1 Discrete Markov random process is used to describe runoff.
1. 1 Describe runoff with Markov process
In order to facilitate calculation and application, the time series is discretized (that is, divided into several periods: months), and there is a dependency between adjacent periods. The runoff relationship of three adjacent periods of reservoir inflow t 1, t2 and t3 is analyzed. X 1, X2 and X3 represent the runoff in three periods, and the correlation between them can be divided into two situations: (1) direct correlation. That is to say, no matter what the value of X2 is (or regardless of the influence of X2 value), X 1 is related to X3, which is called partial correlation, and its degree of correlation is characterized by correlation coefficient, which can be expressed as γ 13. (2) Indirect correlation. That is to say, the size of X 1 affects the size of X2, and then affects the size of X3, because there is a correlation between X 1 and the adjacent time periods of X2, X2 and X3. This correlation is transmitted through the intermediate quantity X2, which is not direct, so it is called indirect correlation.
1.2 Calculate the corresponding conditional probability.
When a year is divided into k time periods (months), the runoff of each time period is represented by the average value, which is denoted as qk (k = 1, 2,3, ..., k).
The conditional probability distribution function QK, QK-1 (as shown in figure1) and the schematic number of conditional probability distribution function fQK- 1 (qk/qk-1) can be determined by applying relevant theoretical analysis. Its conditional probability distribution is two-dimensional, and the conditional probability formula of runoff is derived by using probability theory and hydrological statistics principle.
Figure 1 Runoff in adjacent periods
When studying the correlation of runoff in adjacent periods, the correlation coefficient r and regression equation are used to obtain it.
( 1)
The interval correlation coefficient is:
(2)
Where: Q 1i, Q2i and Q3i are the measured runoff values of adjacent periods in the first year; Is the average; N is the number of years of measured runoff. The correlation of runoff in this period is expressed by autoregressive linear formula by applying linear correlation in correlation:
(3)
Where σ k and σ k- 1 are the mean square deviation of runoff in time periods Tk and Tk- 1 respectively; R 1 is the correlation coefficient between runoff in adjacent periods.
When autoregressive linear correlation is applied to runoff in adjacent periods, the deviation between runoff and regression line in interval periods is the error distribution. After comparing the correlation between rigidity and elasticity, the elastic correlation processing method is adopted, that is, skewed distribution, which is distributed according to Pearson ⅲ curve. What is the flow QPK corresponding to conditional probability? It can be obtained by the following formula:
(4)
Where: conditional coefficient of variation, where Cvk is the coefficient of variation. A year is divided into K periods, and the runoff of each period is divided into M levels (that is, M states), then the transfer probability of adjacent periods is pkij (k = 1, 2,3, ..., k; I, j = 1, 2, 3, ..., m) indicates the probability that the runoff in tk- 1 period is in state I and the runoff in tk period is in state J.
Sum matrix
(5)
Then the transition probability matrix from tk- 1 period to tk period is expressed. Obviously, the sum of the non-negative elements in each row of this matrix is 1, that is:
(6)
In order to calculate the transition probability of Pkij conveniently, take 5% of the probability of 10, 15%, ... 95%, so that the values of the transition probabilities are all 0. 1, and then the flow Qpi corresponding to the conditional probability can be obtained by Equation (4).
2 Dynamic Programming The dynamic programming method was put forward by American mathematician Berman, and it is a mathematical method to study the multi-stage decision-making process. In recent years, it has been widely used in the field of water resources planning and management.
2. 1 mathematical model of dynamic programming
The calculation of reservoir optimal operation chart with stochastic runoff is a multi-stage stochastic decision-making process. The calculation model is as follows.
Stage (1): divide the reservoir operation map into 12 interrelated stages (time periods) every month (or ten days) for solution.
(2) State: Because there is a correlation between the average inflow Qt and Qt+ 1 in two adjacent stages, the reservoir water level at the beginning of flood season and the predicted runoff Qt in this period are taken as state variables St(Zt- 1, Qt).
(3) Decision-making: After the cycle state is determined, the corresponding decision is made, that is, the water supply qt facing the cycle, the water level at the end of the cycle is determined, and the state is transferred. The water level of the reservoir is divided into m levels, so there are m state transitions, which are optimized in the decision-making domain according to the method of 0.6 18. For each state variable St, an optimal water supply quantity qt should be selected, and the relationship curve between ST and Qt is the scheduling line of time period T, and the decision domain is (QDmin, t; Qxmax，t)
When all the states of decision variable water supply qt are optimized, it is necessary to check and judge the water level limit of the reservoir. If the final storage capacity V2 is greater than the maximum allowable water level or limited water level, the water supply will be calculated according to qt flow before the reservoir is full, and the water supply will be based on inflow after the reservoir is full. When the inflow is greater than the maximum excess capacity of the power plant, the excess part is regarded as abandoned water.
(4) State transition: The state of the reservoir is related to the form of the dispatching diagram. Considering the inflow runoff and short-term runoff factors at that time, the reservoir operation year is divided into k periods, and each period is determined by the initial water level z? Initial and periodic flow qt constitute the operation state of the reservoir, and each state has a corresponding decision variable, water supply flow Qt, which is expressed as:
Qt = q (early z, Qt, tk)
(7)
Tk is the number of cycles, each decision has a corresponding water level at the end of the cycle, and the state of the reservoir has shifted. If the water level of the reservoir is divided into Z grades, then the runoff is divided into M grades. For a period of time, there are Z×M reservoirs facing the state, and there are K×Z×M reservoirs operating in the whole year. Reservoir optimal operation diagram is a relational diagram to make corresponding decision variables for various operation States throughout the year.
It can be seen from Formula (7) that the initial water level and runoff of the reservoir in the current period have been fixed, and the optimal decision-making water supply in this period is a constant value, so the initial water level of the reservoir in the next period tk+ 1 (that is, the water level at the end of the period tK) is also a constant value. Because the runoff of the next time period tK+ 1 is not a definite value, but a random value that changes with the runoff Qt of the time period tK, and its value is determined by the conditional probability distribution function (elastic correlation). Therefore, if the reservoir state transition probability of time period tK is in state I and time period tK+ 1 is Pkij, then the matrix Pk=(Pkij) represents the reservoir state transition probability matrix from time period tK to time period tK+ 1, and Pk is completely determined by the dispatching mode of time period tK and the runoff state transition matrix. After years of operation, the reservoir operation state has reached a stable probability distribution.
(5) Benefit function: When the state of the reservoir is transferred, benefit functions (including industrial water, domestic water, irrigation water, power generation water and three-guarantee rate) are generated.
Irrigation water demand: Because the irrigation water demand is different every year, every month and every day, it is a random variable, so it is difficult to compile a computer program to calculate it. Therefore, the concept of "effective rainfall" is introduced into farmland water conservancy for the first time, which greatly simplifies the whole optimization calculation and completely solves the problem of water balance, and the water balance in the whole optimization calculation reaches 100%.
Calculation of effective rainfall: the data Mij obtained by the experimental station in the reservoir irrigation area is the irrigation quota of 1952 ~ 1999 over the years (I = 1952 ~ 1999, where j is the month (or ten days) of the first year (according to the actual crop water demand measured by the experimental station over the years, using the universal Mmax-mij = P0ij, I = 1952, …, 1999, J = 1, …, 12, and calculate them one by one. Add the effective rainfall of each year and month to the inflow Qt of each year and month. Because Mmax is a constant, there is only a random variable Mij. Its mathematical expression is as follows: Cixj=Aixj-Bixj, that is:
(8)
Among them, Cij is the effective rainfall of the I-year series J (month), and aij is the crop water demand of the I-year series J (J can be calculated on a daily basis and then converted into months). Bij is the comprehensive irrigation water quantity of various crops in J period of I-year series.
(6) Objective function: According to the specific situation of insufficient water resources in the reservoir, it is proposed to meet the agricultural water demand as much as possible under the condition of meeting the guarantee rate of domestic water and industrial water. The objective function can be expressed as: under the condition of meeting the water guarantee rate, the water supply is the largest. The calculation of the objective function can be obtained by the following piecewise linear function:
f(st，qt)=qt
Qxmax≥qt≥Qxmin
(9)
f(st，qt)=qt+CA(qt-Qxmin)
Qxmin≥qt≥QDmin
f(st，qt)=Qxmax+CE(qt-Qxmax)
QDmax≥qt≥Qxmax
Qt is the reservoir water supply and QDmin is the lower limit of the system water supply, that is, the lower limit of ensuring urban domestic water and industrial water; Qxmin is the sum of agricultural guaranteed water supply and QDmin; QDmax is the maximum excess water capacity of the power plant; Qxmax is the sum of the upper limit of agricultural water supply and QDmin; CE is the conversion coefficient when the special water quantity for power generation is less than Qxmin, and CA is the conversion coefficient when the water supply quantity is less than Qxmin. In the CAlculation, it CAn be arbitrarily assumed that ca, CE, ca and CE are directly proportional to the guarantee rate of Qxmin. Given a CA and CE, an optimal dispatching diagram can be obtained recursively. According to the dispatching diagram, the reservoir inflow data for many years can be used for diachronic operation calculation. If the assurance rate obtained from the calculation results is lower than the required assurance rate, CA and ce can be modified for recursive calculation (generally 2-3 times can be repeated) to obtain another optimal scheduling diagram, and then diachronic operation is carried out until the obtained assurance rate meets the requirements. That is, the values of CA and CE that meet the requirements of guarantee rate are selected through trial calculation.
2.2 The recursive equation of dynamic programming takes qt as the decision variable in T stage and St(Zt- 1, Qt) as the state variable in T stage, so the optimal recursive equation of inverse time series dynamic programming is:
Ft(St，qt)=max{ft(St，Qt)+Ft+ 1(St+ 1)} Qt∈Qt t = 1，2，…，N
( 10)
Where Ft(St, qt) represents the objective function value of the reservoir from time t in St state to time n at the end of reservoir operation (end of calculation period); Ft(St, qt) indicates the expected value of time period benefit when the reservoir is in St state and the water supply quantity qt is taken; Ft+ 1(St+ 1) represents the expected benefit of the reservoir from the time when t+ 1 is in St+ 1(j state) to the time when the optimal decision is adopted. Qt represents the inflow runoff series used in the calculation of t period; Pi, j is the conditional probability when qt decision is taken at time T. When the system moves from I state St in T stage to J state St+ 1 in T stage, the revenue of Ft+ 1 corresponds to the optimal decision of St+ 1 state.
The constraint conditions of the recursive equation are as follows: ① the reservoir water level is constrained by Vmin, t≤Vt≤Vmax, t, that is, the reservoir water level in each period is not lower than the dead water level Vmin, t, nor can it exceed the maximum allowable water level Vmax, t in this period. ② water balance constraint Vt+ 1 = Vt+(Qt-Qt) Δ t-yt-et, where vt+1and vt represent the water storage at the end and beginning of period t respectively; Qt and Qt represent the average inflow runoff and water supply in t period; Yt is the abandoned water, and Et is the evaporation and leakage loss of the reservoir. (3) water supply constraint and water delivery capacity constraint QDmax, t ≥ Qt ≥ qdmin, and the water supply in t. t period shall not exceed the maximum excess capacity qdmax, t of the turbine, and shall not be less than the lower limit qdmin, t..
2.3 Recursive calculation of dynamic programming adopts recursive calculation of inverse time series dynamic programming by time period, that is, all states are optimized one by one in each time period. The expected revenue generated by several representative inbound traffic values in a period of time. The optimization method is 0.6 18 and the number of search points is 20.
2.4 optimal scheduling diagram Howard? It is proved by the method of Z transformation that the calculation of formula (10) converges with the increase of the number of years t, and the recursive calculation adopts inverse time series recursion, that is, from n period to 1 period. As long as FN(SN) is known, it can be recursively calculated according to formula (10). Firstly, the relationship curve between reservoir water level (storage capacity) and water storage capacity can be used as the initial recurrence line FN(SN). When the best decision is selected for all states in the first time period, one time period can be recursively advanced. After all the recursive calculations in the first year are completed, the recursive calculation in the second year will be carried out. Because the initial recursion FN(SN) is arbitrary, the strategy obtained by the first year's recursion is not a stable optimal strategy, and it is necessary to continue recursion until the recursion lines of each period converge, at which time the strategy obtained is a stable optimal strategy. The convergence criterion of recursive lines is that the difference of recursive lines in the same period of the previous two years is less than the specified relative error ε, that is:
| Ft(Si)n-Ft(Si)(n+ 1)|/Ft(Si)(n+ 1)≤ε
( 1 1)
Where: Ft(Si)n represents the future income value corresponding to the state Si on the recursive line in the nth year T; Ft(Si)(n+ 1) is the future income value of the same state Si on the T recurrence line during n+ 1, and ε is 0.00 1. Generally, it can converge in two years at most, that is, the optimal scheduling line of 12 or 36 (ten days) can be obtained. At this time, the optimal decision of each cycle constitutes an optimal strategy, that is, the optimal scheduling diagram. Obviously, considering the correlation between months (or ten days) and separated months (ten days), that is, using a probability prediction, the economic benefits will be improved accordingly. Because the Markov single-chain elasticity theory is used to deal with runoff, the reservoir dispatching diagram changes from two-dimensional coordinates to three-dimensional coordinates, forming a spatial reservoir optimal dispatching diagram, and then the dispatching diagram becomes an equation set with Qt as the parameter, and the recursive line also changes from one to a group, that is, the optimal dispatching line changes from one to a group, forming a dispatching diagram family, which is convenient for actual dispatching.
2.5 Dynamic programming calculation program The calculation of dynamic programming is a very complicated process. Different planning problems need different calculation programs. According to the mathematical model of opt problem [2], we wrote a calculation program with VISUL C, and used recursive equation to find the optimal solution. The program has been successfully debugged on P ⅱ microcomputer. Practice has proved that it has the characteristics of powerful function, convenient use and fast operation speed, and can automatically draw the optimal operation diagram of three-dimensional space reservoirs and the operation curve diagram of a group of parameters.
Application Example This method has been applied to several large and medium-sized reservoirs such as Bath, Bashan and Huang Qian, and achieved ideal results. Only an example is given to illustrate the application of multi-objective optimal scheduling of bath.
Bath Reservoir is located in Laiyang City, Yantai District, Shandong Province, with a controlled drainage area of 455km2 and a total storage capacity of 654.38+87 million m3. Li Xing's storage capacity10.07 billion m3, with an average annual inflow of 69 million m3. The reservoir supplies 65,438+800,000 m3 of water to Laiyang every year, with an irrigation area of 930,000 hm2. Hydropower stations are divided into East Power Plant and West Power Plant, with installed capacity of 1.80 kW. It is a large-scale water conservancy project that comprehensively utilizes irrigation, flood control, urban industry, domestic water supply, power generation and aquaculture. As shown in figure 2.
In the process of optimal operation of Bath Reservoir, Markov single-chain elasticity correlation theory is used to deal with runoff, water supply flow is taken as decision-making condition, and on the basis of introducing effective rainfall, dynamic programming algorithm is adopted to meet three guarantee rates (domestic water guarantee rate, industrial water guarantee rate and irrigation water guarantee rate), so as to coordinate the relationship among life, industry, irrigation and power generation.
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