World Environmental and Water Resources Congress 2013: Showcasing the Future © ASCE 2013
Source Identification in Water Distribution Systems using the Adjoint Method with Non-Ideal Sensors and Non-Detect Measurements D.E. Wagner1 and R.M. Neupauer2 1
Department of Civil, Environmental, and Architectural Engineering, University of Colorado, UCB 428, ECOT 441, Boulder, CO 80309-0248, FAX (303) 492-7317; email:
[email protected]
2
Department of Civil, Environmental, and Architectural Engineering, University of Colorado, UCB 428, ECOT 441, Boulder, CO 80309-0248, FAX (303) 492-7317; email:
[email protected] ABSTRACT It is important to identify the source of contamination in a drinking water distribution system to discontinue the event and determine ways to prevent future contamination from entering the system. Previous work has demonstrated the effectiveness of an adjoint method using contaminant concentration measurements from system sensors to probabilistically determine the source of contamination in a drinking water system. The method can be applied in systems with either steady-state or transient hydraulics. The method uses publicly available software (EPANET) coupled with a particle backtracking model (BTX) and a conditioning method to probabilistically identify the contamination source and release time. Prior work assumed perfect sensors capable of measuring the exact concentration of contamination. In this work the method is enhanced to account for measurements from more realistic, non-ideal sensors that identify the range of contamination at the sensor (e.g., high, medium, low), rather than the precise concentration. In addition, the method is augmented to use nondetect measurements, i.e., sensor measurements for which the measured contamination is below the limit of detection. INTRODUCTION Water utilities are tasked with providing an uninterrupted supply of potable drinking water to their service populations. The events of September 11, 2001 illuminated the potential for terrorist activities on U.S. soil and increased the focus on protecting utilities and infrastructure from such attacks (DHS 2003). Specifically, efforts have been made to harden the water infrastructure against contamination. While removing the threat of contamination would be the best deterrent, this is not feasible, and so it is important to moderate the effects of contamination. Expedient source identification is one method for reducing the effects of contamination in a water system. Previous work on source identification focused on inverse methods (e.g., Islam et al. 1997; Laird et al. 2005, 2006; Guan et al. 2006; Preis and Ostfeld 2006, 2007). Inverse methods employ advanced algorithms to determine the origin of contamination. Using system observations (e.g., contaminant concentration arriving
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at a sensor) as simulation goals, the algorithms in conjunction with modeling software test multiple contamination scenarios (e.g., source node, release time, or concentration) and run multiple forward simulations in the water distribution system to determine what scenario leads to the observed outcomes. While many researchers have shown that inverse methods are able to correctly determine the contamination scenario, these methods are inherently inefficient as they require a forward simulation for each potential scenario. An alternative method has been developed which uses one simulation to find multiple potential contamination scenarios: the adjoint method (Neupauer et al. 2010). The adjoint method is a tool for directly calculating the sensitivity of a system state at a particular location and time to a system parameter; this sensitivity is called the adjoint state and is the state variable of the adjoint of the forward contaminant transport equations. For source identification, the system state is the concentration observed at a sensor and the system parameter is the mass released at the source. Adjoint methods have been successfully used for source identification in groundwater (e.g., Neupauer and Wilson 1999), but research in water distribution systems has been limited to relatively simple systems with perfect sensors (e.g., Neupauer et al. 2010). Both inverse and adjoint methods are highly dependent on the availability of data within the system (e.g., contaminant concentration, contaminant arrival time, etc.). Although the sensors which are currently being used by water utilities are unable to accurately determine the levels of contaminant concentrations consistently (ASCE, 2004), many researchers assume that precise contaminant concentration data is available. A more likely scenario is using “fuzzy sensors” which are only able to determine the relative ranges of water quality parameters (e.g., low, medium, or high level). Preis and Ostfeld (2008) developed an inverse method which was able to find the true source node and time using data from fuzzy sensors, however their research showed that the fuzzy sensor data led to a lower rate of success and an increased number of potential contamination scenarios. This work uses the adjoint method developed by Neupauer et al. (2010) as the foundation for a new adjoint-based method which uses fuzzy sensor data to determine the contamination scenario. In addition to using fuzzy sensor data, sensor measurements of no contamination (i.e. non-detects) are used to help determine the contamination scenario. This is something that has not previously been integrated into adjoint methods. ADJOINT METHOD The method developed by Neupauer et al. (2010) uses contaminant concentration observations to calculate travel time probability density functions (PDFs) which identify the most likely contamination scenarios in a water distribution system. These PDFs are related to adjoint states of concentration. The adjoint states are obtained by solving adjoints of forward equations.
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Forward transport of a conservative chemical in pipes can be modeled using Ci Q i Ci + =0 t Ai x i where Ci, Qi, Ai, and xi are the concentration, flow rate, cross-sectional area, and distance along pipe i respectively, while t is time.
∂ ∂
∂ ∂
Assuming complete mixing at the nodes, the concentration in the nodes can be determined using ∑i∈di = j Q i Ci |xi =Li + U j C*j = D j + ∑i∈u = j Q i
(1)
(2)
i
where C *j is the concentration of water leaving node j, di is the downstream node of pipe i, Li is the length of pipe i, Uj is the mass loading rate at node j, Dj is the water demand at node j, and ui is the upstream node of pipe i. These equations can be solved for nodal concentrations as a function of time for known contaminant sources, defined by the source node, release time, and source concentrations. In this way, information is propagated downstream from the source to all possible downstream nodes. In the source identification problem, the nodal concentrations are known at one or multiple observation points, i.e., sensor nodes. We solve the adjoints of the forward equations to determine the possible source nodes for a set of observed concentrations. Neupauer (2011) showed that the adjoints to Equations (1) and (2) are ψi Qi ψi =0 τ Ai x i and Q i ψ i x =0 + U *l ∑ i∈u i =l * i ψl = ∑i∈d =l Qi
∂ ∂
∂ ∂
(3)
(4)
i
respectively, where ψ i =dCi/dMℓ is the adjoint state of the concentration in pipe i, which represents sensitivity of the concentration in the pipe i (Ci) to a source mass released at node l (Mℓ), τ is backward time, defined as τ = tf - t where tf is a dC *j ( τ sj ) * reference time, ψ l ( τ; j, τ s ) = is the adjoint state of concentration at Node j, dM l (τ ) which physically represents the marginal sensitivity of nodal concentration at backward time τ to a source release of mass Mℓ at node l, τsj is the backward time at which the concentration is at Node j, U *l is the load term given by U *l = δ (τ − τ sj ) for l =j, and δ(·) is a Dirac delta function.
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The adjoint equation (3) differs from the forward equation (1) in the sign on the Q ∂ψ i ) which is negative instead of positive because the adjoint advection term ( i A i ∂x i state is propagated against the flow of water. Neupauer et al. (2010) showed that the marginal sensitivity of the nodal concentration, as determined by Equation (4), can be used to determine a backward travel time probability density function (BTTPDF) through f T (τ; l, j, τS ) = ψ*l (τ; j, τS ) ∑ Q i (τ) (5) i∈u i =l
where fT(τ; l,j,τS) is the backward travel time probability density function representing the backward time τ that a contaminant particle observed at node j at backward time τs could have been released at node ℓ, which is a potential source node in the water distribution system. When Equation (4) is solved using a single observation (i.e., a specific contaminant concentration observed at a specific node and time), the PDF obtained from (4) identifies all hydraulically connected, upstream nodes as potential source nodes, each with different possible release times. More information can be obtained if two or more observations are used. If two or more observations identify the same source node/time combination, then that combination is a possible solution. If the results show that one observation identifies the node as a potential source node and the other does not, then that node is not a possible source. Combining information from multiple observations is accomplished by calculating the joint BTTPDF, using (Neupauer et al. 2010)
(
(
)
NS
f T τ; l, j1 , j2 ,... jN S , τs = α T ∏ f T (τ; l, j, τs )
)
(6)
n =1
where f T τ; l, j1 , j2 ,...jNS , τs is the joint backwater travel time probability density function, αT is used to ensure that the total probability is unity (as seen in Neupauer et al. 2010), and NS is the number of observations. Neupauer and Records (2009) used Bayes’ theorem to condition the joint BTTPDF on measured contaminant concentrations to produce the following relationship
f T|Cˆ*
(
)
τ | cˆ* ; l = βT ∫
m
NS
∏f j =1
Cˆ *jn | M , T
(cˆ
* jn
)
| m, τ; l f T (τ; l, jn , τsn )dm
(7)
where f T |Cˆ * (τ | cˆ* ; l ) is the conditioned BTTPDF at node ℓ, Cˆ* is the vector of Ns contaminant concentration observation, M is a random source mass, * f Cˆ* |M ,T (cˆ*jn | m, τ; l ) is the PDF of obtaining the measured concentration cˆjn (τsn ) for a jn
given source node ℓ, release time τ, and source mass m, defined as a normal * distribution with a mean of mψ jn (τ; l, τsn ) , and the standard deviation σ of the measured concentration. The value for βT, which ensures that the total probability is unity, is determined using (Neupauer et al., 2010)
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World Environmental and Water Resources Congress 2013: Showcasing the Future © ASCE 2013
β T−1 = ∫
∞
0
NS
∫ ∏f m
j=1
Cˆ*jn |M ,T
(cˆ
)
| m, τ; l f T (τ; l, jn , τsn )dmdt
* jn
(8)
Neupauer et al. (2010) showed that source node and time can be determined by first using (3) and (4) for each observation to obtain the adjoint states which are used in (5) and (7) to obtain the conditioned BTTPDF for each possible source node. The BTTPDF for a particular node represents the backward time at which contamination could have been released at that node to produce the observed concentrations at the sensors. If the set of observed concentration cannot have originated from the particular node, the conditioned BTTPDF for that node will be zero at all times. The magnitude of βT can be used to determine the relative likelihood of each node being the source node (Neupauer et al. 2010).
FUZZY SENSORS The method developed by Neupauer et al. (2010) assumes contaminant concentration measurements from ideal sensors. Fuzzy sensors are unable to provide discrete values for the contaminant concentrations and, therefore, present a problem for the original method. The contaminant concentrations are used to determine the PDF of the measured concentration in (7). Neupauer and Records (2009) used a normal distribution to describe this PDF: ⎧⎪ cˆ* − c* 2 ⎫⎪ 1 * f Cˆ* | M , T cˆ jn | m, τ; l = exp⎨− jn 2 jn ⎬ (9) 2 jn 2σ jn ⎪ 2Πσ jn ⎪⎩ ⎭ * * where is cˆ jn the measured concentration at the observation node, c jn is the true concentration at the observation node, and σ2 is the variance of the measurement error.
(
(
)
)
Since fuzzy sensor data only measures the contaminant concentration qualitatively, we cannot use this relationship. Instead, we calculate the cumulative distribution function, FCˆ * | M , T (cˆ*j | m, τ; l ) , which is equal to the probability that the measured j
* concentration is less than or equal to the concentration bound cˆ j . For example, the following relationship can be used to determine the probability for a middle-range reading
(
)
(
ˆ ≤c )=F * P(clower < C ˆ | M , T c upper | m, τ; l − FCˆ * | M , T c lower | m, τ; l upper C j
=∫
c upper
c lower
f Cˆ | M , T (cˆ j | m, τ; l )dcˆ j j
j
) (10)
where clower and cupper are the lower and upper bounds of the middle-range reading. The final expression is differentiated with respect to cˆ j and the results are used in place of f Cˆ *
jn
|M ,T
(cˆ
* jn
NON-DETECTS
)
| m, τ; l in (7).
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World Environmental and Water Resources Congress 2013: Showcasing the Future © ASCE 2013
Contaminant concentration observations below the limit of detection may be referred to as non-detect results or non-detects. Depending on the scenario, there are far more contaminant concentration measurements below the limit of detection (non-detects) than above (detects). Inverse methods use both non-detects and detects when determining the source of contamination in a water distribution system, however current adjoint-based methods (e.g., Neupauer et al. 2010) do not use non-detect results. We developed an adjoint-based method which uses non-detect measurements to help determine the source of contamination. We use a similar approach as is used with detects, but instead of resulting in a probability that the contaminant came from an upstream node, we determine the probability that the contaminant did not come from an upstream node. The probability that a node is not the source node (PNS) = 1-PS where PS is the probability that the node is the source. The cumulative distribution function (CDF) is calculated using t
PNS = FT (τ; l, j) = ∫ f T (τ; l, j)dτ 0
(11)
where FT (τ; l, j) is the CDF representing the probability that a contaminant was not at node l at or before time t, τ is the backward time, t is the forward time for the final non-detect measurement being evaluated, and f T (τ; l, j) is the adjoint travel time PDF. We use the CDF calculated to determine the probability that the node is a source node ( 1 − FT (t; l, j) ) and differentiate this value with respect to time to obtain a PDF of the source, which has the same meaning as the PDF in (7) and therefore can be evaluated with the PDFs from positive observations (i.e., detects).
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World Environmental and Water Resources Congress 2013: Showcasing the Future © ASCE 2013
REFERENCES ASCE. (2004). Interim voluntary guidelines for designing an online contaminant monitoring system. Reston, VA: American Society of Civil Engineers. DHS. (2003). Homeland Security Presidential Directive 7: Critical Infrastructure Identiification, Prioritization, and Protection. Washington, D.C.: Department of Homeland Security. Guan, J., Aral, M., Maslia, M., & Grayman, W. (2006). Identification of contaminant sources in water distribution systems using simulation-optimization method: Case study. Journal for Water Resources Planning and Management, 252262. Islam, M. R., Chaudhry, M. H., & Clark, R. M. (1997). Inverse Modeling of Chlorine Concentration in Pipe Networks Under Dynamic Conditions. Journal of Environmental Engineering, 1033-1040. Laird, Biegler, L., van Bloemen Waanders, B., & Barlett, R. (2005). Contamination source determination for water networks. Journal for Water Resources Planning and Management, 125-134. Laird, C., Biegler, L., & van Bloemen Waanders, B. (2006). Mixed-integer approach for obtaining unique solutions in source inversion of water networks. Journal for Water Resources Planning Management, 242-251. Neupauer, R. (2011). Adjoint sensitivity analysis of contaminant concentrations in water distribution systems. Journal of Engineering Mechanics. Neupauer, R. M., & Records, M. K. (2009). Conditioned backward probability modeling to identify contamination sources in a water distribution system. Proceedings, World Environmental and Water Resources Congress 2009. Reston, VA: ASCE. Neupauer, R. M., & Wilson, J. L. (1999). Adjoint method for obtaining backward-intime location and travel time probabilities of a conservative groundwater contaminant. Water Resources Research, 3389-3398. Neupauer, R., Records, M., & Ashwood, W. (2010). Adjoint model to identify contaminant sources in water distribution systems. Journal for Water Resources Planning and Management, 587-591. Preis, A., & Ostfeld, A. (2006). Contamination source identification in water systems: A hybrid model tree-linear programming scheme. Journal of Water Resources Management, 263-273. Preis, A., & Ostfeld, A. (2007). A contamination source identification model for water distribution system security. Engineering Optimization, 941-951. Preis, A., & Ostfeld, A. (2008). Genetic algrorithm for contaminant source characterization using imperfect sensors. Civil Engineering and Environmental Systems, 29-39.
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