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Adjoint-based Method for Contaminant Source Identification in Complex Distribution Systems Using Realistic Sensor Data D.E. Wagner1 and R.M. Neupauer2
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1
Department of Civil, Environmental, and Architectural Engineering, University of Colorado, UCB 428, ECOT 441, Boulder, CO 80309-0428, FAX (303) 492-7317; email:
[email protected] 2
Department of Civil, Environmental, and Architectural Engineering, University of Colorado, UCB 428, ECOT 441, Boulder, CO 80309-0428, FAX (303) 492-7317; email:
[email protected] ABSTRACT Water distribution systems sensors are becoming increasingly more efficient and effective at discovering water quality changes in water distribution systems. We are attempting to develop an efficient and effective method to determine the source of the water quality change. We use publicly available software (EPANET) coupled with a particle backtracking model (BTX) and a conditioning method to probabilistically locate the contamination source. In prior work, we have shown that this method is effective for both steady-state and transient flow conditions in a simple water distribution system (i.e., containing one reservoir, one pump, and one tank). In this work, we demonstrate the effectiveness of this approach in more complex water distribution systems (i.e. containing multiple tanks, reservoirs, valves, and/or pumps) with more realistic sensor conditions (e.g., “fuzzy sensors”, non-detect results, sampling uncertainty). INTRODUCTION Finding and abating sources of drinking water quality changes has always been a goal of water utilities. Increased awareness of the potential for deliberate injection of biological or chemical contaminants has extended the interest in drinking water contamination to other public safety and security organizations (Ostfeld, 2006). Increasing the ability to efficiently and effectively determine the source of deleterious contamination can significantly reduce both the population affected by water contamination (with subsequent loss of service) and the resources required to moderate the spread of the contamination. Water contamination and/or loss of service have a clear impact on public welfare (both physical and mental). When developing a method to determine the source of water quality changes in a water distribution system, it is important to appropriately account for the inherent complexities. Structures such as pumps, valves, and storage tanks play a significant role in the spread of contamination and greatly increase the difficulties associated with determining the source of contamination. Many methods currently in use utilize water distribution system modeling software such as EPANET (Rossman, 2000) as
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the abacus for computing how a water distribution system will react to varying conditions. This software allows the user to input the characteristics of the water distribution system and model how the system will react under different conditions. Through this method, the observation locations of water quality changes can be used to run simulations and determine where the contamination entered the system, most typically through the use of an inverse process. Many researchers have employed this or similar methods to accomplish source identification [Laird et al., 2005; Laird et al., 2006; Guan et al., 2006; Preis and Ostfeld, 2006; Preis and Ostfeld, 2007]. These methods are highly dependent on the availability of sensor data within the system. Generally, more sensors and more accurate data lead to better results. The current state of technology is unable to consistently, accurately determine the levels of contaminant (ASCE, 2004). Instead, the sensors are best utilized for determining when certain water quality parameters are outside specified ranges; essentially acting as a presence/absence test (i.e. water quality is “good” or “not good”). This information can be used along with the hydraulic data of the system to determine where the contamination originated. With a well-defined model of the system, simulations can be run with varying input parameters until the outputs determined by the sensors are replicated. Processing data in this way is often referred to as an inverse method. The user takes outputs of the system to calculate what the inputs must/could have been. Various inverse methods have been employed in the past [Dawsey et al., 2006; Preis and Ostfeld, 2008; De Sanctis et al., 2010]. De Sanctis et al. (2010) developed a source identification method which utilized the particle backtracking algorithm of Shang et al. (2002). In their method, they used both positive and negative sensor outputs to determine all possible sources and times of contaminant intrusion into the water distribution system. Neupauer et al. (2010) used a different approach. Instead of running multiple forward simulations with varying input parameters, they mathematically derived an adjoint solution to the forward transport equation. The adjoint equations allowed them to backtrack the contamination through the system to determine potential sources. Additionally, Neupauer et al. (2010) were then able to use an algorithm to estimate the probability of each potential source node being the actual source node. METHODS For this research, the simulations were conducted utilizing EPANET 2.0 (Rossman, 2000) software in conjunction with a particle backtracking extension (BTX) developed by Shang and Uber (2009). The primary PBA relationship is: =
−
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In this equation c0 = the concentration; N = number of flow paths between the observation node and the (potential) source node; j = flow path index; γj = water quality impact coefficient; sj = water quality input strength for path j; T0 = observation time; and, tj = water travel time for flow path j. Neupauer et al. (2010) developed a similar relationship for the sensitivity of the observation node concentration at backward time τ to a mass M released at the source node at time = 0: ∗
∗
:
=
In this equation φ*j is the adjoint state; τ is the backward time; l is the (potential) source node; C*j is the concentration at the observation node; and, Ml is the mass at the (potential) source node. Neupauer et al. (2010) used the adjoint state to find the backward travel time probability density function (PDF) as shown by: ; ,
=
∗
; ∈
In this equation, fT is the backward travel time PDF at the (potential) source node given the results at the observation node and Qi is the flow rate in pipe i. With some minor adjustments, the impact coefficient (Shang et al., 2002) can be used in place of the adjoint state to determine the backward travel time PDF. The backward travel time PDF can then be conditioned using the following relationship (Neupauer et al., 2010): |
∗
;
=
; ,
∗
|
,
̂∗ |
, ;
In this equation, Ĉ* is the vector of measured concentrations; ml is a random source mass at the (potential) source node; and, jm is the node number of mth observation. The following relationships are also true: ∗ 1. = ∗ , ∗ ,…, ∗ = the vector of measured concentrations. 2. α is used for normalization. 3. ; , is the unconditioned travel time PDF for an observation at Node jm. ∗ ∗ ̂∗ | , ; = = the PDF of measured 4. , | , 2 concentration. In this equation, σ is the measurement of the error variance. 5. If tm is the observation time of mth observation, then C*jm (tm) is the true concentration of mth observation.
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Proper utilization of this method requires using data from two or more observation events (i.e. different nodes or different times) to calculate probable source nodes and times for the contamination. In order to accomplish this, the PBA code (BTX) developed by Shang and Uber (2009) is used to calculate impact coefficients for each potential source node. The impact coefficient is then conditioned to determine the PDFs for each potential source node (Neupauer and Wagner, 2011). DISCUSSION Initially, there were some difficulties with this method. Observation data sets were created by running a forward simulation in EPANET. This simulation created observation data at various nodes based on the input parameters. The observation data was then used as the inputs for BTX and the results were used in the adjoint equations to determine where the contamination had originated. In this way, the results could be compared to the true inputs of the system and the validity of the method could be established. However, when the method was employed, the results did not completely validate the method as established. Even for relatively simple, transient flow systems, the results for this method did not accurately predict the true inputs (source node and time). While, the results often agreed that the actual source node was at least a potential source node, the predicted time of contaminant introduction was generally inaccurate. The primary reason for the discrepancy involved dispersion. Running the forward EPANET simulation produced observation data that included numerical dispersion. This numerical dispersion equates to the longitudinal dispersion that would be found in a real water distribution system. BTX is a particle backtracking algorithm and does not include dispersion, so the BTX results were highly sensitive to the observation data point used. In order to overcome this obstacle, sets of observation data were used instead of single points. This was accomplished by creating impact coefficient vectors. Each impact coefficient vector would be calculated by finding the individual impact coefficients for each observation in a set of observations. A set of observations consists of consecutive observations which begin and end with a positive observation. This idea is illustrated using Figure 1. If the impact coefficients are calculated for each non-zero (positive) observation, there are 15 or more non-zero results. On the other hand, if these observations were grouped into sets, there are 3 sets of observations (denoted by the numbers “1”, “2”, and “3”). Each set of observations begins with a non-zero observation preceded by a non-observation (zero or below an established threshold) and ends with a non-zero observation followed by a nonobservation.
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450
2
400
Concentration (mg/L)
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350
1
300 250 200
3
150 100 50 0 310
330
350
370
390
410
430
Time (min)
Figure 1. Observation Node data. Once the observations are separated into sets, the impact coefficients are calculated for each observation in the set. For instance, Set #1 in Figure 1 contains approximately 5 observations, so at least 5 impact coefficients would be calculated (more if multiple potential source nodes were being evaluated). For each potential source node, the impact coefficients are combined into a vector of impact coefficients. For example, if there were 2 observations in the set and the impact coefficient vector was [0.0; 0.8; 0.0] for the first observation and [0.3; 0.0; 0.0] for the second, then the combined impact coefficient vector would be [0.3; 0.8; 0.0]. By creating these combined impact coefficient vectors for each set of observations, the method presented here is best able to produce results consist with natural processes (e.g. dispersion).
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REFERENCES ASCE. (2004). “Interim voluntary guidelines for designing an online contaminant monitoring system.” American Society of Civil Engineers, Reston, Virginia. Dawsey, W.J., Minsker, B.S., and VanBlaricum, V.L. (2006). “Bayesian belief networks to integrate monitoring evidence of water distribution system contamination.” J. Water Resour. Plann. Manage., 132(4), 234-241. De Sanctis, A.E., F. Shang, and Uber, J.G. (2010). “Real-time identification of possible contamination sources using network backtracking methods.” J. Water Resour. Plann. Manage., 136(4), 444-453. Guan, J., Aral, M.M., Maslia, M.L., and Grayman, W.M. (2006). “Identification of contaminant sources in water distribution systems using simulationoptimization method: Case study.” J. Water Resour. Plann. Manage., 132(4), 252-262. Laird, C.D., Biegler, L.T., van Bloemen Waanders, B.G., and Barlett, R.A. (2005). “Contamination source determination for water networks.” J. Water Resour. Plann. Manage., 131(2), 125-134. Laird, C.D., Biegler, L.T., and van Bloemen Waanders, B.G. (2006). “Mixed-integer approach for obtaining unique solutions in source inversion of water networks.” J. Water Resour. Plann. Manage., 132(4), 242-251. Neupauer, R.M., Records, M.K., and Ashwood, W. (2010). “Backward Probabilistic Modeling to Identify Contaminant Sources in Water Distribution Systems.” J. Water Resour. Plann. Manage., 136(5), 587-591. Neupauer, R.M. and Wagner, D.E. (2011). “Adjoint-based probabilistic characterization of contaminant sources in water distribution systems under realistic flow and sampling conditions.” Proceedings, World Environmental and Water Resources Congress, American Society of Civil Engineers, Palm Springs, California. Ostfeld, A. (2006). Enhancing water-distribution system security through modeling. J. Water Resour. Manage., 209-210. Preis, A. and Ostfeld, A. (2006). “Contamination source identification in water systems: A hybrid model trees-linear programming scheme.” J. Water Resour. Manage., 132(4), 263-273. Preis, A. and Ostfeld, A. (2007). “A contamination source identification model for water distribution system security.” Engineering Optimization. 29(8), 941951. Preis A., and Ostfeld, A. (2008). “Genetic algorithm for contaminant source characterization using imperfect sensors.” Civil Engineering and Environmental Systems, 25(1), 29-39. Rossman, L.A. (2000). EPANET 2: User’s Manual. U.S. Environmental Protection Agency, EPA/600/R-00/057, Cincinnati, Ohio. Shang, F., Uber, J.G., and Polycarpou, M.M. (2002). “Particle backtracking algorithm for water distribution system analysis.” J. Environ. Eng., 128(5), 441-450. Shang, F., and Uber, J. G. (2009). EPANET Backtracking Extension (BTX) User's Manual (Version 1.0). Cincinnati, OH, USA: University of Cincinnati.
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