rule for a value of the storage coefficient of 1.7 and BOUND = » is Rt = 0.304 Ifc - 0.053 St + 256542. The averages of these annual reliabilities for the 750 replicates are shown in Fig.2

Annual Reliability % 10

S t o r a g e C o e f f i c i e n t

FIG.2 Annual reliability for Gunpowder River.

as a function of the storage ratio (capacity/mean annual runoff volume) and draft ratio D (draft rate/mean annual flow). The bounds used in the generation of the release rules have little or no effect on the annual reliability curves and are not shown in the figure.

The annual reliability is seen to depend more strongly on the draft ratio than on the storage coefficient. This is due to the fact that the safe range of the releases of 0.4 MAF (from 0.8 to 1.2 MAF) corresponds to 1.22 standard deviations of the annual flows and a large percentage of the flows is within this range. As expected, the reliability of 50% is obtained with a draft ratio equal to unity and the reliability increases uniformly as the draft ratio decreases.

The curves are different from those given by Klemes et al. (1981) in that their slopes change for reliabilities less than 0.5. This is due to the loss function and safe range used in generation of

release rules. Fig.2 shows that as the reservoir capacity increases there is more control and the probability of high drafts (floods) and low drafts (droughts) is reduced. For example, for a storage coefficient of 0.2 the prob(1.2 MAF > D > 0.8 MAF) = 0.74 - 0.25 = 0.49 whereas for a storage coefficient of 1.7 the

prob(1.2 MAF > D > 0.8 MAF) = 0.88 - 0.13 = 0.75; this shows the increased probability of being in the safe range with the large reservoir.

MONTHLY RELIABILITY

The disaggregation model for Mejia & Rousselle (1976) was used to simulate the monthly flows. It may be written in the form

Y_ = AX + Bt + £Z (13) where Y is the matrix of the monthly disaggregated standardized

flows of dimension 12 x 1, A is a 12 x 1 column matrix of parameters, X is the value of the current standardized annual series previously generated, B is a 12 x 12 parameter matrix, E is 12 x 1 column matrix of random components, C is a 12 x 1 column matrix of parameters, and Z is the flow value of the last month of the previous year. The procedure for the estimation of the parameters and of generation are given in Salas et al. (1980, chapter 8 ) .

The loss function for monthly flow is of the same form as shown in equation (8) but with RUP = 1.2 (mean monthly flow), RLOW = 0.8 (mean monthly flow). The safe range is thus within 20% of the mean monthly flow (averaged over the 12 months). The values of A and B are the same as before and result in a loss of 10 units when the release is zero or twice the mean monthly flow.

The release rules for monthly flows are of the same form as in equation (10) where It and Rt represent the inflow and the release during month t and St is the storage at the beginning of month t.

As before, the release rules were obtained by regression of the optimal release vs. the optimal storage (resulting from the discrete dynamic program) and the current inflow (Karamouz & Houck, 1981) . As an example, the release rule for a value of the storage

coefficient of 1.0 and BOUND = 0.15 is Rfc = 0.168 It + 0.061 Sfc + 16035.

Four hundred years of monthly reservoir operations have been computed routing the monthly flows through the reservoir in

accordance with the release rules. The average of the 4800 months' reliabilities are plotted in Fig.3 as a function of the storage ratio and of the draft ratio for the several bounds used in the definition of the release rule. The bounds are seen to affect the reliability. For some value of the BOUND the probability of having a release in the safe range is maximum and as the storage

coefficient increases this probability increases.

Plots of the annual and monthly volume reliability may also be obtained in a similar manner. The monthly volume reliability may be of particular interest, as it is important to know not only the number of failure months but also the amount of the shortages.

Further uncertainty in the reservoir operation may be introduced

Uncertainty in reservoir operation 15

0.2 04 06 08 10 1.2 14

FIG.3 Monthly reliability for Gunpowder River.

by the uncertainty in the parameters of the models.

Similar calculations of the yearly and monthly reliabilities have been prepared for the Blacksmith Fork, Utah, and Osage River, near Bagnel, Missouri and the Tygart River, near Belington, West Virginia.

These results may be found in Karamouz & Houck (1981) .

ACKNOWLEDGEMENTS This material is based upon work supported by the National Science Foundation under Grant no. CME 7916819. The writers wish to thank Professor M.H.Houck for his assistance throughout the research.

REFERENCES

Akaike, H. (1974) A new look at the statistical model identification.

IEEE Trans. Automat. Control AC-19(6), 716-723.

Box, G.E.P. & Cox, D.R. (1964) An analysis of transformations. J.

Roy. Statist. Soc. Ser. B, 26, 211-252.

Box, G.E.P. & Jenkins, G.M. (1976) Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco, California, USA.

Eastman, J. (1972) Extension of the linear decision rule in reservoir design and operation. MS Thesis, Johns Hopkins Univ., Baltimore, Maryland, USA.

Feller, W. (1951) The asymptotic distribution of the range of sums of independent random variables. Ann. Math. Statist. 22, 421-432.

Hashino, M. & Delleur, J.W. (1981) Investigation of the Hurst

coefficient and optimization of ARMA models for annual river flow.

Report CE-HSE-81-1, Civil Engng, Purdue Univ., West Lafayette, Indiana, USA.

Hipel, K.W. & McLeod, A.I. (1980) Perspectives in stochastic

hydrology. In: Time Series (Proc. of Int. Conf. held at Nottingham Univ., March 1979) (ed. by O.D.Anderson), 73-102.

North Holland Publ. Co., Amsterdam.

Hurst, H.E. (1951) Long term storage capacity of reservoirs. Trans.

Am. Soc. Civ. Engrs 116, 770-808.

Karamouz, M. & Houck, M.H. (1981) Generation of monthly and annual operating rules. Tech. Report CE-HSE-16, School of Civil Engineering, Purdue Univ., West Lafayette, Indiana, USA.

Kitagawa, G. (1971) On a search procedure for the optimal ARMA order.

Ann. Inst. Statist. Math. 29, part B, 319-332.

Klemes, V. (1974) The Hurst phenomenon - a puzzle? Wat. Resour. Res.

10(4), 675-688.

Klemes, V., Srikanthan, R. & McMahon, T.A. (1981) Long-memory models in reservoir analysis: what is their practical value? Wat.

Resour. Res. 17(3), 737-751.

Mandelbrot, B.B. & Wallis, J.R. (1969) Some long-run properties of geophysical records. Wat. Resour. Res. 5(2), 321-340.

McLeod, A.J. & Hipel, K.W. (1978) Simulation procedures for Box- Jenkins models. Wat. Resour. Res. 10(5).

Mejia, J.M. & Rousselle, J. (1976) Disaggregation models in hydrology revisited. Wat. Resour. Res. 12(2), 185-186.

O'Connell, P.E. (1974) A simple stochastic modelling of Hurst's law. In: Mathematical Models in Hydrology (Proc. Warsaw Symp., July 1971), vol. 1, 169-187. IAHS Publ. no. 100.

Potter, K.W. (1979) Annual precipitation in the northeast United States: long memory, short memory or no memory? Wat. Resour.

Res. 15(2), 340-346.

Salas, J.D., Delleur, J.W., Yevjevich, V. & Lane, W.L. (1980) Applied Modeling of Hydrologie Time Series. Water Resources Publications, Littleton, Colorado, USA.

Yevjevich, V.M. (1963) Fluctuations of wet and dry years, 1,

research data assembly and mathematical models. Hydrol. Pap. 1, Colorado State Univ., Fort Collins, Colorado, USA.

Young, G.K., Jr (1967) Finding reservoir operating rules. J.

Hydraul. Div. ASCE no. HY6, 297-321.