Probabilistic Approach to Risk Analysis of Chemical Spills at Sea

International Journal of Automation and Computing 2 (2006) 117-124Probabilistic A pproach to Risk Analysis ofChemical Spills at SeaMagda Bogalecka*, Krzysztof KolowrockiGdynia Maritime University, Gdynia 81-225, PolandAbstract: Risk analysis of chemical spills at sea and their consequences for sea environment are discussed. Mutualinteractions between the process of the sea accident initiating events, the process of the sea environment threats, and theprocess of the sea environment degradation are investigated. To describe these three particular processes, the separatesemi-Markov models are built. Furthermore, these models are jointed into one general model of these processes interactions.Moreover, some comments on the method for statistical identification of the considered models are proposed.Keywords: Risk analysis, dangerous chemicals, sea accidents, semi- Markov processes.1 IntroductioninitiatinghazardousthreatdegradatiorSea transport of dangerous chemicals sometimeseventschemicalsstateseffctsan become dangerous because of accidents and maygenerate marine environment threats. Unexpected re-leases of toxiAammable, explosive, carcinogenic, andFig. 1 Interrelations of the environment degradationother substances into the sea environment as the resultfactorsof sea accidents are often inevitable.The sea transport of dangerous goods is pretty safetions in the marine environment may occur during theat normal environmental conditions. However， thesea transport of dangerous chemicals. These incidentsnay bring abouttransported goods may be swept overboard as a re-sult of bad weather, hard sea conditions, collisions oricals into the sea environment and, as a result, mayship sinking, failing devices, human errors, and ran-possibly have disastrous influence on the human healthdom incidents. The released chemicals may create theand the marine environment. Such incidents are namedthreat to the crew and the ship as well as pollute thethe initiating event of dangerous situations in the shipseawater and coast. Therefore, besides prevention, itoperation surroundings.is necessary to develop the tools that will help to de-We fix the time interval (0,T)， T > 0, where Ttermine the kinds of accidental spills, their range, andis the time of a ship operation. Also, we distinguishn1，n1∈N, events initiating the dangerous situa-their infuence on the sea environment.The risk analysis of the environment threats com-tion of the marine environment and mark them bying from the transport of hazardous chemicals is basedE1, E1,., En. Moreover, we introduce the followingon fixing and interrelating the sea accident initiatingseevents E, the hazardous chemicals H and their influ-E={e: e=[e,e,-,en], ei∈{0,1}}ence on the marine environment degradation, the statesof environment threats s, and the environment degra-wheredation effects R, as shown in Fig. 1.( 1，if initiating event E; occurs2 Modeling of chemical substances in-i={if initiating event E; does not occurfluence on marine environment degrada-tionDefinition 1. A function E(t) defined on the timeSome kinds of incidents connected with ship opera-interval (0,T〉 and having values from the set E, forexample,Manuscript received September 21, 2005; revised January 7,中国煤化工2006.*Corresponding author. E- mail address: kota@am.gdyniaplis called a processJMHCNMHG118International Journal of Automation and Computing 2 (2006) 117-124The vectors from the set E are called the states ofwherethe process E(t), while the set E is called the set ofVk= 1,2，..v，hK()=0.states of the process E(t).Under these assumptions about the process of the .We number the states of the process of initiatinginitiating events E(t), its selected parameters can beevents E(t) and have assumed that this process has Vfound.different states from the set. As such, we assume thatNamely, the expected values E[0kl] and variancesE={e,e,.,e"}D[0k] of variables 0hl are determined byE[0%]= th"(t)dt(2)e*=le',e.,,ef]， k= 1,2..,0D(0%=|~ (t- E()h2()dtandej∈{0,1}，j= 1,2..,n1.= E(0(k)2]- EB2[0的]。(3)Furthermore, we assume a semi-Markov model1] ofUnconditional distribution functions of sojournthe process E(t) and denote by θkl its random condi-times θk of the process E(t) in statesek, k= 1,2...,v,tional sojourn time in the state ek，while its next tran-are determined bysition will be done to the state e'. kh,l = 1,2,..,U,k:≠l. Then, the process is described by the vector ofVk= 1,2..0，H*()= >p"Hk(t)probabilities of its initial states at the moment t = 0[p(0)] = [p(0),2(20),，，p"(0)] .and their corresponding density functions are given byand by the matrix of probabilities of transitions be-tween the statesVk=1,2..,o，h*(t)= Sphk(t).1=p21The expected values E[0*] and variances D[0k] of[们=1)variables 0k are given respectively byVk=1,2..,u，E(0%]=>pE[0}] (4)Vk= 1,2，..,v，phk=0.Moreover, this process is defined by the matrix of-12.,,-.. D[的1=二D州conditional distribution functions of sojourn times 0klof the process E(t) in the state ek while its next tran-where E[0k] and D[0k] are defined by (2) and (3).sition will be done to the state e', k,l = 1,2,.., U,Boundary values of the instantaneous probabilitiesk:≠l,of the process of initiating events E(t) in its particularstates「Hl"(t) H12(t)Hl"(t)H21(t) H22(t)H20(t)p"(t)= P(E(t)=e*)，k=1,2,..,0(5)[Hk(t)]=are calculated from the formula。H"l(t) H2(t) ...HO"()」p*= limp"(t)=-πk E[0幻]-， k= 1,2,.,0 (6)Vk= 1,2..,0，Hk*(t)=0.乙r' E[0]1=1This matrix is complied with the matrix of condi-tional densities of sojourn times Oet of the process E(t)where probabilities πk satisfy the system of the follow-in the state ek , while its next transition will be done toing equations .the statee', k,l= 1,2,，.,0, k≠l,[n"]= [π*]p們]「h1(t) hl2(t) .... h20(t)]h21(t) h22(t) .... h2(t)[h()]=中国煤化工[hl(t) h"2(t) ... h0(t)」JYHCNMH G .M. Bogalecka and K. Kolowrocki/Probabilistic Approach to Risk Analysis of Chemical Spills at Sea119and [pkI] is given by (1whereThe asymptotic distribution of the sojourn totalif a factor fj does not havetime Ok of the process E(t) in the time interval (0, 0),θ > 0, in the state ek is normal with the expected valueinfuence on the sub-region Dxthreat by the substance HE[0k]=p°0= .π*E[0k] A>n' E[0]()=fjit，ifa factor fj has influence on=1the sub-region Dk threat bywhere E[k] are given by (4).the substance H; and this factoris in the range fjt,l= 1,2,.,mjical substances that may cause the sea environmentdegradation and denote them by H1, H,，.,Hn2. Wfor i= 1,2,..n2, j= 1,2,..,n4, k= 1,2,.,n3, isalso distinguish n3, nz∈N environment sub-regionscalled the threat state of the sub-region Dk, caused byD1, D2，.，Dns of the considered environment regiona group of chemicals Hi.Definition 3. A setD= D1∪D2∪...∪Dn3S()={S{%): s{%)∈ {S{}], i= 1,2..,m,that may be degraded by the distinguished chemicalj=1,2,-,n3}, l= 1,2.0k}substance groups. Degrading influence of the chemi-cal substance groups on the distinguished environmentsub-regions is presented in Fig. 2.s{)≠S{%) forl≠l2, 1,l2= 1,2....kHis called the set of the environment threat states of( D\the sub-region Dk, k= 1,2,., n3, while a number VkH2is called the number of the threat states of this sub-region.Definition 4. A function S(k)(t) defined on thetime interval (0, T〉and having values in the threatstates set S(k), such thatHmDnsS(k):<0,T)→s()，k=1,2,..,nsFig. 2 Hazardous chemicals infuence on the marineis called the process of the environment threats of theenvironment sub-regions degradationsub-region Dk, k= 1,2,, n3,Definition 5. A function sl{?)(t) defined on thetime interval (0,T〉 and having values in the threatThe threat level of these sub- regions depends on n4,n4∈N, factors f1, f2,.., fn4; characterised particularchemical substances. Simultaneously, the environments():<0,T)-S(k)，k=12，n3， 1=1,，,0threat by the chemical substance groups is dependenton some of these parameters.is called the conditional sub- process of the environmentDifferent ranges of these factors generating vari-threats of the sub-region Dk, k= 1,2,-., n3, while theous scales of the sea sub-region environment threatsprocess of initiating events E(t) is in the state e'.are also distinguished. Namely, the factor fj; j =We assume a semi-Markov model of the sub- process1,2...,n4，may assume the values in mj rangesS{%)(t), k= 1,2，,n3,l= 1,2,..,0, and denote byfj+,2..,fjims."n its random conditional sojourn times in the stateDefinition 2. A matrixS{x) while its next transition will be done to the stateS{%)s}),i,j= 1,2,..vk,i≠j. Then, this sub-process is .s{X) sdefined by the vector of probabilities of its initial states[S{]nxna=|'at the moment 1中国煤化工Sh2ma」1YHCNMH G[pe(0)]( ' irki ~)]120International Journal of Automation and Computing 2 (2006) 117-124and by the matrix of probabilities of transitions be-and their density functions are given bytween the statesVi=1,2.,Uk，硫(t)= 二唱h2(t).j=1[喝]=7)The expected values E[ni] and variances D[nu] ofvariables n%l are determined respectively byP%whereVi=1,2,.,Uk，Elmial= > P唱E[n2] .Vi= 1,2，..,Vk，p呢=0.(10)Moreover, this process is defined by the matrixv-=-2-,v，D刻=二喝D咖of conditional distribution functions of sojourn times唱of the process Si{%(t) in the state S{) while itswhere E[nZl and D[n] are defined by (8) and (9).next transition will be done to the state S, i,j =Boundary values of the instantaneous probabilities1,2,.,Uk,i≠jof the sub-process of the environment threats S[{")(t)” H1(t) Hk2(t)in its particular statesH2(t) H器(t)H2U"(t)pa(t)= P(S{%)(t)=s{})，i= 1,2,.,Uk (11)[院(t)] =are calculated from the formulaH2+(t) HRp2(t) ... HYKUR(t)」hl= lim pi(t) =π E[nui= 1,2,..,VkVi= 1,2,.,Vk， H楚(t)=0=1and compiling it with a matrix of conditional densities(12)of sojourn times呢of the sub-process S%(t) in thewhere probabilities π硫l satisfy the system of the follow-state S{h) while its next transition will be done to theing equationsstate s}),i,j= 1,2,.,vk,i≠j,[rz] = [ilDn「隰() 号() ... 18*(t) 1仁硫=1[贺()]=h资(t) h啜(t) .... h272(t)[h贺"(t) h%2(t) ... h"()」[Tk]= [,前_,谈and [p龍] is determined by (7).The asymptotic distribution of the sojourn totalVi= 1,2..,Uk,H%(t)= 0.time呢of the process s({?)(t) in the time interval <0, 0),Under these assumptions， the expected valuesθ > 0, in the state S{k) is normal with the expectedE[n2] and variances D[n2] of variables 唱are deter-valuemined byEnu]=poθ= -πi En]0， i= 1,2,，,Vk .El篇]= . th鼠(t)dt8)D[喝g= | (t- E[喝)2h赢()dtwhere E[ni] are given by (10).Afterwards, applying the expression for total prob-= E(n唱)门] -品.9)ability (5) and (11), we can find unconditional proba-bilities of the process of the environment threats of theUnconditional distribution functions of sojournsub-region Dk, k= 1,2，..,n3, in its particular statestimes n系of the process S{")(t) in states S(%, i =according to the following formula1,2,..., Uk, are determined byp咬(t)= P(S(k| 中国煤化工Vi=1,2...,vk，H%()= 2喝HE(t)=9P(lMYHCN MH G()=e)M. Bogalcka and K. Kolowrocki/Probabilistic Approach to Risk Analysis of Chemical Spills at Sea212 P()=e)P(S{%)=s{!})Definition 7. A matrixl=R()(t) R{)(t)R\(t) :R{?)(t) R(?)(t)Ri2()Er(t)i(t), t≥0, i-=2..-.R(t)=(13)_R(?)(t) R?s)(t)Rims)(t)」Hence, for sufficiently large t the boundary proba-bilities of the environment threats process of the sub-whereregion Dk, k: = 1,2,..,n3, in its particular states aregiven byR()(t), k= 1,2,.ns, m= 1,2,.mk，t∈<0,T)are the degradation processes of the environments of呢= lin P(S(k)(t)= s{!)the sub-regions Dk defined on the time interval (0,T)and having their values in the state sets￥》p'pie，i=1,2...,Uk(14)=1R() = {R(), R(),R(k)where p' and pi are defined respectively by (6) andwhere ki = 1,2,..,n3, m = 1,2,...,mk is the process(12).of the environment degradation of the region DSince the particular states of the process of the en-D1∪D2∪...∪Dnsvironment threats S(k)(t) of the sub-region Dk, ki =Definition 8. A function Rlm)(t) defined on the1,2,..,n3， may lead to dangerous effcts degradingtime interval (0, T〉and having values in the degrada-the environment at this sub-region, we assume thattion effect states set R(k) such thatthere are mk different dangerous degradation effects forthe environment sub-region Dx,k= 1,2,..,n3 and weR{M):<0,T)→R()，k=1,2..,n3mark them by RS) , R，, Rimm. This way the setm= 1,2，..,mk， l= 1,2,..,vkR(x)= {R{),R2)，., R()}s the conditional sub-process of the environmentis the set of degradation effects for the environment ofdegradation of the sub-region Dx,k = 1,2,..., n3 whilethe process of environment threats S()(t) of the sub-the sub-region Dx.These degradation effects mayattainregion Dk is in the state S[x),l= 1,2...,Uk.We assume a semi- Markov model of the sub-processferent levels. Namely， the degradation effectRm= 1,2,-,mk， may reach v'm levelsRiml(t)， k = 1,2,.,n3， m = 1,2,.,mk， l =R()，m= 1,2，.., mk, are the states1,2, . Vk, and denote n1imhe by its random conditional .sojourn times in the state Rim), while its next transi-of this degradation effect.The settion will be done to the state Rih), i,j= 1,2..,um),i≠j. Then, this sub-process is defined by the vectorR(N) = {RM), R.......}，m= 1,2,,mkof probabilities of its initial states at the momentt= 0is called the set of states of the degradation effect for[Dmke(0)] = pl)()((..,2(0)]he environment of the sub-region Dk，where m =1,2...,mk.nd by the matrix of probabilities of transitions be-Definition 6. A vectortween the statesR(x()= [R{)(), (),, R()()]，t∈ (0,T)「pmkl p唱[Dpink]= |pmkl呢(15)R(%)(t)，m= 1,2,..,m，t∈ <0,T)are the processes of degradation effects for the environ-Vi= 1,2...,fh)， pxu=0.ment of the sub-region Dk defined on the time interval(0,T) and having their values in the state sets Rlh,Moreover, this中国煤化工”。the ma-m= 1,2,., mk, is the degradation process of the en-rix of conditionYHCNMH(of sojournvironment of the sub-region Dk.times nink1 of the .-m-.9 state Rm122International Journal of Automation and Computing 2 (2006) 117-124while is next transition will be done to the state Rimi,hi,j=1,2,..,uM),i≠j,Vi= 1,2...,vf%， D[mkl= > PineDI[imk]j=0HHhe() Hiar(t) ...Hng(t)where E[ne] and D[nmkl are given by (16) and (17).[Hix(t)]=H2hkr(t) H2?r(t) ..Hmk (tabilities of the sub-process of the environment degra-[H%IL(t) H%i2() ... Hml!”(t)」dation RM)(t) of the sub-region Dx in its particularstateswhereVi= 1,2.-,0(),Hme(t)= 0.Pmk(t)= P(R(M}(t)= R(M)，i=1,2,-.vM) (19)and complied with it's matrix of conditional densitiesare calculated from the formulaof sojourn times nmke of the sub-process RlMl(t) in thestate.while its next transition will be done to thepmkl= lim pink(t)=ThineBlnme (20)state Rimj; i,j= 1,2,...Vm,i≠j,2 rmkl ELrmka]hmke(t) hihie(t) .... hng,"where probabilities π京satisfy the system of the follow-[hie(t)]=h?he(t) h2?e(t) .. h2m2()ing equations(t) h(t) .... hmi盟?(); ][mel = [me]lpmk]2 πmkl=1Vi= 1,2...,v(h)，hme(t)=0.、j=1Under these assumptions， the expected valuesE[n'nki] and variances D[n'nx1] of variables nihel are de-termined by[πmnk]= mmkfee,tonx]，i= 1,2，，oih)und [pink] are determined by (15).EI品a]=。thie(t)dt(16)The asymptotic distribution of the sojourn totaltime nikl of the sub-process Ri2(t) in the time in-Dhmel= [" (- Elanlpnel()dtterval (0,0), θ > 0, in the state RM is normal with the= E[(nmk)2]- E2nmnel.(17)expected valueUnconditional distribution functions of sojournElmkl=pmnko=TunsElmk_ -0, i= .,..9)times nkl of the sub-process Rm{(t) in states Rih),πmkl Emkli= 1,2,.., vm, are determined byVi= 1,2,,u[N)， Hme(t)= > pnmeHinr(t)Finally, applying the expression for total probabil-j=1ity (13) and (19), we can find unconditional probabil-ities of the degradation process in its particular statesand their density functions are given byaccording to the following formulapime(t)= P(R()()= R(k)Vi= 1,2...,uh)， himkr(t)= >mhmke().=1=之P()()= s{").P(Rl) = (5(0()=(!)The expected values E[minki] and variances D[nkel]of variables Mmel are determined respectively by= 2 P(S(k)(t)= s(). P(R()= R(%)中国煤化工Vi=1,2...u)，E[nmkul= 2 PrmnieElnimal= Z()pin(,MYHC N MH Gu'm.j=0 .(18M. Bogalcka and K. Kolowrocki/Probabilistic Approach to Risk Analysis of Chemical Spills at Sea123Hence, for suficiently large t, the boundary proba-where C%e(O) are given by (23).bilities of the process of the degradation efects Rm (t),3 Statistical identification of environ-m= 1,2,, mk, in its particular states are given byment degradation general modelPink= lim P(R{然)(t)= Rm)≌2 phPmkl (21)The quantitative analysis of threats caused by ac-cidents at the sea involving dangerous chemical sub-stances is based on the tree of threatsl2. The treefor i= 1,2,of threats is a graphical presentation of the initiatingPimkl are defined respectively by (14) and (20).events，dangerous chemical substances, and environ-Hence the sojourn total time nink of the process ofment degradation effects interrelated with one another,the degradation efects Rl)(t), m = 1,2...,mk, inas shown in Fig. 3.the time interval (0,0), θ > 0, in the state Rl), hasnormal distribution with the expected valueEJE:LE[EJEsELE]LEJ[E En国E'nim=pimuo，i= 12..Spill of chemicalwhere pink are given by (21).Furthermore, if we denote byCime(t)，i= 1,2, o,h)↓↓↓↓↓↓↓↓↓;he function of environmental losses caused by theRR[RRRJ[R][R[R]process of the degradation effectFig. 3 Tree of threatsR()(t)，m= 1,2.mof the sub-regionIn the tree of threats analysis, we distinguish n1 =1 events initiating dangerous situations E, n2 = 9Dk，k:= 1,2,,n3groups of chemical substances H; that may cause n4 =in the degradation state9 degradation effects Ri in n3 = 4 kinds of environmentsub-regions D)31R(k)，i= 1,2,,vfh)during the timet, t≥0, then the expected value of thetiating events E(t), we firstly fix the number of states Uenvironment losses in the time interval <0,0), θ > 0,of the process E(t) and define the states e',e2,caused by the process of the degradation effect Rn (t)of the set E. Furthermore, we fix the vector of real-isations n*(0), k = 1,2,，..,v, of the numbers of theof the sub-region Dk is given byprocess E(t) transients in the particular states ek atthe initial moment t= 0Cmk(0)≌2 Cnk(rimoO)(22)i=1[n(0)] = [n'(0), n2(0),.,n"(0)]for m = 1,2,.,mk, k = 1,2,.,n3， where Pink areand we fix the matrix of realisations nkl k,l =given by (21), the expected value of the environment1,2,，..,0 of the numbers of the process E(t) transi-losses in the time interval <0, 0) in the sub-region Dk is tions from the state ek into the state e' during thegiven byexperimental timeCk(0)≌2 Cm:(日)， k= 1,2，-,n3(23)m=1[n则=of the total environment losses in the time interval (0, 0>in the region D is given byHaving these numbers, we estimate the vector of3realisations中国煤化工C(0)≌2 Ck(0)(24)MHCNMHG-i124International Journal of Automation and Computing 2 (2006) 117-124of the initial probabilities of the process E(t) transientsin the particular states ek at the moment t = 0 accord-that this approach can give a valuable and useful the-ing to the formulaoretical tool to analyse and to determine the risk ofdangerous chemicals accidents at sea and their conse-p*()=，k=12,,0quences.n(0)Additionally, practical identification of the inves-tigated relations using the proposed models and realdata could be possible to obtain as the next steps ofn(0)=2nz(0)the research.k=1is the total number of the process E(t) realisations atReferencest=0Next, we evaluate the matrix of realisations [p%]山FCrabski. Semi-Markov Modes oft System Reliabity andOperations.. Systemns Research Institute, Polish Academy ofk,l = 1,2，..,0 of the transitions probabilities of theM. Bogalecka, M. Rutkowska. The tree of threats - theprocess E(t) from the state eh into the state e' duringthe sea environment pollu-the experimental time according to the formulation.In ProceedingsofSafetvandRelabilityInternationalConference (KONBiN 2001). Air Force Institute of Technol-ogy Press. Warsaw, vol. 3, pp. 17- -27. 2001 (in Polish).p"(0)= nk,l=1,2，..,0，k≠lM. Bogalecka, K. Kolowrocki. Preliminary approach to risk” analvsis of chemical spills at sea.In Proceedings of Safetvplk=0，k= 1,2,.,0and Reliability International Conference - (KONBiN 2003).Air Force Institute of Technology Press, Warsaw, vol. 1, nowhere3, pp. 37-44, 2003.n*=Snk， k=1,2..,0l≠kMagda Bogalecka is an AssistantProfessor at the Department of Chem-is the realisation of the total number of the process E(t)istry and Industrial Commodity Sctransitions from the state ek: during the experimentalence at the Faculty of Business Ad-timeministration of Gdynia Maritime Uni-versity. Her research interests includeFurthermore, we formulate and verify the hypothe-the transport of dangerous chemicals,ses about the conditional distribution functions of thesea accidents, and emergency responseprocess E(t) lifetime0kl,k,l= 1,2，.,v,k≠l, in theprocesses at sea. She has publishedstate ek, while the next transition is the state e' on themore than 40 papers in journals andconference proceedings.base of their realisations 时l, γ= 1,2,We proceed in an analogous way in statistical eval-uation of unknown parameters of the remaining twoKrzysztof Kolowrocki is a Professorsemi-Markov models of the sub-processes of the envi-and the Head of the Mathematics De-partment at the Faculty of Navigationronment threats Si~(t) and the sub-processes of theof Gdynia Maritime University. Hisenvironment degradation RlM{(t) of the of the sub-research interests include mathemati-regions Dk, k= 1,2...n3.cal modelling of safety and reliabilityof complex systems and processes. He4 Conclusionhas published more than 190 books, re-ports and papers in journals and con-ference proceedings. He is the Vice-Currently, there are no general and complete meth-President of the Polish Safety and Re-ods and results identifying the risk involved in the sealiability Association.中国煤化工MHCNM HG