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Firm outflow from multiannual reservoirs with skew and autocorrelated inflows
Journal of Hydrology, 38 (1978) 93--112 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
93
[3] F I R M O U T F L O W F R O M M U L T I A N N U A L R E S E R V O I R S W I T H SKEW AND AUTOCORRELATED INFLOWS
L. VALADARES TAVARES
Department of Civil Engineering, Technical University of Lisbon (I.S.T.) * and Centre of Urban and Regional Systems of the Universities of Lisbon (I. N.I.C.), Lisbon (Portugal) (Received June 15, 1977; revised and accepted December 22, 1977)
ABSTRACT
Valadares Tavares, L., 1978. Firm outflow from multiannual reservoirs with skew and autocorrelated inflows. J. Hydrol., 38:93--112 Similarity analysis is used to generalize the simulation results on the risk associated with firm outflow from multiannual reservoirs with skew and autocorrelated inflows. General conclusions on the relationship of this risk with the guaranteed outflow, the reservoir capacity and the inflow features (mean, dispersion, skewness and autoeorrelation) are obtained and presented herein.
INTRODUCTION T h e p u r p o s e o f a m u l t i a n n u a l reservoir is t o r e t a i n f o r use in d r y years the excess w a t e r c o l l e c t e d in w e t years in o r d e r t o have f o r y e a r t a firm o u t f l o w , C(t), w h i c h is g u a r a n t e e d w i t h a p r o b a b i l i t y (1 - P). Obviously f o r a specific risk, t h e highest i n t e r a n n u a l u n i f o r m i t y is achieved w i t h C ( t ) = C, w h e r e C is a positive c o n s t a n t and t h e l o w e s t o n e with C(t) equal t o the inflow during y e a r t, Z ( t ) , w h i c h o c c u r s w h e n t h e r e is n o reservoir at all. T h e r e f o r e , a m o r e general d e f i n i t i o n o f C(t) is:
C(t)=8.Z(t)+(1-8).e
with0<8
<1
(1)
and t h e s t u d y o f t h e risk with eq.1 can be m a d e assuming t h a t t h e reservoir has inflows equal t o (1 - 8)Z(t) and t h a t C(t) is equal t o C(1 - 8) because 5 Z ( t ) can always be g u a r a n t e e d . Thus, t h e s t u d y o f t h e risk f o r m u l t i a n n u a l reservoirs can b e m a d e f o r a c o n s t a n t firm o u t f l o w w i t h o u t a n y loss o f generality. T h e s t u d y o f t h e risk as a f u n c t i o n o f C f o r a n y reservoir w i t h stochastic inflows is a m a j o r and c o m p l e x p r o b l e m o f t h e t h e o r y o f reservoirs w h i c h has to be solved in o r d e r to d e f i n e p r o j e c t priorities and reservoir sizes in w a t e r resources planning. With this aim several m e t h o d s have b e e n p r o p o s e d , b u t the p r o b l e m is n o t y e t c o m p l e t e l y solved because t h e m o d e l s a d o p t e d and t h e h y p o t h e s e s as*Address: I.S.T., C.E.S.U.R., Av. Novisco Pais, Lisbon, Portugal.
94
sumed are often unacceptable and because several real factors are not considered, e.g., the skewness and the serial interdependence of the inflow process. The objective of this paper is to study how relevant these factors are in determining the risk associated with C. Simulation methods and similarity analysis are used for this purpose. METHODOLOGY The proposed methods can be classified into four major groups: Markov chain models, analytical deduction of the range, mathematical programming models and experimental methods.
Markov chain models Usually, the inflow process is assumed to be a sequence of independent and identically distributed random variables and the reservoir is described through a small number n of states (Lloyd, 1970; Moran, 1959; Yeo, 1975). The number of parameters to be estimated increases with n and therefore the chosen value of n is usually low, but then the precision of the results is also low. The steady-state distribution of the reservoir water level has been obtained for independent inputs following geometric or gamma laws (Moran, 1959; Prabhu, 1958) and also for Markovian geometric inflows (Pyke and Phatarfod, 1976), but usually the theoretical solutions are quite complicated and are based on unrealistic reservoir operation rules. Other theoretical studies on reservoirs with Markovian inputs (Ali Khan and Gani, 1968; Phatarfod and Mardia, 1973; Collings, 1975; Lloyd, 1977; Lloyd and Odoom, 1965) have been made, but unfortunately several difficulties arise in obtaining and applying explicit results for realistic models of reservoirs.
Analytical deduction o f the range The analytical deduction (Yevjevich, 1965) of the range (Rn(C)), as defined by eq.4 for a period of N units of time and outflow of C, is not feasible for N > 4, but in practical problems N has to be much higher than 4 (usually N >/ 30). The expected value and the variance of Rn(C) have been obtained by several authors for stationary sequences of independent and Gaussian inflows with mean p equal to C. Feller (1951) and Anis and Lloyd (1953) have deduced their asymptotic limits and their exact values, respectively. The statistical properties of the range, assuming that the output is linearly dependent upon storage, has also been studied by several authors, e.g., Buras (1972) and Melentijevich (1965).
Mathematical programming models Mathematical programming (usually linear programming) has also been used
95
to determine the minimum reservoir capacity for which the risk is not greater than some specific limit for a given release policy (Revelle and Gundelach, 1975; Luthra and Arora, 1976), but the exact risk cannot be determined from these studies because it is assumed in the balance equation that 0(t) can always be released and that no spill ever occurs. Dynamic programming has also been used intensively to obtain numerical solutions for specific problems (Buras, 1972), but often no general conclusions can be drawn from these results.
Experimental methods Owing to the shortcomings of previously available models, experimental methods have had to be used, initially with real data (Rippl, 1883), but since computers and Monte Carlo techniques have become available the reservoir operation is simulated with synthetic data for sufficiently long periods, and the statistical properties of the risk related to C are estimated from the computed outflows. The major drawback of this method is the lack of generality of the results obtained, because for each particular set of data a complete simulation experiment has had to be repeated. The methodology adopted in this paper is to apply similarity analysis to experimental results in order to obtain more general conclusions. PROBLEM FORMULATION
To formulate this problem several variables have to be defined (see Notation for symbols used): NOTATION List of symbols used a, b, C O, k , k l , k ~ , k 3
A(t), A*(t) B(t), B*(t) C, C* Cv
E[-] F(A) i,i,M,N,n L,L* l+.,,(i), l~.,,(i) P Q
rz(i) ~z(i) RN(C)
real parameters available affluency during t i m e unit t reservoir stock at the beginning o f t i m e unit t firm o u t f l o w coefficient o f variation o f Z e x p e c t e d value of cumulative distribution f u n c t i o n of A(t) for a generic t positive integer variables reservoir effective capacity c o n f i d e n c e limits for a significance level of 0.95 shortage probability F(A) quantile for period i
estimated mean of F(A ) quantile a u t o c o r r e l a t i o n f u n c t i o n of z for lag i estimate of rz(i) a u t o c o r r e l a t i o n f u n c t i o n of z* range with c o n s t a n t o u t f l o w equal to C and for a period o f N units of time
96 NOTATION (continued)
s(t), s*(t)
surplus at the end o f time unit t m a x i m u m S(t) during a period o f N units of time m i n i m u m S(t) during a period of N units o f time standard error o f time integer variable variance of real variables inflow time-series inflow time-series for year t shortage frequency shortage frequency for period i expected a, a * total deficit total deficit during period i expected ~,~* positive constant skewness coefficient outflow deficit during year t Z* mean and Z mean Z* variance and Z variance estimate of u, a
s} s~ SE(.) t VAR[. ] X,Y Z, Z* Z(t), Z*(t) 0~ Ot ~ Oti, Ot~
#,#* 5,h A(t) 0 2 * ^
a 2
(1) Inflow time-series: z = (z(t),
1 < t < +oo ).
(2)
where Z ( t ) is the inflow received b y the reservoir R during year t. Obviously, the nonexistence of losses due to inadequate distribution of inflow during each year has to be assumed. This is a general hypothesis implied b y any analysis of stock theory (Hadley and Whitin, 1963). (2) Reservoir effective capacity L [L is the reservoir capacity that can be used to store inflows (L > 0)]. (3) Firm outflow C. (4) Reservoir stock at the beginning of time unit t, B(t). (5) Range for a period of duration N and with outflow C, R N ( C ) . Defining: t S ( t ) = S(1) + ~ [Z(i) - C] i=l
with t/> 1
(3)
+
S g and S N are the maximum and minimum values of S ( t ) with t = 1,...,Nand:
RN(C)
=
+ sN
- s~
(4)
if infinite capacity is assumed (6) Shortage period, M. For a period of N units, M is the number of time units during which C cannot be satisfied. (7) Available affluency during unit t, A ( t ) . For any t > 1: A(t) = B(t) + Z(t)
(5)
97
(8) Outflow deficit A(t). For any t i> 1 one has: A(t) = 0
if
(6)
A ( t ) >i C
A(t) = C - A ( t )
if
(7)
A(t) < C
because the stock theory hypothesis (Hadley and Whitin, 1963) is assumed. (9) Reservoir operation rules. The risk of failure to satisfy 0(t) = 8 Z o ( t ) + (1 - 8). Co is the same as t h a t obtained for a reservoir with Z ( t ) = (1 - 8 ) ' Z o ( t ) and a firm outflow C = (1 - 8)C0 as shown in eq.1. Therefore, only this last case will be studied here. The following reservoir operation rules are adopted (a) If:
[A(t) - C] < O,
O(t) = A ( t )
and
A(t) = C - A ( t )
(S)
because C cannot be satisfied in year t and therefore the outflow during t[0(t)] is made equal to A ( t ) . (b) If: 0 < [A(t) - C] < L
0(t) = C
and
A(t) = 0
(9)
because the risk is m i n i m u m with this rule. (c) If: [A(t) - C] > L
O(t) = A ( t ) - L
and
A(t) = 0
(10)
because an additional discharge has to occur: [A(t) - (C + L)]. Therefore: 0(t) = C + A ( t ) - (C + L )
(11)
(10) Measures of risk associated with C. This is a debatable issue and several ways of measuring the risk can be proposed for a period with duration N: (a) Shortage frequency ~, and total deficit/3. Measures of how often (a) and of how much (/3) shortages occur are given by: =M
and
N ~ = ~ A(t)
(12), (13)
t=l
The probability P of C not being satisfied can be determined by: P = lim ( M / N ) (b) Statistical distributions of X = {3/N, of Y = a / N and of A ( t ) for any unit of time t (t >I 1). In this paper the measures studied are a, ~ and the A ( t ) distribution, because X and Y can be easily determined if a and/3 are known. Therefore the following quantities are estimated: (i) Cumulative distribution function of A ( t ) for any unit t (t >1 1), F ( A ) _ [This function is estimated through the mean values of its seven quantiles Q and their standard errors SE(Q).] (ii) Expected values of a(~) and of/3(~) for N = 1000 and their standard errors SE(~) and SE(j~).
98
SIMILARITY ANALYSIS O n a reservoir R, similarity analysis is applied to this problem to investigate h o w ~, ~ and F(A) depend on the adopted values for firm outflow C, effective capacity L, initial stock B(1), inflow time-series m e a n p and variance a 2 . Obviously, C, L and B(1) cannot be negative, and/~ and a 2 must have values in which the probability of Z(t) <~ 0 with t >I 1 has to be either zero or negligible for the assumed inflow time-series model because Z(t) < 0 has no physical meaning. For this purpose, a standard reservoir R * is defined by C*, L*, B*(1), p* and o *2 as follows:
(15)
Z(t) = a + bZ*(t)
with: b = a/a*
and
a = p - ap*/a* = (pa* - ap*)/a*
(16), (17)
For the transformation shown by eq.15, Z ( t ) a n d Z * ( t ) have the same autocorrelation function. If: L = bL*
(18)
C = a + bC*
(19)
S(1) = b .B*(1)
(20)
one has: S(t) = b .S*(t)
and
(21), (22)
B(t) = b .B*(t)
Thus, the following equivalence relations are obtained: a = ~*
(23)
= ~*
(24) (25)
A ( t ) = a + b . A *(t)
with: a = (pa*
-
op*)/a* = P
-
(l~*/a*)'a
and
b = a/a*
(26), (27)
Then, R and R * are named "equivalent reservoirs". Usually, C, L and B(1) are defined as fractions of g : C = k,u
(28)
L = k2p
(29)
B(1) = k3u
(30)
99
and the equivalent reservoir R * is represented by: (kl - 1)p
C* =
alo*
+ p* = ~* +
p * ( k l - 1)
(31)
C~
k2 k2o* ~--o/o* Cv
(32)
L*= --
k3 -
-
o*
B*(1) = olo* p"
~_.
m
k3 Cv
(33)
for specified values of p* and o*. Without loss of generality p* and o* can be made equal to 0 and 1, respectively, and the following equivalence equations are obtained: C* = (k, - 1)/Cv
(34)
L* = k2/Cv
(35)
B*(1) = k3/Cv
(36)
Here, a, ~ and F ( A ) can be easily derived from values of a*, ~* and F ( A * ) because: = a*
(37)
= o~*
(38) (39)
A = ~ + oA*
Therefore, if a small number of combinations of (kl - 1)/Cv, k2/Cv and k3/Cv values is studied the results obtained can be generalized to a much larger number of cases defined by p, o, L, C and B(1). If the simulated period N is sufficiently large, a, ~, F ( A ) are n o t sensitive functions of B(1) because B(t) is independent of B(1) for t >I to (to is the first B(t) zero). EXPERIMENTAL RESULTS
The inflows process {Z(t), 1 ~< t < o¢ ) is assumed to be a stationary sequence of autocorrelated normal (or lognormal) variables with:
p(k) = k lkl
0 ~< X < 1
(40)
because an exponential function fits the estimated autocorrelation (Yevjevich, 1973) of annual r u n o f f for m a n y rivers [rz(0]. Usually ~z(1) :~ 0.5 and for several rivers the annual r u n o f f is serially independent (Quintela, 1967). The skewness coefficient of the annual runoff: A
E [ Z ( t ) - E { Z ( t ) } ]3 %/~ =
U3
o
100
is usually within the interval (0; 1.6) and 0.7 is quite a c o m m o n value for some regions, such as Portugal [the hypothesis of x/7 = 0.7 is not rejected for most Portuguese rivers (Quintela, 1967) such as the rivers SabSr at Quinta das Laranjeiras, Z~zere at Cabril, Odivelas at Odivelas, Corona at Moinho do Bravo, Rabag~o at Venda Nova, etc.] Therefore the six models were considered (see Table I). For each model 18 cases were studied, using TABLEI Model
Inflow distribution
~
Model
Inflow distribution
A B C
normal normal log-normal
0 0. 5
D
log-normal ~/~/ = 0. 7
0.5
0
E
log-normal
0
~/-y= o 7
~/~, = 1 6 F
log-normal
0.5
x/~/ = 1 . 6
__-xEl/ 480
440
1 l | l
400 360 320 \
~ = 0
280 \ ;\
240
".o °:. 200 k 160
"..?%. "..
\:,, "~.
..........
V~-~= 0 7
. . . .
~/-~'~: 1.6
",~'..... ,,. "~..?-,,
120
'°°.oO°°o 80 40
®
C
1/2
t
C = -0.1
: -0.2
2
4
16
L
101
520%
\ •
%
\ ~o~
~... \
380"
"~ ~ °. %
%
,\\
320-
- -
%%
VT--o~ 0.7
%. %'%
240-
°..
200-
,~
160-
) 2080-
":Z::2".~... ........
C=-0.1
-':0~
.... ~%"::'"::""".:C= -0.2
®
112 1
2
,~
~6
Fig.1. Estimated ~* as a function of L* and C* for: ~_ Uncorrelated (k = 0), normal (~/~ = 0) and log-normal inflows (x/~ = 0.7; 1.6). B. Autocorrelated (X = 0.5), normal (x/~ = 0) and log-normal inflows (x/r~ = 0.7; 1.6).
C * = p * - k , a * a n d L * = k ~ o * with p* = 0, o* = 1, C* = 0, - 0 . 1 , - 0 . 2 and L* = 1~
1, 2, 4, 8, 16. For each case 10 periods o f 1 0 0 0 yr were simulated and the Q* 's ~* and ~*, as well as their standard errors, were estimated, using: lO
1
I
lO
~* = ~ Z~ Q* 10 i=l
qV2
and SE(Q*)=L 10~:-~ ~j
i=1
~* - - I0
_ _
(41), ¢42)
10
and
i=1
~* = ~
10
(43), ( 4 4 )
102 SE(~*) =
I10 i=1
l 1/2
-
10(10 - 1)
i 10 SE(~,) =
and
i 1/2
i=1 10(10
-
1) (45), (46)
where a*, ~* and Q* are the estimated a*, ~* and Q , for period i(i = 1 , . . . , 10). From eqs. 34 and 35 one may conclude that these results are applicable to any case with: kl = 1 + C*'Cv
and
k2 = L * . C v
(47), (48)
for a pair studied (L*, C*). In Figs.l--4
~*, ~* and
Q* are shown as functions of L* (reservoir capacity),
300
250
200 A = 0
150
100
•,
% -°
\-,
\
50
C
=0
%%. -. %.. C
(•)
1/2
I
C ' ~ = _ 0.1
=-0.2
2
4.
8
16
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..." ....'" ...""
I
~, , ~ . i ~ f
/
16
•
~*
2
o°°
,°
•••
o• •
I
j f
3
~
~ t I
.." "'"
/
oO•
. , o o•°
/
/
•
6
I
•..,-•
1 t ~"
..°
/
. •• . • •
,-'"
..'""'"
I
o°
.""
4
/ .; ..".." i .. /
0
oO°°° ~ o° ~ o° ~
r
'/J
Fig.4A. For caption see p.109.
®
0.005
0.010
0.050
!~/~."~//"/' "" / /
0.I00
::1/.'/
I~ , r . I/.: //: / /
rl/ I/" /,"
ll.' l,f / /
112 1
~1~." / / " " / / / / r#.."//...
L'W
0150
0.250
0.500
0.750
F(~C
•o•
,,o
i
/
8
•
•°
°•
'"
j,-
9
I
o0•
/
°
~.
,s
........
I
13
. . . .
/
I"
/ •0
0.0
•
#s
C~ : -
0.2
C~= - 0 . 1
C "w =
0.•
/
/
16 /
•o
/
•°
/ •,
:
O.
:
0
-
0.150
0.005
0.010
0.050 -
0.100"
-
0.250
0500
0.750
F{
j
2
." /
."
~'t
' ,°" /
/."
• ~.
/I/i/,'f-
t/I
0
r/,,t ~ ,. r./~l"" ."
, '
Ill II
""
/I//L / ~
IW/.
/.
1
/
"
"""
3
...."
..'"
....'"
."
"
t
.'"
'" ~
•
-'"
...."
.'"
.~
.........
.:7.. ...-"°
..~"
~..-
'# o"
8
... ,,"-""
..."
.. -"
/
flS2,......';..":"" "
." ~. ~'°
1
°° lII: /,.'.,:l~" /",,://,.~...'" ,". ... ~
1/2
/ ~ , " .i. , " . . , "
L~
9
i
.-"
11
..
'
.....
13
. . . .
" 7.....
.,.."**
,,"" .
o _ 0.2
~. : _ o . ,
~ : oo
- . ."" .'"
S , , *~ oO, "°
16
Gi.
o
Oo
-
I
..
."
.'"
•
/.'"
j
I
, • o" • •
.,. " "
3
."""
/'"
""
i
//
,
0
."
•
f
4
~'
j
...."
~ J f ."" //
2
/,/:.
1/2 1
~ °.°"
/,,
;o. //
# o.. "
j
i
L .x.
."
,, oO
/
• "" " " "
/,
°°"
...
I 6
,,"
..
.,-''
8
Fig.4. Estimated mean quantiles Q* with C* = 0; 0.1; - 0 . 2 for: A. Autocorrelated (k = 0.5), normal inflows (~/~ = 0). B. Autocorrelated (k ffi 0.5), log-normal inflows (x/~ = 0.7 ). C. Autocorrelated (k = 0.5), log-normal i n f l o w s ( ~ ffi 1.6).
0.005 -
o.olo
0.050 -
O. 100 -
0.150
0.250
0.500
F (~.}.w-
oO°°•°°°°°°°
,,~ ~,°o,°
°°°°°
oO
°
........
. . . .
o°
o •oo
,4"
~ °o"
,°
_ 0.i
C :-0.2
C'~=
C'w'= 0.0
° ° o°
,,p
16 oO
110
°~ o ~ .2
99.9 99.8 99.5 99
/
98
v o
95
f
90 80
/
70 60 60
.J
S
L'",.:i" 10
2" J
f
0.1
Annua.( 0
200
/.00
600
800
1000
1200
1 4 0 0 1 6 0 0 1800
2000
2200
2400
2600
runoff |mm)
Fig. 5. Cumulative normal distribution of annual runoff of River Rabag~o at Venda Nova (1940/41--1962/63). CONCLUSIONS
The behaviour of a stochastic reservoir with autocorrelated and skew inflows was studied b y simulation and the estimated results were generalized by similarity analysis. General conclusions on the estimated shortage frequency (~/1000), the overall o u t f l o w deficit for a period of N units of time {~ with N =_1000) and the cumulative distribution function of the available affluency F (Q ) are shown in Figs.l--4 for several levels of firm outflow and several values of the coefficients of skewness ~/7 and autocorrelation k. From these graphs several conclusions can be drawn: (1) For uncorrelated inflows, ~* and ~* are only slightly sensitive to the skewness of the inflows. However, with k = 0.5, x/~/is a very significant parameter. (2) Autocorrelation is important in estimating ~* and ~*: a* and ~* are significantly increased b y a positive k if inflows follow a skew distribution,
111
i: z
{i)
10 09 0.8 0.7 06 0.5
•.~~).95(i)
0/, 0.3 0.2 0.1 0
I
I
t
I
~
-0.1
1
2
3
4
5
) i
-02 -0,3 -04 -05 -06
,'~ 095
(i)
-0.7 -08 -0.9 -1.0
Fig. 6. Estimated correlogram [~z (i)] for annual runoff of River Rabag~o at Venda Nova (1940/41--1962/63).
(3) d~*/dL*, d~*/dL* < 0 and d2~*/dL .2, d2~*/dL .2 i> 0. (4) A(t) is not normally distributed except when L* = 1/i, 1, and x/7 = 0. A practical example of the application of these results to a Portuguese multiannual reservoir at Venda Nova (River Rabag~o) is also considered.
REFERENCES Ali Khan, M.S. and Gani, J., 1968. Infinite dams with inputs forming a Markov chain. J. Appl. Probab., 5: 72--83. Anderson, R.L., 1941. Distribution of the serial correlation coefficient. Ann. Math. Stat., 13: 1--13. Anis, A.A. and Lloyd, E.H., 1953. On the range of partial sums of a finite number of independent normal variables. Biometrika, 40: 35--42. Buras, N., 1972. Scientific Allocation of Water Resources. American Elsevier, New York, N.Y. Collings, P.S., 1975. Dams with autoregressive inputs. J. Appl. Probab., 12: 541--553. Feller, W., 1951. The asymptotic distribution of the range of series of independent random variables. Ann. Math. Stat., 22: 427--432. Hadley, G. and Whitin, T.M., 1963. Analysis of Inventory Systems. Prentice-Hall, Englewood Cliffs, N.J.
112
Lloyd, E.H., 1970. Transient and asymptotic behaviour of reservoirs with non-persistent seasonal inflows. J. Hydrol., 10: 243--258. Lloyd, E.H., 1977. Reservoirs with seasonally varying Markovian inflows, and their first passage times. Int. Inst. Appl. Syst. Anal., Laxenburg. Lloyd, E.H. and Odoom, S., 1965. A note on the equilibrium distribution of levels in a semi-finite reservoir subject to Markovian inputs and unit withdrawals. J. Appl. Probab., 2: 215--222. Luthra, S. and Arora, S.R., 1976. Optimal design of single reservoir systems using 6 release policy. Water Resour. Res., 12: 606--612. Melentijevich, M.J., 1965. The analysis of range with output linearly dependent upon storage. Colo. State Univ., F o r t Collins, Colo. Hydrol. Pap., No.11. Moran, P.A.P., 1959. The Theory of Storage. Methuen, London. Phatarfod, R.M, and Mardia, K.V., 1973. Some results for dams with Markovian inputs. J. Appl. Probab., 10: 166--180. Prabhu, N.U., 1958. Some solutions for the finite dam. Ann. Math. Stat., 29: 1234--1243. Pyke, T. and Phatarfod, R.M., 1976. Some exact results for dams with Markovian inputs. J. Appl. Probab., 13: 329--337. Quintela, A., 1967. Recursos de ~guas superficiais em Portugal continental. In: A. Quintela (Editor), Surface Water Resources in Portugal, Lisbon. Revelle, C. and Gundelach, J., 1975. Linear decision rule in reservoir management and design. 4. A rule that minimizes output variance. Water Resour. Res., 11: 197--203. Rippl, W., 1883. The capacity of storage reservoirs for water supply. Proc. Inst. Civ. Eng., 71: 270--278. Yeo, G.F., 1975. A finite dam with variable release rate. J. Appl. Probab., 12: 205-211. Yevjevich, V., 1965. The application of surplus, deficit and range in hydrology, Colo. State Univ., F o r t Collins, Hydrol. Pap. No.10. Yevjevich, V., 1972. Stochastic processes in hydrology. Water Resour. Publ., F o r t Collins, Colo.
93
[3] F I R M O U T F L O W F R O M M U L T I A N N U A L R E S E R V O I R S W I T H SKEW AND AUTOCORRELATED INFLOWS
L. VALADARES TAVARES
Department of Civil Engineering, Technical University of Lisbon (I.S.T.) * and Centre of Urban and Regional Systems of the Universities of Lisbon (I. N.I.C.), Lisbon (Portugal) (Received June 15, 1977; revised and accepted December 22, 1977)
ABSTRACT
Valadares Tavares, L., 1978. Firm outflow from multiannual reservoirs with skew and autocorrelated inflows. J. Hydrol., 38:93--112 Similarity analysis is used to generalize the simulation results on the risk associated with firm outflow from multiannual reservoirs with skew and autocorrelated inflows. General conclusions on the relationship of this risk with the guaranteed outflow, the reservoir capacity and the inflow features (mean, dispersion, skewness and autoeorrelation) are obtained and presented herein.
INTRODUCTION T h e p u r p o s e o f a m u l t i a n n u a l reservoir is t o r e t a i n f o r use in d r y years the excess w a t e r c o l l e c t e d in w e t years in o r d e r t o have f o r y e a r t a firm o u t f l o w , C(t), w h i c h is g u a r a n t e e d w i t h a p r o b a b i l i t y (1 - P). Obviously f o r a specific risk, t h e highest i n t e r a n n u a l u n i f o r m i t y is achieved w i t h C ( t ) = C, w h e r e C is a positive c o n s t a n t and t h e l o w e s t o n e with C(t) equal t o the inflow during y e a r t, Z ( t ) , w h i c h o c c u r s w h e n t h e r e is n o reservoir at all. T h e r e f o r e , a m o r e general d e f i n i t i o n o f C(t) is:
C(t)=8.Z(t)+(1-8).e
with0<8
<1
(1)
and t h e s t u d y o f t h e risk with eq.1 can be m a d e assuming t h a t t h e reservoir has inflows equal t o (1 - 8)Z(t) and t h a t C(t) is equal t o C(1 - 8) because 5 Z ( t ) can always be g u a r a n t e e d . Thus, t h e s t u d y o f t h e risk f o r m u l t i a n n u a l reservoirs can b e m a d e f o r a c o n s t a n t firm o u t f l o w w i t h o u t a n y loss o f generality. T h e s t u d y o f t h e risk as a f u n c t i o n o f C f o r a n y reservoir w i t h stochastic inflows is a m a j o r and c o m p l e x p r o b l e m o f t h e t h e o r y o f reservoirs w h i c h has to be solved in o r d e r to d e f i n e p r o j e c t priorities and reservoir sizes in w a t e r resources planning. With this aim several m e t h o d s have b e e n p r o p o s e d , b u t the p r o b l e m is n o t y e t c o m p l e t e l y solved because t h e m o d e l s a d o p t e d and t h e h y p o t h e s e s as*Address: I.S.T., C.E.S.U.R., Av. Novisco Pais, Lisbon, Portugal.
94
sumed are often unacceptable and because several real factors are not considered, e.g., the skewness and the serial interdependence of the inflow process. The objective of this paper is to study how relevant these factors are in determining the risk associated with C. Simulation methods and similarity analysis are used for this purpose. METHODOLOGY The proposed methods can be classified into four major groups: Markov chain models, analytical deduction of the range, mathematical programming models and experimental methods.
Markov chain models Usually, the inflow process is assumed to be a sequence of independent and identically distributed random variables and the reservoir is described through a small number n of states (Lloyd, 1970; Moran, 1959; Yeo, 1975). The number of parameters to be estimated increases with n and therefore the chosen value of n is usually low, but then the precision of the results is also low. The steady-state distribution of the reservoir water level has been obtained for independent inputs following geometric or gamma laws (Moran, 1959; Prabhu, 1958) and also for Markovian geometric inflows (Pyke and Phatarfod, 1976), but usually the theoretical solutions are quite complicated and are based on unrealistic reservoir operation rules. Other theoretical studies on reservoirs with Markovian inputs (Ali Khan and Gani, 1968; Phatarfod and Mardia, 1973; Collings, 1975; Lloyd, 1977; Lloyd and Odoom, 1965) have been made, but unfortunately several difficulties arise in obtaining and applying explicit results for realistic models of reservoirs.
Analytical deduction o f the range The analytical deduction (Yevjevich, 1965) of the range (Rn(C)), as defined by eq.4 for a period of N units of time and outflow of C, is not feasible for N > 4, but in practical problems N has to be much higher than 4 (usually N >/ 30). The expected value and the variance of Rn(C) have been obtained by several authors for stationary sequences of independent and Gaussian inflows with mean p equal to C. Feller (1951) and Anis and Lloyd (1953) have deduced their asymptotic limits and their exact values, respectively. The statistical properties of the range, assuming that the output is linearly dependent upon storage, has also been studied by several authors, e.g., Buras (1972) and Melentijevich (1965).
Mathematical programming models Mathematical programming (usually linear programming) has also been used
95
to determine the minimum reservoir capacity for which the risk is not greater than some specific limit for a given release policy (Revelle and Gundelach, 1975; Luthra and Arora, 1976), but the exact risk cannot be determined from these studies because it is assumed in the balance equation that 0(t) can always be released and that no spill ever occurs. Dynamic programming has also been used intensively to obtain numerical solutions for specific problems (Buras, 1972), but often no general conclusions can be drawn from these results.
Experimental methods Owing to the shortcomings of previously available models, experimental methods have had to be used, initially with real data (Rippl, 1883), but since computers and Monte Carlo techniques have become available the reservoir operation is simulated with synthetic data for sufficiently long periods, and the statistical properties of the risk related to C are estimated from the computed outflows. The major drawback of this method is the lack of generality of the results obtained, because for each particular set of data a complete simulation experiment has had to be repeated. The methodology adopted in this paper is to apply similarity analysis to experimental results in order to obtain more general conclusions. PROBLEM FORMULATION
To formulate this problem several variables have to be defined (see Notation for symbols used): NOTATION List of symbols used a, b, C O, k , k l , k ~ , k 3
A(t), A*(t) B(t), B*(t) C, C* Cv
E[-] F(A) i,i,M,N,n L,L* l+.,,(i), l~.,,(i) P Q
rz(i) ~z(i) RN(C)
real parameters available affluency during t i m e unit t reservoir stock at the beginning o f t i m e unit t firm o u t f l o w coefficient o f variation o f Z e x p e c t e d value of cumulative distribution f u n c t i o n of A(t) for a generic t positive integer variables reservoir effective capacity c o n f i d e n c e limits for a significance level of 0.95 shortage probability F(A) quantile for period i
estimated mean of F(A ) quantile a u t o c o r r e l a t i o n f u n c t i o n of z for lag i estimate of rz(i) a u t o c o r r e l a t i o n f u n c t i o n of z* range with c o n s t a n t o u t f l o w equal to C and for a period o f N units of time
96 NOTATION (continued)
s(t), s*(t)
surplus at the end o f time unit t m a x i m u m S(t) during a period o f N units of time m i n i m u m S(t) during a period of N units o f time standard error o f time integer variable variance of real variables inflow time-series inflow time-series for year t shortage frequency shortage frequency for period i expected a, a * total deficit total deficit during period i expected ~,~* positive constant skewness coefficient outflow deficit during year t Z* mean and Z mean Z* variance and Z variance estimate of u, a
s} s~ SE(.) t VAR[. ] X,Y Z, Z* Z(t), Z*(t) 0~ Ot ~ Oti, Ot~
#,#* 5,h A(t) 0 2 * ^
a 2
(1) Inflow time-series: z = (z(t),
1 < t < +oo ).
(2)
where Z ( t ) is the inflow received b y the reservoir R during year t. Obviously, the nonexistence of losses due to inadequate distribution of inflow during each year has to be assumed. This is a general hypothesis implied b y any analysis of stock theory (Hadley and Whitin, 1963). (2) Reservoir effective capacity L [L is the reservoir capacity that can be used to store inflows (L > 0)]. (3) Firm outflow C. (4) Reservoir stock at the beginning of time unit t, B(t). (5) Range for a period of duration N and with outflow C, R N ( C ) . Defining: t S ( t ) = S(1) + ~ [Z(i) - C] i=l
with t/> 1
(3)
+
S g and S N are the maximum and minimum values of S ( t ) with t = 1,...,Nand:
RN(C)
=
+ sN
- s~
(4)
if infinite capacity is assumed (6) Shortage period, M. For a period of N units, M is the number of time units during which C cannot be satisfied. (7) Available affluency during unit t, A ( t ) . For any t > 1: A(t) = B(t) + Z(t)
(5)
97
(8) Outflow deficit A(t). For any t i> 1 one has: A(t) = 0
if
(6)
A ( t ) >i C
A(t) = C - A ( t )
if
(7)
A(t) < C
because the stock theory hypothesis (Hadley and Whitin, 1963) is assumed. (9) Reservoir operation rules. The risk of failure to satisfy 0(t) = 8 Z o ( t ) + (1 - 8). Co is the same as t h a t obtained for a reservoir with Z ( t ) = (1 - 8 ) ' Z o ( t ) and a firm outflow C = (1 - 8)C0 as shown in eq.1. Therefore, only this last case will be studied here. The following reservoir operation rules are adopted (a) If:
[A(t) - C] < O,
O(t) = A ( t )
and
A(t) = C - A ( t )
(S)
because C cannot be satisfied in year t and therefore the outflow during t[0(t)] is made equal to A ( t ) . (b) If: 0 < [A(t) - C] < L
0(t) = C
and
A(t) = 0
(9)
because the risk is m i n i m u m with this rule. (c) If: [A(t) - C] > L
O(t) = A ( t ) - L
and
A(t) = 0
(10)
because an additional discharge has to occur: [A(t) - (C + L)]. Therefore: 0(t) = C + A ( t ) - (C + L )
(11)
(10) Measures of risk associated with C. This is a debatable issue and several ways of measuring the risk can be proposed for a period with duration N: (a) Shortage frequency ~, and total deficit/3. Measures of how often (a) and of how much (/3) shortages occur are given by: =M
and
N ~ = ~ A(t)
(12), (13)
t=l
The probability P of C not being satisfied can be determined by: P = lim ( M / N ) (b) Statistical distributions of X = {3/N, of Y = a / N and of A ( t ) for any unit of time t (t >I 1). In this paper the measures studied are a, ~ and the A ( t ) distribution, because X and Y can be easily determined if a and/3 are known. Therefore the following quantities are estimated: (i) Cumulative distribution function of A ( t ) for any unit t (t >1 1), F ( A ) _ [This function is estimated through the mean values of its seven quantiles Q and their standard errors SE(Q).] (ii) Expected values of a(~) and of/3(~) for N = 1000 and their standard errors SE(~) and SE(j~).
98
SIMILARITY ANALYSIS O n a reservoir R, similarity analysis is applied to this problem to investigate h o w ~, ~ and F(A) depend on the adopted values for firm outflow C, effective capacity L, initial stock B(1), inflow time-series m e a n p and variance a 2 . Obviously, C, L and B(1) cannot be negative, and/~ and a 2 must have values in which the probability of Z(t) <~ 0 with t >I 1 has to be either zero or negligible for the assumed inflow time-series model because Z(t) < 0 has no physical meaning. For this purpose, a standard reservoir R * is defined by C*, L*, B*(1), p* and o *2 as follows:
(15)
Z(t) = a + bZ*(t)
with: b = a/a*
and
a = p - ap*/a* = (pa* - ap*)/a*
(16), (17)
For the transformation shown by eq.15, Z ( t ) a n d Z * ( t ) have the same autocorrelation function. If: L = bL*
(18)
C = a + bC*
(19)
S(1) = b .B*(1)
(20)
one has: S(t) = b .S*(t)
and
(21), (22)
B(t) = b .B*(t)
Thus, the following equivalence relations are obtained: a = ~*
(23)
= ~*
(24) (25)
A ( t ) = a + b . A *(t)
with: a = (pa*
-
op*)/a* = P
-
(l~*/a*)'a
and
b = a/a*
(26), (27)
Then, R and R * are named "equivalent reservoirs". Usually, C, L and B(1) are defined as fractions of g : C = k,u
(28)
L = k2p
(29)
B(1) = k3u
(30)
99
and the equivalent reservoir R * is represented by: (kl - 1)p
C* =
alo*
+ p* = ~* +
p * ( k l - 1)
(31)
C~
k2 k2o* ~--o/o* Cv
(32)
L*= --
k3 -
-
o*
B*(1) = olo* p"
~_.
m
k3 Cv
(33)
for specified values of p* and o*. Without loss of generality p* and o* can be made equal to 0 and 1, respectively, and the following equivalence equations are obtained: C* = (k, - 1)/Cv
(34)
L* = k2/Cv
(35)
B*(1) = k3/Cv
(36)
Here, a, ~ and F ( A ) can be easily derived from values of a*, ~* and F ( A * ) because: = a*
(37)
= o~*
(38) (39)
A = ~ + oA*
Therefore, if a small number of combinations of (kl - 1)/Cv, k2/Cv and k3/Cv values is studied the results obtained can be generalized to a much larger number of cases defined by p, o, L, C and B(1). If the simulated period N is sufficiently large, a, ~, F ( A ) are n o t sensitive functions of B(1) because B(t) is independent of B(1) for t >I to (to is the first B(t) zero). EXPERIMENTAL RESULTS
The inflows process {Z(t), 1 ~< t < o¢ ) is assumed to be a stationary sequence of autocorrelated normal (or lognormal) variables with:
p(k) = k lkl
0 ~< X < 1
(40)
because an exponential function fits the estimated autocorrelation (Yevjevich, 1973) of annual r u n o f f for m a n y rivers [rz(0]. Usually ~z(1) :~ 0.5 and for several rivers the annual r u n o f f is serially independent (Quintela, 1967). The skewness coefficient of the annual runoff: A
E [ Z ( t ) - E { Z ( t ) } ]3 %/~ =
U3
o
100
is usually within the interval (0; 1.6) and 0.7 is quite a c o m m o n value for some regions, such as Portugal [the hypothesis of x/7 = 0.7 is not rejected for most Portuguese rivers (Quintela, 1967) such as the rivers SabSr at Quinta das Laranjeiras, Z~zere at Cabril, Odivelas at Odivelas, Corona at Moinho do Bravo, Rabag~o at Venda Nova, etc.] Therefore the six models were considered (see Table I). For each model 18 cases were studied, using TABLEI Model
Inflow distribution
~
Model
Inflow distribution
A B C
normal normal log-normal
0 0. 5
D
log-normal ~/~/ = 0. 7
0.5
0
E
log-normal
0
~/-y= o 7
~/~, = 1 6 F
log-normal
0.5
x/~/ = 1 . 6
__-xEl/ 480
440
1 l | l
400 360 320 \
~ = 0
280 \ ;\
240
".o °:. 200 k 160
"..?%. "..
\:,, "~.
..........
V~-~= 0 7
. . . .
~/-~'~: 1.6
",~'..... ,,. "~..?-,,
120
'°°.oO°°o 80 40
®
C
1/2
t
C = -0.1
: -0.2
2
4
16
L
101
520%
\ •
%
\ ~o~
~... \
380"
"~ ~ °. %
%
,\\
320-
- -
%%
VT--o~ 0.7
%. %'%
240-
°..
200-
,~
160-
) 2080-
":Z::2".~... ........
C=-0.1
-':0~
.... ~%"::'"::""".:C= -0.2
®
112 1
2
,~
~6
Fig.1. Estimated ~* as a function of L* and C* for: ~_ Uncorrelated (k = 0), normal (~/~ = 0) and log-normal inflows (x/~ = 0.7; 1.6). B. Autocorrelated (X = 0.5), normal (x/~ = 0) and log-normal inflows (x/r~ = 0.7; 1.6).
C * = p * - k , a * a n d L * = k ~ o * with p* = 0, o* = 1, C* = 0, - 0 . 1 , - 0 . 2 and L* = 1~
1, 2, 4, 8, 16. For each case 10 periods o f 1 0 0 0 yr were simulated and the Q* 's ~* and ~*, as well as their standard errors, were estimated, using: lO
1
I
lO
~* = ~ Z~ Q* 10 i=l
qV2
and SE(Q*)=L 10~:-~ ~j
i=1
~* - - I0
_ _
(41), ¢42)
10
and
i=1
~* = ~
10
(43), ( 4 4 )
102 SE(~*) =
I10 i=1
l 1/2
-
10(10 - 1)
i 10 SE(~,) =
and
i 1/2
i=1 10(10
-
1) (45), (46)
where a*, ~* and Q* are the estimated a*, ~* and Q , for period i(i = 1 , . . . , 10). From eqs. 34 and 35 one may conclude that these results are applicable to any case with: kl = 1 + C*'Cv
and
k2 = L * . C v
(47), (48)
for a pair studied (L*, C*). In Figs.l--4
~*, ~* and
Q* are shown as functions of L* (reservoir capacity),
300
250
200 A = 0
150
100
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Fig.4. Estimated mean quantiles Q* with C* = 0; 0.1; - 0 . 2 for: A. Autocorrelated (k = 0.5), normal inflows (~/~ = 0). B. Autocorrelated (k ffi 0.5), log-normal inflows (x/~ = 0.7 ). C. Autocorrelated (k = 0.5), log-normal i n f l o w s ( ~ ffi 1.6).
0.005 -
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Annua.( 0
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600
800
1000
1200
1 4 0 0 1 6 0 0 1800
2000
2200
2400
2600
runoff |mm)
Fig. 5. Cumulative normal distribution of annual runoff of River Rabag~o at Venda Nova (1940/41--1962/63). CONCLUSIONS
The behaviour of a stochastic reservoir with autocorrelated and skew inflows was studied b y simulation and the estimated results were generalized by similarity analysis. General conclusions on the estimated shortage frequency (~/1000), the overall o u t f l o w deficit for a period of N units of time {~ with N =_1000) and the cumulative distribution function of the available affluency F (Q ) are shown in Figs.l--4 for several levels of firm outflow and several values of the coefficients of skewness ~/7 and autocorrelation k. From these graphs several conclusions can be drawn: (1) For uncorrelated inflows, ~* and ~* are only slightly sensitive to the skewness of the inflows. However, with k = 0.5, x/~/is a very significant parameter. (2) Autocorrelation is important in estimating ~* and ~*: a* and ~* are significantly increased b y a positive k if inflows follow a skew distribution,
111
i: z
{i)
10 09 0.8 0.7 06 0.5
•.~~).95(i)
0/, 0.3 0.2 0.1 0
I
I
t
I
~
-0.1
1
2
3
4
5
) i
-02 -0,3 -04 -05 -06
,'~ 095
(i)
-0.7 -08 -0.9 -1.0
Fig. 6. Estimated correlogram [~z (i)] for annual runoff of River Rabag~o at Venda Nova (1940/41--1962/63).
(3) d~*/dL*, d~*/dL* < 0 and d2~*/dL .2, d2~*/dL .2 i> 0. (4) A(t) is not normally distributed except when L* = 1/i, 1, and x/7 = 0. A practical example of the application of these results to a Portuguese multiannual reservoir at Venda Nova (River Rabag~o) is also considered.
REFERENCES Ali Khan, M.S. and Gani, J., 1968. Infinite dams with inputs forming a Markov chain. J. Appl. Probab., 5: 72--83. Anderson, R.L., 1941. Distribution of the serial correlation coefficient. Ann. Math. Stat., 13: 1--13. Anis, A.A. and Lloyd, E.H., 1953. On the range of partial sums of a finite number of independent normal variables. Biometrika, 40: 35--42. Buras, N., 1972. Scientific Allocation of Water Resources. American Elsevier, New York, N.Y. Collings, P.S., 1975. Dams with autoregressive inputs. J. Appl. Probab., 12: 541--553. Feller, W., 1951. The asymptotic distribution of the range of series of independent random variables. Ann. Math. Stat., 22: 427--432. Hadley, G. and Whitin, T.M., 1963. Analysis of Inventory Systems. Prentice-Hall, Englewood Cliffs, N.J.
112
Lloyd, E.H., 1970. Transient and asymptotic behaviour of reservoirs with non-persistent seasonal inflows. J. Hydrol., 10: 243--258. Lloyd, E.H., 1977. Reservoirs with seasonally varying Markovian inflows, and their first passage times. Int. Inst. Appl. Syst. Anal., Laxenburg. Lloyd, E.H. and Odoom, S., 1965. A note on the equilibrium distribution of levels in a semi-finite reservoir subject to Markovian inputs and unit withdrawals. J. Appl. Probab., 2: 215--222. Luthra, S. and Arora, S.R., 1976. Optimal design of single reservoir systems using 6 release policy. Water Resour. Res., 12: 606--612. Melentijevich, M.J., 1965. The analysis of range with output linearly dependent upon storage. Colo. State Univ., F o r t Collins, Colo. Hydrol. Pap., No.11. Moran, P.A.P., 1959. The Theory of Storage. Methuen, London. Phatarfod, R.M, and Mardia, K.V., 1973. Some results for dams with Markovian inputs. J. Appl. Probab., 10: 166--180. Prabhu, N.U., 1958. Some solutions for the finite dam. Ann. Math. Stat., 29: 1234--1243. Pyke, T. and Phatarfod, R.M., 1976. Some exact results for dams with Markovian inputs. J. Appl. Probab., 13: 329--337. Quintela, A., 1967. Recursos de ~guas superficiais em Portugal continental. In: A. Quintela (Editor), Surface Water Resources in Portugal, Lisbon. Revelle, C. and Gundelach, J., 1975. Linear decision rule in reservoir management and design. 4. A rule that minimizes output variance. Water Resour. Res., 11: 197--203. Rippl, W., 1883. The capacity of storage reservoirs for water supply. Proc. Inst. Civ. Eng., 71: 270--278. Yeo, G.F., 1975. A finite dam with variable release rate. J. Appl. Probab., 12: 205-211. Yevjevich, V., 1965. The application of surplus, deficit and range in hydrology, Colo. State Univ., F o r t Collins, Hydrol. Pap. No.10. Yevjevich, V., 1972. Stochastic processes in hydrology. Water Resour. Publ., F o r t Collins, Colo.