Yıllık akımların gidiş özelliklerini benzeştiren bir matematik modelin araştırılması

Abstract

Günümüzde hazne kapasitesi beldr^ işletilmesi problemleri gidecek önem kazanmaktadır. Bunun sebebi nüfus artışı, endüstrinin gelişmesi ve yaşam seviye sinin yükselmesi olmaktadır* Bazne kapasitelerinin.belirlen - me ve haznelerin İşletilme problemlerinin sağlıklı çözümleri İçin âözkonusu akarsuya alt uzun -süreli akım kayıtlarına Mfâiyaç vardır^ Ancak gerçekte eilmizBe -bâgbir: zaman yeteri mzTin'lükta 3caglt bulunamamaktadır. Su thırum hidrolojide mkzm '::serilerlmln sentetik olarak ıtMretilme çalışmalarına yol aç mıştır. Yıllık akım serilerinin türetllmeşinde en yaygım.xHarak kullanılan metod birindi derece.Markov (MR) modelle ridir. Bir matematik modelin bir stokastik sürecin tüm özel liklerini henzeştirebilmesi mümkün olamamaktadır. Birinci derece Markov modeli de âkım sürecinin ortalama, standart Sapma ve otökorelasyon katşayısi parametrelerini korumakta, ancak kurak 'devre 'özelliklerini benzeştlrebllmekte başarısız olmaktadır. Bu çalışmada yıllık akımların gidiş (kurak ve sulak devre) özelliklerini benzeştirebilecek bir matematik modelin araştırılması amaçlanmıştır. Bu nedenle aynı amaca yönelik, II yani stokastik süreçlerin gidiş özelliklerini korumak üzere geliştirilmiş çalışmalar incelenmiştir. Ağrıca gidiş özellik lerine ait bir literatür araştırılması, yapılmış ve ilgili bilgiler ba^ğımlı ve bağımsız süreçler için verilmiştir. Bu bilgilerin ışığı altında önce gidiş^uzunlukları sü reci simüle edilmeye çalışılmıştır. Daha sonra gidis-tpplam- ları süreci, gidiş-uzunlukları süreci ve bir regresyon mode li kullanılmak: suretiyle simüle edilmiştir. Bu işlem bağım sız ve bağımlı süreçler İçin yapılmıştır. Ancak bağımsız ve bağımlı süreç durumlarında kullanılan regresyon modeli fark- 1 1 olmaktadır. Son aşamada ise simüle edilmiş gidiş- toplamları, kendi lerine karşı gelen gldiş-ruzunlukları kadar akışa ayrıştırı larak gerçek akış süreci elde edilebi İnektedir. Gerçek akış sürecinin hangi parametreleri koruyup, hangilerinin koruya madığı sunulmaktadır. Son bölümde ayrıştırılmış bağımsız ire bağımlı akış sü reçleri def işit açısından incelenmiş bu akış süreçlerine karşı gelen defislt parametreleri literatürde mevcut def isi t parametreleri ile karşilaştırılmıştır. Defislt İle gidiş özellikleri arasındaki İlişki elde edilmiştir

The development of water resources gains importance due to increase in population and living standards. In order to supply a constant amount of water with regard to time, construction of reservoirs are essential. Both for the construction of a reservoir and then for its operation recorded flow data of sufficient length is needed. However almost always the case is not so, and therefore the problem of simulation of flow series arises. In practice the most commonly used flow model is the first order Markov model, an AR model which needs only three parameters (the mean, the standard deviation, the first order autocorrelation coefficient of the historical flows) in order to be used. A survey of literature shows that the first order Markov model is frequently insufficient in simulating the run properties of the historical record. The aim of this study, is to develop a mathematical model for annual flows which would be capable of simulating the run properties. Surely a mathematical model cannot be expected to simulate all the parameters of a historical flow record which arises from a very Complex phenomenon. IV A short summary of literature related to mathematical modelling of flow series that simulate run properties has been given. If we cross a standard flow series at a crossing level, the number of years during which the flows are higher (or lower) than the crossing level is known as run- length. The sum of flows corresponding to the above defined run-length is known as run-sum. The statistical properties of the run-lengths and the run-sums constitute the properties of runs. The knowledge gathered from literature concerning the statistical properties (parameters and distributions) of the run-lengths and run-sums Of standard flow series is given for two cases: independent and dependent flow series. Throughout the study, normally distributed standard flow sequences are investigated at a median crossing level. The median crossing level for a standard sequence crosses the sequence at zero level. A flow model is proposed for independent flows which would simulate the run properties of an independent flow series. The general outline of the model is as follows. The theoretical distribution of run-lengths is known for an in*- dependent series. Therefore a series of run-lengths can be simulated using this distribution. Then a series of run- sums can be simulated using a regression model of the first order Markov model type. In the simulation of the run-sums by the regression model, independent standard variables which are distributed with a certain skewness are required V together with the expected value and Standard deviations of the run-lengths and the run-sums and the correlation coefficient between the run-? lengths and the sun-sums. The simulated run-length and run-sum sequences are then analysed with regard to their parameters and distributions. The parameters show a very good fit to the theoretical parameters. The distributions are compared to the theoretical distributions and the distributions obtained from observed data i these distributions show a very good fit also. The proposed flow model was applied for depend flow series with the first order autocorrelation coefficient were chosen since they were frequently observed values in nature. The run-length simulation was done in a similar way to the independent case. Naturally the distributions corresponding to depend case were used. The theoretical parameters and the parameters calculated from the simulated run-length sequences were compared. This comparison showed that the ^simulated run-length parameters were in good accordance with the theoretical values of the same parameters. The run-sum simulation for the dependent case was made similar to the run-şUm simulation for the independent case. That is to say a regression model was used, and the run-rsums were simulated by regression VI upon the formerly s imiulated run-length values.However this method did not work out Well fox the simulation of the run-sums of a dependent sequence. The run-sum values should carry the same sign, so that they could be successively thought as positive and negative rvm^sums one after the other. The run-^sum sequence simulated using *fte regression model sometimes carried different signs, which was not desired. Mn order to overcome this problem, a different method for the simulation of run-sums has been applied. The new method was simulating ithe run-sum sequence by using the expression öf the first order autoregresslve model for the gamma distributed variables. This simulation has been done for two dependent cases, p = 0.3 and p = 0.5. The parameters of the run^sums simulated by the new method are compared with the theoretical parameter values. The distributions of the run-sums for p= = 0.3 and p =0.5 can only be compared to the distributions obtained from observed data for the two cases, since there is no theoretical knowledge concerning the distributions of run-sums for dependent sequences. Finally both for the independent case and the two dependent cases (p = 0.3 and p= 0.5) the run-sums were disaggregated into the real flow values. That is to say a run-sum was disaggregated so as to give the number of flows which is the length of the corresponding run- length. TOT The applied method was the Vâlencia^Söhaake disaggregation method. For the real flow series simulated for the independent case the mean, the standard deviation and the autocorrelation coefficient parameters of the flow sequence were preserved. The flows were normally distributed} the run properties were ^preserved also. For the dependent cases, the mean, standard deviation and run properties of, the flows were preserved hut the autocorrelation structure of the sequence and the normal distribution were deformed. in the determination of the capacity of a reservoir two concepts are widely used. One of these concepts is the range which is the capacity of an ideal reservoir which would supply a certain yield along a given «period of time without allowing any spilling. The other one of these concepts is the deficit which İs the capacity of an ideal reservoir supplying a certain yield along a given period of time Jbut allowing rspiiling. The parameters of deficits for full regulation and partial regulation cases have been given in literature. These parameters (the expected rvalue and standard deviations) of the deficits have been given as functions of the first-order autocorrelation coefficients and design periods. In the last part of the study an analysis of the simulated flow series has been made with regard to deficit. The first and second moments of the deficits obtained from the simulated flow series are compared to the same moments of deficits present in literatüre. The relationship between deficit and run- properties has been given.