2.1. Location
The ArjoDedessa Dam Project is located in Ethiopia, within the boundaries of Jimma and East Wollega Zones in the Oromia Regional State. The project area is located between 80°30'00'' to 80°40'00'' N latitude and 36°22'00'' to 36°43'00'' E longitude. Dedessa river is one the tributaries of Abay River basin. Arjo Dedessa dam is constructed across Dedessa River at specified location of the coordinate point 80 31' 12” N latitude and 36o 40' 1.4” E Longitude.
2.2. The method used for analysis
2.2.1. Reservoir Yield
The reservoir storageyield function determines the minimum active storage capacity needed to maintain a constant release rate for a specific sequence of reservoir inflows (Anagnostopoulos and Papantonis, 2007; Nkwonta et al., 2017). The analysis of reservoir yield capacity is conducted to determine the minimum storage volume needed to meet specified demands with a predetermined level of reliability during the planning stage. Conversely, it can also be used to reevaluate the water demand that an existing reservoir can meet. Mass diagrams, sequent peak analyses, and optimization are three methods that can be used to define these functions (Field and Lund, 2006; Loucks and van Beek, 2017b; Mukheibir, 2013). An optimization model based on linear programming (LP) was utilized to establish the yield function for the Arjo Dedessa reservoir's storage capacity. The linear programming (LP) model developed to maximize the reliable yield (Y) of a reservoir with a given active storage capacity is as follows:
Objective function
Maximize Y
This maximum yield is constrained by the water available in each period, and by the reservoir capacity. Two sets of constraints are needed to define the relationships among the inflows, the reservoir storage volume, yields, excess release, and the reservoir capacity. The first set of continuity equations equate the unknown final reservoir storage volume St + 1 in period t to the unknown initial reservoir storage volume St plus the known inflow Qt, minus the unknown yield Y and excess release, Rt, if any, in period t.
The constraint on the balance of mass flow or continuity equation is written for each period t as
\({S}_{t}+{Q}_{t}Y{R}_{t}={S}_{t+1} \forall t=\text{1,2},\dots .12\) Eq. 2.1
\({S}_{t} \le K \forall t=\text{1,2},\dots .12\) Eq. 2.2
Where: 
St + 1 = the final reservoir storage volume at period t.
St = the initial storage volume at period t.
Qt = Inflow to the reservoir at period t.
Rt = Excess release from the reservoir at period t.
K is the active storage capacity of the reservoir at different level
Y = the yield of the reservoir
2.3. Approximation of the hydropower potential using the optimization model.
Optimization methods are fundamental tools that are valuable in reservoir management education. The problem of optimum reservoir operation involves finding the optimal release, reservoir storage, and downstream reachrouting flows based on predicted inflows. The period phase in these models can be hourly, daily, weekly, monthly, or yearly. When hydropower processes are combined with flood control or other uses, daily and hourly time steps are advantageous. To model the optimization, the following steps were taken: determination of the reservoir's yield capacity, estimation of the probabilities of reservoir inflow exceedance, establishment of a relationship between the generating head and reservoir storage, development of an optimization model for the binary probable arrangements shown in Figs. 2 and 3, and evaluation of the model's output by considering various scenarios.
2.3.1. Estimation of the reservoir inflow with various probabilities of exceedance.
The reservoir inflow was fit with probability distribution method based on the monthly mean of the historical data and the extent of data. The reservoir inflow at the dam site, the best probability distribution method was evaluated using Easy fit software. The reliability of 50%, 75%, and 90% of the probability of inflow to the reservoir is determined using the equation below.
\(Q= \stackrel{}{Q}+\delta K\) Eq. 2.3
Where:
Q = flow of a particular month (Mm3),
\(\stackrel{}{Q}\) = Mean flow (Mm3)
σ = Standard deviation (Mm3) and
K is a constant, depending on the probability, reservoir inflow volume of 50%, 70%, and 90%, and the probability of exceedance.
2.3.2. Generating the head as a function of the reservoir’s storage equation.
The elevation and storage data from the topographical map of an Arjo Dedessa reservoir’s impounding area and the assumed tail race elevation will use to obtain the relationship between the head and reservoir storage. The tailrace elevation will be deducted from the reservoir elevation to obtain a generating head.
2.3.3. Formulation of the problem of reservoir operations
1. System and description of the problem
The main features of the reservoir system can be briefly summarized as follows:

An Arjo Dedessa reservoir has a catchment area of 5632.64km2 and a live storage capacity of 1515.8 Mm3.

The purposes of this reservoir system are irrigation, water supply, and ecological releases

The hydropower was proposed as an additional scheme. (i.e., integration of a hydropower turbine for energy generation).

Energy production requires water to drive the turbine and can be released to serve the purposes the reservoir will be designed.
2. Expansion of a longrange operational guide for the Arjo Dedessa Dam
The linear programming technique is one of the further most extensively used mathematical programming methods in water resources planning and management due to its suitability, particularly in the optimal allocation of scarce resources for various purposes. The objective function will be the expansion of energy, while the reservoir characteristics, irrigation requirements, ecological needs, and the non – negative of the hydropower releases are included in the constraints.
3. System and problem description for the scenario of independent hydropower release
In this scenario, the Arjo Dedessa reservoir; the irrigation, hydropower system and ecological release are positioned in the plan below. The release assigned for the hydropower in this scenario is independent of releases for other uses. This plan is portrayed in in figure below, However, in this case, the hydropower is to be integrated so that a dispersed release of water is assigned for the hydropower scheme.
Objective Function
The objective function, along with the constraints, constitutes the linear programming formulation. The objective function of this optimization process is the maximization of the total annual energy generation TE,
\(TE=Max \sum _{t}^{T}\left({E}_{t}\right)\) Eq. 2.4
The total annual energy generation is a summation of the twelvemonthly energy generations Et and Et, which are calculated
\({E}_{t}=2.73{HP}_{t}{H}_{t}e \left(MWH\right)\) Eq. 2.5
Where: 
HPt = Release for hydropower generation (Mm3)
Ht = Generating head (m)
T = Monthly period t = 1, 2, T = 12
e = overall efficiency of the plant.
Technically available power is obtained by including losses due to conveyance, plant losses such as entrance loss, rack loss, generator, and turbine loss, etc. For SHP, the overall efficiency, e, of 50% is multiplied with the theoretical power to obtain the technically available power. The low overall efficiency is as a result of the following losses(Frey and Linke 2002)(Ellabban, AbuRub, and Blaabjerg 2014)(Kaygusuz and Sarı 2003)(Paish 2002)(Zarfl et al. 2015)
Power output is obtained after all these losses are considered.
Power output = 0.5 * power input
Therefore, overall efficiency, e for SHP = 0.5. Due to the above reason, the overall efficiency of the power plant is 50% used for this study.
Model Constraints
The maximization of the total annual energy generation is subject to the following constraints:
I. Constraints on release (yield) from the reservoir
The upper limit (Monthly maximum yield from storage yield analysis)
\({TR}_{t}={IR}_{t}+{ER}_{t}{+HP}_{t}\le Upper limit. \forall t=\text{1,2},\dots .T, T=12\) Eq. 2.6
Lower limit (Ecological demand, minimum release)
\({TR}_{t}={IR}_{t}+{ER}_{t}+{HP}_{t}\ge Lower limit. \forall t=\text{1,2},\dots .T, T=12\) Eq. 2.7
Where:  TRt: Total monthly releases (Mm3)
ERt: Monthly ecological releases (Mm3)
IRt: Monthly irrigation Releases (Mm3)
Hpt: Monthly Releases for hydropower generation (Mm3)
II. Ecological release constraint
The release of ecological requirements should be sufficient to meet the ecological demands
\({ER}_{t}\ge {EC}_{t}. \forall t=\text{1,2},\dots .T, T=12\) Eq. 2.8
Where:
ECt: Monthly ecological demand (Mm3)
III. Irrigation release constraint
The release for irrigation released during a particular month from the reservoir should be enough to meet the crop water requirement. It should also be greater than the minimum irrigation required.
\({IR}_{t}\ge {I}_{t}. \forall t=\text{1,2},\dots .T, T=12\) Eq. 2.9
Where:
It: Monthly irrigation demand (Mm3)
IV. The non negative constraint for hydropower release
The amount of water discharged for hydropower generation should be nonnegative
\({HR}_{t}\ge 0. \forall t=\text{1,2},\dots .T, T=12\) Eq. 2.10
Where: 
HPt: Monthly hydropower releases (Mm3)
V. Reservoir storage capacity constraint
The available water in the reservoir should not exceed the reservoir’s life storage capacity but must be greater than the dead storage capacity (Mm3) for the whole time period. The live storage in the reservoir should be less than or equal to the maximum capacity for all time periods.
\({D}_{sc}\le {S}_{t}\le {L}_{sc} \forall t=\text{1,2},\dots .T, T=12\) Eq. 2.11
Where: 
St: Monthly reservoir storage (Mm3) during release.
Lsc: Life storage capacity of the reservoir (Mm3)
Dsc: Deadly storage capacity of the reservoir (Mm3)
VI. Continuity of the mass balance constraint
The mass balance between the inflow into the reservoir and the releases from the reservoir make up the continuity constraints. These constraints relate to the total release, reservoir storage, and inflows into the reservoir, overflows, and evaporation losses for all time periods. The reservoir storage continuity relationship is expressed as:
\({S}_{t+1}={S}_{t}+{Q}_{t}+{P}_{t}{TR}_{t}{E}_{t}{SP}_{t}. \forall t=\text{1,2}\dots .T, T=12\) Eq. 2.12
Where: 
St+1: Final reservoir storage at the next of the previous month (Mm3)
St: Initial reservoir storage at the starting of the month (Mm3)
Qt: Monthly stream inflow into the reservoir (Mm3)
Pt: Monthly direct rainfall over the reservoir (Mm3)
TRt: Total monthly releases (Mm3)
Et: Monthly evaporation losses (Mm3)
SPt: Monthly overflow (Mm3)
4. System and problem description for scenarios of Complimentary Hydropower release
In this scenario, hydropower is to be integrated as designated, so that the whole of the available flow is used to turn the turbine, after which diversion for various other uses can be achieved. This is the optimum plan since the turbine only needs water for turning purposes and can be fully released for other users. This plan must be combined into the main design at the beginning of the project for real operation.
Objective function
The objective function is the maximization of the total annual energy generation TE, as presented in Equations 2.4 and 2.5. This has also been adopted, and the constraints for the scenario complimentary hydropower releases are given as follows:
Model constraints
The constraints in equations (2.8) – (2.12) are also applicable in this scenario, but the constraints on the upper and lower releases change
Due to the complimentary releases added and presented in equations 2.13 and 2.15, respectively. Moreover, there is a need to incorporate a constraint on release (equilibrium) from the reservoir as presented in the equation
Upper Limit (Monthly maximum yield from the storage yield analysis)
\({TR}_{t}={IR}_{t}+{C}_{t}{+HP}_{t}\le Upper limit. \forall t=\text{1,2},\dots .T, T=12\) Eq. 2.13
Lower Limit (Ecological demand, minimum release)
\({TR}_{t}={IR}_{t}+{C}_{t}{+HP}_{t}\ge Lowe limit. \forall t=\text{1,2},\dots .T, T=12\) Eq. 2.14
Constraints on release (equilibrium) from the reservoir
\({C}_{t}{+HP}_{t}\ge {ER}_{T}. \forall t=\text{1,2},\dots .T, T=12\) Eq. 2.15
Where:
Ct: Monthly complimentary release (Mm3)