1,
2,
3 stand for their
averages; s1, s2, s3 stand
for their mean square errors.
Xiaohua Liu
Benzhong Xu
Shandong University
PO Box 250061,73 Jingshi Road, Jinan, Shandong, China
Tel.：0531-2618017, E-mail：lbxbz@jn-public.sd.cninfo.net
Abstract: Taking one year as its period, the continuos runoff process can be discretization into processes in a series of months (or ten days), which called month time series. The runoff process will accord with Markov chain if a dependent relationship exists among adjacent month time series. When the conditional probability is got, transferable probability matrix of the adjacent time intervals could be calculated, and the corresponding conditional flow can be obtained.
Keywords: reservoir, Markov chain, conditional probability, transferable probability
Reservoir multipurpose optimum operation is affected by many factors, such as reservoir’s water coming, reservoir’s feature, water using behavior (for industry, livelihood and irrigation) as well as mechanical and electrical features of power station etc. Among them, reservoir runoff is primary; water using for irrigation is secondary. The description accuracy for runoff will directly influence the benefit of reservoir optimum operation.
Researches on runoff should be carried out from various aspects according to the features of runoff. As it’s periodicity, the river runoff in China can be considered as a continuous stochastic process which takes a year as it’s period. As it’s variance, the discharge in a river varies from time to time. That is to say, if the reservoir runoff is observed at a fixed time, it will be a random variable. As it’s un-repeating, for a fixed observation time, it cannot reoccur completely and the processes are not the same too. It is a function of time. As it’s ergodicity, at an enough long observation time, various discharges can occur within the whole scope of discharge variation.
In order to use mathematical method to simplify the problem, for convenience, the continuous runoff stochastic process can be disperse by times and states, and changed into another corresponding variable with an certain kind of distribution. The runoff description method of Mu Yu reservoir by Markov chain is discussed in this paper.
Markov process can be described as follows: if one state of the random variable at the time of t with an expression of X (t)=i is known, the probability of it’s another state at the time of t+1 with an expression of X (t+1) = j will not vary with the increasing information of the process before time t. That means the probability of the state of the process at the time of t +1 with the expression of X (t+1) = j has relations only to the state at it’s just last time t, and has no any relation to the state before time t. Thus, between X(t) and X(t+1), X(t-1) and X(t), … , X(2) and X(1), there seems exist a chain to link them up. For this reason, Markov process is also called Markov chain.
X(N), ¬ … , ¬X(t+1), ¬ X(t), ¬ X(t-1), ¬ … ¬ X(3), ¬ X(2), ¬ X(1)
If the water coming series of a reservoir accords with this pattern, the reservoir runoff can be described by Markov process.
Taking the runoff relations of water
coming of a reservoir at three adjacent time intervals as an example, the
discrimination for simple Markov process is discussed as follows. X1,
X2, X3 stand for the runoff at the three time intervals
respectively;
1,
2,
3 stand for their
averages; s1, s2, s3 stand
for their mean square errors.
a3=a3
(X1, X2) =E (X3|X1, X2)
(1)
a3
stands for the conditional average value of X3 corresponding to one
certain value of X1, X2. In another word, the linear
regression equation of X3 relying on X1, X2 is:
(2)
P11, P31, P32 are the determinant complement minor of the correlation coefficient matrix P.
Let
Then the linear regression equation of X3 relying on X1, X2 can be described as:
a3 -
3 = b31 (
X1 -
1 ) + b32 (
X2 -
2 )
(3)
According to this relation, the correlation status of X1, X2 to X3 can be shown by Fig.1 (a). In Fig.1 (b), the influence of X1 on X3 is carried out along two routes. The upper route stands for the direct influence. That is, no matter what values X2 takes (or ignore the effect of X2), the correlation of X1 to X3 is termed as deviation correlation. The correlation coefficient (r13 - r12 · r23) can be used to express their direct interrelated degree. Obviously, it is different from the correlation coefficient r13, which is calculated directly from the original data of X1 and X3 . The lower route stands for indirect influence. That is, because of the correlation existence among adjacent time intervals of X1 to X2, X2 to X3 , the values of X1 affect the values of X2 , and further affect X3. For example, if the value of X1 is pretty large, because of the correlation between X1 and X2, the possibility of X2 taking a large value will be higher. And because of the correlation between X2 and X3 , if X2 takes a large value, the possibility of X3 taking a large value will also be higher. This kind of interrelation is transmitted by the median variable X2, not directly from X1. So it is called indirect correlation or transmission correlation. This part of correlation can be expressed by r12·r23 which is derived as follows.
Under the case of simple linear interrelation, there is:
(4)
for X2 and X1 , there further is:
(5)
Where x3 and x2 stands for stochastic variables for regression error. Put Equation (5) into Equation (4), we can get
(6)
The correlation coefficient between X1 and X3 for the part of indirect correlation r1-3 is:
(7)
For (X1 -
1 )2 = s1 2 , x3 and
x 2 are relatively independent random
variables to X1, so E [x 2(X1 -
1)] = E [x 3(X1 -
1)] = 0. Put these relations into Equation (7), we can get r1-3
= r12 · r23.
Obviously, the direct correlation coefficient can be gotten by subtracting the indirect correlation coefficient (or transmitted influence) from the correlation coefficient calculated directly from the data.
That is ( r13 - r12·r23 ). If
r13 - r12·r23 = 0 (8)
The above equation means that no direct influence can be deemed exists. Now b31 = 0, b32 = b = r23 (s3 / s2). The relation shows in Fig.1 (a) can be simplified to that showed in Fig.1(c). That is to say, the influence of X1 , X2 on X3 can be comprehensively shown by the influence of X2 on its adjacent X3.
Therefore, it is obviously that the formula r13 - r12 · r23 = 0 can be taken as the discriminate for simple Markov process. By this discriminate equation, weather a reservoir runoff series can be described by Markov process should be decided.
In the case of linear interrelation among different time interval runoffs, the deviated value of the latter time interval runoff to the regression equation, that is the error distribution, can be treated by two methods. One is rigid interrelation, and another is elastic interrelation. The former is simple. For example, interrelating Xn , Xn+1 with linear interrelation, the regression equation is:
(9)
x is the error of the interrelation equation. The random variable Xn consists two parts: settled part and random part x. If the probability distribution function f (x) of random x with 0 average value and sx mean square error is known, the above equation can be changed into:
(10)
two sides
square of the above equation, then carry out a mathematical exception to it. With an attention that (Xn+1 -
n+1 ) is independent to x, the following equations can be
derived:
(11)
(12)
(13)
is a constant, which shows that the probability distribution of the error x has nothing to do with Xn+1.
In another word, when Xn+1varies, the shape of the distribution curve
of the error x doesn’t change. This simple kind of
interrelation is called rigid interrelation. The distribution of the error x belongs to a normal distribution.
The other is elastic interrelation. In this kind of interrelation, the mean square deviation of the error x has relation to whether its former time interval runoff is large or small. Namely:
(14)
where
n ·Xn+1 is the conditional mean value of Xn on Xn+1.
The existence of this relation can be explained by the physical and random
nature of rainfall and runoff. The relations between the means of Xn
and Xn+1 when showed by regression equation depend on the ratio of
the rainfall means, and the consuming status of water stored in the drainage
area.
For elastic interrelation, the distribution of the error usually takes the type of P- III distribution.
Generally, when deciding the distributions of errors for normal runoff values, the linear regression and rigid interrelation are often used. But according to the physical features of hydrology, the elastic interrelation is more suitable for hydrology analysis. So the elastic interrelation was used in Mu Yu reservoir.
If a runoff can be described by Markov process, the transition of the runoff states between two adjacent time intervals n and n+1 will be studied as follows.
First, the infinite runoff states need to be expressed through definite runoff states. The runoff in each time interval is divided into M states. By using above equations about conditional probability analysis, the transition probabilities of the runoff states between two time intervals can be calculated.
Pnij ( n = 1, 2, …, N i, j = 1, 2, …, M)
It stands for the probabilities of run-off which takes state j at time interval n when the runoff takes state i at time interval n+1.
( n = 1,
2, …, N)
(15)
It means the transition probability of runoff states from time interval n to time interval n+1. Obviously, the sum of the elements in each row will be 1.
( n = 1, 2, …,
M)
(16)
Using P(n) i to stand for the probabilities of the runoff taking state i in time interval n, P(n) , which stands for a row rector making up by the elements of P(n) i , can be expressed as follows:
P(n) = (P(n) 1 , P(n) 2 ,…, P(n) m)
According to the all-probability law, P(n) i can be decided:
(
n = 1, 2, …, N)
(17)
Commonly, Pn = Pn+1 ·Pn ( n = 1, 2, …, N). This equation shows the dependent relationship of the runoff state at time interval n+1 to that at time interval n. By their recursion relations, we can get:
P(n) = P(n+2)·P(n+1)·Pn (18)
P(n) = P(n+3)·P(n+2)·P(n+1)·Pn
(19)
P(n) = P(n+4)·P(n+3)·P(n+2)·P(n+1)·Pn
(20)
These equations show the probability dependent relationship among runoff states at time interval 1, 2, and 3. They also show that the dependent relationship becomes more and more week with the apart time interval increase.
In reservoir’s operation process, the runoff flowing into the reservoir at a certain time interval is known. The runoff at the next or the confront time interval can be forecasted by means of short term forecasting. If the time interval is short enough, the forecast runoff can be deemed as accurate and a known date. The long term forecasting is only an approximate tendency estimation in which there exist some errors that cannot be ignored. If adequate forecasting data of every calendar year and their corresponding statistical errors are available, by using the above method, they can be counted into the transition probabilities of runoff states. And these transition probabilities can be used in reservoir optimum operation.
The runoff of reservoir Mu Yu is analyzed in this paper. The calculation is programmed by FORTRAN 77 Language and has run on VAX-8350 computer. The program can deal with runoff correlation analysis and conditional discharge calculation of adjacent time intervals. The calculated conditional discharge is memorized in the computer as data libraries, which can be used by reservoir multi-purpose optimum operation.
The calculation process is explained as follows:
(1) Input the real observed runoff data QI, the frequency value YY (27) of P-III curve, deviation coefficient XX (14), and variation coefficient WW (14.27);
(2) Let Q II = QIT, bring into matrix Q Ip whose discharges precede that of matrix Q I by one time interval. Then the conditional transition probability discharge is F (Q II / Q Ip);
(3) Calculate the average discharge of each interval Q MO, variation coefficient Cvc, interrelation coefficient R1 between adjacent time intervals and interrelation coefficient R between apart time intervals;
(4)
Arrange the discharge Q Ip in big or small, find the corresponding
discharge matrix Q II;
(5) From matrix Q II, calculate the conditional average discharge and conditional deviation coefficient for each subintervals of that time interval. At last the probability distribution curve of conditional discharge for each subinterval can be calculated.
upper line:
upper line: r13 - r12·r23
b = r23 (s3 /
s2)
lower line:
lower line: r12·r23
(a) (b) (c)
Fig. 1 Relations Among X1, X2
and X3
References
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Miao Zhao You, Yang Wen Kong, Probability Theory
and Mathematical Statistics, Press of Beijing University of Techers,
1988,125~140.