Risk management of hazardous material transportation

Transcription

1 Maagemet of atural Resources, Sustaable Developmet ad Ecologcal azards 393 Rs maagemet of hazardous materal trasportato J. Auguts, E. Uspuras & V. Matuzas Lthuaa Eergy Isttute, Lthuaa Abstract I recet years o Lthuaa roads a cosderable amout of hazardous materals have bee trasported, especally ol products. owever, there s o commo methodology whch could assess the rs of such trasportato. Lthuaa has accepted drectves, orms ad other acts of law related to rs ad hazard assessmet ad preveto that are vald the Europea Uo. I relato to ths stuato, hazard assessmet ad aalyss has eve greater sgfcace. The ovelty of the wor s assocated wth employg the Marov process to descrbe a hazard dstrbuto mechasm ad to determe a lmted hazard dstrbuto the odes of etwors. Keywords: azard, rs, rs sources, rs dstrbuto, Marov process. Itroducto azard detfcato ad assessmet are rather complcated tass, whch have receved atteto the lterature (Adams []). Rs of ay actvty or process s ofte defed as a set made of pars of frequecy of hazardous evets ad ther outcomes. Sometmes these values are multpled. azard measuremet s less clearly defed ad here such qualtatve evaluatos as hgh hazard level, medum hazard level, low hazard level, etc. are used. I certa cases, quattatve expressos are also used. Durg trasportato of hazardous materals or at the outspread of commucable dseases, etc. hazard s dvded, moved from oe place to the other, dstrbuted amog varous structures. I the lterature, much atteto s devoted to the vestgato of varous hazard dstrbuto mechasms. Oe of the most wdely explored amog these s the dstrbuto of dfferet polluto WIT Trasactos o Ecology ad the Evromet, Vol 99, 2006 WIT Press ISS (o-le) do:0.2495/rav06039

2 394 Maagemet of atural Resources, Sustaable Developmet ad Ecologcal azards materals the atmosphere, water ad sol, spread of commucable dseases amog people, amals ad plats (Lefevre ad Pcard [2]) ad shppg hazardous materals wth dfferet meas of trasportato (Purdy [3]). Latter wors provde detaled aalyses of dstrbuto mechasms, speed, process durato, etc. of hazardous materals ad dseases. It s also obvous that together wth the mprovemet of meas of trasportato ad the crease the quatty ad sze of the loads, the assessmet of hazard dstrbuto becomes more promet the systems of trasportato. It has to be oted that the maorty of the scetfc research artcles ad wors o the hazard dstrbuto assessmet the etwor systems has bee made durg the last several decades ad ths topc s stll uder actve vestgato. The ma am of the paper s the aalyss of hazard dstrbuto the etwor systems. I the cotext of the aalyss, hazard s uderstood as the amout of hazardous materals, dsease cocetrato, etc. azard ca be dstrbuted through the chaels of varous etwors ad cocetrated the odes of the etwors. azard trasmsso through the chaels that coect etwor odes ca tae place may ways: for example, hazard ca be trasmtted to a sgle or to several odes, as a udvded value or dvded to parts. Each ode ca also have certa protecto or mmuty agast hazard, whch blocs ts trasmsso or dmshes t. The paper has a purpose to preset the aalyss ad mathematcal model of hazard dstrbuto the odes of the etwor. 2 Deftos of hazards, trasfers ad other cocepts As t was already metoed, hazard ths paper s equalled to such umercal values as the quatty of hazardous materals, the testy of formatoal trasfer, etc. azard wll be oted as. We wll ow defe several terms that wll be used the paper: azard source. It s oe of the etwor odes whch hazard ca arse or occur. Pot source of hazard. It s a source of hazard whch hazard occurs oly oce. Iftve source of hazard. It s a source of hazard whch hazard arses perodcally, for a ftve umber of tmes. Addtve hazard. It s a sort of hazard, whe hazards the odes of the etwor ca be added to or a part of hazard moved to the other odes. The examples of the addtve hazard are: collecto of hazardous materals, trasport testy, etc. o-addtve hazard. It s a sort of hazard whe the sum of hazards s equal to the maxmum of those hazards: + = max 2 { ; 2}. Trasfer testy coeffcet betwee the etwor odes. It s a coeffcet q that mars the part of the hazard the ode, that wll be trasmtted to the ode. It s clear that q,, = WIT Trasactos o Ecology ad the Evromet, Vol 99, 2006 WIT Press ISS (o-le)

3 Maagemet of atural Resources, Sustaable Developmet ad Ecologcal azards 395 here - a umber of etwor odes. The trasfer testy to the ode s q = q ad from the ode s q =, q., = etwor. It s system defed as a oreted graph, whch hazard from oe ode ca be trasmtted oly to oe ode durg a cycle. etwor ode mmuty. It s a coeffcet I that mars whch part of the hazard s trasmtted to the ode ( 0 I ). ode mmuty ca be created by the securty systems, at-vrus computer software, etc. Trasfer probablty. It s a probablty p,, that durg oe cycle hazard from the ode wll be trasmtted to the ode. Trasfer probablty has the followg feature: p =., = azard trasfer cycle. azard trasfer the etwor from oe ode to the other s regarded as oe hazard trasfer cycle. azard etwor odes after cycles. azard that s accumulated the ode. after cycles, wll be mared as Margal hazard the etwor odes. Margal hazard the ode s a steady = lm. = hazard after a ftve umber of cycles. 3 Addtve hazard dstrbuto the etwor The dstrbuto of hazard that ca be dvded or added the etwor odes wll be aalysed. Two hazard dstrbuto methods wll be aalysed separately. I the frst case t wll be assumed that hazard ca be trasferred from every ode oly to oe of the possble odes, whle the secod case, let us allow the hazard spreadg though the etre etwor. 3. azard dstrbuto Marov chas Let us suppose that we have a etwor wth odes. azard from the ode ca be trasferred oly to oe ode, whch s selected accordg to trasfer probablty P. Thus, durg each cycle, hazard ca occur oly oe etwor ode. I the paper a assumpto wll be made that trasfer probabltes have Marov propertes. Thus, f the hazard that exsts ode after cycles wll be mared as X(), so P = P( X = X( ) = ) = () = P( X = X( ) = ; X( 2 ) = ;...; X( ) = ) 2 WIT Trasactos o Ecology ad the Evromet, Vol 99, 2006 WIT Press ISS (o-le)

4 396 Maagemet of atural Resources, Sustaable Developmet ad Ecologcal azards Ths way the process X() wll be Marov cha wth fte set of the states {;2; ;}. The homogeeous Marov cha should also be dscussed sce P s ot depedet o. Let us mar hazard occurrece probablty the ode after cycles π. It s clear that π =. = ow t ca be retured to the hazard calculato each ode after cycles. each ode aturally, t s possble to determe oly average hazard sce hazard after steps s a radom value. If we would also mae a assumpto that all the etwor le flows are equal to, the followg would be obtaed: = π( ) = π( ) = = (2) ere - the hazard that has occurred oe of the etwor odes durg zero step,.e., we hold that ths ode s a pot source of the hazard. From the theory of the Marov chas we ow that state probabltes after cycles are descrbed usg recursve formulas Or, to put t smpler, [ π (), π (),..., π ()] = [ π (0), π (0),..., π (0)] [ P ] (3) 2 2 π () = π (0)P here P = [ P ] - trasfer probablty matrx ad π ( 0) = [, 0, 0,..., 0] (4), f we mae a assumpto that the pot source of the hazard s located the frst ode. The t follows: 2 π(2) = π() P= ( π(0) P) P= π(0) P,, =,2,..., (5) Gve that π ( 0) = [,0,0,...,0] π, we receve: [ ] ( 0) = π P = π P = P, P,..., P Thus, we ca calculate the average hazard the ode after cycles recursvely, usg the followg formula: here, P,2, (6) ( ) = (7) = ( ) P s the elemet of the frst le of matrx P. Accordg to the eq. (2), t s ot dffcult to prove the theorem of the margal dstrbuto of the hazard average, whe coverge to fty. WIT Trasactos o Ecology ad the Evromet, Vol 99, 2006 WIT Press ISS (o-le)

5 If the Marov cha wth states ad trasfer probablty matrx P = [ P ] s lm π = π, =,2,...,, so margal hazard average values ergodc,.e., all the odes of the etwor also exst. Whe the eq. (2) we reach the lmt whe coverge to fty we get: lm = lm π ( ) (8) = As lm π ( ) = π, so there s vashg fucto ( ) π ( ) = π + ε, where lm ε = 0, =,2,.... The Maagemet of atural Resources, Sustaable Developmet ad Ecologcal azards 397 ε whch s lm = lm ( ()) π + ε = = = lm π lm + { } Let us select ε () = max ε ; ε 2 ;...; ε. The: = ε () (9) 0 lm lm = 0 ( ε () ε ) () (0) = Let lm ε () = 0, ad, therefore, = π. = Thus, the margal average hazard exsts every ode, besdes, t s equal to the product of the tal ad the margal ode probablty. 3.2 The dstrbuto of the addtve hazard the etwor odes durg the trastoal perod I ths secto, a hazard whch s characterzed by a value that ca be summed or dvded as a real umber wll be aalysed. The examples of such hazard are cocetratos of sgfcat amouts of hazardous materals, cocetrato of polluto materals, the amout of water that rses the reservor, etc. For calculato of hazard each etwor ode durg the trastoal perod, systems of equato were made, codtos for margal hazard exstece were specfed ad for the calculato of margal hazards the etwor systems, systems of equato were formed. WIT Trasactos o Ecology ad the Evromet, Vol 99, 2006 WIT Press ISS (o-le)

6 398 Maagemet of atural Resources, Sustaable Developmet ad Ecologcal azards We aalyse a etwor system whch hazard from each ode ca be of the trasferred to other odes durg oe cycle, by dvdg hazard ode proporto to the flows q, whe =,2,..., ad q. Frst of all, let us assume that oe etwor ode, for example, the frst oe, s a pot source of the addtve hazard, whch hazard ( 0) occurs. Thus, at the zero step we have the followg hazard dstrbuto the odes: 0 = (0) () [ ] Durg the followg cycles, hazard modfcato wll occur each ode. From that ode hazard wll be trasferred to other odes by flows q. The total trasfer wll be: q q q, + q, q = q (2) = ( + +, ) part of hazard. I the ode t wll rema q = q (3) part of hazard. The hazard q q wll be respectvely trasferred from = other odes to the ode. Thus, after cycles, we wll have the followg hazard the ode : ( + ) = q + 2 q q, where =,2,...,. After defg etwor trasfer matrx Q = [ q ], we ca wrte the system of equatos the form of matrx: + = Q (4) Thus, we have receved hazard dstrbuto the teratve process. As the process s statoary,.e. matrx Q s ot depedet o the umber of cycles, so rrespectve of the tal hazard dstrbuto, ths process coverges oly whe all matrx Q ow values wll be less tha oe. Ths s as well the oblgatory ad suffcet codto for the margal dstrbuto of the addtve hazard the etwor systems. It s easy to ascerta that f the sum of the elemets of the les square A = s = the les matrx matrx [ ] a a =, where a <, the the sum of the elemets of A s also equal to, ad ts elemets are < a. WIT Trasactos o Ecology ad the Evromet, Vol 99, 2006 WIT Press ISS (o-le)

7 The teratve process of hazard dstrbuto coverges whe the umber of cycles s ad t s ot depedet o the tal hazard dstrbuto, f all the flows are 0 q <. < It s mportat to aalyse rs dstrbuto after the certa umber of teratos. From the eq. (4) follows that: 2 + = Q Q = Q =... = 0 Q + (5) Therefore Q + ( + ) = ( 0) (6) Ths equalty allows employg the deas that are used whe provg the ergodc theorems of Marov s cha states. We shall mar the elemets of matrx Q ths way: q, ad q =,, =,2,...,. We wll frst otce that q () because q (7) = ( ) m ( ) = m ( ) q q q q q q l l l l l l l l= l= q l l= = = m q ( ). Ths feature s also correct wth q, for whch. Thus, m q m q (8) By aalogy, t s possble to show that max q max q (9) Let us evaluate q q s < : Maagemet of atural Resources, Sustaable Developmet ad Ecologcal azards 399 l, for all, l,2,...,, =. Of course, for ay l r r lr r r= r= r= q q = q s q s q s q s = = q s q s q s r r r Postve dffereces q () s qlr ( s) ( r ) oes as β ( ). As l r wll be mared as ( r ) ( + ) l (20) β ad egatve WIT Trasactos o Ecology ad the Evromet, Vol 99, 2006 WIT Press ISS (o-le)

8 400 Maagemet of atural Resources, Sustaable Developmet ad Ecologcal azards q ( s) = q ( s) = r (2) lr r= r= Thus, ( r [ ] ) ( r q s q s ) r lr βl βl (22) 0= = + = 0 r= r r Let us mar As all the () s > 0 q, so ( r ) ( r ) (23) υ = β + = β l l ( r) ( r) υ ( r β ) ( + ) < q = l (24) r ( r) r= Therefore, 0 <. Let us mar υ = max υ. The 0 υ <. It s ow clear that:, ( r ) r l βl r βl r r= r= r q q = + q s q s ( r ) ( r β m β ) (25) max q s + q s r l r l r r= r= υ υ max q s m q s max q s q s r r l r r, l For all, l =,2,...,. The, max q q υ max q q (26) l l l, l, After usg ths recursve equalty for s of tmes we get: s max υ max l l l, l, q q q s q s s s From prevous statemets 0 < q ( s) <, so q ( s) q ( s) From that we get max l, l. l s (27) q q υ (28) WIT Trasactos o Ecology ad the Evromet, Vol 99, 2006 WIT Press ISS (o-le)

9 The q q lm max = 0. l, l Let us remember the equaltes of ths proof eq. (8) ad eq. (9). We get { q } that flows m ad ( 0 < < q q { q } max are mootoous ad defte ), whch meas that they have lmts: * = ** lm max q ad q = m q From the eq. (28) we obta that there s a lmt q q. = = It s clear that lm Maagemet of atural Resources, Sustaable Developmet ad Ecologcal azards 40 = = lm. q = lm q = q = q. * ** Let us retur to the eq. (4). Whe we come to the lmt of, we mar Q + = Q lm + =, ad get: lm + = lm 0 Q + = 0 Q (29) lm, Or Thus, q q... q 2 q q... q 2 = ( 0) Q = [ ] q q... q 2 = q q q [ ] 2... Ths dstrbuto does ot deped o tal codtos. (30) (3) 4 The ma results ad coclusos The ma am of the paper s to preset the developed hazard dstrbuto mathematcal model ad ts aalyss. ere was aalysed the mechasm of hazard propagato etwor systems the case of sgle hazard, evolved oe of the etwor odes ad case whe hazard arse durg each cycle. Few etwor system cases were aalysed. The most mportat cases are hazard propagato Marov chas ad etwor systems, where peas have a mmuty or resstace to the hazard characterstc. Refereces [] Adams, J., Rs, UCL Press Ltd, Lodo, 995. WIT Trasactos o Ecology ad the Evromet, Vol 99, 2006 WIT Press ISS (o-le)

10 402 Maagemet of atural Resources, Sustaable Developmet ad Ecologcal azards [2] Lefevre, C., Pcard, P., A epdemc Model wth Fatal Rs, Mathematcal Bosceces, Y, 7, pp , 993. [3] Purdy, G., Rs Aalyss of the Trasportato of Dagerous goods by Road ad Ral, Joural of azardous Materals, 33, pp , 993. WIT Trasactos o Ecology ad the Evromet, Vol 99, 2006 WIT Press ISS (o-le)