In parallel systems, the failure of any one or more elements does not necessarily mean the failure of the system. A parallel system fails when all of its components fail (Melchers 1999).
If the criterion for acceptable performance of a WDN is defined as at least one node must have adequate pressure, then the network fails if all nodes of the network fail. In this case, WDN is treated as a parallel system. Therefore the probability of failure of water distribution network as a parallel system (Pf)n,p composed of n nodes can be written as: (Pf)n,p=P(f1∩f2∩…∩fj∩…∩fn) (4.5) Similar to Equation 4.1, fj (j=1, n) is the failure event of node j represented by limit state equation 4.2.
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The following steps explain MCS for assessing the reliability of WDN as a parallel system. All steps except Step three are similar to the methodology of series system.
1- The limit state function is established for each node of the network using methodology explained in Sections 3.2.1 of Chapter 3.
2- Random variables and their PDFs are identified, and random variates are then generated using the methodology explained in Section 3.2.3 of Chapter3.
3- The probability of network failure is computed as follows:
The network will fail when limit state is violated at all nodes; namely, if all nodal pressures drop below the minimum required pressure, the network will fail.
4- Steps 1 to 3 are repeated for a large number of trials i.e., 1000 vectors of random nodal demands and pipe roughness coefficients,
5- The probability of network failure as a parallel system Pf,n,p can be calculated:
(Pf)n,p=n/N (4.6) where N is the total number of trials and n is the number of trials in which limit state equation 4.2 is violated for all nodes.
6- The reliability of the network as a parallel system Rn,p is then computed as a complement of the probability of failure (Pf)n,p:
Rn,p=1–(Pf)n,p (4.7)
4.2.3 Numerical Example
The numerical example shown in Figure 4.1 is used to demonstrate the methodology explained in Sections 4.2.1 and 4.2.2.
WDN as a Series System
The probability of failure of the water distribution network shown in Figure 4.1 as a series network (Pf)n,s composed of three Nodes 1, 2, and 3 is as:
(Pf)n, s = P (f1 U f2 U f3) (4.8) In which f1, f2, and f3 are nodal failure events at Nodes 1, 2 and 3, respectively, therefore:
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(Pf)n, s = P ( P1 <30 U P2 <30 U P3 <30) (4.10) Then MCS is applied to compute the probability of network failure as follows:
1- Firstly uncorrelated random nodal demands and pipes roughness coefficient, and then correlated random nodal demands and pipes roughness coefficient are generated as explained in Section 3.2.3.
The independent case is: X d1 =290.22 X d2 = 264.17 X d3 = 221.84 X C1 =133.91 X C2 =116.36 X C3 =144.72 X C4 = 110.22
The correlated case is: X d1 = 179.50 X d2=246.01 X d3= 276.94 X C1 =102.62 X C2 = 111.46 X C3 = 84.52 X C4 = 151.12
2- Then limit state is established for Nodes 1, 2 and 3 as:
G1 = P1– 30≥ 0, for Node 1, (4.11a)
G2 = P2– 30≥ 0, for Node 2, (4.11b)
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3- The Limit state at Nodes 1, 2 and 3 are then checked to assess if the network fails as a series system. Three nodal pressures P1, P2 and P3 under independent random vector X are as:
P1= 38.57 m P2 = 19.57 m P3 = 30.2 m Therefore: G1 = 38.57 –30>0, for Node 1, And G2 = 19.57 – 30<0 , for node 2, And G3 = 30.2 – 30> 0 , for node 3,
Since Node 2 failed, the network is also considered to have failed. 4- Then Steps 1 to 3 are repeated 1000 times.
5- After generating 2000 random vectors of nodal demands and pipe roughness coefficients (1000 vectors of uncorrelated and 1000 vectors of correlated variables), the probability of network failure as a series network (Pf)n,s can be calculated as follow:
If no correlation is considered between nodal demands and no correlation between pipe roughness, the number of trials n in which at least one nodal pressure Pj is below 30m of water or G(X) = Pj– 30< 0 is n =441,
Therefore,
(Pf)n,s =441/1000 =0.441 (4-11d) 6- Then the reliability of the network as a series system under un-correlated variables
can be computed as:
Rn,s =1– (0.441) = 0.559 (4-11e) However, if correlation is considered between all nodal demands, also all pipe roughness coefficients are treated as the correlated variables, then the number of trials n in which at
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least one nodal pressure Pj is below 30 m of water (0.2942 MPa) or G(X) = Pj – 30< 0 is n =461,
Therefore:
(Pf)n,s =461/1000 = 0.461 (4-11f) And network reliability as a series system under correlated variables is:
Rn,s =1– (0.461) = 0.539 (4-11g) As can be seen from the results, when WDN is treated as a series system, the network reliability Rn,s under correlated variables is smaller than the network reliability under independent variables.
WDN as a Parallel System
The probability of the failure of the water distribution network, shown in Figure 4.1 as a parallel system (Pf)n,p composed of three Nodes 1,2, and 3 is:
(Pf)n, p = P (f1 ∩ f2 ∩ f3) (4.12) In which f1, f2, and f3 are nodal failure events at Node1, 2 and 3, therefore:
(Pf)n,p = P (G1 < 0 ∩ G2 < 0 ∩ G3 < 0 ) (4.13)
Or
(Pf)n, p = P ( P1 <30 ∩ P2 <30 ∩ P3 <30 ) (4.14) The MCS is then applied to compute the probability of network failure as follows:
1- The independent and correlated random nodal demands and pipe roughness which were already generated are applied.
2- The limit state is established for Nodes 1, 2 and 3 as:
G1 = P1– 30≥ 0, for node 1, (4.15a)
G2 = P2– 30≥ 0, for node 2, (4.15b)
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3- Then limit state at Nodes 1, 2 and 3 are controlled to assess whether network fails as a parallel system. Three nodal pressures P1, P2 and P3 under correlated random vector X are: P1= 16.5 m P2= -8.93 m P3= -8.1 m Therefore: G1 =16.5 –30<0, for node 1 And G2 = -8.93 –30<0, for node 2 And G3= -8.1 –30<0, for node 3 Since all nodes are failed, the network (as a parallel system) is considered to be failed.
4- Then Steps 1 to 3 are repeated 1000 times.
5- After generating 2000 random vectors of nodal demands and pipe roughness coefficients (1000 vectors of uncorrelated and 1000 vectors of correlated variables), the number of trials n in which all P1 and P2 and P3 are below 30 m of water (0.2942 MPa) are determined to compute the failure of WDN as a parallel system (Pf)n,p,.
If nodal demands and pipe roughness are assumed un-correlated, the number of trials n in which all nodal pressures Pj are below 30 m of water or G(X) = Pj– 30< 0 is n =243, Therefore:
(Pf)n,p =243/1000 =0. 243 (4.15d) Then the reliability of the network as a parallel system is:
Rn,p =1– (0. 243) = 0.757 (4.15e) However, when correlation is considered between nodal demands, and also between pipe
roughness, the number of trials n in which all nodal pressures Pj are below 30 m of water or G= Pj– 30< 0 is n =283,
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Therefore:
(Pf)n,p =283/1000 = 0. 283 (4.15f) 6- Then network reliability is:
Rn,p =1– (0. 283) = 0.717 (4.15g) Similar to a WDN as a series system, when a WDN is treated as a parallel system Rn,p , the
network reliability under correlated variables is smaller than network reliability under independent variables.