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Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design

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Minimize cost of instrumentation while satisfying the constraints on attributes like Accuracy Precision Reliability Residual Accuracy etc… The Sensor Network Design Problem

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Minimize Cost of instrumentation such that accuracy of S3= 7% S7= 8% Similarly we can have constraints on residual accuracy, reliability, precision etc.. The Sensor Network Design Problem

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Tree Enumeration Procedure At each node calculate accuracy (and other attributes mandated by the constraints) compare with thresholds. If node is feasible, stop; explore sister nodes. If infeasible, go down. How to find optimal solution?

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The Tree enumeration procedure can be made computationally effective by using cutsets instead of streams (Bagajewicz and Gala, 2006(a)). The efficiency is further more increased by decomposing the graph into subgraphs, (Bagajewicz and Gala, 2006(b)) Gala M and M. Bagajewicz. (2006b). “Rigorous Methodology for the Design and Upgrade of Sensor Networks using Cutsets. Industrial and Engineering Chemistry Research”. Vol 45, No 21, pp. 6679-6686. Gala M and M. Bagajewicz. (2006b) “Efficient Procedure for the Design and Upgrade of Sensor Networks using Cutsets and Rigorous Decomposition”. Industrial and Engineering Chemistry Research, Vol 45, No 21, pp. 6687-6697. Modified Tree Enumeration Procedure

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Accuracy has been conventionally defined as the sum of absolute value of the systematic error and the standard deviation of the meter (Miller, 1996). Since the above definition is of very less practical value, accuracy of a stream can defined as the sum of the precision and the maximum induced bias in the respective stream, Bagajewicz (2005). Software Accuracy

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-Software Accuracy -Precision -Maximum induced bias The maximum induced bias in a stream ‘i’ due a gross error in ‘s’ is given by, (using maximum power measurement test) Where, ‘A’ is the incidence matrix and ‘S’ is the variance covariance matrix of measurements Software Accuracy

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In the presence of n T gross errors in positions given by a set T, the corresponding induced bias in variable ‘i’ is We have to explore all the possible combinations of locations of gross errors. Thus the problem can be stated using a binary vector as Software accuracy in the presence of ‘nt’ gross errors

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When there is more than one gross error, two gross errors may be equal in magnitude but opposite in sign which tend to cancel each other. Gross Error Equivalency S1 S2 S3

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Residual Accuracy of order ‘k’ is the software accuracy when ‘k’ gross errors have been found out and the measurements have been eliminated. Residual Accuracy

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Probability with which a variable ‘i’ can be estimated using its own measurement or through material balance equations in the time interval [0, t]. Estimation Reliability

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Cutset is the set of edges (streams) when eliminated, separates the graph into two disjoint subgraphs. Deletion of a subset of the edges in cutset does not separate the graph into two subgraphs. Streams 8, 6, 2 is a cutset. Streams 2, 3 is another cutset. There are several others. Cutset

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x m = [1, 2, 3]; x m is also a cutset P{S1}= P{S2}= P{S3}= 0.9 Probability of estimating S1= Probability of S1 working or Probability of S2, S3 working simultaneously. Calculation of Estimation Reliability- Example R S1 = P{S1} υ [P{S2}∩P{S3}] R S1 = P{S1} υ [P{S2}×P{S3}] R S1 = P{S1}+ [P{S2}∩P{S3}]- [P{S1}×P{S2}×P{S3}] R S1 = 0.9+0.81-0.9×0.81 When x m = [2, 3]; S1 becomes non redundant and so it can be estimated only by its material balance relations. Thus, R S1 = P{S1}.P{S2} = 0.81 S2S5 1 2 3 S1 S3 S4 4 S6

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If the variable is measured, then its estimation is directly the service reliability of the sensor measuring it. If the variable is not measured, Estimation Reliability for Non Redundant Variable

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Generate all the cutsets that has the variable of interest ‘i’. Removing the variable ‘i’ from those yields the reduced cutsets. Estimation Reliability for Redundant Variable S2S5 1 2 3 S1 S3 S4 4 S6 x m = [1, 2, 3]; Since the variable of interest is S1, the reduced cutset would be [2,3]. Let this be denoted by Z j (i), where ‘i’ is the variable of interest- here it is S1.

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x m = [1, 2, 3, 4, 5]; [1, 2, 3], [1, 4, 5] are two cutsets. [2, 3] and [4,5] are reduced cutsets. P{S1}= P{S2}= P{S3}= P{S4}= P{S5}= 0.9 Probability of estimating S1= Probability of S1 working or Probability of S2, S3 working simultaneously or P { S4 and S5} working simultaneously Calculation of Estimation Reliability- Example R S1 = P{S1} υ [P{S2}∩P{S3}] υ [P{S4}∩P{S5}] R S1 = P{S1} υ [P{S2}×P{S3}] υ [P{S4}×P{S5}] R S1 = [ P{S1}+ [P{S2}∩P{S3}]- [P{S1}×P{S2}×P{S3}] ] υ [P{S4}×P{S5}] R S1 = 0.981+0.81- 0.981×0.81 When x m = [1, 3]; S1 becomes non redundant and so it can be estimated only by its direct measurement. Thus, R S1 = P{S1} = 0.9 Z 1 (1)- reduced cutset Z 2 (1) S2S5 1 2 3 S1 S3 S4 4 S6 ENV

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For a measured variable, For a unmeasured variable, Estimation Reliability for Redundant Variable

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Computation of estimation reliability of unmeasured variable- Sum of disjoint products It can be proved that,

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Input Data: 1.Binary vector of measured streams at each node. 2.Service reliability of sensors. 3.Variables of interest. Steps to be performed: 1.Generate all the cutsets that has the variable of interest. 2.Choose only those reduced cutsets that have measured streams for reliability calculation. Other cutsets are useless as they do not make the variable of interest observable. 3.If no such cutset for unmeasured variable exist, then node is infeasible. Implementation in the Program

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Check if the variables of interest are non redundant. If so we got three cases. Case 1: The variable is measured, then estimation reliability is the sensor service reliability itself. Case 2: The variable is not measured, the estimation reliability is product of service reliabilities of sensors in the reduced cutset. Case 3: The variable of interest is not observable, then node is infeasible, go down the tree. Implementation in the Program Non redundant variable

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Case 1: variable is measured too. Case 2: unmeasured variable. We have already discussed the computational method for above equations. Implementation in the Program Redundant variable

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Compare the obtained reliability with the specifications/ requirements/ thresholds, If node is feasible, transfer control to appropriate statement, which explores sister nodes If infeasible, go down the tree. Implementation in the Program Comparison with threshold

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Let there be ‘n’ sensors when calculating reliability. Assume one of the sensors has malfuntioned and the measurement eliminated, the estimation reliability we now have is “Residual Reliability of order one” The sensor that has a gross error or the malfunctioned sensor can be identified. This helps to know which measurement is eliminated. Residual Reliability

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Input Data: 1.Binary vector of measured streams at a node. Say ‘ns’ streams are measured. 2.Sensor Service Reliability 3.Reduced Cutset Information Calculation of Residual Reliability

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Steps Involved: 1.Choose reduced cutsets from already available information. Eliminate those who have streams with the malfunctioning sensor. 2.Calculate Reliability the same way. Calculation of Residual Reliability- Order One.

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Example 1 23 45 6 710118 9 5 16 21 22 2324 12 14 13 19 2017 15 6 7 9 4 1810 3 1 2 8 Madron and Veverka (1992)

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Instrumentation Details- Madron and Veverka (1992) StreamFlowSensor cost Sensor Precision (%) StreamFlowSensor Cost Sensor Precision 1140192.51310122.5 220172.51410122.5 3130132.51590172.5 440122.516100192.5 510252.5175 2.5 645102.518135182.5 71572.51945172.5 81062.52030152.5 91052.52180152.5 10100132.52210132.5 1180172.5235132.5 1240132.52445132.5

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Software Accuracy when all streams are measured StreamSoftware Accuracy 16.9201 227.8008 35.6573 415.1941 552.7085 615.1186 736.4044 852.6420 952.6686 108.1288 117.4660 1214.2693

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Software Accuracy when all streams are measured 1351.7501 1451.7501 156.6338 165.9791 17101.7665 185.4671 1912.7847 2018.6534 217.4659 2252.6091 23101.7665 2412.7847

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Requested Software Accuracy The software accuracy requested were. Three gross errors were allowed and no feasible nodes were found. Computed Reliability values were 90% for all streams when two gross errors are allowed. StreamThreshold Accuracy 1010% 168% 1810% 1915% 2419%

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Solution of Madron and Veverka (1992) Cost of the NodeStreams Measured Values of Accuracy of requested streams in percentage 1853, 4, 5, 6, 7, 8, 9, 10, 15, 16, 19, 20, 23, 24 S10= 9.00 S16= 5.46 S18=7.53 S19=13.85 S24=13.85 2023, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 19, 20, 23, 24 S10= 9.00 S16= 5.46 S18=7.53 S19=13.85 S24=13.85 2203, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 18, 19, 20, 23, 24 S10= 7.63 S16= 5.48 S18=5.49 S19=12.91 S24=12.91 2273, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 19, 20, 23, 24 (node- a) S10= 9.00 S16= 5.46 S18=7.53 S19=13.85 S24=13.85

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2273, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 19, 20, 23, 24 (node- b) S10= 9.00 S16= 5.46 S18=7.53 S19=13.85 S24=13.85 2393, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 19, 20, 23, 24 S10= 9.00 S16= 5.46 S18=7.53 S19=13.85 S24=13.85 2573, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24 S10= 7.64 S16= 5.48 S18=5.49 S19=12.91 S24=12.91 2751, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 23, 24 S10= 8.91 S16= 5.48 S18=7.00 S19=13.20 S24=13.20 2931, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24 S10= 7.62 S16= 5.48 S18=5.49 S19=12.20 S24=12.20

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Thank You

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