## Determining The Capacity Of Dam Using Probabilistic Approach

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Determining The Capacity Of Dam Using Probabilistic Approach.

Chapter One

Prelude

1.1 Introduction

Dam, barrier, commonly across a watercourse, to hold back water, often forming a reservoir or lake; dams are also sometimes used to control or contain rockslides, mudflows, and the like in regions where these are common. According to the definition given by Federal Emergency Management Agency (FEMA), “Dam means an artificial barrier, including dikes (a barrier blocking a passage, especially for protection), embankments, and appurtenant works- that impounds, diverts, water or is designed to impound or divert water or a combination of water and any other liquid or material in the water”.

Important structure or machinery incident to or annexed to a dam that is built to operate and maintain a dam is called the appurtenant structure. Important appurtenant structures of a dam may include spillways- to allow safe passage of flood flows, tunnels- to control releases for irrigation and / or power generation, power station, and canal outlet (s). Spillway means water in or about a dam, designed for the discharge of water. Impoundment means the water held back by a dam.

Dams are made of timber, rock, earth, masonry, or concrete or of combinations of these materials. The height of a dam may vary from less than 50 feet to more than 1,000 feet. The reservoir or lake created by a dam may also vary in size; its area may vary from a few acres to more than 100 square miles.

In a dam, water arrives which is termed as input, and water is stored in the associated reservoir for some time. Then a fraction of water stored is released from the dam to meet the demand. The ‘release’ is usually called ‘draft’. The release of water from the dam is made according to some policy called the ‘release policy’. Thus, the inflow of water (input) and the policy of release are the two main features of what we call the ‘operating rules’ or ‘operating policy’. A dam is maintained and operated according to the operating rules.

When a dam is constructed a storage reservoir is formed automatically. Efficient operation of a dam requires meeting the demand of water from the dam properly. And to meet the demand properly, the dam should be of that size that can contain as much water to supply. For efficient operation of a dam, researchers have been studying for long. They related the storage of a dam to the inventory problem since the basic feature of a dam is storing water until it is released. But it was Professor P.A.P. Moran who for the first time gave the probabilistic formulation of a storage model for the dam in 1954 and called the theory a “Probability theory of Dams”. In a dam situation water arrives into the dam continuously which is random. Water is released from the dam according to certain policy which is usually deterministic. The random input in the dam process makes the storage function

where,

amount of water in the dam at time t,

is the input or inflows into the dam during time interval t, and

is the release in the corresponding time.

a stochastic process, the behavior of which must depend on the input and release pattern. Here the amount of water into the dam at different points of time is called the storage function. As the input into the dam is random, the dam process is a stochastic process and thus statistical techniques can be applied in studying the dam process.

There are four main problems in reservoir theory where statistics and applied probability theory are involved. These are:

(1) Studying the nature of input pattern and to augment the short record values,

(2) Finding the stationary level of the dam content,

(3) Finding the probability distribution of first emptiness time, and

(4) Determination of capacity.

These problems are actually connected to each other, since a method of solution of the fourth problem automatically leads to studying the nature of input pattern of the historical flow and then finding the stationary level of the content of the dam. The first problem is essentially a problem in time-series analysis and hence, statistical tools can be directly applied in it. We have to obtain the characteristics of the historical flows, such as- mean, variance, coefficient of variation, coefficient of skewness, and serial correlation coefficient. Experience shows that consideration of lag one serial correlation is enough although more lags can be considered. We, in our study considered lag one serial correlation coefficient.

Researchers focused mainly on the fourth problem in which relationship between capacity, reliability (probability of failure or probability of emptiness) and release policy is studied. This fourth problem is essentially related to applied probability because the concept involved in determining the reliability in terms of probability is the core matter of statistics. Now inflow pattern may be independent or dependent to each other. The dependence pattern (if any) depends on many factors and conditions. But until Lloyd’s work in 1963, in which he considered Markov dependent inflows, researchers considered the inflows as being independently and identically distributed.

Ideally, the assumption of independent and identically distributed inflows is not quite realistic as flow pattern is not naturally independent of each others because of “persistence” of the hydrologic series (e.g., water flows series, stream flow series, snowfall etc.). Persistence is a non-random characteristic of hydrologic series that indicates how one event is influenced by the other. For example, a month with high flow will tend to be followed by another high flow rather than a low flow. That is, one event is related to or characterized by another event adjacent to it. This feature of hydro-series is quantitatively characterized by the serial correlations coefficient. It indicates how strongly one event is affected by a previous event. Thus the behavior of inflow is something like the following- the flow at a given time (say a day or a month) may depend on the inflow of the previous time period and not further on the past. Therefore, inflow pattern is Markova and hence, flows are not serially independent. Assumption of serial ‘dependence’ is rather than realistic.

As in the dam situation, inflow follows either independent or Markova dependent pattern, and inflows are stored in a dam which are released according to some release rule; the process is similar to the inventory process-an important branch of stochastic process. So, the selection of inflow model and the determination of capacity can very well be studied using statistical techniques.

In our study, we will focus on determining the capacity of a reservoir using various techniques. Before going for further description, we need to know what the ‘capacity of a dam’ is and why determination of capacity is so important.

1.2 Capacity of Dam

The capacity of a dam is defined as the volume capable of being impounded at the top of the dam. We will use the terms ‘reservoir capacity’ and ‘capacity of the dam’ synonymously. We will also use the terms ‘reservoir’ and ‘dam’ synonymously.

Active capacity of reservoir is the capacity normally usable for storage and regulation of reservoir inflows to meet established reservoir operating requirements. It is also the total capacity less the sum of the inactive and dead storage capacities.

Total capacity is the reservoir capacity below the highest of the elevation representing-

(i) The top of exclusive flood control capacity [i.e., the reservoir capacity assigned to the sole purpose of regulating flood inflows to reduce possible damage downstream. In some instances, the top of exclusive flood control capacity is above the maximum controllable water surface elevation.],

(ii) The top of joint use capacity [i.e., the reservoir capacity assigned to flood control purposes during certain periods of the year and to conservation purposes during other periods of the year], or

(iii) The top of active conservation capacity [i.e., the reservoir capacity assigned to regulate reservoir inflow for irrigation, power generation, municipal and industrial use, fish and wildlife, navigation, recreation, water quality maintenance, and other purposes. It does not include exclusive flood control or joint use capacity.]

Total capacity is fixed at a certain level. Water in the lower elevation of a reservoir that is unavailable for use is called dead storage level. The reservoir storage capacity up to dead storage level is called dead storage capacity and is provided to accommodate the incoming sediments. The number of years taken to fill the dead storage capacity is called reservoir life. The reservoir capacity between the dead storage and live storage levels is called the live storage capacity and represents the amount of water which could be stored during the flood season(s) and released during the low flow season(s). Live storage capacity of a dam is reduced day by day because of sedimentation. Therefore, a replacement storage dam is required if we wish to maintain current levels of water availability. The storage capacity has the units of volume and is generally measured in million acre-feet (MAF). An acre-foot is the volume of water required to cover one acre of land (43,560 square-feet) to the depth of one foot and is equal to 43,560 cubic feet. For a dam designed to safely pass the maximum probable floods arriving at a time when the dam is already at maximum storage level, additional storage capacity is provided above the maximum storage level and spillways of adequate capacities are provided to handle the flood flows.

1.3 Importance of Capacity Determination

Water is one of the continuously renewable natural resources of the globe. It establishes a connection between the other spheres of the earth and it is an important component of the human environment. It is generally agreed perception that water is increasingly becoming an issue of primal significance. ECOSOC committee on Natural Resources in a recent strategy paper indicated that as many as 52 countries with a population of more than three billion will be “Water Stressed” or face chronic water scarcity by the year 2025 (Hussain, 2000). The growing problem has as much to do with the availability of fresh water in the overall global context as with the fact that such resources, even when available, are in the wrong places or available at wrong times. That’s why effective management of water resources is a no denying fact for proper utilization of this resource. For the beneficial use of the water resource, it can be managed in various ways one of which is by constructing storage reservoirs. The concept of storage reservoir is not only for beneficial use of water resource but also to help develop structures to control seasonal natural flooding which causes much damage to an economy. Now let us explore some purposes of a dam from which we would be able to realize the importance of a proper capacity dam.

The purposes may be manifold- from simply storing the water for use in lean period to constructing reservoir for multiple purposes. Dams are built for specific purposes. In ancient times, they were built only for water supply or irrigation. One of the earliest large dams for this purpose was a marble structure built c.1660 in Rajputana (Rajasthan), India. The main purpose of a dam is to use water in the most efficient way. A dam may be constructed to meet some specific functions. Specially, the purposes may be-

(i) To provide water for irrigation, to aid flood control and hence improve the navigability of waterways, and especially to furnish power for hydroelectric plants. Notable dams built to provide hydroelectric power include the Aswan Dam, 3 mi (4.8 km) south of the city.

(ii) To impound water is often called a barrage; the largest such barrage is the Syncrude Tailings Dam in Canada, which impounds 540 million cubic meters of water.

(iii) To store water during the wet seasons for using in dry seasons (or during period of low flows).

(iv) To store water in a dam during high flow (flood) season and release during the low flow season to supplement the natural flows to meet the irrigation requirements.

(v) For generating hydro-electricity.

(vi) To manage multipurpose water demand of a particular basin.

(vii) To divert/ transfer water from a stream to another stream to augment the later stream to maintain smooth navigation.

(viii) To address usual hydraulic problems of a particular basin (e.g., flooding, river instability, sea tides and salinity) having a distributory type drainage pattern (e.g., the region of Ganga basin, mostly in Bangladesh deltaic region).

(ix) Sometimes, dams are constructed to obtain multifarious problems of management of water resources such as optimal and multiple utilization of resources of an international river.

(x) Storage reservoirs are built for harnessing the water resources to mutual benefit of all the co-basin regions.

(xi) To produce fish and protect and improve environment, and

(xii) For recreation and so on.

These are the few specific purposes of a dam. To meet the purposes mentioned above, constructing a dam is not the only solution to the problem. It should be appropriate size so that the purposes, for which it is constructed, are fully served. For example, for a flood control dam, the capacity determination is very important because of its purpose. Generally, a flood control dam should be as large as possible to be helpful to control flood- even for the worst flood in the respective rivers or streams. But, making of a large dam involves lot of money and manpower and so, the economic side should also be taken into consideration. The dam should not be too small to serve the purposes, or it should not be extravagantly large so that its capacity is rarely or never utilized. A dam which is constructed for irrigation purpose should be of that capacity which can store enough water needed for irrigation. Similarly, a dam, constructed for storing water during wet season (flood period), should be of that capacity so that the stored water can be used during dry period without failure (deficit). That is why the capacity determination of a dam is so important.

An important consideration in determining reservoir capacity is the minimum annual runoff. The available storage determines the magnitude of demand that can be met during a period of low runoff.

1.4 Objective of the Study

The main objective of the study is to determine the capacity of dam using probabilistic approach.

The specific objectives are

(i) To review the existing methods

Ø Methods using engineering consideration (Mass Curve method, Sequent Peak Algorithm etc)

Ø Methods using statistical techniques (Dincer’s method, Gould’s Gamma method etc)

(ii) To generate inflow data using various model.

(iii) To modify or develop new techniques for determining of capacity of the dam using probabilistic approach.

(iv) Comparison of various methods for determining the capacity of the dam.

(v) Application of methods to simulated data.

1.5 Organization of the Study

The study is organized into five chapters.

The first is the introductory chapter discussing introduction of the study and also objectives are introduced. The theoretical concepts such as general description of dam and dam system used in the study are discussed.

In the next chapter, a brief description of early works in determination of capacity of dam has been given. The early works have been divided into three categories, namely, probabilistic approaches and the Moran related methods in determining the capacity, methods based on Mean emptiness time, methods in which the capacity is determined by Linear Programming. Finally, the methods based on generated data are described.

Third chapter begins with brief introduction about simulation which we have used in our analysis. In this chapter, we have explored the characteristics of the historical inflows and its distribution. Also some models in data generation are also given there.

Chapter four is the main part of the study. Here we have generated annual and monthly data by simulation. Also we have developed a new approach to determine the capacity of a dam. We have suggested considering the level of dam content to obtain the required capacity. Capacity by this approach has found to have no emptiness and also the estimated capacity ensured that there will be no overflow.

In chapter five, some of the well known methods such as Mass curve technique, Sequent Peak Algorithm, Gould-Dincer’s Normal method, Gould-Dincer’s Log-normal method, Gould-Dincer’s Gamma method and Gould’s Gamma technique to determine the storage capacity of dam are described.

In chapter six, the main stream of the work begins. Using existing techniques to obtain capacity of dam, we have applied the generated data. Also we considered various drafts viz. 70%, 75%, 80%, 85%, 90%, 95% and 98% of mean inflow. Some tables and graphs are presented in this chapter. Also results are discussed. Capacity determined using the existing techniques are then compared with the capacity determined by the developed technique.

The final concluding chapter includes the major study findings. Conclusion has been drawn on the basis of the estimated capacity along with the limitations faced in this study have been described in chapter seven. A brief description of scope of further research in this field is also given at the end of this chapter.

Chapter Two

Review of Literature

2.1 Introduction

In this section we shall concentrate on a review of the earlier works related to the subject matter of this project. There have been a number of studies to determine the capacity of dam. In early days the capacity of a dam was determined by using the historical data and empirical method of problem solving. In those days, in determining the capacity, a quantity was assumed as the capacity and further it was assumed that the dam was initially full at the beginning of drought. By adding the monthly inflow to the dam and subtracting the monthly demand, the quantities left in the dam at the end of each month are calculated for a period of one year. Should the quantity show a deficiency (i.e., the initial dam content assumed, appearing a negative quantity), the capacity originally assumed for the dam was increased and the calculation was repeated.

2.2 Earlier Methods in Determining Capacity

Systematic investigation for determining the capacity of a dam dates back from the work of Rippl (1883). Rippl’s method assumed that during an interval and at unit interval of time the historical flows and corresponding releases are known and given by and respectively. Let Then T is plotted against time gives a curve which is called the mass curve (Rippl, 1883).

Rippl used the mass curve and took as the capacity; where, is the peak and is the trough of the mass curve. This mass curve method for determining the capacity is based solely on the historical record, which is often very short in length and is likely to differ from the economic life of the proposed dam. Also, the same flow might not occur in the future time period. The mass diagram has many limitations which will be discussed later.

To overcome the inadequacy of the short term records, Hazen (1914) used the annual flow records of 14 rivers. He combined the 14 records of various lengths to form a continuous record of 300 years. To calculate the risk of water shortages, different storage occurring with each size were counted (Hazen, 1914). The disadvantage of this method is that, Hazen did not consider the probable correlation among the flows resulting in correlation between the successive sections of the combined data.

Sudler (1927) for the first time described a method of producing synthetic stream flow. He selected 50 representative annual stream flows and wrote each on a card. The cards were well shuffled and drawn one by one until all the 50 records were used. By repeating this procedure, he compiled a 1,000-year record. The record was then subdivided into shorter records of duration equal to the economic life of the dam. These were then analyzed by the mass curve method (Sudler, 1927).

The serious defect of Sudler’s method was that the individual records were collectively the same from one sequence to another. Hence, moments and extreme values were always the same which is an unrealistic assumption for future sequences. This method also did not consider the possible correlation between the successive flows.

Following Sudler (1927), Barnes (1954) used the stochastic simulation to design a dam on the upper Yarra river in Australia. He generated a 1,000 year flow sequence by using a table of standardized normal variables by preserving the mean and variance as found in the historical data. A residual mass curve was then obtained by plotting against where is the mean inflow. Straight lines of various uniform draft rates were drawn on the residual mass diagram and capacity, for a particular draft rate, is then taken as the maximum vertical distance between the draft line and the residual mass curve.

Hurst (1951, 1956) also determined the capacity by using the residual mass curve technique. He took the range of the cumulative sums of departures form the mean of inflow, , as the capacity for a dam.

Langbein (1958) determined the capacity by using an analogy between a dam and finite capacity queue. He assumed that inputs are normally and independently distributed with mean and variance and the release in any period is given by

where, is the content of the dam. He then determined that capacity required to maintain a target draft and is some fraction, say, 0.5, 0.6, etc..

Bryant (1961) considered several models for optimal design of dams for specified target release.

2.3 Probabilistic Methods in Determining the Capacity

The probability theory of storage systems formulated by P.A.P. Moran in 1954 has now developed into an active branch of applied probability. An excellent account of the theory, describing results obtained up to 1958 is contained in Moran’s (1959) monograph. Considerable progress has since been made in several directions- the study of the time- dependent behavior of stochastic processes underlying Moran’s original model, modifications of this model, as well as the formulation and solution of new models.

2.3.1 Moran’s Model

In Moran’s Monograph, he considered three main purposes for the construction of a dam of which the third one is ‘to provide a storage which will be filled during the wet season and used during the dry season’. In order to obtain a tractable theory,he supposed that all the inputs occur during the ‘wet’ period (i.e., the period of high flows or flood season) and that all the output occur during the ‘dry’ season (i.e., period of low flow). This is, of course, not the usual situation to be considered, but he considered it for simplicity and considered the process as occurring at a discrete series of time intervals which he considered as years. Thus, the amount of water which flows into a dam (called input) will vary from time to time, and will have a probability distribution. Apart from a possible overflow, which may occur if the dam is of finite capacity,

this water is stored, and released according to a specific rule. The stored water is used for generation of hydro-electric power, and the released water (the output) is used for irrigation purposes. The central characteristic of the system is the storage function, giving the amount f water stored in the dam at various points of time.

In the basic storage model considered by Moran (1954), the content of a dam of finite capacity , is defined at discrete times by the recurrence relation

where,

a) denotes the amount of water which has flowed into the dam during the time interval (say, the year), and it is assumed that are mutually independent and identically distributed random variables.

b) the term represents a possible overflow at time , which occurs if and only if the content of the dam being after the overflow. and

c) the term indicates a release policy of “meeting the demand if physically possible”, according to which, at time , an amount, of water is released, unless the dam contains less than, in which case the entire available amount is released.

Moran’s approach can be subdivided into three main groups:

(i) Those in which time and volumes are considered as continuous variables.

(ii) Those in which time is discontinuous but water volumes are continuous. In this approach, Moran derived the following integral equations describing a mutually exclusive situation (McMahon & Mein, 1978).

For

For

where,

inflows

reservoir capacity

constant release during unit period

inflow function, and

probability function of storage content plus inflow during unit period.

Gani and Prabhu (1957), Prabhu (1958a) and Ghosal (1959, 1960) derived solutions for particular inflow distribution and release rules.

(iii) Those in which time and water volumes are both discrete variables. This approach was given by Moran (1954) in his paper and Ghosal (1962) and Prabhu (1958b) followed him. This approach involves sub-dividing the reservoir volume into a number of parts, thus creating a system of equations which approximates the integral equations given earlier [Equations 2.3.2, 2.3.3]. this approximation primarily affects the results at the storage boundaries (that is; full and empty) but is satisfactory if the storage volume is fine enough.

Two main assumptions can be made about the characteristics of inflow and outflow which occur at discrete time intervals. Moran (1954) assumed that the inflow and outflow do not occur at the same time. He termed this type of model as “mutually exclusive model”. In this model the unit period is sub-divided into a wet season (i.e., all inflow, and no outflow) followed by a dry season (i.e., all release but no inflow). The other assumption is only a simple further development to Moran’s assumption, i.e., the inflows occur simultaneously. This is called the “simultaneous model”. The mutually exclusive model and the simultaneous models are given below.

2.3.1.1 A Simple Mutually Exclusive Model

For the mutually exclusive model we have:

where,

inflow during period, and

capacity of the reservoir,

constant volume released at the end of the unit period.

stored water at the beginning of the period

stored water at the end of the period or the beginning of the period.

Given the information about capacity, draft and inflows, the first step is to set up a “transition matrix” of the storage contents. A transition matrix shows the probability of the storage finishing in any particular state at the end of a time period for each possible initial state at the beginning of that period. Because of mutually exclusive assumption of inflows, the reservoir can never be finished in the full condition.

2.3.1.2 A Simple Simultaneous Model

For the simultaneous model, we have:

That is, and occurs simultaneously. Here the notations bear the same meaning as in the case of mutually exclusive model.

In contrast to the mutually exclusive model, it is now possible for the reservoir to finish in a full condition at the end of a period.

The mutually exclusive model described earlier, overestimates both the probability of failure and the probability of spill. This is because of the assumption that inflows always precede outflow- thus the reservoir can never be full at the end of a time period. The simultaneous model on the other hand, is more realistic as it is more representative of reservoir inflow and outflow conditions.

2.3.2 Other Probabilistic Methods

In 1955, Moran modified the discrete model to deal with seasonal flows. Transition matrices were prepared for each season and were multiplied together to yield an annual transition matrix. However, the seasonal flows were assumed to be independent. Lloyd and Odoom (1964) adopted a somewhat similar model.

Harris (1965) gave a worked out example of Moran’s seasonal method applied to a British catchment. He found the flows to be seasonal and independent and he prepare wet and dry season transition matrices which were multiplied together to get the annual transition matrix.

Lloyd (1963) partly became the independence assumption in Moran’s approach by assuming that the inflows are represented by a bivariate distribution rather than a simple histogram. In effect, this squared the number of equations to be solved.

Dearlove and Harris (1965) made the techniques more applicable by combining Lloyd’s approach with Moran’s seasonal method, but computationally the problem was large and therefore its use was limited. However, Doran’s recent work (Doran, 1975) on the divided interval technique for solving the transition matrix may overcome this limitation.

Venetis (1969) developed monthly bivariate transition matrices from generated flows using Rossener and Yevjevich’s model (1966). Following Moran, and Dearlove and Harris, he multiplied the matrices together to get an annual transition matrix.

Gould (1961) modified the simultaneous Moran-type model to account for both seasonality and serial correlation of inflows. He did this by using the transition matrix with a yearly time period, but accounting for within-year flows by using behavior analysis. Thus monthly flow variations, monthly serial correlations and draft variations can be included.

McMahon (1976) took 156 Australian rivers and used Gould’s modified procedure to estimate the theoretical storage capacities for four draft conditions () and three probability of failure values (). These capacities were related by least squares analysis to the appropriate coefficient of variation of annual flows by the following simple relationship:

where,

Storage capacity in volume units

Mean annual flow in volume units

Reservoir capacity divided by mean annual flow,

Coefficient of variation of annual flows, and

Empirically derived constants (McMahon & Mein, 1978)

Langbein (1958) gave probability routing method which is similar to the Moran’s (1954) probability matrix method except that Langbein modified his technique to deal with correlated annual flow. Both the streamflow regime and reservoir storage were divided into low, medium and high sub-regimes. By classifying each flow into the same streamflow regime as its predecessor, three separate streamflow histograms were obtained. Thus setting up his system of equatioms describing the cumulative probability of reservoir contents, Langbein used the inflow distribution appropriate to the state of the reservoir.

Hardison (1965) generalized Langbein’s probability routing procedure using theoretical distributions of annual flows and assuming serial correlation to be zero. This is equivalent to Moran’s model except that Hardison used a simultaneous model rather than the mutually exclusive model adopted by Moran. The annual storage estimates were shown graphically for Log-normal, normal and Weibull distributions of annual flows. The percentage change of deficiency shown in his graphs was defined by Hardison as the percentage of years that the indicated storage capacity would be insufficient to supply the target draft. In this technique, first the mean, standard deviation and skewness of both annual flows and the common logarithms of annual flows are needed. Then selecting the appropriate distribution of the flows (which is selected by the value of the skewness coefficient), capacity was determined graphically for a given chance of deficiency and variability.

Melentijevich (1966) obtained expressions for both time dependent and steady state distributions of reservoir content assuming an infinite storage and independent normal inflows. In considering finite reservoirs, Melentijevich used a random model and a behavior analysis of 100,000 random normally distributed numbers. From the analysis, he obtained an expression for the density function of the stationary distribution of storage contents. The solution is complex and limited in use because of the assumption of normality, independence and neglect of seasonality.

Klemes (1967) in his method of determining the capacity was able to reduce the probability of failure within a limited period to the classical occupancy problem. His technique was restricted to a uniform release or a randomly varying one. The major limitation is that the unit period is of one year. Consequently, it is not possible to distinguish a failure within a year.

Phatarfod (1976) suggested another method in determining the capacity which is based on random walk theory and is concerned with finding the probability of the contents of a finite reservoir being equal to or less than some value lC where lC > 0 and is the reservoir capacity. The physical process of dam fluctuations can be linked to a random walk with impenetrable barriers at full supply and empty conditions. Phatarfod used Wald’s identity which is an approximate technique to solve the problem with absorbing barriers and a relation connecting the two kinds of random walks (McMahon & Mein, 1978). Phatarfod considered annual flows are gamma distribution and is based on a fixed draft.

2.3.3 Khan’s Suggested Methods for Determining the Capacity

2.3.3.1 Capacity Based on Mean Emptiness Time

Suppose that at time the dam contains an amount of water and let be the first subsequent time at which it becomes empty. Then is called the ‘wet period’ of the dam and the time at which the dam becomes empty is called the ‘Emptiness Time’. If a dam situation is observed in a simulation study i.e., if the same capacity dam is observe for various inflow values (inflows are randomly generated), we will have different Emptiness Times. The average value of those is what we call ‘Mean Emptiness Time’. M.S.H. Khan (1992) in his studies determined the capacity of a dam by considering mean emptiness time. Khan’s methods can be described as:

Assume that the input rate is less than the demand rate and that a certain quantity “”, the initial dam content, is stored into the dam capacity before supply actually commences. In this case, emptiness will be certain and the mean emptiness period will be a function of the capacity. We may expect that the mean emptiness period increases with the increase in the capacity and then becomes stationary and does not increase any further by increasing the size of the dam and therefore any further increase in the size may not economically or otherwise be justified. Thus, we can find the optimal capacity of the dam from its mean emptiness period such that if the capacity is increased further, mean emptiness time will not increase significantly (e.g., further increase in the capacity will not guarantee a longer period of functioning) provided that an initial content ‘’ is available for storage or supply. The value of may be taken as the average rate of supply. Khan (1992) studied for Geometric and Exponential inflows.

2.3.3.2 Capacity Based on Mean Emptiness Time for Geometric and Exponential Inputs

Suppose, is the input process. Consider that a dam of capacity starts functioning with an initial storage . To determine the optimal capacity , suppose the input process is a discrete independent increment process and the release is at unit rate at the end of each unit time interval

For geometric input with

the mean emptiness period can be shown to be [For details, see (Khan, 1992)]

The mean will be finite provided that .

For exponential input with probability density function

And constant unit release, the mean emptiness time can also be shown to be

It may noted that is finite for And is the unique non-zero solution of (Khan,1992)

Table 2.1 shows the mean emptiness period for geometric input with unit release and table 2.2 shows the mean emptiness period for exponential input distribution which are computed using equations 2.12 and 2.13 respectively. It can be seen in table2.1 that, for geometric input with , the optimal capacity would be . If the capacity is increased further, the mean emptiness time will not increase. Similarly, for exponential input with unit release, it is seen in table 2.2 that, for and unit release, the optimal capacity is . If the capacity is increased further, the mean emptiness period will not increase.

Table 2.1: Mean Emptiness Period for Geometric Input and Unit release

 Capacity 1

2

2

4

5

6

7

8

9

10

11

12

15

201.000

1.111

1.123

1.124

1.125

1.125

1.125

1.125

1.125

1.125

1.125

1.125

1.125

1.125

1.1251.000

1.250

1.321

1.328

1.332

1.333

1.333

1.333

1.333

1.333

1.333

1.333

1.333

1.333

1.3331.000

1.428

1.612

1.698

1.724

1.739

1.745

1.748

1.749

1.749

1.749

1.749

1.750

1.750

1.7501.000

1.666

2.111

2.407

2.604

2.736

2.824

2.883

2.922

2.948

2.965

2.976

2.993

2.999

3.0001.000

1.818

2.487

3.035

3.483

3.850

4.150

4.397

4.596

4.760

4.895

5.005

5.228

5.400

5.500  5

6

7

8

9

10

12

20

505.484

5.609

5.623

5.624

5.625

5.625

5.625

5.625

5.6256.222

6.555

6.638

6.659

6.664

6.666

6.666

6.666

6.6667.456

8.195

8.512

8.648

8.706

8.731

8.746

8.750

8.7509.790

11.526

12.684

13.456

13.970

14.313

14.695

14.988

14.99911.824

14.674

17.006

18.917

20.475

21.752

23.652

26.727

27.499

Khan (1992) obtained mean emptiness period for arbitrary input also. He used probability relations and probability generating function to obtain the mean of the first emptiness period (allowing overflow before emptiness) for an arbitrary discrete independent input distribution. Mean of first emptiness period can be defined as the average time period at which the dam becomes empty for the first time during its life.

Table 2.2: Mean Emptiness Period for Exponential Input and Unit Release

 Capacity 1

2

3

4

5

6

7

8

10

12

15

20

501.421

1.249

1.249

1.250

1.250

1.250

1.250

1.250

1.250

1.250

1.250

1.250

1.250

1.2501.372

1.426

1.428

1.428

1.428

1.428

1.428

1.428

1.428

1.428

1.428

1.428

1.428

1.4281.506

1.649

1.664

1.666

1.666

1.666

1.666

1.666

1.666

1.666

1.666

1.666

1.666

1.6661.952

2.322

2.442

2.481

2.493

2.498

2.499

2.499

2.499

2.499

2.499

2.500

2.500

2.5002.832

4.266

4.538

4.710

4.817

4.885

4.927

4.971

4.982

4.991

4.998

4.999

5.000

5.0008.441

8.721

8.985

9.181

9.339

9.467

9.567

9.719

9.714

9.852

9.222

9.973

9.999

10.000

2.3.3.3 Capacity Based on the Stationary Level of the Dam Content

M.S.H. Khan (1979) determined the capacity using stationary level of the dam content. According to Khan, if the successive inputs are mutually independent and identically distributed, then it is known from Phatarfod (1976) that,

; for continuous input and constant unit release (2.3.6)

; for discrete input and unit release

Where is the probability that the dam content is less than or equal to , is the probability that starting with a quantity u the dam of capacity gets emptybefore overflow. Then capacity of the dam can be determined such that

(2.3.7)

Where, is a specified fraction of the capacity and is also given. The value of should depend on the input rate.

As an illustration, the stationary distribution of the dam content for exponential input with probability density function

And with unit release per unit of time [Khan (1979)], is given by

(2.3.8)

For general arbitrary input and for constant release of M units, g(u) can approximately be given by

Hence from (2.3.6)

for continuous inputs,

for discrete inputs, (2.3.9)

Where is the unique non-zero solution of being the moment generating function (mgf) of the net input in unit time.

For markovian inputs, Equation 2.14 also remains valid if the markov sequence is reversible. In this case, is the non-zero solution of , where is such that for exponential results, may be taken equal to one. Then equation 2.17 will give approximate distribution function once is found.

Table 2.3: Capacity for Exponential Input and Unit Release

 Mean Input 1/4 1/3 3/4 P=0.1 P=0.05 P=0.1 P=0.05 P=0.1 P=0.05 1.2

2.07.12

2.3010.07

3.068.65

2.6411.93

3.4925.99

7.3233.83

9.53

To determine the capacity, we have from equation 2.15 and equation 2.16 when the input is exponential that

(2.3.10)

Where is the real root (other than 1) of



And when . Table 2.3 gives the capacity of the dam for various values of , and the mean input rate.

2.3.3.4 Capacity by Specifying the Probability of Overflow

In a multipurpose dam1 where a fraction of the incoming water into the dam is allocated for power generation and the remainder is used for irrigation and other purposes. Suppose the input into the dam is continuous and water is released continuously for power generation at a fixed rate, say, , where is the mean input rate. The dam contents are then leftover for using it for other purpose. Suppose that there is always a demand for water from the dam. Then the maximum of the dam contents during a given period may be taken as the capacity of the dam. In this case there will be no overflow and therefore 100% of the available water could be utilized. If we take a particular value, say, , of the dam content equal to the capacity of a dam and during a given interval of time if it is observed that in 95% cases the level of the dam content is below the assumed value and only in 5% cases the level exceeds , then is the capacity with 0.05 as the probability of overflow. The capacity can thus be determined by specifying probability of overflow.

As an illustration for determining the capacity by this method, Khan (1979) first generated an input series of Gamma-Markov type by using

where are random errors with mean zero and variance one. The distributional form of determines the behavior of and it can be shown that if is normal, is also normal. But it has been observed that the theoretical flows are quite close to the observed data if the errors are assumed to be the standard gamma variates rather than the standard normal variates. The deviates can be generated by using the transformation of Wilson and Hilferty (1931) [Ref. Fiering (1967)]. With mean , standard deviation skewness and first-order autocorrelation coefficient

Using i.e., a 50% utilization for power generation, and an input series of length 25 we compute in Table 2.4, page-29, the capacity by specifying the probability of overflow. It will be seen that if we take 4.54 as the capacity, then there will be no overflow; and if the capacity is taken to be 4.118, the probability of overflow is 0.04. From the Table 2.4 we find the capacity as 4.54 with 100% utilization of available water and 4.118 with probability of overflow equal to 0.04. To use this method for determining the capacity of a dam a long sequence of inflow data is to be used and any release rule may be followed.

Table 2.4: Capacity by Specifying the Probability of Overflow (Gamma type inputs )

 Content after release Content in order of magnitude 3.3837 0.5823 1.8890 0.7753 1.6168 0.0068 4.1180 1.2890 1.0068 1.6168 2.1927 1.6250 1.6250 1.6492 2.5982 1.8890 2.5748 1.9283 2.4011 2.0658 3.3682 2.1316 2.0658 2.1011 0.7753 2.5363 4.5400 2.5748 2.5363 2.5805 2.5805 2.5982 1.2890 2.7344 2.1316 2.7974 1.9683 2.9127 1.6492 3.3682 2.7344 3.3837 0.5823 3.4892 3.8762 3.8765 2.7974 4.1180 3.4892 4.5400

Capacity =4.54 with 0.0 as the probability of overflow

Capacity =4.118 with 0.04 as the probability of overflow

2.4 Approaches Based on Linear Programming

The method of linear programming has been applied to water-resources design by Masse and Gibart (1957, 1962), Lee (1958), Castle (1961), Heady (1961) and Dorfman (1961). The principle of such applications can be illustrated [as given in Chow 91961)] by a simple example as:

A single multi-purpose reservoir is subject to analysis by liner programming for its beneficial use of water. The hydrological data used for inflow to the reservoir are in corrected inflow-hydrograph for estimated evaporation and leakage. The initial reservoir content is given or so chosen that the reservoir is full, empty or an optimal condition of operation. The duration of the analysis was assumed to be one year. It was divided into a number of equal interval, say, 12 months. Let be the respective volumes of monthly inflow, be the initial storage capacity and be the volumes of water planned to be released in the respective months. The twelve volumes of water used monthly add up to the total volume of outflow from reservoir, including any unavoidable spills in an average year. Since the total outflow up to the month cannot exceed the sum of inflow volumes up to the month plus the initial storage, the following inequality can be written

Where Also, the total volume of water in storage at any time cannot exceed the maximum useful storage capacity of the reservoir. Thus

Where when , the above inequalities become the equation

The theory of inventory and the theory of storage are often used to determine the optimal policies in water resource. In the United States, Karlin and Koopmans have applied the inventory theory to determine water-storage policies in hydroelectric systems.

“Little used the functional equation approach of the dynamic programming to inventory problems and thus to formulate stochastic dynamic programming model for determining the optimal water-storage policy for an electric generating system.” (Chow, 1964).

Hall and Howell (1970) optimized the size of a single-purpose reservoir by applying dynamic programming to sequentially generated data (Hall & Howell, 1970).

2.5 Approaches Based on Simulated Flows

In many situations the historical flow sequence are not long enough to rely on to determine the long term capacity of the dam and the need arises to generate synthetic flows by simulation which are statistically in distinguishable the historical flows. The term “simulation” is used to mean empirical sampling when the process sampled from the population is a close model of the real system. The benefit of the simulated series is that is possible to generate long series of inflows keeping the historical characteristics fixed. Thus the long sequence will contain more extreme events than the observed values. The generated flows should have same population mean, variance, skewness and correlation coefficient, as of their historical values. The length of the input sequence for determining the dam capacity, should depend on the desired economic life of the dam. Capacity of the dam can then be determined by some established technique.

To generate inflows by simulation, we need some model which generates inflows maintaining the same properties of the historical flows. Several methods for generating synthetic flows have been proposed. In prescribing a model for data generation, it is important to study the behavior of historical data. Flow records at a gagging station in a given time interval may be considered as a hydrologic time series. Many hydrologic time series have no important smooth trend and that can be found using statistical analysis. In case of monthly inflow data, it is reasonable to assume that the series has some seasonal effects or seasonality. Seasonality means, that the characteristics of inflow changes with seasons (or months). For example, with seasonal variation in inflows, mean flow and standard deviation of January will differ from that March. Similarly, inflow during rainy seasons will obviously be different from inflow during winter. This is how seasonality affects inflow mean and standard deviation.

Inflow follows some particular distributions. For monthly inflows, normal and gamma type distribution are common. Annual flows usually found to be log-normal or Weibull or gamma-type distributions. A wide variety of distributions for generating input series have been used and it is found that Gaussian, log-normal and gamma type distribution fit well in most of the cases. Dependence in the input series have been considered by Fiering (1964), Rosesner and Yevjevich (1966) and others. In hydrology, log-normal and the gamma distributions have been the most popular for simulating input sequences.

The use of simulation analysis of water-resources systems began in 1953 by the U.S. Army Corps of Engineers on the Missouri River (US Army, 1957). In this analysis, the operation of six reservoirs on the Missouri River was simulated on the Univac-I computer to maximize the power generation; subject to constraint for navigation, flood control and irrigation specifications.

Other simulation analysis were made by Britain (1960) and Fiering (1962). Britain dealt with the integration of an energy-producing Glen Canyon Dam into already existing power system on the Colorado River in order to approximate maximum return. Earlier, in 1961, Britain conducted probability analysis to the development of a synthetic hydrology for the Colorado River (Britain, 1961). Fiering proposed a method for the optimal design of a single multi-purpose resevior by computer simulation studies of a simple coded model. (Chow, 1964)

Fiering (1963, 1965, 1967) and Svanidze (1964) many investigators have employed stochastic streamflow models to examine the probability distribution of over-year reservior storage capacity.

A variety of monthly stochastic streamflow models have been developed to investigate the combined within-year storage-reliability-yield (S-R-Y) relationship by Lawrance and Kottegoda (1957); Hirsch (1979); Klemes et al. (1981); Stedinger and Taylor (1982a, b); Stedinger et al. (1985).

Hazen (1914) developed the first relationship between the over-year design storage capacity and provide tables of over-year reservoir storage capacity based upon of the coefficient of variation of the inflows and level of development. The limitation is that tables developed for one region are not necessarily applicable to another.

Hurst (1951) developed algebraic expressions which relate the required over-year storage S, to the mean and variance of the inflows as well as the level of development . The relationship of the form:

(2.5.1)

(2.5.2)

Where are constants and k is the Hurst coefficient.

Hurst applied the single-cycling sequent peak algorithm to sequences of streamflow, precipitation, temperature, tree ring and varve records to obtain single estimates of S which in turn were used to obtain estimates of the constants a, b and k using graphical curve fitting procedures. The expressions on equations (2.5.1) and (2.5.2) are only reasonable approximations over the range considered by Hurst. Actually m could be any non-negative number since are non-negative and the demand as a fraction of the mean annual flow is usually in the interval (0, 1).

Gomide (1975) and Troutman (1978), both derived the probability density function of S and its first two moments , which result from application of the single-cycling sequent peak algorithm to realistic models of annual streamflows. For example, when the annual streamflows, Q, are normally distributed and follow a first-order autoregressive model:

where the are independent normal random disturbances with mean 0 and variance 1.

Gomide (1975) derived the pdf of S and its first moment. The resulting expressions were so complex that Gomide presented his results graphically for equal to 0.0, 0.2, 0.5; m equal to 0 (full regulation) and N ranging from 0 to 100 years.

Troutman (1978) derived the of the asymptotic distribution of S when m=0 (full regulation) and inflows are described by an AR(1) log-normal model:

where , are independent normal disturbances with mean 0, variance 1 and are the mean, variance and serial correlation of the log transformed streamflows. No analytical expressions have been developed for the pdf or moments of the stedy-state required storage obtained using the double-cycling sequent peak algorithm.

Vogel and Stedinger (1987) presented general over-year Storage-Reliability-Yield (S-R-Y) relationship in an analytical form. They provided approximate but general expressions for evaluating the quantiles of the distribution of over-year storage capacity as a function of the inflow parameters the demand level , and the planning period for log normal inflows.

Oguz and Bayazit (1991) studied with the properties of Critical Period (CP). The probability distribution function of the length of the critical period, defined as the time interval during which an initially full reservoir is completely emptied, can be determined using the storage theory when the inflows are considered as discrete variables. For continuous flows, Oguz and Bayazit solved it by simulation. Simulation experiments with normally and log-normally distributed inflow have shown that the mean and standard deviation of the CP length increased rapidly (more than linearly) with the reservoir size and with the skewness of the inflows. Mean and standard deviation of CP increases with the level of regulation (release) for normally distributed flows, but decreases for log-normal flows. They decreased only slightly as the serial dependence of the inflow increases.

Bayazit and Bulu (1991) determined the probability distribution and parameters by simulation using streamflow series generated by four differents models.

1. Normal first-order autoregressive

2. Log- normal first-order autoregressive

3. Normal first-order autoregressive moving average

4. Log- normal first-order autoregressive moving average

They showed that the reservoir capacity standardized by subtracting the mean inflow and dividing by the standard deviation of inflow follows the three parameter log-normal distribution with a constant lower bound -2.0.

Phein (1993) estimated storage capacity of dams where annual flows are assumed to be gamma distributed, following the existing studies on normal and log-normal inflows. Particularly, Phein (1993) described the gamma first-order autoregressive GAR (1) model along with suitable methods to produce the required exponential, Poisson, and gamma variates. He obtained the approximate formula for mean and standard deviation of capacity for a wide range of reservoir life n, t