STOCHASTIC RESERVOIR THEORY: AN OUTLINE
O F THE STATE O F THE ART UNDERSTOOD
BY A P P L I E D P R O B A B I L I S T S
A.A. A n i s E.H. L l o y d
September 1975
R e s e a r c h R e p o r t s a r e p u b l i c a t i o n s r e p o r t i n g o n t h e w o r k of t h e a u t h o r s . A n y v i e w s o r c o n c l u s i o n s a r e t h o s e of t h e a u t h o r s , and do n o t n e c e s s a r i l y r e f l e c t t h o s e of I I A S A .
Stochastic Reservoir Theory: An Outline of the State of the Art as Understood
by Applied Probabilists 1 A.A. Anis and E.H. Lloyd 2 3
1. Introduction
The relation between water-storage systems in the real world and the system encompassed by the basic theory (due to Moran [15,16]may be described as follows in Table 1.
~t first sight this model would appear to rest on simpli- fications that are so drastic as to render it devoid of
mathematical interest or of practical potentialities. In fact, however, this is not the case. As far as purely mathematical developments are concerned, the interest that these simpli- fications have aroused is amply illustrated by the so-called
"Dam Theory," in which the basic Moran theory was transformed largely by Moran [I 41 himself and by D. G. Kendall [ I 01 into a sophisticated corpus of pure mathematics dealing with con- tinuous-state, continuous-time stochastic processes. (For comprehensive contemporary surveys see Gani [61, Moran [151, and Prabha [ 22
I .
It is true that the "reservoirs" in Dam Theory are of infinite capacity, that the release occurs at unit rate, and that the "water" involved posesses no inertia and no correla- tion structure of any kind, so that the inflow rate at t
+
Bt is statistically independent of the inflow rate at time t, however small the increment Bt. This lack of realism does not detract from the beauty of the mathematics involved, but it does limit the possibilities of applying the theory to the real world..- -
his
report is an expanded version of a seminar pre- sented by the authors at IIASA, in i4ay 1975.in
Shams University, Cairo, E r ~ y ~ t . 3~niversitY of Lancas ter, UK.- 2 -
Table 1.
A second avenue of development, which is directed specif- ically towards engineering applicability, is also possible. This development views the basic Moran theory as an extremely inge- nious abstraction from reality, so constructed as to allow
modifications which are capable of removing practically all the restraints listed in Table 1 above. In particular, seasonality in inflow and outflow processes may be accommodated; the errors involved in approximating continuity by discreteness may be re- duced to an acceptable level by working in appropriate units;
Basic Moran theory
Single reservoir
Discrete in state and time
Non-seasonal
Independent incre- ments
Ignored
Constant
Discrete in state and time
Constant rate of release
Storage mechanism:
Inflow process
Losses
Capacity
Release procedure
Real World Situation System of interconnected reservoirs
Continuous in state and time
Seasonal
Autocorrelated
Evaporation, seepage
Time-dependent due to silting
Continuous in state and time
Related to past, current and predicted future contents
flexible release rules may be built in; and, most important, the original requirement of mutual independence in the
sequence of inflows may be abandoned, and realistic auto-
correlation structures incorporated by using Markovian approx- imations of arbitrary complexity (Kaczmarek [8]
,
Lloyd [I 21 ).
What this development of the theory produces is the
probabilistic structure of the sequence of storage levels and overspills in the reservoir
--
both structures being obtained in terms of the size of the reservoir, the inflow characteris- tics and the release policy. (See for example Odoom andLloyd [1 91
,
Lloyd [I 21,
Anis and El-Naggar [2],
Gani [51,
Ali Khan [1
1 ,
Anis and Lloyd [3],
Phatarfod and Mardia [21] ;and review articles by Gani [7] and Lloyd [1 31
.
) Thus theeffect of varying the release policy may be determined, with a view to optimizing various performance characteristics.
The theory can therefore be said to have reached a fairly satisfactory form, and regarded as a probabilistic model.
The same cannot perhaps be said of the statistical estimation and modelling procedures needed for the practical application of the theory, and since the usefulness of the theory must depend on the reliability of the numerical estimates of the probability distribution and autocorrelation structure of the inflow process, it is the authors' opinion that particular attention ought now to be concentrated on these statistical problems. This is discussed further in Section 9.
2. The Basic Moran Model: Inde~endent Inflows
In this section we describe the simplest version of the model, as outlined by the following diagram of the reservoir
(see Figure 1). The llscheduling" or "programming" required by the fact that continuous time is being approximated by discrete time is shown in Figure 2.
Available Inflows {xt}(1ID) Additional
"intermedi- ate Storage"
W
Effective Capacity
!
Contents {zt}Desired release Dt
0
= w Actual release Dt = Wt-
< wFigure 1.
time epochs
Figure 2.
Here Xg denotes the acceptable part of the inflow, in the light of the restraints imposed by the finite capacity of the reservoir. Similarly Dg ( = W ) is the feasible part of
t
the desired outflow w, taking into account the fact that the reservoir may contain insufficient water to meet the whole of the demand.
In the Moran sequencing illustrated above, the outflow W is supposed to occur after the inflow has been completed.
t
The inflow quantities Xt = 0.1
,...,
the outflow quantities Wt = 0 ~ 1 , ...
,w, and the storage quantities Zt = 0 , l,...
,care all quantized, all being expressed as integer multiples of a common unit. The "intermediate storage" indicated in the figure is required for the operation of the program.
The inflow process {X is supposed to be IID: that is, t
the random variables
are supposed to be mutually
-
independent, and-
identically - dis- tributed.Taking account of the finite size of the reservoir and of the sequencing imposed by the "program" we may formulate the following stochastic difference equation for the storage process
{zt1:
which we shall abbreviate when necessary in the form
Here the constant, w, represents the desired outflow Dt.
The actual outflow Wt is given by
s o t h a t
The a c t u a l l y a c c e p t e d i n f l o w i n t o t h e r e s e r v o i r d u r i n g ( t , t
+
I ) , a s d i s t i n c t from t h e a v a i l a b l e i n f l o w X t , i sand t h e q u a n t i t y l o s t t h r o u g h o v e r f l o w i s
I t i s worth p o i n t i n g o u t t h a t , w h i l e p r e s e n t a t i o n s o f t h i s t h e o r y o f t e n c o n c e n t r a t e on t h e d e t e r m i n a t i o n of t h e s t o r a g e p r o c e s s { z t l , b o t h t h e o u t f l o w p r o c e s s {wtl and t h e s p i l l a g e p r o c e s s {St} a r e p r o b a b l y more i m p o r t a n t i n a p p l i - c a t i o n s , p a r t i c u l a r l y when i t i s d e s i r e d t o o p t i m i z e t h e r e l e a s e p o l i c y , s i n c e t h e f u n c t i o n t o b e o p t i m i z e d w i l l nor- m a l l y depend d i r e c t l y on t h e s e two p r o c e s s e s .
(An a l t e r n a t i v e "programming,11 which- may b e d e s c r i b e d a s a n e t i n f l o w scheme, and which d i s p e n s e s w i t h t h e need t o
-
i n t r o d u c e a n i n t e r m e d i a t e s t o r a g e zone, i s shown i n F i g u r e 3.
Here t h e t o t a l a c c e p t a b l e i n f l o w X z t h a t o c c u r s d u r i n g ( t , t
+
1 ) and t h e t o t a l o u t f l o w Wt a r e assumed t o b e s p r e a d o v e r t h ee n t i r e i n t e r v a l ( w i t h s u i t a b l e m o d i f i c a t i o n s when a boundary i s r e a c h e d ) b o t h t a k i n g p l a c e a t c o n s t a n t r a t e d u r i n g t h a t i n t e r v a l , s o t h a t t h e y combine t o form a s i n g l e " i n f l o w " of magnitude Xt
-
W t , t h i s b e i n g n e g a t i v e i f Xt < W t . Thes t o c h a s t i c d i f f e r e n c e e q u a t i o n f o r { z t l i s t h e n
which coincides with the equation (1) obtained for Moran's own program. Similarly the actual outflow Wt and the spillage S are the same as under Moran's programming.)
t
Lt Lt+l
f)
timeI I
Figure 3.
3 . The Storage Process
{ztI
for the Basic Moran ModelAS before, we take
{xtI
to be an IID process, and D+ = w.Then
{zt]
is a lag-1 Markov Chain. For. since (by (2))-
where
the information relating to Zt,Zt-l,..., is suppressable on account of the assumed structure of
{xtI.
Since (4) depends on s but does not depend on sl,s",,.., the result follows. Thuswhere
and
q ( r , s ) = P ( Z t + , = r ( z t = s )
,
f o r r , s = 0.1,...,
c.
I n a n o b v i o u s m a t r i x n o t a t i o n t h i s may b e w r i t t e n
5
-
- t + l - Q C t t = 0,1,...
whence
Thus t h e d i s t r i b u t i o n v e c t o r
L~
o f s t o r a g e a t t i m e t i s d e t e r m i n e d i n t e r m s o f t h e i n i t i a l c o n d i t i o n s 5-0 ' I n a l l
" r e a l i s t i c " s i t u a t i o n s t h e t r a n s i t i o n m a t r i x
-
Q may b e assumed t o be e r g o d i c , whence, f o r s u f f i c i e n t l y l a r g e v a l u e s of t , 5 = 5 where-t
-
5 b e i n g t h e " e q u i l i b r i u m d i s t r i b u t i o n " v e c t o r which i s t h e
-
u n i q u e p o s i t i v e n o r m a l i z e d s o l u t i o n of t h e honogeneous l i n e a r a l g e b r a i c system
The m a t r i x
-
Q , o f o r d e r ( c+
1 ) x ( c+
1 ),
h a s a s i t s ( r , s ) e l e m e n tand f o r a n y g i v e n f u n c t i o n h ( * ) , t h i s i s e a s i l y o b t a i n e d i n terms o f t h e i n £ low d i s t r i b u t i o n P (Xt = j ) = f ( j )
,
s a y , j = O , l , . . . , L . 3A s a s i m p l e example, t a k i n g w = 1 , w e h a v e :
( f o r r = 1 , 2 , . .
.
, c ) and4 . The Y i e l d P r o c e s s i n t h e B a s i c Iloran PIodel
The y i e l d i s t h e a c t u a l q u a n t i t y r e l e a s e d from t h e r e s e r v o i r , d e f i n e d by ( 2 ) a s
Wt = min ( Z t
+
X t , w )Because of a) the "lumping" of Z-states implied by this function, and b) the addition process Zt
+
Xt, the yield process {W is not Markovian. For most purposes it is bestt -
studied as a function defined on the llarkov Chain {zt), with conditional probabilities
This is, for each r and s, a well-defined function of the distribution of Xt, which allows us to evaluate (for example) the expected yield at time t as
For other purposes it may be convenient to use the fact that the pair {Zt,Xt} forms a bivariate lag-1 Markov Chain.
5. The Basic Moran Model with Flexible Release Policy
If one modifies the release policy Dt so that, instead of being a fixed constant w, Dt is, for example, a function of the current values of Z and of X--say a monotone non- decreasing function of Z
+
Xt--the effect of this on thet
theory outlined in Section 5 is merely to modify the transi- tion matrix - Q, without altering the Markovian structure of
{zt}.
Equation (I), whether under Moran programming or net- inflow programming, is replaced byZt+l = min (zt
+
Xt,c+
Dt)-
min (Zt + XttDt) (1 2)say. This still holds good, and Zt maintains its simple
Markovian character, if Dt also depends on some additional random variable Yt which may be correlated with Xt, provided that the sequence Yt consists of mutually independent elements.
The actual outflow Wt is given by
the analogue of (2), and the spillage becomes
st
= max (zt+
Xt-
Dt-
c,O).
(1 4)If for example Dt is defined by the following Table 2:
Table 2.
we may construct Table 3 with entries such as the following (for, say, c = 8) as found in Table 3 below. Row (a) indicates a situation giving no spillage, (b) one where five units
are spilled, (c) one where several combinations of Z and Xt t
lead to the same value of Zt+l, in which case we have
whereas q(O,O) = f(O), a single term.
6. The Basic Moran Model Operating Seasonally
The effect of working with a multi-season year may be
adequately illustrated in terms of a two-season year (See Table 4):
Table 3 .
Dt Zt+l
Table 4 .
Epoch:
S t o r a g e d i s t r i b . v e c t o r s
Year t
+
2- - -
T r a n s i - t i o n m a t r i x
Year t
i
Season 0
Year t
+
1Season 1
Season 0
Season 1
Here
where Q = Q Q and z (t) is the storage distribution vector
0 1, -
at the beginning of season 1 of year t. Clearly the year-to- year storage sequence at this season is a homogeneous llarkov Chain, with transition matrix Q = QoQ1, and the preceding theory applies.
7. Reservoir Theory with Correlated Inflows: Basic Version F4oran's basic theory is applicable as a first approxima- tion, more or less without modification, to a reservoir providing year-to-year storage, in which there is a well-defined
"inflow season" with no outflow, followed by a well-defined and relatively short "outflow interval," a situation approxi- mated by the conditions on the Nile at Aswan. This approxi- mation holds only to the extent that inter-year inflow correla- tions can be neglected, which may not be totally unreasonable for a one-year time scale. The approximation becomes increas- ingly inaccurate if one reduced the working interval from a year to a quarter, or a month, or less, and it becomes essen- tial to provide a theory that allows the inflow process to have an autocorrelation structure. It was pointed out simultaneously in independent publications in 1963 by Kaczmarek [8] and by
Lloyd that this could be done by approximating the actual inflow process by a Markov Chain.
In the simplest form of this theory we may consider a discrete-state/discrete-time reservoir, with stationary inflow process ( ~ ~ 1 , where Xt is a finite homogeneous lag-1
Markov C h a i n w i t h e r g o d i c t r a n s i t i o n m a t r i x B = b s a y , w h e r e brs = P(Xt+l =
r l x t
= s ) , w i t h r , s = 0 , 1 , . . . , n( X t b e i n g assumed
-
< n f o r a l l t ) a n d t h e d i s t r i b u t i o n v e c t o r o f Xt i s-
5 , w h e r e-
5 i s t h e u n i q u e n o n - n e g a t i v e n o r m a l i z e d s o l u t i o n o f t h e homogeneous s y s t e mWith a r e l e a s e p o l i c y Dt w h i c h may b e a f u n c t i o n o f Z t ,
Z t - l , Xt a n d X t - l , t h i s s t o c h a s t i c d i f f e r e n c e e q u a t i o n f o r { z t ) i s t h e same a s i n ( 1 2 ) , w i t h t h e s u p p l e m e n t a r y i n f o r m a t i o n t h a t { x t ) i s a Markov C h a i n . I t i s e a s y t o show, b y a n a r g u m e n t e n t i r e l y a n a l o g o u s t o t h a t employed i n S e c t i o n 3 , t h a t { z t ) i s n o l o n g e r a Markov C h a i n , b u t t h e o r d s r e d p a i r { z t , x t l
-
f o r m s a b i v a r i a t e l a g - 1 ~ a r k o v C h a i n , t h a t i s , t h a t
i s i n d e p e n d e n t o f i ' , j l , i " , j " ,
...,.
The v e c t o r :w h e r e
represents the joint distribution vector of Z and Xt, and this t
is determined by a vector equation of the form
where M
-
is the relevant transition matrix, of order(c
+
1 ) (n+
1 ) x (c+
I ) (n+
1 ).
The structure of M is obtained-
from the stochastic equation for
{zt)
and the inflow transition matrix. If for example we take the simple release policyDt = 1, the equation (17) in partitioned form becomes
where the M (r,
-
s) are submatrices of order (n+
3 ) (n+
1 ),
which may most easily be defined in terms of the following representation of the inflow transition matrix B. Let
where b represents the s-column of B, that is
-S
-
and let
Then, in formal agreement with (8), we find
for r = 1,2,. ..,c and
Equation (17) has as its solution:
giving the joint distribution vector of Z and Xt in terms of t
the initial vector L, this being the analogue for Markovian inflows of the equation (5) for independent inflows. The analogue of (6) is given by the statement that, for large values of t,
T~
23
where-
~r is the "joint equilibriumdistribution" vector which is the unique normalized non-negative solution to the homogeneous system
(M-I)
-
Tr = 0.
One extracts the distribution Z from the joint distri- t
bution by using the result that
where, for given r, the terms P(Zt = r r X t = s) are elements of
t h e v e c t o r ~ ( r ) o f ( 3 6 ) , t h i s i n t u r n b e i n g a s u b v e c t o r o f t h e v e c t o r
zt
g i v e n by ( 21 ).
8 . R e s e r v o i r Theory w i t h C o r r e l a t e d I n f l o w s : E l a b o r a t i o n s The t h e o r y d e s c r i b e d i n S e c t i o n 7 r e f e r s t o a s t a t i o n a r y lag-1 Markovian i n f l o w p r o c e s s , and a c o n s t a n t - v a l u e r e l e a s e p o l i c y . The r e p l a c e m e n t of a c o n s t a n t r e l e a s e by a r e l e a s e Dt depending on Z t , X t , Z t - l and Xt-l and p o s s i b l y f u r t h e r
random e l e m e n t s Y t and Yt-l i s a c h i e v e d by s u i t a b l y m o d i f y i n g t h e m a t r i x M. T h i s d o e s n o t a l t e r t h e s t r u c t u r e o f t h e
{ Z , X t } p r o c e s s . ( A s h a s been p o i n t e d o u t by Kaczmarek [9]
t
t h e s t o r a g e p r o c e s s { z t } becomes a lag-2 Markov c h a i n p r o v i d e d t h a t t h e e q u a t i o n ( 1 2 ) c a n be s o l v e d t o g i v e a u n i q u e v a l u e o f X t f o r e a c h p a i r ( Z t , Z t + l )
.
)F u r t h e r , t h e i n t r o d u c t i o n of s e a s o n a l l y v a r y i n g i n f l o w and o u t f l o w p r o c e s s e s may be a c h i e v e d by u s i n g t h e m a t r i x p r o d u c t t e c h n i q u e d e s c r i b e d i n S e c t i o n 6. A s i n t h e c a s e o f i n d e p e n d e n t i n f l o w s , once one h a s o b t a i n e d t h e d i s t r i b u t i o n of Z t o n e may o b t a i n t h a t of t h e a c t u a l r e l e a s e Wt and t h e
s p i l l a g e S t , and u t i l i z e t h e s e i f d e s i r e d i n o p t i m i z a t i o n s t u d i e s .
We have spoken s o f a r o f a lag-1 Markovian i n f l o w .
G e n e r a l i z a t i o n s t o a m u l t i - l a g Markovian i n f l o w a r e immediate:
w i t h , f o r example, a l a g - 2 Markov i n f l o w one c o n s i d e r s t h e t h r e e - v e c t o r - q ( t ) = { Z X X
.
T h i s w i l l b e a l a g - 2t ' t ' t - 1
( t r i v a r i a t e ) Markov c h a i n . S i m i l a r l y we may accommodate a m u l t i v a r i a t e i n f l o w p r o c e s s . Suppose f o r example we have a b i v a r i a t e i n f l o w p r o c e s s {xt )
,
Xt ( 2 )1 ,
i n which t h e components a r e c r o s s - c o r r e l a t e d a s w e l l a s s e r i a l l y c o r r e l a t e d . T h i s i n f l o w p r o c e s s w i l l be s p e c i f i e d by a n a p p r o p r i a t e t r a n s i t i o n m a t r i x whose e l e m e n t s r e p r e s e n t t h e c o n d i t i o n a l p r o b a b i l i t i e su s i n g any s u i t a b l e o r d e r i n g c o n v e n t i o n , f o r example t h e f o l l o w i n g T a b l e 5 .
T a b l e 5.
v a l u e a t t i m e t + l
I n t h i s example t h e t r i p l e t { z ~ , x : ' )
,xi2)
1 would be a( t r i v a r i a t e ) lag-1 Markov c h a i n (Lloyd [ I 1 ]
,
A n i s and Lloyd [ 4 1.
x ( l )
0
1
n
9 . C l o s i n s Remarks: P e r s ~ e c t i v e s
I n t h e a u t h o r s ' o p i n i o n , t h e . p r o b a b i l i s t i c framework 0
1
.
n
0
n
-
o
1
n
Values a t t i m e t
0 1
. . .
n0 l . . . n 0 l . . . n
...
I I I
- -
0 l . . . n
- - - -
I I
I I I I
-
I I
I I
I
I
I
provided by the model described above is adequate for practical purposes, and the next steps ought to be concerned with the statistical problems of specifying families of few-parameter inflow transition matrix models and estimating their parameters.
In the case of a nonseasonal univariate lag-1 inflow process, for example, it is likely that the available information will be best adapted to estimating the inflow distribution vector p (which is of course a standard and well-understood statis- -
tical procedure) and the first few autocorrelation coefficients, say P 1 and P2* A model involving these directly would be
particularly welcome. In the case of where an exponential autocorrelation function
were thought to be appropriate, the Pegran [ 2 0 1 transition model
that is
where
and
1, r = S
cS(r,s) =
0, otherwise
satisfies these requirements. Further development along these lines, making m a discrete-state and discrete-time framework, and avoiding the difficulties inherent in the use
of transformations of the normal autoregressive model, (Moran, [ 1 7 1 ) , would be highly desirable, particularly in the case of multivariate inflows.
Acknowledgements
The authors wish to express their indebtedness to IIASA, whose guests they were whilst this report was being prepared.
-21
-
R e f e r e n c e s
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