Stochastic Reservoir Systems with Different Assumptions for Storage Losses

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1 American Journal of Operaions Research, 26, 6, hp:// ISSN Online: ISSN Prin: Sochasic Reservoir Ssems wih Differen Assumpions for Sorage Losses Carer Browning, Hillel Kumin 2 Halliburon Drill Bis and Services, Olahoma Ci, OK, USA 2 School of Indusrial and Ssems Engineering, Universi of Olahoma, Norman, OK, USA How o cie his paper: Browning, C. and Kumin, H. (26) Sochasic Reservoir Ssems wih Differen Assumpions for Sorage Losses. American Journal of Operaions Research, 6, hp://dx.doi.org/.4236/ajor Received: Augus 5, 26 Acceped: Sepember 25, 26 Published: Sepember 28, 26 Coprigh 26 b auhors and Scienific Research Publishing Inc. This wor is licensed under he Creaive Commons Aribuion Inernaional License (CC BY 4.). hp://creaivecommons.org/licenses/b/4./ Open Access Absrac Moran considered a dam whose inflow in a given inerval of ime is a coninuous random variable. He hen developed inegral equaions for he probabiliies of empiness and overflow. These equaions are difficul o solve numericall; hus, approximaions have been proposed ha discreize he inpu. In his paper, exensions are considered for sorage ssems wih differen assumpions for sorage losses. We also develop discree approximaions for he probabiliies of empiness and overflow. Kewords Sochasic Sorage Ssems, Sorage Losses, Probabili of Empiness and Overflow. Inroducion Moran [] [2], Prabhu [3] [4] and Ghosal [5] all considered a finie dam whose inpu in a given inerval of ime is a coninuous random variable. Inegral equaions are hen developed ha give he probabili of empiness and overflow. I is difficul o obain exac numerical resuls from hese equaions. An analic soluion has onl been obained for an Erlang inpu. Klemes [6], Locher and Phaarfod [7], Phaarfod and Srianhan [8] and ohers have obained approximaions for hese probabiliies b discreizing he inpu. Following Bae and Devine [9], we consider reservoir ssems wih differen assumpions for sorage losses. We hen obain inegral equaions as above for he probabili of empiness and overflow, and develop discree approximaions o obain numerical resuls for he probabiliies of overflow and empiness. Moran considered a sorage model of a dam in discree ime, =,, 2,. Le Z be he level of he dam before inpu X, where he X s are i.i.d. random variables. Le DOI:.4236/ajor Sepember 28, 26

2 Y be he release a he end of he ime period ( +, ), where he Y s are i.i.d. random variables independen of he X s, and le < be he capaci of he ssem. If Z + X >, hen here is an overflow of X + Z. If Z + X hen no overflow occurs. A he end of he period, if here is an overflow, hen Z+ = Y. If here is no overflow, hen eiher Z = Z + X + Y or Z + = if he sorage ssem is emp. Lindle [] showed ha if cerain independence condiions are saisfied hen { } s F+ ( ) = Pr sorage level Z+ of ( + ) period = Pr{ Z+ = } + Pr{ < Z+ } = Pr{ Z + X Y } + Pr{ < Z + X Y } = Pr { Z + X Y }. where [, ]. Furher, define H ( ) o be he c.d.f. of Then, b convoluion U = X Y ( ) ( ) ( ) U where F+ = F x d H x, < Since he limiing disribuion F( ) of Z is independen of ime in he sead sae, for he semi-infinie case (bounded below), we have: which is equal o ( ) F F( x) dh( x) if = () if < ( ) ( ) ( ) F = F x d H x, (2) Equaions () and (2) are nown as Lindle s equaions. Numerical soluions for specific inpu disribuions o Lindle s equaions are difficul o obain. In Moran s original wor, a soluion for exponenial inpus was found, bu was sricl limied o ha disribuion. I is no an eas as o obain probabiliies for empiness and overflow in coninuous ime. In his regard. Moran [6] proposed a discree approximaion in order o obain numerical resuls for he probabilies of empiness and overflow. Modificaions o his approach have been developed b Klemes [3], Locher and Phaarfod [5], Phaarfod and Srianhan [8]. In his paper, we model energ sorage ssems wih differen assumpions abou sorage losses, and develop similar discree approximaions o calculae he probabiliies of eminess and overflow. 2. Finie Model Moran s model ields he following Marov chain: if Z + X Y Z+ = Z + X Y if Y < Z + X < Y if Z + X 45

3 For he case of a finie ssem of capaci <, F( ) = for and Equa- ion (2) becomes: ( ) = ( ) d ( ) d ( ) (3) F F x H x H x ( ) ( ) ( ) If he ssem has a consan release, hen = H F x dh x (4) Y = m< and if Z + X m Z+ = Z + X m if m< Z + X < m if Z + X Thus, Z+ min { m, max [, Z X m] } = +. Figure illusraes a single ime period of he previous Marov chain: Now, le G (.) be he c.d.f. of X, hen ( ) = Pr{ } = Pr{ } = ( + ) H u X Y u X m u G u m Since F( x ) = for x m, Equaion (3) becomes ( ) = ( ) d ( + ) F F x G x m m 3. Losses from Sorage Model ( ) d ( ) d ( ) = F x G x+ m G x+ m m ( ) ( ) ( ) = G + 2m F x d G x+ m, m The mos basic case of leaage occurs when a fixed amoun q leas from sorage afer he release Y a he end of each ime inerval. This pe of ssem is shown below in Figure 2: (see Bae and Devine [9]): And Then, we have Pr m { Z } Pr{ Z X Y q } + = + Figure. Simple sorage ssem. 46

4 Figure 2. Fixed leaage from sorage. ( ) ( ) ( ) F = F x d H + q x, q When he oupu is fixed, i.e., he previous equaion gives ( ) d ( ) ( ) d ( ) = F x H + q x F x H + q x q ( 2 ) ( ) d ( ) q = H + q F x H + q x Y (5) = m, define U = X ( m+ q). Appling his o m q ( ) = + 2( + ) ( ) d ( + + ) F G m q F x G x m q (6) Equaions (5) and (6) indicae ha leaage ma be reaed as a par of he oupu; hus, no separae analsis is needed in his case. The second case represens a variable leaage whereb a quani proporional o he amoun sored is los a he beginning of each ime inerval. Le e denoe he fracion of Z los in each period (i.e., e is a measure of he sorage efficienc). The Marov chain corresponding o his case is Figure 3 illusraes his pe of ssem: We hus have if ez + X Y Z+ = ez + X Y if Y ez + X < Y if ez + X ( ) = Pr{ + } F ez X Y + { X Y e x Z x} Pr{ Z x} = Pr = = ( ) d ( ) = H ex F x ( ) = ( ) ( ) d ( ) F H e F x H e x (7) 47

5 Figure 3. Fixed leaage from sorage. When he oupu is fixed, i.e., Then, he limiing c.d.f. of Model 2 Y Z is given b = m, we again have F( x ) = for x m. m ( ) = + ( + ) ( ) d ( + ) F G e e m F x G e x m (8) This model describes a ssem in which he inpu passes hrough a process having an efficienc e before enering sorage, and he quani released from sorage passes hrough an oupu process having an efficienc e 2 before leaving he ssem. The inpus X are independen random variables following a given c.d.f. and each Y = m. A diagram of his ssem is given in Figure 4. Now we define X m if X m S = (9) oherwise m X if X < m Ti = () oherwise When boh he inpu and oupu devices have efficienc facors, he schemaic changes o (Figure 5). I is apparen from (9) and () ha eiher S = es or T Similar o previous models, we also define ( ) G = he c.d.f. of X Uˆ = S T ( ) Ĥ( u ) and G ( ) are hen given b As before, ( ) H u H = he c.d.f. of Uˆ ( ) = = T = mus be. e2 u Hˆ eu if u G + m if u e ˆ u H if u G( eu 2 m) if u < + < e2 48

6 Figure 4. Inpu and oupu efficiencies. Figure 5. In-ou devices efficiencies. ( ) = lim ( ) = ( ) d ( ) F F F x H x () ( ) ( ) ( ) = H F x d H x, (2) If ( ) { } Hˆ u = Pr S T u (3) hen ( ) ˆ ( ) F( ) = F x dh x 4. Mehodolog F( x) dhˆ ( x) F( x) dhˆ ( x) F( x) dhˆ ( x) F( x) dhˆ ( x) F( x) dhˆ ( x) lim F( x) dhˆ ( x) ( ) d ˆ ( ) ( ) d ˆ ( ) lim ˆ ( ) ˆ ( ) ( ) ( ) d ( ) ( ) d ( ) = Hˆ F x Hˆ x F x Hˆ x = N = N = F x H x F x H x H N + H = H F x H e x e2 N ( ) d ( ( )) ( ) 2 2 x F x dh e 2 x = G F( x) dg( e ( x) ) F( x) dg e e We develop a discree analogue b defining 49

7 G F = Z i (4) i i Pr ( ) Pr ( ) g = X = i (5) i j = i= j < j g if j (6) if Thus if i < m Fi= Gi+ 2m + gi+ m jfj if i =,,, m j= if i m where he probabiliies of empiness and overflow are { } = { Z } = F( ) = F Pr empiness Pr { } = { } = ( ) = Pr overflow Pr Z m F m F m 4.. Fixed Leaage When he sorage has a fixed quani q ha leas as given in Equaions (5) and (6), he discree analogue is given b if i < m q Fi= G 2( ) + gi m q jfj if i,,, m i+ m q + + = j= if i m where he probabiliies of empiness and overflow are { } = { Z } = F( ) = F Pr empiness Pr { } = { Z m} = F( m ) = F m Pr overflow Pr 4.2. Sorage Leaage A discree analogue of (8) is if i < m Fi= G ( ) + g if,,, i e m e i+ m e jfj i m + = j= if i m where he probabiliies of empiness and overflow are { } = { Z } = F( ) = F Pr empiness Pr { } = { } = ( ) = m Pr overflow Pr Z m F m F 5. Consan Oupu Model Using he definiions (4)-(6), we have 42

8 if i < Fi= Gi+ m + gi+ m jfj if i =,,, j= if i where he probabiliies of empiness and overflow are 6. Numerical Resuls { } = { Z } = F( ) = F Pr empiness Pr { } = { Z } = F( ) = F Pr overflow Pr Figure 6 and Figure 7 represen he impac of sorage leaage on he basic ssem. The capaci of he following ssems is deermined in relaion o he release amoun. Figure 6 below gives he empiness probabiliies for an average inpu of 3.75 wih a sandard deviaion of, and a sorage efficienc e, of 75%. Figure 6. Comparaive empiness probabiliies. Figure 7. Comparaive overflow probabiliies. 42

9 Figure 8 represens he difference beween Figure 6 and Figure 7. For boh ssems, here is an inpu of 4, wih a release beween 2 and, and a capaci deermined b he relaion of m= 3. Figure 9 represens he impac of a varing sandard deviaion on an inpu. In his ssem, he inpu is 5, release is 5, and capaci ranges from 7 o 26. Addiionall, each probabili is deermine wih a sandard deviaion of.75,, 2 and 4. Figure represens he impacs of efficienc on he inpu and oupu process on he model obained for he probabili of overflow. For his ssem, he inpu is 5 wih a sandard deviaion of 4. The capaci for his ssem is 3. Addiionall, he release changes from 7 o 2. Series has inpu/oupu efficienc of, Series 2 has efficienc of.5/, Series 3 has efficienc of /.5, and Series 4 has efficienc of.7/ Sensiivi Analsis Figure 6 and Figure 7 represen he impac of sorage leaage on he basic ssem Figure 8. Comparaive simple and consan oupu probabiliies. Figure 9. Represen he impac of a varing sandard deviaion on an inpu. 422

10 Figure. Impac of inpu/oupu efficienc. when he basic discree ime model is considered. The capaci of he following ssem is deermined in relaion o he release amoun. The inpu is 3.75 wih a sandard deviaion of, and a sorage efficienc, e, of 75%. Figure 8 represens he difference beween Figure 6 and Figure 7. For boh ssems, here is an inpu of 4, wih a release beween 2 and, and a capaci deermined b he relaion of m= 3. References [] Moran, P.A.P. (954) A Probabili Theor of Dams and Sorage Ssems. Ausralian Journal of Applied Science, 5, [2] Moran, P.A.P. (959) The Theor of Sorage. Mehuen and Co, London. [3] Prabhu, N.U. (958) Some Exac Resuls for he Finie Dam. Annals of Mahemaical Saisics, 29, hp://dx.doi.org/.24/aoms/ [4] Prabhu, N.U. (985) Sochasic Sorage Processes: Queues, Insurance Ris, Dams, and Daa Communicaion. Springer-Verlag, New Yor. [5] Ghosal, A. (969) Some Aspecs of Queueing and Sorage Ssems. Lecure Noes in Operaions Research and Mahemaical Ssems, No. 23, Springer-Verlag, New Yor. [6] Klemes, V. (977) Discree Represenaion of Sorage for Sochasic Reservoir Operaion. Waer Resources Research, 3, hp://dx.doi.org/.29/wr3ip49 [7] Locher, P. and Phaarfod, R.M. (979) On he Problem of Discreizaion in Dam Theor. Waer Resources Research, 5, hp://dx.doi.org/.29/wr5i6p593 [8] Phaarfod, R.M. and Shrianhan, R. (98) Discreizaion in Sochasic Reservoir Theor wih Marovian Inflows.Journal of Hdrolog, 52, hp://dx.doi.org/.6/22-694(8)97- [9] Bae, H.M. and Devine, M. (978) Opimizaion Models for he Economic Design of Windpower Ssems. Solar Energ, 2, hp://dx.doi.org/.6/38-92x(78)964-6 [] Lindle, D.V. (952) The Theor of Queues wih a Single Server. Mahemaical Proceedings of he Cambridge Philosophical Socie, 48, hp://dx.doi.org/.7/s

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