Stochastic Reservoir Systems with Different Assumptions for Storage Losses

Moran considered a dam whose inflow in a given interval of time is a continuous random variable. He then developed integral equations for the probabilities of emptiness and overflow. These equations are difficult to solve numerically; thus, approximations have been proposed that discretize the input. In this paper, extensions are considered for storage systems with different assumptions for storage losses. We also develop discrete approximations for the probabilities of emptiness and overflow.


Introduction
Moran [1] [2], Prabhu [3] [4] and Ghosal [5] all considered a finite dam whose input in a given interval of time is a continuous random variable.Integral equations are then developed that give the probability of emptiness and overflow.It is difficult to obtain exact numerical results from these equations.An analytic solution has only been obtained for an Erlang input.Klemes [6], Lochert and Phatarfod [7], Phatarfod and Srikanthan [8] and others have obtained approximations for these probabilities by discretizing the input.Following Bae and Devine [9], we consider reservoir systems with different assumptions for storage losses.We then obtain integral equations as above for the probability of emptiness and overflow, and develop discrete approximations to obtain numerical results for the probabilities of overflow and emptiness.
Moran considered a storage model of a dam in discrete time, 0,1, 2, t = . Let t Z be the level of the dam before input t X , where the X's are i.i.d.random variables.Let t Y be the release at the end of the time period ( ) , 1 t t + , where the Y's are i.i.d.random variables independent of the X's, and let k < ∞ be the capacity of the system.If occurs.At the end of the period, if there is an overflow, then − .If there is no overflow, then either Lindley [10] showed that if certain independence conditions are satisfied then ( ) Then, by convolution F y of t Z is independent of time t in the steady state, for the semi-infinite case (bounded below), we have: which is equal to Equations ( 1) and (2) are known as Lindley's equations.Numerical solutions for specific input distributions to Lindley's equations are difficult to obtain.In Moran's original work, a solution for exponential inputs was found, but was strictly limited to that distribution.
It is not an easy task to obtain probabilities for emptiness and overflow in continuous time.In this regard.Moran [6] proposed a discrete approximation in order to obtain numerical results for the probabilies of emptiness and overflow.Modifications to this approach have been developed by Klemes [3], Lochert and Phatarfod [5], Phatarfod and Srikanthan [8].In this paper, we model energy storage systems with different assumptions about storage losses, and develop similar discrete approximations to calculate the probabilities of emtiness and overflow.

Finite Model
Moran's model yields the following Markov chain: For the case of a finite system of capacity k < ∞ , ( ) 1 F y = for y k ≥ and Equa- tion (2) becomes: If the system has a constant release, then t Y m k = < and . Figure 1 illustrates a single time period of the previous Markov chain: The most basic case of leakage occurs when a fixed amount q leaks from storage after the release t Y at the end of each time interval.This type of system is shown below in  ( ) ( ) ( ) When the output is fixed, i.e., t Y m = , define ( ) Equations ( 5) and (6) indicate that leakage may be treated as a part of the output; thus, no separate analysis is needed in this case.
The second case represents a variable leakage whereby a quantity proportional to the amount stored is lost at the beginning of each time interval.Let 0 1 e − denote the fraction of t Z lost in each period (i.e., 0 e is a measure of the storage efficiency).The Markov chain corresponding to this case is  When the output is fixed, i.e., t Y m = , we again have Then, the limiting c.d.f. of t Z is given by ( ) ( ) ( ) ( ) This model describes a system in which the input passes through a process having an efficiency 1 e before entering storage, and the quantity released from storage passes through an output process having an efficiency 2 e before leaving the system.The inputs t X are independent random variables following a given c.d.f. and each t Y m = .A diagram of this system is given in Figure 4.

Now we define
When both the input and output devices have efficiency factors, the schematic changes to (Figure 5).
It is apparent from ( 9) and ( 10) that either As before,

Methodology
We develop a discrete analogue by defining ( ) ( ) where the probabilities of emptiness and overflow are

Fixed Leakage
When the storage has a fixed quantity q that leaks as given in Equations ( 5) and ( 6), the discrete analogue is given by ( ) where the probabilities of emptiness and overflow are Pr emptiness Pr 0 0

Storage Leakage
A discrete analogue of ( 8) is ( ) where the probabilities of emptiness and overflow are

Constant Output Model
Using the definitions ( 14)-( 16), we have where the probabilities of emptiness and overflow are     Figure 9 represents the impact of a varying standard deviation on an input.

Numerical Results
In this system, the input is 15, release is 15, and capacity ranges from 17 to 26.
Additionally, each probability is determine with a standard deviation of 0.75, 1, 2 and 4.
Figure 10 represents the impacts of efficiency on the input and output process on the model obtained for the probability of overflow.
For this system, the input is 15 with a standard deviation of 4. The capacity for this system is 30.Additionally, the release changes from 7 to 21. "Series 1" has input/output efficiency of 1, "Series 2" has efficiency of 0.5/1, "Series 3" has efficiency of 1/0.5, and "Series 4" has efficiency of 0.7/0.9.

Figure 6 and
Figure 6 and Figure 7 represent the impact of storage leakage on the basic system.The capacity of the following systems is determined in relation to the release amount.

Figure 6
Figure 6 below gives the emptiness probabilities for an average input of 3.75 with a standard deviation of 1, and a storage efficiency 0 e , of 75%.

Figure 8
Figure 8 represents the difference between Figure 6 and Figure 7.For both systems, there is an input of 4, with a release between 2 and 10, and a capacity determined by the relation of 3 k m − = .

Figure 6 andFigure 8 .
Figure 6 and Figure 7 represent the impact of storage leakage on the basic system

Figure 9 .
Figure 9. Represent the impact of a varying standard deviation on an input.

Figure 8
Figure 8 represents the difference between Figure 6 and Figure 7.For both systems, there is an input of 4, with a release between 2 and 10, and a capacity determined by the relation of 3 k m − = .