Resilience of water systems in wake of disruptions Lina Sela Dept. of Civil, Architectural & Environmental Engineering, UT Austin 1/48
The Bad News Water crisis in Flint, Mich., L.A.’s aging water pipes; federal state of emergency a $1 ‐ billion dilemma January, 2016 February, 2015 2/48
The Bad News 3/48
The Good News: Smart Cities 4/48
The Good News: Smart Homes 5/48
Infrastructure systems Reduce: � Water loss � Water quality � Energy requirements � Infrastructure failures � Supply interruptions 6/48
Sensor placement Objective � Sensor placement for detection and location identification of failures Approach 1. Influence model � Network and sensing models 2. Combinatorial optimization � The minimum test cover (MTC) problem � Augmented greedy solution algorithm - L. Sela and S. Amin. ““Robust sensor placement for pipeline monitoring: Mixed integer and greedy optimization.” Advanced Engineering Informatics , 2018. - L. Sela, W. Abbas, X. Koutsoukos, and S. Amin. “Minimum test cover approach for fault location identification in flow networks.” Automatica , 2016. - W. Abbas, L. Sela, X. Koutsoukos, and S. Amin. “An efficient approach to fault identification in urban water networks using multi-level sensing.” ACM BuildSys 2015 . 12/48
Influence model Sensing: Detection: 100 � Pressure signal - p ( t ) 1 if ξ ( p i , t − ˆ p i , t ) ≥ ε, 80 y S i ( t , ℓ j ) = 0 otherwise . 60 40 Node 2 Node 5 20 Fault signature: 0 2 4 6 8 10 12 14 16 18 Sensor state - y ( t ) � 1 1 if y S i ( t , ℓ j ) = 1 , for any t > 0 , y S i ( ℓ j ) = 0 otherwise . Node 2 Node 5 0 0 2 4 6 8 10 12 14 16 18 Fault matrix: Time [s] y S ( ℓ 1 ) L = { ℓ 1 , . . . , ℓ n } – set of n y S ( ℓ 2 ) failure events M ( L , S ) = . S = { S 1 , . . . , S m } – set of m . . sensor locations y S ( ℓ n ) 13/48
Influence model Example: S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 100 Pressure signal - p ( t ) ℓ 1 1 1 1 0 1 0 0 0 80 ℓ 2 1 1 1 1 0 1 0 0 ℓ 3 1 1 0 1 1 0 0 1 60 ℓ 4 1 0 1 1 1 1 1 0 40 Node 2 ℓ 5 1 0 1 1 0 1 1 0 Node 5 M ( L , S ) = 20 0 2 4 6 8 10 12 14 16 18 ℓ 6 0 1 1 1 1 0 1 1 ℓ 7 0 0 1 1 1 1 1 1 ℓ 8 0 1 0 1 1 0 1 1 Sensor state - y ( t ) 1 ℓ 9 0 0 1 1 0 1 1 1 ℓ 10 0 0 0 1 1 1 1 1 Node 2 Node 5 0 0 2 4 6 8 10 12 14 16 18 Time [s] ℓ 3 ℓ 8 2 5 8 ℓ 6 ℓ 10 ℓ 7 X ℓ 1 4 7 ℓ 4 ℓ 9 1 3 6 ℓ 2 ℓ 5 14/48
Detection as MSC Detection The detection problem is to select the minimum number of sensors S ⊆ S , such that when a detectable event occurs, at least one sensor in S detects the event. Minimum set cover (MSC) Let L be a finite set of elements, and C = { C i : C i ⊆ L} be the collection of given subsets of L . The minimum set cover is to find C s ⊆ C with the minimum cardinality such that � C i = � C j . C i ∈C C j ∈C s Proposition The detection problem is equivalent to the MSC problem where � � � � � f D ( C S ) = C i � is the detection function, C i ⊆ L is the set of link � � � � C i ∈C S � failure events detected by the sensor S i , i.e., C i = { ℓ j ∈ L| y S i ( ℓ j ) = 1 } . 15/48
Solving the MSC The greedy approach ◮ In each iteration select: (a) Select C i ∗ ∈ C covering the most uncovered elements in L . (b) Add to current set C ∗ ← C ∗ ∪ { C i ∗ } . (c) Repeat until all elements in L are covered or no new element can be covered by any C i ∈ C . ◮ Best approximation ratio of O (ln n ). ◮ Running times O ( mn ). Can be made faster by reducing the number of function evaluations exploiting the submodularity property. Lazy greedy (Krause et al 2008). 16/48
Solving the MSC The greedy approach ◮ In each iteration select: (a) Select C i ∗ ∈ C covering the most uncovered elements in L . (b) Add to current set C ∗ ← C ∗ ∪ { C i ∗ } . (c) Repeat until all elements in L are covered or no new element can be covered by any C i ∈ C . ◮ Best approximation ratio of O (ln n ). ◮ Running times O ( mn ). Can be made faster by reducing the number of function evaluations exploiting the submodularity property. Lazy greedy (Krause et al 2008). 16/48
Identification as MTC Identification The identification problem is to select the minimum number of sensors S ⊆ S that uniquely detect the events in L . Pair-wise event { ℓ i , ℓ j } is detectable , if there exists a sensor that gives different outputs for ℓ i and ℓ j , ∃ S p ∈ S : y S p ( ℓ i ) � = y S p ( ℓ j ). Minimum test cover (MTC) The MTC is to find C t ⊆ C with the minimum cardinality such that if for a pair of elements { ℓ u , ℓ v } ∈ L , there exists C i ∈ C that contains either ℓ u or ℓ v but not both, then there exists some C j ∈ C t that also contains either ℓ u or ℓ v , but not both. Proposition The problem of identification of link failures in networks is equivalent to the MTC problem. 17/48
Example cont.: Detection: { S 2 , S 4 } Identification: { S 1 , S 2 , S 3 , S 5 } 1 1 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 0 1 1 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 1 0 0 1 1 1 1 0 1 1 0 1 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 1 0 0 1 1 0 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 ◮ All events are detected ◮ All events are detected ◮ Only three unique sensor outputs ◮ All events are uniquely identified 18/48
Solving the MTC Greedy solution 1. Input: C = { C 1 , · · · , C m } , C i ⊆ L . 2. Transform: the MTC to the equivalent MSC ◮ Create a new set of events : L t = { ℓ t 12 , · · · , ℓ t ( n − 1) n } . For each unordered pair { ℓ i , ℓ j } , define a new element ℓ t ij . ◮ Create a new sets of sensors’ outputs : C t = { C t 1 , · · · , C t m } , where C t v = { ℓ t ij : |{ ℓ i , ℓ j } ∩ C v | = 1 } , ∀ k ∈ { 1 , · · · , m } . 3. Solve: using greedy algorithm i ∗ ∈ C t covering the most uncovered elements in L t . (a) Select C t (b) Add to current set C ∗ ← C ∗ ∪ { C i ∗ } . (c) Repeat until all elements in L t are covered or no new element in L t can be covered by any C t i ∈ C t . 4. Output: MTC, C ∗ ⊆ C . 19/48
Solving the MTC Greedy solution 1. Input: C = { C 1 , · · · , C m } , C i ⊆ L . 2. Transform: the MTC to the equivalent MSC ◮ Create a new set of events : L t = { ℓ t 12 , · · · , ℓ t ( n − 1) n } . For each unordered pair { ℓ i , ℓ j } , define a new element ℓ t ij . ◮ Create a new sets of sensors’ outputs : C t = { C t 1 , · · · , C t m } , where C t v = { ℓ t ij : |{ ℓ i , ℓ j } ∩ C v | = 1 } , ∀ k ∈ { 1 , · · · , m } . 3. Solve: using greedy algorithm i ∗ ∈ C t covering the most uncovered elements in L t . (a) Select C t (b) Add to current set C ∗ ← C ∗ ∪ { C i ∗ } . (c) Repeat until all elements in L t are covered or no new element in L t can be covered by any C t i ∈ C t . 4. Output: MTC, C ∗ ⊆ C . 19/48
Solving the MTC Greedy solution 1. Input: C = { C 1 , · · · , C m } , C i ⊆ L . 2. Transform: the MTC to the equivalent MSC ◮ Create a new set of events : L t = { ℓ t 12 , · · · , ℓ t ( n − 1) n } . For each unordered pair { ℓ i , ℓ j } , define a new element ℓ t ij . ◮ Create a new sets of sensors’ outputs : C t = { C t 1 , · · · , C t m } , where C t v = { ℓ t ij : |{ ℓ i , ℓ j } ∩ C v | = 1 } , ∀ k ∈ { 1 , · · · , m } . 3. Solve: using greedy algorithm i ∗ ∈ C t covering the most uncovered elements in L t . (a) Select C t (b) Add to current set C ∗ ← C ∗ ∪ { C i ∗ } . (c) Repeat until all elements in L t are covered or no new element in L t can be covered by any C t i ∈ C t . 4. Output: MTC, C ∗ ⊆ C . 19/48
Solving the MTC Greedy solution 1. Input: C = { C 1 , · · · , C m } , C i ⊆ L . 2. Transform: the MTC to the equivalent MSC ◮ Create a new set of events : L t = { ℓ t 12 , · · · , ℓ t ( n − 1) n } . For each unordered pair { ℓ i , ℓ j } , define a new element ℓ t ij . ◮ Create a new sets of sensors’ outputs : C t = { C t 1 , · · · , C t m } , where C t v = { ℓ t ij : |{ ℓ i , ℓ j } ∩ C v | = 1 } , ∀ k ∈ { 1 , · · · , m } . 3. Solve: using greedy algorithm i ∗ ∈ C t covering the most uncovered elements in L t . (a) Select C t (b) Add to current set C ∗ ← C ∗ ∪ { C i ∗ } . (c) Repeat until all elements in L t are covered or no new element in L t can be covered by any C t i ∈ C t . 4. Output: MTC, C ∗ ⊆ C . 19/48
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