Research Article Volume 3 Issue 6
^{1}Department of Crop Production, Niger State College of Agriculture Mokwa, Nigeria
^{2}Department of Agricultural and Bioresources Engineering, Federal University of Technology Minna, Nigeria
^{3}Department of Agricultural and Bio resources Engineering, Federal University of Technology Bauchi, Nigeria
Correspondence: Mohammed J Mamman, Department of Crop Production, Niger State College of Agriculture Mokwa, Nigeria
Received: October 24, 2019  Published: November 20, 2019
Citation: Mamman MJ, Matins OY, Abdullahi AS, et al. Application of vogel and stedinger (V–S) empirical procedure to develop storage reliability yield relationships for Kainji reservoir system. Int J Hydro. 2019;3(6):461465. DOI: 10.15406/ijh.2019.03.00211
Storage reservoirs tend to be large and complex systems requiring equally complex mathematical models to simulate their behaviour. What are lacking are simple, reasonably accurate methods which give insight into a wide range of reservoir storage system characteristics and reliability indices. Such methods would be useful for the education of water supply analysts. This study tends to apply the Vogel Stedinger (VS) procedure to simulate and develop a mathematical model to predict reservoir storages of a within year system. Inflows, out flows and reservoir levels of Kainji reservoir were obtained between 1991 and 2014. Reservoir storages were determined and Vogel Stedinger parameter variables (S_{p} is the p^{th} quartile of the distribution of required reservoir capacity for 100% failurefree operation over a specified planning period N, Z_{p} is the standardised Normal variate at p%, is the standard deviation of annual stream flows, and are mean and standard deviation of the logarithms of the storages and is the lower bound of the storage) determined and were used to develop an empirical model for predicting storages and other VS parameters. The R^{2} Value (0.72) of the model indicates the strength and reliability of the model. The model could be useful to reservoir managers in overcoming extreme events of flood and drought.
Keywords: reservoir capacity, mathematical model, simulate storages, and reliability
The dynamics in hydrology which results from climate change, leads to variation in key reservoir variables like the inflow, storage and outflow are some problems that constitute reasons why, water managers face the challenge of making available adequate quantities of water for drinking, agricultural and other uses and also because of the geometrically increasing population pressure and socioeconomic development, increase the needs and demands for particular water flows.^{1} Challenges remain widespread and reflect severe problems in the management of water resources in many parts of the world. These problems will intensify unless effective and concerted actions are taken.^{2} The role of waterstorage reservoirs, therefore, is to impound water during periods of higher flows, thus preventing flood disasters, and then permit gradual release of water during periods of lower flows.
Storage Yield Relationships (S Y R) are the traditional tool used by water resource engineers to determine the required capacity of a storage reservoir to maintain a prespecified reservoir release. Essentially, two schools of thought exist regarding the development of Storage Reservoir Yield (SRY) relationships. In the USA, SRY relationships are usually based on an interpretation of reliability which depends upon the most critical draw down period of a reservoir over its planning horizon. According to Vogel,^{3} basically, these methods utilize the automated equivalent of Ripl's mass curve approach (M C A), known as the sequent peak algorithm,^{4} in conjunction with stochastic streamflow models to obtain the probability of nofailure reservoir operations,(p) corresponding to a specific reservoir capacityyield combination. Alternatively, in Australia and elsewhere, a common approach to estimating the S. R.Y relationship is to determine the steadystate probability (S S P) of failure, (q) corresponding to a specific reservoir capacity yield combination.^{5,6} Essentially, reliability gives a measure of certainty that a given yield can be met without failure. The reliability index is between 0 and 1 and can be expressed in one of the three ways: (1) annual, (2) time based (3) volumetric. Understanding the reliability and uncertainty associated with water supply yields derived from surface water reservoirs is central for planning purposes.^{7}
The Voggel and Stedingar empirical procedure estimates the expected valued and variance of reservior storage capasity assuming both the inflows to the reservior and the standardised storages are lognormaly distributed. Although the method is based on empiricall procedure, its application is straightforward.^{8,9 }While behaviour (or simulation) analysis is a simple and visual procedure to estimate storage capacity and is not restricted by the characteristics of the inflows; unlike some of the analytical approaches, evaporation and operating rules that are a function of reservoir storage levels can be easily taken into account.^{8}
Vogel and Stedinger,^{9} showed that for a reservoir system fed by AR (1) lognormal stream flows, the standardised storage C (capacity divided by the standard deviation) for a failure free operation (S.P.A approach) is a random variable described by a three parameter Lognormal distribution. The form of the V.S relationship is:
${S}_{p}=\sigma [{\vartheta}_{s}+exp({\mu}_{1}+{Z}_{p}{\sigma}_{1})]$ (1)
Where S_{p} is the p^{th} quartile of the distribution of required reservoir capacity for 100% failurefree operation over a specified planning period N, Z_{p} is the standardised Normal variate at p%, is the standard deviation of annual stream flows, and are mean and standard deviation of the logarithms of the storages and is the lowerbound of the storage.
The Vogel and Stadinger empirical procedure estimates the expected value and variance of sequent peak algorithm (SPA) reservoir storage capacity assuming both the inflows to the reservoir and the standardized storages are lognormally distributed. Although the method is based on six equations and 22 parameters its application is straightforward.^{8,9} Being an empirical procedure as reported by McMahon^{8} and Vogel,^{9} it should only be appllied within the range of values of m, Cv, N and q that were used to define the 22 parameters. Faith and Richard^{10} reported that understanding the reliability and uncertainty associated with water supply yields derived from surface water reservoirs is central for planning purposes. In their sudy they used a global dataset of monthly river discharge, to introduce a generalized model for estimating the mean and variance of water supply yield, Y, expected from a reservoir for a prespecified reliability, R, and storage capacity, S assuming a flow record of length n. The generalized storage–reliability–yield (SRY) relationships reported by them have numerous water resource applications ranging from preliminary water supply investigations, to economic and climate change impact assessments.
According to Longobardia et al., a number of studies presented in the recent past, have demonstrated that the combination of a simulation approach coupled with a performance assessment via indices evaluation is a valuable tool to measure the sensitivity of reservoirs to climate variability and prolonged droughts.^{11‒13} An example indicates how Faith and Richard^{10} used generalized SRY relationship combined with a hydroclimatic model to determine the impact of climate change on surface reservoir water supply yields. They document that the variability of estimates of water supply yield are invariant to characteristics of the reservoir system, including its storage capacity and reliability. Standardized metrics of the variability of water supply yields were shown to depend only on the sample size of the inflows and the statistical characteristics of the inflow series. Abdesselam and Sylvain^{14} used empirical models to compares the two main statistical approaches to calculate the effective discharge (the empirical method based on histograms of sediment supply by discharge classes and an analytical calculation based on a hydrological probability distribution and on a sediment rating curve) to a very simple proxy: the halfload discharge, i.e. the flow rate corresponding to 50% of the cumulative sediment yield. Three types of discharge subdivisions were tested. In the empirical approach, two subdivisions provided effective discharge close to the halfload discharge.
According to Thomas et al.,^{15} Annual and monthly streamﬂows for 729rivers from a global data set were used to assess the adequacy of ﬁve techniques to estimate the relationship between reservoir capacity, target draft (or yield) and reliability of supply. The techniques examined were extended deﬁcit analysis (EDA), behaviour analysis, sequent peak algorithm (SPA), Vogel and Stedinger empirical (lognormal) method and Phien empirical (Gamma) method. In addition, a technique to adjust SPA using annual ﬂows to account for withinyear variations were assessed. Of their nine conclusions the key ones were, that, EDA is a useful procedure to estimate streamﬂow deﬁcits and, hence, reservoir capacity for a given reliability of supply. Secondly, that the behaviour method is suitable to estimate storage but has limitations if an annual time step is adopted. Thirdly, that in contrast to EDA and behaviour analysis which are based on time series of ﬂows, if only annual statistics are available, the Vogel and Stedinger empirical method compares favourably with more detailed simulation approaches.
Materials
The hydrological data employed for this study was reservoir storages.
The study area
Geographically, Kainji hydroelectric dam is located in New Bussa town now headquarter of Borgu local government area of Niger State, Nigeria. The lake created behind the dam span between latitude 9°8’ to 10°7’ and between longitude 4°5’ to 4°7’ E with reference point 9.54N and 4.38E northwest of the Federal Capital Territory (FCT, Abuja).^{16}
Hydrology of the Niger river system
The average rainfall at the headwaters of Niandan and Milo rivers at the source of the Niger at the Fouta Djallon Mountains in Guinea and its exit to the sea in Nigeria is 2200mm. The river flow regime is characterized by two distinct flood periods occurring annually namely the White and Black floods. The black flood derives its flow from the tributaries of the Niger outside Nigeria (flow lag October to May) and arrives at Kainji reservoir (Nigeria) in November and lasts until March at Jebba after attaining a peak rate of about 2,000m^{3}/sec in February.^{17} The White flood is a consequent of flows from local tributaries especially the SokotoRima and Malendo river systems. The White flood is heavily laden with silts and other suspended particles (flow lag June to September) and arrives Kainji in August in the preKainji Dam River Niger having attained a peak rate of 4,000 to 6,000m^{3}/sec in SeptemberOctober in Jebba. The critical low flow period into the Kainji reservoir is March and July each year (Figure 1).
Data base
The data base for this study constitutes reservoir storages from 1991 to 2014.
Vogel stedinger (VS) empirical procedure
The VS procedure was employed to develop SRY relationships, it is based on the premise that a reservoir system fed by AR (1) lognormal inflows, the standardized storage capacity C for a failure free operation is a random variable. The general VS relationship was employed:
${S}_{p}=\sigma [{\vartheta}_{s}+exp({\mu}_{1}+{Z}_{p}{\sigma}_{1})]$ (2)
Where:
${u}_{1}=\mathrm{ln}\left[({\mu}_{s}{\vartheta}_{s})\left(1+\frac{{\sigma}_{s}^{2}}{{({\mu}_{s}{\vartheta}_{s})}^{2}}\right)0.5\right]$ (3)
${\sigma}_{l}^{2}=\mathrm{ln}\left[({\mu}_{s}{\vartheta}_{s})\left(1+\frac{{\sigma}_{s}^{2}}{{({\mu}_{s}{\vartheta}_{s})}^{2}}\right)\right]$ (4)
Where Sp is the p^{th} quantile of the distribution of required reservoir capacity for 100% failurefree operation over a specified planning period N, z_{p} is the standardised Normal variate at p%, is the standard deviation of annual streamflows, and are mean and standard deviation of the logarithms of the storages, is the lower bound of the storage (Table 1).
Month 
Inflow (m3/sec) 
Antecedent reservoir storage (Mm3) 
Antecedent reservoir level(m) 
Antecedent reservoir release (Mm3) 
Apr 
131.95 
12634.34 
137.13 
1036.625 
May 
81.46 
12587.79 
134.94 
925.5417 
Jun 
122.26 
12560.91 
132.85 
786.3333 
Jul 
655.99 
12556.17 
132.07 
626.375 
Aug 
1862.46 
12696.2 
133.21 
766.25 
Sep 
2743.94 
13141.93 
136.88 
1246.917 
Oct 
2030.27 
13803.16 
140.26 
1443.167 
Nov 
1548.73 
14271.61 
140.64 
1112.417 
Dec 
1708.73 
14602.58 
140.75 
1138.583 
Jan 
1617.77 
12000 
141.21 
1186.5 
Feb 
1334.49 
12331.41 
140.37 
1177.042 
Mar 
642.57 
12570.94 
139.71 
1064.542 
Table 1 Reservoir working table for Kainji Dam
Vogel stedinger (V–S) relationships
The result of the Vogel Stedinger relationship is as presented below:
The VS procedure was employed to develop SRY relationships, it is based on the premise that a reservoir system fed by AR(1) lognormal inflows, the standardized storage capacity C for a failure free operation is a random variable. The developed SRY model was used to simulate the reservoir storages. Figure 2 depicts the observed and simulated storages. The developed models’ extent of correlation was also tested with a value of 0.72 imperatively showing strong correlation between the observed storages and the simulated storages. It also shows that the model is strong (Table 2).^{18}
Model type 
Developed model 
R2 
VS 

0.72 
Table 2 Developed models for SRY relationships
VS, Vogel–Stedinger; Sp, Required storage capacity; Zp, standardize normal variate.
The flowing conclusions are drawn from the study;
The authors declare that there are no conflicts of interest.
None.
©2019 Mamman, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work noncommercially.