23 Reservoir sizing

Deeksha Katyal

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Introduction

 

The term “Reservoir” has its origin from French word reservoir which means a storehouse. A reservoir is constructed with the objective of storing water, which can be used to serve several purposes. A reservoir may exist naturally or constructed artificially using dam or bar to stock water. Demand of water at certain point may be less than the total inflow (at times of excess flow), and sometimes demand may be higher than the inflow (at times of low flow) resulting in the imbalance between demand and supply. Reservoirs are constructed with the objective to balance surplus and deficit periods of inflow. During the excess water flow periods, reservoirs store water to prevent flooding and when there is an inadequate natural water flow that cannot meet water demands then water is released. Therefore, we can say that “A reservoir is storehouse of water in periods of excess water flow so that it can meet certain level of demands at times of inadequate water flow.

 

The solution of the problem of reservoir sizing does not depend on single factor but on multiple factors such as variations in the stream flow, size of the demand target group and the efficiency of the reservoir to meet the demand. Even though there are various ways to design the problem of reservoir sizing but it can be simply compacted as to find out the association between inflows characteristics, reservoir capacity, controlled release and aspired performance. There are various terminologies associated with the reservoir storage analysis, which are as follows:

 

Active storage: active storage, also known as storage capacity is the volume of water stored above the level of minimum offtake. It is the volume of water in the reservoir subtracted from the dead storage.

 

Within year storage: many small reservoirs fills up and spill on the average several times a year. These reservoirs are constructed to provide water over a short drawdown period of only a month or two of a low flows. The estimation of storage in this case is termed as within year storage.

 

Carryover storage: when the reservoir fills up and spill only every few years on the average, the water stored at the end of an year is carried forward to the another subsequent year and termed as carryover storage.

 

Dead storage: It is the volume of water present below the level of lowest offtake.

 

Yield: Yield can be defined as the synchronized flow supplied from the water reservoir during a given period of time.

 

Demand: Demand is the desirable yield of the reservoir.

 

Spill: It is the term used to signify the unrestrained discharge of water from the reservoir. It used to happen when storage of water exceeds the total demand.

 

Release rule or operating rule: Generally, the volume of water released from the reservoir is the function of total demand or water requirement of the consumers. Nevertheless, there are certain periods when either the storage volume is so low that could not meet the certain demands or that only the part of requirement is released from the reservoir. The way in which releases are controlled or regulated is called release rule or operating rule of the reservoir.

 

The reservoir capacity estimated on the basis of mass curve analysis has to be adjusted to account for dead storage, evaporation losses and carryover storage. Life of the reservoir is mainly dependent on the silting capacity. According to Khosla et al, silting capacity in Indian conditions may be taken roughly as 3.6 ha-m per sqKm/year. For Ramganga dam in U.P. the estimated silting rate is 4.3 ha-m/ sqkm/year. This means that there will be a loss of 19% in the dead storage and loss of 4.3% in the live storage.

 

It is extremely important to know, that what should be the optimum size of the reservoir so that it can meet the given level of demands. The reservoir sizing problem includes determination of requisite storage capacity of reservoir when inflow sequence and demand sequence is known. For example, if we consider a stream with known sequence of inflow and we want to construct a reservoir to meet known demands, then what should be the minimum storage capacity that can meet given demand? Let’s consider for the time period t, flow sequence is Qt and demand sequence is Dt and If Qt is always more than Dt for all t, then it is obvious that there is no need of reservoir, we can directly withdraw water from stream and can be used as per our demand. The need of reservoir comes up when we want to redistribute the water among numerous time periods. In some periods Qt is equal to or less than Dt and therefore there is a need to store water. Generally, in redistributing the flow, we store surplus inputs and supply it during deficit periods. In order to identify the storage capacity of reservoir, it is essential to identify the periods in which the maximum deficit occurs and this deficit is addressed with the storage.

 

If total supply for all time periods is greater than or equal to total demand, i.e. ƩQt ≥ ƩDt, therefore it is possible to meet the required demand by proving certain capacity of reservoir so that the time distribution is maintained. If total demand for all time periods is more than the supply i.e. ƩQt ≤ ƩDt, therefore in this condition, it is not feasible to fulfill the demand during all the time periods regardless the storage capacity of reservoir.

 

The hitch of reservoir sizing can be addressed by determination of requisite storage competence of the reservoir for known inflows and demands in a series of periods. Reservoir capacity can be determined by using various methods; some of them are as follows:

 

1.      Ripple Diagram or mass curve method: Mass curve method is the conventional and most widely used method for the estimation of reservoir’s size. It was developed by W. Rippl in 1883. A mass inflow or cumulative inflow curve is the plot of cumulative inflow versus time and demand curve or the outflow curve is the plot of cumulative demand versus time. This method works well if the rate of outflow is constant. The method involves following steps:

 

(a)   Find out the cumulative inflow (supply) by adding inflow values for the considered period and plot the cumulative inflow with respect to time in scale.

 

(b)   Find out the cumulative outflow (demand) by adding demand values and plot cumulative outflow with respect to time in scale. Outflow per unit time can only be plotted if there is a constant outflow throughout the considered period.

 

(c)    Draw lines parallel to the demand curve tangential to the mass curve at the starting and end of the deficit period.

 

(d)   Required storage capacity of the reservoir is the maximum of the vertical intercepted ordinates the two tangent drawn to a peak and through consecutively.

 

Thus, mass curve method involves the estimation of highest positive difference between the sequence of outflow and inflow and the maximum deficit volume associated with inflow and outflow is the required capacity of the reservoir. There are certain limitations associated with this method as

  • It consider outflow to be constant which is not possible every time because demands are often seasonal.
  • Storage capacities estimated by mass curve procedure increases with increasing length of record. Therefore, it is difficult to relate storage size to economic life.
  • It is not possible to calculate a storage size for a given probability of failure.
  • The method does not take into account the net evaporation losses.

 

Mass curve method is suitable when small time period data is considered.

 

2.      Sequent peak analysis: Sequent peak analysis is the modified form of mass curve analysis which is suitable for inflow data of large time period. In sequent peak algorithm also, rule remains the same. In sequent peak analysis, Critical period (period in which maximum deficit occurs) is captured and maximum deficit is calculated. The method involves the calculation of maximum collective deficit over adjacent series of critical periods and then finds out maximum of these collective deficits. It may be possible in the large data sets, that critical periods may occur at the end of data sets and then in that case an assumption of recurrence of inflow sequence has to be taken into account.

 

The mathematical expression of sequent peak algorithm is defined as:

 

Kt = Kt-1+Dt-Qt (if positive)

Otherwise Kt=0

 

If Qt is very high as compared to Dt, and expression gives negative result then it denotes that demand (Dt) can be met with available inflow (Qt) and there is no need of extra storage.

 

Here,

 

Kt is the requisite storage capacity of the reservoir at the end of period t.

Kt-1 is the requisite storage capacity of the reservoir at the end of previous period.

Dt is the demand or release during the time period t.

Qt is the inflow during time period t.

Dt-Qt collectively denotes the deficit during time period t.

 

Setting K0 equals to 0, the method involves the calculation of Kt using above mentioned expression for up to twice the total length of record time period. The maximum of all Vt is the requisite storage capacity for the given demands (Dt) and inflows (Qt).

 

The calculation of second cycle is not required if the value of Kt is zero at the end of the last period of first cycle.

 

Limitations and attributes:

 

The sequent peak algorithm method was constructed for use with replicates of generated data rather than a single historical record therefore there are several limitations associated with it. Contrasting the mass curve procedure, it can handle variable drafts so long as they can be specified without reference to the reservoir content, but at the same time it does not allow to calculate capacities other than those that satisfy the rank 1 low flow sequence. The use of generated flows allows a probability of failure to be indirectly obtained. Non inclusion of evaporation losses and non adaptability of algorithm to multi reservoir system are certain limitations associated with sequent peak algorithm method.

 

Example: The inflow and demand data for six period sequences is given below; compute the required capacity of a reservoir.

Period (t) 1 2 3 4 5 6
Inflow(Qt) 5 7 6 4 2 0
Demand (Dt) 6 0 4 7 2 5

 

Solution: First we will check the total inflow and total demand and if total demand is higher than the total inflow, then it is impossible to satisfy the demands regardless the size of reservoir. Here, the total inflow is equal to the total demand i.e. 24 units.

 

Therefore, we have to redistribute the inflows in order to meet the demands in all time periods. We calculate required storage (Kt), considering Kt-1 as 0 at starting:

If at the end of first cycle Kt would comes out 0 then we need not to repeat the next cycle but in this case it is 8 therefore, we have to repeat the cycle considering that the critical period would have occurred at the end of sequence. The maximum value of Kt is the required capacity of reservoir, which in this case is 9.

 

Therefore,

 

Reservoir capacity= Max{Kt} = 9

 

3.      Linear programming method: It is the substitute and elegant method to sequent peak analysis. It takes in to account the evaporation losses and varying demand. Whenever we talk about reservoir systems, the storage continuity problem needs to be met from one time period to another time period and this is the main problem in LP (Linear programming)

 

Optimization model for active storage is= Ka (active storage capacity) Subject to

 

(A) Mass balance

 

St + Qt– Dt-Lt = St+1

 

Where,

 

St is the storage at the beginning of the time period t.

 

Qt is the inflow during time period t

 

Dt is the demand or release during time period t

 

Lt is the losses taken place during the time period t, and

 

St+1 is the storage at the end of time period t+1

 

(B)  Maximum active storage

 

St ≤ Ka (storage should always be less than or equal to capacity)

 

(C)  Non negativity St ≥ 0, Ka≥ 0

 

Example: Let’s consider the same example as we solve in sequent peak algorithm Linear programming problem can be written as-

 

Objective function is to minimize K

 

Subject to continuity equations (St-1=St+Qt-Rt) as follows:

 

For t1S1 + 5 – 6 = S2

 

t2S2 + 0 + 7 = S3

 

t3S3 + 6 – 4 = S4

 

t4S4 + 4 – 7 = S5

 

t5S5+ 2 – 2 = S6

 

t6S6 + 0 – 5 = S1

 

Considering set of constraints, St ≤ Ka (storage at any time period must be less than or equal to K that we have obtained from optimization problem)

 

In this case, we can see

 

K = 9 (solution same as sequent peak algorithm)

 

S1 = 1 (optimal solution)

S2 = 0

S3 = 7

S4 = 9

S5 = 6

S6 = 6

 

Therefore, reservoir capacity constraints are satisfied as

 

S1 ≤ K; S2 ≤ K; S3 ≤ K; S4 ≤ K; S5 ≤ K; S6 ≤ K

 

Large LP problems can be solved very efficiently using LINGO – Language for INteractive General Optimization, LINDO Systems Inc, USA.

 

Reservoir performance criteria: Performance estimation of the artificial reservoir is extremely important in order to know how well a reservoir will perform under expected supply demand and hydrologic conditions during its operational life. The aim of the methods implied to account for the expected performance of a reservoir is to illustrate particular phase of the inadequate operating conditions particularly during low flow periods.

 

To measure the performance of the reservoir, various methods have been used but the most simple and widespread method is reliability. The definitions used to describe the term reliability are diverse but it can be simplified as the assessment of the constancy of the system’s requirement being met. The most common definition of reliability (Rt) explains it as the percentage of time units in which the specific demand is met and can be given as:

Rt = (1 − Nr/N) ∗ 100

 

Where,

 

Nr = number of periods where specified demand is not met

N = total number of time periods under consideration, and

Nr/N Collectively can be called as the empirical probability of failure of demand being met

 

However, this impression agrees with the empirical probability and does not account for theoretical probability. This concept of reliability can also be called as time based reliability.

 

The information we get from the above mentioned expression is uncertain as it hides two important aspects governing the behavior of the system i.e. the duration of the water shortage period and extent deficit of the water associated with each shortage period. To cope up with these problems an alternative definition of reliability is given by McMahon & Mein and McMahon and Adeloye which is defined by the following expression:

 

 

Where,

 

Dt = target demand

D’t = Actual supply during the tth time period

N = Total number of time period under consideration, and

Nr = Number of periods where specified demands are not met.

 

An additional measure to evaluate reservoir performance is vulnerability which (here) is the function of severity of the shortfalls and signifies that the period with the largest shortfall is the most severe in terms of its impacts on water supply. Vulnerability (V), of a reservoir can be defined as the average of the maximum shortfall occurring in the end of incessant failure period sequence.

Where,

 

Nfs = Total number of failure sequence

Max(Shk) = Maximum water shortage in the Kth failure sequence, and

Dk = Supply demand during the same sequence.

 

The recovering ability of the reservoir from failure is called as resilience (Φ). The most widely used definition of resilience was proposed by Hashimoto et al, (1982):

 

Φ= Nfs/Nr

Where,

 

Nfs = Total number of failure sequence and

Nr = Number of periods where specified demands are not met

 

Various operational characteristics of the reservoirs are the function of time based reliability, volumetric reliability, vulnerability and resilience. For example, time based reliability illustrates the rate of occurrence of failures, volumetric reliability and vulnerability are the function of degree of the shortfall volumes and resilience explains the interval of the water shortfall events.

 

It is necessary to consider all abovementioned characteristics while assessing the performance of the reservoir. Here, it is essential to cite that not all the storage yield measure methods should be subjected to an analysis under these characteristics. For example, cumulative flows curve method which intends a reservoir for the full supply of the target demand.

 

Uncertainty coupled with estimation of storage capacity:

 

If the stipulated outline of a reservoir is higher than the definite percentage of the average twelve-monthly flow, the reservoir will need resort to over-year regulation. This signifies that the intended storage capacity should permit for the inflows of a wet year to be relocated to dry years. The incidence of over-year regulation means s that within-year storage is not only the function of the inflow of that year, but also of the storage that is passed on from the previous year. As a result, an inflow time series make up a solitary event in a reservoir storage trouble. Under this understanding, the use of a single stream flow series, as is the historical sample, cannot result in more than one estimate of the reservoir capacity.

 

The application of a storage-yield procedure to each inflow series, having defined a specific yield and performance, produces a single estimate of the reservoir storage capacity. A collection of estimates of such capacities enables the application of statistical analysis to the definition of the design storage capacity. By this approach, the probability distribution of the storage volumes can be identified. Then, to each collection of estimates of the storage capacity, C, has a probability density function, f(c), and a probability distribution function F(c) = P(C ≤ c), the reliability is the probability that the required storage capacity required not be greater than c∗, hence reliability = F(c∗). The probability of failure is the probability that the storage capacity required will be greater than C∗, hence risk = 1 − F(c∗).

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References

  • Adeloye, A. J., Montaseri, M., and Garmann, C., 2001. Curing the misbehaviour of reservoir capacity statistics by controlling shortfall during failures using the modified sequent peak algorithm. Water Resources Research, 37(1), 73–82.
  • Bayazit, M., and Bulu, A., 1991. Generalised probability distribution of reservoir capacity. Journal of Hydrology, 126, 195–205.
  • Hashimoto, T., Stedinger, J. R., and Loucks, D. P., 1982. Reliability, resiliency and vulnerability criteria for water resource system performance evaluation. Water Resources Research, 18(1), 14–20
  • McMahon, T. A., and Adeloye, A. J., 2005. Water Resources Yield. Littleton: Water Resources Publications.
  • McMahon, T. A., Adeloye, A. J., and Zhou, S.-L., 2006. Understanding the performance measures of reservoirs. Journal of Hydrology, 324, 359–382.
  • Montaseri, M., and Adeloye, A. J., 1999. Critical period of reservoir systems for planning purposes. Journal of Hydrology, 224, 115–136
  • Stedinger, J. R., Vogel, R. M., and Foufoula-Georgiou, E., 1993. Frequency analysis of extreme events, Chapter 18. In Maidment, D. R. (ed.), Handbook of Hydrology. New York: McGraw-Hill.