Stochastic Reservoir Systems with Different Assumptions for Storage Losses ()

Carter Browning^{1}, Hillel Kumin^{2}

^{1}Halliburton Drill Bits and Services, Oklahoma City, OK, USA.

^{2}School of Industrial and Systems Engineering, University of Oklahoma, Norman, OK, USA.

**DOI: **10.4236/ajor.2016.65038
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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.

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Browning, C. and Kumin, H. (2016) Stochastic Reservoir Systems with Different Assumptions for Storage Losses. *American Journal of Operations Research*, **6**, 414-423. doi: 10.4236/ajor.2016.65038.

1. 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 approxi- mations to obtain numerical results for the probabilities of overflow and emptiness.

Moran considered a storage model of a dam in discrete time,. Let be the level of the dam before input, where the X’s are i.i.d. random variables. Let be the release at the end of the time period, where the Y’s are i.i.d. random variables independent of the X’s, and let be the capacity of the system. If, then there is an overflow of. If then no overflow occurs. At the end of the period, if there is an overflow, then. If there is no overflow, then either or if the storage system is empty.

Lindley [10] showed that if certain independence conditions are satisfied then

where. Further, define to be the c.d.f. of where

Then, by convolution

Since the limiting distribution of is independent of time t in the steady state, for the semi-infinite case (bounded below), we have:

(1)

which is equal to

(2)

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 assum- ptions about storage losses, and develop similar discrete approximations to calculate the probabilities of emtiness and overflow.

2. Finite Model

Moran’s model yields the following Markov chain:

For the case of a finite system of capacity, for and Equa- tion (2) becomes:

(3)

(4)

If the system has a constant release, then and

Thus,. Figure 1 illustrates a single time period of the previous Markov chain:

Now, let be the c.d.f. of, then

Since for, Equation (3) becomes

3. Losses from Storage

Model 1

The most basic case of leakage occurs when a fixed amount q leaks from storage after the release at the end of each time interval. This type of system is shown below in Figure 2: (see Bae and Devine [9] ):

Then, we have

And

Figure 1. Simple storage system.

Figure 2. Fixed leakage from storage.

(5)

When the output is fixed, i.e., , define. Applying this to the previous equation gives

(6)

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 denote the fraction of lost in each period (i.e., is a measure of the storage efficiency). The Markov chain corresponding to this case is

Figure 3 illustrates this type of system:

We thus have

(7)

Figure 3. Fixed leakage from storage.

When the output is fixed, i.e., , we again have for. Then, the limiting c.d.f. of is given by

(8)

Model 2

This model describes a system in which the input passes through a process having an efficiency before entering storage, and the quantity released from storage passes through an output process having an efficiency before leaving the system. The inputs are independent random variables following a given c.d.f. and each. A diagram of this system is given in Figure 4.

Now we define

(9)

(10)

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 or must be 0.

Similar to previous models, we also define

and are then given by

As before,

Figure 4. Input and output efficiencies.

Figure 5. In-out devices efficiencies.

(11)

(12)

If

(13)

then

4. Methodology

We develop a discrete analogue by defining

(14)

(15)

(16)

Thus

where the probabilities of emptiness and overflow are

4.1. 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

4.2. Storage Leakage

A discrete analogue of (8) is

where the probabilities of emptiness and overflow are

5. Constant Output Model

Using the definitions (14)-(16), we have

where the probabilities of emptiness and overflow are

6. Numerical Results

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

Figure 6. Comparative emptiness probabilities.

Figure 7. Comparative overflow probabilities.

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.

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

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.

7. Sensitivity Analysis

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

Figure 8. Comparative simple and constant output probabilities.

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

Figure 10. Impact of input/output efficiency.

when the basic discrete time model is considered. The capacity of the following system is determined in relation to the release amount. The input is 3.75 with a standard deviation of 1, and a storage efficiency, , of 75%.

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.

Conflicts of Interest

The authors declare no conflicts of interest.

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