Safety and resilience of Global Baltic Network of Critical Infrastructure Networks related to cascading effects

Transcription

1 Blokus-Roszkowska Agneszka Dzula Przemysław Journal of Polsh afety and Relablty Assocaton ummer afety and Relablty emnars, Volume 9, Number, Kołowrock Krzysztof Gdyna Martme Unversty, Gdyna, Poland afety and reslence of Global Baltc Network of Crtcal Infrastructure Networks related to cascadng effects Keywords crtcal nfrastructure, crtcal nfrastructures network, cascadng effect, mpact, safety Abstract The man am of the paper s analyss of safety of the Global Baltc Network of Crtcal Infrastructure Networks (, takng nto account nteractons among partcular crtcal nfrastructure networks ths network conssts of. The safety functon and other safety characterstcs: the rsk functon and mean values and the standard devatons of the lfetmes n the safety state subsets are determned assumng the partcular crtcal nfrastructure networks have exponental safety functons. Fnally, the coeffcents of cascadng effect mpact on the ntenstes of degradaton of the, and the ndcator of that network reslence to cascadng effect mpact, are presented.. Introducton The Baltc ea s the area coverng relatvely hgh concentraton of dfferent knd of nstallatons, that n case of ther malfunctons, can cause sgnfcant negatve nfluence on socetes and natural envronment wthn the regon and ashore around. ome of nstallatons and systems, showng nterconnectons, nterdependences and nteractons, can be qualfed as Crtcal Infrastructure Networks (CI networks, defned as a set of nterconnected and nterdependent crtcal nfrastructures, nteractng drectly and ndrectly at varous levels of ther complexty and operatng actvty [U-CIRCL Report D., U-CIRCL Taxonomy, 5. Works performed wthn the scope of U-CIRCL Report [U-CIRCL Report D., 6, allowed to dstngush eght CI networks operatng n the Baltc ea regon. The networks, abbrevated as BCIN (Baltc Crtcal Infrastructure Networks, have been descrbed and analysed n the report. Further, ther operaton process model has been developed and ntroduced n reports [U-CIRCL Report D3.3, 6 and [Kołowrock, et al., 7. Recognsed networks, operatng wthn the Baltc ea regon, and ther nterconnectons and nterdependences, lead to qualfy them as formng a Network of CI Networks. Ths resulted wth a concept of the Global Baltc Network of Crtcal Infrastructure Networks (. The artcle s amng to analyse above mentoned nterconnectons and nterdependences, by ntroducng an approach to show nfluence of one of networks malfunctons, on the other networks. Basc assumpton adopted s partcular CI network departure from the safety state subsets, causes decreasng lfetmes of the other CI networks. The mpact of each of dstngushed CI networks, on the other CI networks, has been llustrated by specfyng of respectve coeffcents. Above mentoned assumptons allowed to determne basc safety ndcators of the, n case one of CI networks malfunctons have nfluence on other CI networks functonng e.g. ther lfetmes n the safety state subsets: the safety functon, the rsk functon, the mean values and the standard devatons of the lfetmes n the safety state subsets and the ntenstes of the departure from the safety state subsets. 79

2 Blokus-Roszkowska Agneszka, Dzula Przemysław, Kołowrock Krzysztof afety and reslence of Global Baltc Network of Crtcal Infrastructure Networks related to cascadng effects. Global Baltc Network of Crtcal Infrastructure Networks and Its afety Parameters The s a network comprsng eght ( n nterconnected and nterdependent Baltc Crtcal Infrastructure Networks (BCINs, dstngushed and analysed n [Dzula, Kołowrock, 7a-b, [U-CIRCL Report D., 6: BCCIN the Baltc lectrc Cable Crtcal Infrastructure Network BGPCIN the Baltc Gas Ppelne Crtcal Infrastructure Network BOPCIN the Baltc Ol Ppelne Crtcal Infrastructure Network 3 BWFCIN the Baltc Wnd Farm Crtcal Infrastructure Network 4 BORCIN the Baltc Ol Rg Crtcal Infrastructure Network 5 BPCIN the Baltc Port Crtcal Infrastructure Network 6 BCIN the Baltc hppng Crtcal Infrastructure Network 7 BTPOICIN the Baltc hp Traffc and Port Operaton Informaton Crtcal Infrastructure Network. To assess the safety and reslence of the, t has been assumed that: all,,...,, and the under consderaton have the safety state set,..., z z the safety states are ordered, the safety state s the worst and the safety state z s the best, T (,,..., are ndependent random, varables representng the lfetmes of, n the safety state subset u, u,..., z whle they were n the safety state moment t, z at the T ( s a random varable representng the lfetme of a n the safety state subset u, u..., z whle t was n the safety state z at the moment t, the BCINs and the states degrades wth tme t, s (t,,...,, s the safety state at the moment t, t, gven that t was n the safety state z at the moment t, s (t s the safety state at the moment t, t, gven that t was n the safety state z at the moment t. The above assumptons mean that the safety states of the wth degradng BCINs may be changed n tme only from better to worse [Kołowrock, 4, [Kołowrock, oszyńska-budny,,, [Xue, 95, [Xue, Yang, 995. Further, followng four safety states z 3, of the and each of BCINs have been dstngushed: / BCINs state of full ablty: z 3, / BCINs mpendency over safety state: z, state of / BCINs unrelablty: z, state of full nablty of the / BCIN : z. The crtcal safety state of the and the BCINs s r. The probablty that the s n the safety state subset u, u,..., z at the moment t, t, whle t was n the safety state z 3 at the moment t, determned as the safety functon of, s a vector (t,,,, 3 t,,...,, ( where P ( s ( t u s ( 3 P( T ( t, u,,,3. ( Under ths defnton,...,. Further, t s assumed that the BCINs have dentcal exponental safety functons (t,,,, 3 t,,...,, (3 wth the coordnates exp[ ( t, u,3. (4

3 Journal of Polsh afety and Relablty Assocaton ummer afety and Relablty emnars, Volume 9, Number, The wll be analysed under the assumpton t s a multstate seres network. That means that the s n the safety state subset { u, u,..., z }, f and only f all ts BCIN networks are n ths subset of safety states. As t has been prevously stated, there are four safety states and the best state z 3. The BCIN lfetmes n the safety states are expressed n years and they have the exponental safety functons (3-(4 wth the ntenstes of departure from the safety subsets, by the assumpton, gven by (., (.5, ( 3.,,...,. (5 3. Interactons between Baltc Crtcal Infrastructure Networks Baltc Crtcal Infrastructure Networks are nterconnected and nterdependent [U-CIRCL Report D3., 6. They are nteractng each other so as the BCIN departure from the safety state subsets causes decreasng lfetmes of the remanng BCINs. Then, f the network ( =,,, leaves the safety state subset {u,u+,,3} (u =,,3, then the safety parameters of the remanng BCIN networks worsen dependng on the type of the network wth the coeffcents of the BCIN mpact on the other BCINs. Namely, the BCINs lfetmes and ther mean values n the subset {υ,υ+,,3} (υ = u,u-,, and u =, decrease accordng to the formulas T ( [ q(, T ( / [ / ( T [ q(, [ T (, (6,,,, (7 where q(,, are the coeffcents of the network networks and mpact on the functonng of other ( =,,,,, q(,,,,...,, ( q(,, (9 for =,,,, =,,,, υ = u, u,, and u =,. These coeffcents, exstng n (6-(7, take by the assumpton followng values q (,,.5, q (,,.,,3,, ( q (,,., q (,,.5, 3,4,, ( q (,, 3., q (,, 3.,,4,5,6,7,, ( q (,, 4.3, q (,, 4.,,3,5,6,7,, (3 q (,, 5., q (,, 5.,,3,4,6,7,, (4 q (,, 6.35, q (,, 6.5,,3,4,5,7,, (5 q (,, 7.4, q (,, 7.,,3,4,5,6,, (6 q (,,.5, q (,,.5,, 7, (7 and ( holds. Consequently, the safety functon of ( =,, network after the departure of ( =,,, from the subset {u,u+,,3} (u =,,3, s defned as a vector / [,, /, / / t (,3, t,,,,, ( wth the coordnates gven by / P( T / ( t, u, u, u, (9 / P( T / ( t P( T ( t u 3, u. (

4 Blokus-Roszkowska Agneszka, Dzula Przemysław, Kołowrock Krzysztof afety and reslence of Global Baltc Network of Crtcal Infrastructure Networks related to cascadng effects Under the assumpton about the exponental dstrbuton, the condtonal ntenstes of the network departure from the subset {υ,υ+,,3} after the departure of the network, by (7, are ( (, ( q(, / for,,,, u, u, u. Thus, consderng (4, (-( and (, the networks ( =,,, after the departure of ( =,,, from the safety subset have the safety functons ( wth the coordnates exp[ q( ( / t, u, u, u, ( exp[ (, u 3, / t u. (3 4. afety and Reslence of Global Baltc Network of Crtcal Infrastructure Networks Assumng the s a multstate seres network and dependence between BCINs, expressed n (6-(7, n case the BCINs have exponental safety functons (3-(4 and consderng (-(3, the safety functon of the related to cascadng effects s gven by the vector [Blokus-Roszkowska, Kolowrock, 7a-b C [,,, 3, (4 where C C C exp[ ( ( [exp[ q(, exp[ ( ( (, ( C ( t ( t (, q(, C (, exp[ (3 (3 [exp[ q(, exp[ ( q(, t, ( (, (3 (, (3 t ( t, t ( (5 (6 3 exp[ (3 t, (7 C for t. Next, applyng (4-(7 and substtutng the assumed values of BCIN ntenstes (5 and the abovementoned values of coeffcents of the BCIN mpact on other networks functonng (-(7, the coordnates of the safety functon are C (t, exp[.4t [exp[.3t exp[.54t exp[.76t exp[.46t exp[.95t exp[.435t exp[.64t exp[.44t exp[.63t exp[.43t exp[.35t exp[.475t exp[.53t exp[.493t exp[.3t exp[.54t, ( C (t, exp[.t [exp[.4t exp[.t exp[.4t exp[.t exp[.439t exp[.39t exp[.47t exp[.7t exp[.44t exp[.4t exp[.46t exp[.6t exp[.4t exp[.t exp[.57t exp[.97t, (9 (t,3 exp[.t. (3 C The safety functon coordnates of the related to cascadng effects, gven by (-(3, are llustrated n Fgure.

5 Journal of Polsh afety and Relablty Assocaton ummer afety and Relablty emnars, Volume 9, Number, Fgure. The graphs of the safety functon coordnates. If r s the crtcal safety state, then the second safety ndcator of the related to cascadng effects s the rsk functon r ( t P( T ( r t, t, (3 C (t, (t, (t, (t, (t,3 (t,3 C C C r C (t 5 5 Fgure. The graph of the rsk functon. Other safety characterstcs of the related to cascadng effects are the mean values and the standard devatons of the lfetme n the safety state subsets {,,3}, {,3}, {3}. In case the BCINs have exponental safety functons (3-(4 and consderng (-(3 for assumed model of dependency, the mean values can be counted from the formulae defned as a probablty that the related to cascadng effects s n the subset of safety states worse than the crtcal safety state r whle t was n the best safety state z 3 at the moment t = [Kołowrock 4, [Kołowrock, oszyńska-budny, and gven by r C ( t, t, (3 C where C (t, s the coordnate of the safety functon gven by (9. Hence, the moment when the rsk functon exceeds a permtted level, s C r (, (33 C where r ( t, f exsts, s the nverse functon of C the rsk functon r (t, gven by (3. For the assumed value., the moment of exceedng an acceptable level equals C.55 years days. (34 The graph of the rsk functon, called the fraglty curve, s llustrated n Fgure. C C ( [/ q(, /[ ( ( ( ( ( (, q(, C ( (, ( [/ q(, /[ (3 (3 (3 (, ( ( (, q(, (3 (, (, (35 (36 (3 (37 C (3 3

6 Blokus-Roszkowska Agneszka, Dzula Przemysław, Kołowrock Krzysztof afety and reslence of Global Baltc Network of Crtcal Infrastructure Networks related to cascadng effects and the standard devatons from C for u, where n C ( u nc ( [ C ( (3 ( [ [ /[ /[ q n C q(, ( ( ( (, ( ( ( (,, ( [ [ /[ /[ q(, q(, (3 (3 (3 (, ( ( ( (3 (,, ( (, (39 (4 and ( 3. (4 C (3 Accordng to (35-(37 and substtutng the values of coeffcents (-(7 and ntenstes (5, the mean lfetmes of the n the subsets {,,3}, {,3}, {3}, n years, are: ( 5.5, (.3, C C ( 3.5. (4 C mlarly, applyng (3-(4, the standard devatons of the lfetmes can be determned and ther values n years are: ( 4.96, (.5, C C ( 3.5. (43 C The mean values of the lfetmes n the partcular states,,3, by (4, n years are: (.75, (.5, C C ( 3.5. (44 C Other safety ndces are the ntenstes of departure from the safety state subsets {,,3}, {,3}, {3},.e. the coordnates of the vector λ [,, λ (t,, C λ C λ (t,3, t. (45 C These ntenstes can be determned from the formula λ C dc dt, u,3, (46 C where C, u,3, are gven by (-(3, and after some transformaton they take form λ C (t, {.4.5 [.6 exp[.t. exp[.4t.76 exp[.4t.46 exp[.6t.95 exp[.5t.435 exp[.35t.64 exp[.36t.44 exp[.4t.63exp[.37t.43exp[.3t.35 exp[.65t.53exp[.47t 4.93exp[.93t}.475 exp[.75t /{.5 [exp[. t exp[.4t exp[.4t exp[.6t exp[.5t exp[.35t exp[.36t exp[.4t exp[.37t exp[.3t exp[.65t exp[.75t exp[.47t exp[.93t}, t, (47 λ C (t, {..5 [.976 exp[.3t.776 exp[.t.4 exp[.3t. exp[.t.439 exp[.36t.39 exp[.39t.47 exp[.393t.7 exp[.7t.44 exp[.396t.4 exp[.4t.46 exp[.33t.6 exp[.6t.57 exp[.3t.97 exp[.7t} /{.5 [ exp[.3t C 4

7 Journal of Polsh afety and Relablty Assocaton ummer afety and Relablty emnars, Volume 9, Number, exp[.t exp[.3t exp[.t exp[.36t exp[.39t exp[.393t exp[.7t exp[.396t exp[.4t exp[.33t exp[.6t exp[.3t exp[.7t }, t, (4 λ 3., t, (49 C and ther graphs are llustrated n Fgure Fgure 3. The graphs of the ntenstes. Usng these ntenstes, the coeffcents of cascadng effect mpact on the ntenstes of departure from the safety state subsets {,,3}, {,3}, {3}, can be estmated. Then, the coordnates of the vector ρ [,, ρ (t,, C ρ C ρ (t,3, t, (5 C are gven by λc ρ C, u,3, (5 λ u where λ, u,3, are the ntenstes of the departure from the safety state subset {,,3}, {,3}, {3}, wthout of cascadng effect mpact. nce the s consdered as a seres network, the ntenstes of the departure wthout of cascadng effect mpact are λ (, t, u,3, (5 u and from (5 they take values λ.6, λ.4, C λ 3.. (53 (t, λ (t, (t, λ (t, (t,3 λ (t,3 C C C Then, applyng (5 and from (47-(49 and (53, the coeffcents of the cascadng effect mpact on the ntenstes of degradaton are ρ ρ C C 6.5λ, C.5λ, C ρ 3 t, (54 C where λ C (t,, λ C (t,, are respectvely gven by (47 and (4. The ndcator of the reslence to cascadng effect mpact s defned by RI C r, t, (55 ρ r C where ρ r s the coeffcent of C cascadng effect mpact on the ntenstes of degradaton gven by (54 and the crtcal safety state s r. Respectve graph s llustrated n Fgure Fgure 4. The graph of the reslence ndcator. Fnally, the statonary coeffcent of cascadng effect mpact on the ntenstes of departure from the safety state subset not worse than a crtcal safety state r.e. {,3}, s determned usng (4 and (53 ~ (.5 ρ ( C.7 (56 (.3 C RI C (t, 5 5 and the statonary ndcator of the reslence to cascadng effect mpact for a crtcal safety state r, by (56, takes value 5

8 Blokus-Roszkowska Agneszka, Dzula Przemysław, Kołowrock Krzysztof afety and reslence of Global Baltc Network of Crtcal Infrastructure Networks related to cascadng effects ~ R I C ( ~.9 9%. (57 ρ ( 5. Conclusons C Analyss of safety of the, takng nto account nteractons of partcular BCINs, has allowed to determne the safety functon and other safety characterstcs: the rsk functon, the mean values and standard devatons of lfetmes n the safety state subsets, the coeffcents of cascadng effect mpact on the ntenstes of degradaton of the Global Baltc Network of Crtcal Infrastructure Networks, and the ndcator of that Network reslence to cascadng effect mpact. Future researches n ths feld wll nclude nfluence of clmate-weather change process, and the operaton process of the. Moreover, researches results wll be enhanced wth consderaton of crcumstances when malfunctons of one of networks have varous nfluence on the other networks. Acknowledgements The paper presents the results developed n the scope of the U- CIRCL proect ttled A pan uropean framework for strengthenng Crtcal Infrastructure reslence to clmate change that has receved fundng from the uropean Unon s Horzon research and nnovaton programme under grant agreement No References [ Blokus-Roszkowska, A. & Kolowrock, K. (7a. Modellng relablty of multstate systems wth dependent components and subsystems. Journal of Polsh afety and Relablty Assocaton, pecal Issue on U- CIRCL Proect Crtcal Infrastructure Impact Models for Operaton Threats and Clmate Hazards, Part, Crtcal Infrastructure Connectons, (3, 3-4. [ Blokus-Roszkowska, A. & Kolowrock, K. (7b. Modelng safety of nterconnected crtcal nfrastructure network cascadng. Journal of Polsh afety and Relablty Assocaton, pecal Issue on U-CIRCL Proect Crtcal Infrastructure Impact Models for Operaton Threats and Clmate Hazards, Part, Crtcal Infrastructure Connectons, (3, [3 Dzula, P. & Kołowrock, K. (7a. Modellng safety of Global Baltc Network of Crtcal Infrastructure Networks. Journal of Polsh afety and Relablty Assocaton, ummer afety and Relablty emnnars, (3, [4 Dzula, P. & Kołowrock, K. (7b. Integrated Impact Model on Global Baltc Network of Crtcal Infrastructure Networks afety Related to Its Operaton Process. Journal of Polsh afety and Relablty Assocaton, ummer afety and Relablty emnnars, (4, 3-. [5 U-CIRCL Report D.. Identfcaton of exstng crtcal nfrastructures at the Baltc ea area and ts seasde, ther scopes, parameters and accdents n terms of clmate change mpacts, 6. [6 U-CIRCL Report D3.. Baltc ea area crtcal nfrastructures global network assets and nterconnectons ( network of networks approach, 6. [7 U-CIRCL Report D3.3. The Baltc ea area crtcal nfrastructures global network ( network of networks safety modellng, dentfcaton and predcton (wthout clmate-weather change nfluence, 6. [ Kołowrock, K. (4. Relablty of Large and Complex ystems. Amsterdam, Boston, Hedelberd, London, New York, Oxford, Pars, an Dego, an Francsco, ngapore, dney, Tokyo. [9 Kołowrock, K., Blokus-Roszkowska, A., Bogalecka, M., Dzula, P., Guze,. & oszyńska- Budny, J. (7. WP3-Task3.3-D3.3-Part - Inventory of Crtcal Infrastructure Impact Assessment Models for Clmate Hazards, U- CIRCL Report. [ Kołowrock, K. & oszyńska-budny, J. (. Relablty and afety of Complex Techncal ystems and Processes: Modelng - Identfcaton - Predcton Optmzaton. London, Dordrecht, Heldeberg, New York. [ Kołowrock, K. & oszyńska-budny J. (. Introducton to safety analyss of crtcal nfrastructures. Journal of Polsh afety and Relablty Assocaton, ummer afety and Relablty emnars 3, 73-. [ Xue, J. (95. On mult-state system analyss. I Transacton on Relablty, 34, [3 Xue, J. & Yang, K. (995. Dynamc relablty analyss of coherent mult-state systems. I Transacton on Relablty, 4(44,