City University of New York (CUNY) CUNY Acadeic Works International Conference on Hydroinforatics 8-1-2014 Experiental Design For Model Discriination And Precise Paraeter Estiation In WDS Analysis Giovanna Darvini Follow this and additional works at: http://acadeicworks.cuny.edu/cc_conf_hic Part of the Water Resource Manageent Coons Recoended Citation Darvini, Giovanna, "Experiental Design For Model Discriination And Precise Paraeter Estiation In WDS Analysis" (2014). CUNY Acadeic Works. http://acadeicworks.cuny.edu/cc_conf_hic/214 This Presentation is brought to you for free and open access by CUNY Acadeic Works. It has been accepted for inclusion in International Conference on Hydroinforatics by an authorized adinistrator of CUNY Acadeic Works. For ore inforation, please contact AcadeicWorks@cuny.edu.
11 th International Conference on Hydroinforatics HIC 2014, New York City, USA EXPERIMENTAL DESIGN FOR MODEL DISCRIMINATION AND PRECISE PARAMETER ESTIMATION IN WDS ANALYSIS GIOVANNA DARVINI (1) (1): Departent ICEA, Università Politecnica delle Marche, via Brecce Bianche, Ancona, 60131,Italy In water distribution syste (WDS) analysis, hydraulic odels ay be used for design, optiization and control purposes. The hydraulic odel of an existing network is built on the basis of the quality and quantity of the collected data. During this first stage the correction of input data errors should be copleted, since different odels of the network could provide results consistent with available experiental data. After choosing the best odel aong those available, one ust calibrate the selected one. Very frequently, distinct procedures are applied for discriination of rival odels and estiation of precise odel paraeters, leading to different saple designs. To conciliate the objectives of both experiental design procedures, the present paper proposes the use of a ulti-objective optiization ethod. Different odels of a water distribution network have been copared and calibrated by using available easureents of nodal pressure. As observed in the analyzed exaples, the analysis of the Pareto set provides the identification of the optial location where install additional pressure sensors for adequate odel discriination and paraeter estiation. INTRODUCTION In WDS analysis, the initial odel building stage is significantly coplicated by the uncertainties in the collected data (network topology, inforations about pipes, asset data, deand allocation, etc.). As a consequence, ore than one odel ay be taken to siulate the WDS, being each of these different odels capable of representing the available experiental data with siilar perforances. Consequently, additional experients ust be designed to select the best odel aong the available alternatives. In addition, further experients ay be needed to iprove the precision of the paraeter estiates, which will iprove also odel predictions and perforance [1] Experiental design procedures for odel discriination and for estiation of precise odel paraeters are usually treated as independent techniques and generally lead to different experiental designs, although odel discriination and reduction of variances of paraeter estiates are closely related to each other [2]. In the present paper the use of a ulti-objective optiization ethod, derived by the field of the cheical engineering, is proposed for WDS analysis [3]. Different odels of an existing network are juxtaposed with one another. During calibration, available easureents of nodal pressure are copared with results obtained by each hydraulic odel to deterine the values of the pipe roughness, assued as odel
paraeters. Starting fro an initial nuber of pressure data, the procedure allows to identify optial locations for additional easures. In turn, these new data are added to ones previously available and used to perfor new siulations though which it is possible siultaneously obtain a suitable odel discriination and precise paraeter estiation. The results are presented as Pareto fronts where the two different objectives can be copared with each other to find the optial solution of the sapling design proble. EXPERIMENTAL DESIGN Experiental designs are alost always perfored iteratively. Based on previous experiental observations, new experients are designed and perfored in order to optiize the perforance index. Then, the new observations are included in the experiental data set and the desired objectives are analyzed. If the obtained results do not eet the required perforance indexes, new experients are designed and realized [1]. In the following a brief suary of the sequential experiental design for precise paraeter estiation and rival odels discriination is presented separately. Precise paraeter estiation The sequential experiental design for precise paraeter estiation is usually perfored through the optiization of a nor of the posterior covariance atrix of paraeter estiates, described as the expected covariance atrix of paraeter estiates after the inclusion of the new set of observations in the experiental data set. The posterior covariance atrix of paraeter estiates can be defined as follows [2,4]: 1 ~ N K 1 1,,,, T V Bi Vi Bi V (1) i N 1 where V, is the current covariance atrix of paraeter estiates of odel (obtained with the available N experients), Ṽ, is the posterior covariance atrix of paraeter estiates (after inclusion of the new K observations), V i is the covariance atrix of experiental uncertainties at experiental condition i and B i, is the atrix of sensitivities of odel which contains the first derivatives of the responses of odel with respect to the odel paraeters at the ith experiental condition, defined as b y r r, p (2) θ p that is, the derivative of the response r with respect to the paraeter p (for a specific odel and experiental condition i). Model discriination To perfor the discriination of rival odels, experiental conditions are norally designed for axiization of soe easure of the difference of the responses obtained with the distinct probable odels. Following Schwaab et al. [5], after perforing N preliinary experients, a new experiental condition should be selected in order to axiize the odel discriination function, defined as n Z T 1 P P y y V y y D, (3) n n, n n
where y is a vector of responses of odel at experiental condition x with odel paraeters θ estiated fro the available N experients. In the case of M odels, the discriinant can be coputed considering all pairs of odels and n. The atrix V,n is defined as V, n 2V V Vn (4) where V is the covariance atrix of experiental deviations and V is the covariance atrix of odel prediction deviations calculated fro odel. In turn, the covariance atrix of odel prediction for odel can be calculated as [2,4]: ~ B. (5) V BV, T In Eq. (3) Z is a paraeter used to odulate the relative iportance of the rival odels: if Z is greater than 1, odel prediction differences are agnified; if Z is saller than 1, odel prediction differences are iniized. The odel probabilities P and P n used in Eq.(3) can be calculated with the help of standard statistical tools and are not sensitive to the ordering of available experients [5]. MULTI-OBJECTIVE PROCEDURE The ulti-objective optiization proble consists in the siultaneous optiization of S objective functions (f 1 (x), f 2 (x),, f S (x), S 2), where x is the feasible set of decision vectors. In ulti-objective optiization, there does not typically exist a feasible solution that iniizes all objective functions siultaneously. Therefore, attention is paid to Pareto optial solutions, constituted by solutions that cannot be iproved in any of the objectives without degrading at least one of the other objectives. By following Alberton et al. [1], the ulti-objective optiization procedure is defined here as the siultaneous axiization of design criteria used for estiation of precise odel paraeters and discriination of rival odels, assuing that a single experient ust be designed each tie. The experiental design criterion used for precise paraeter estiation is the iniization of the deterinant of the posterior covariance atrix of paraeter estiates. To iprove the quality of the result presentation, the objective function is noralized to give values in the range [0,1], as follows: FE F ~ 1 ~ 1 detv det V. (6), in, In analogous anner, Eq. (3) is rewritten as relative objective functions, as D D, n D. (7) ax As a consequence, functions expressed by Eqs. (6) and (7) ust be axiized in the search region. ILLUSTRATIVE EXAMPLE The hypothetical network proposed by Greco and Del Giudice [6] was used to test the procedure for discriination aong three different odels. The WDS schee is illustrated in Figure 1.
A 1 1 6 5 9 6 16 11 5 8 15 2 4 7 7 13 10 2 3 3 4 14 12 8 11 9 10 B Figure 1. Network schee [6] The pipe characteristics and the nodal deand for the odel 1 are reported in Tables 1 and 2 respectively. Model 2 and 3 differs fro the first one just for the diaeter of one pipe. In the odel 2 the diaeter of pipe 5 is changed fro 80 to 100 ; while, in the odel 3 the diaeter of pipe 15 is odified fro 150 to 175. In the initialization of the procedure, it is assued that odel 1 is the actual odel. Given the nodal deands of Table 2, the siulation odel uses the roughness values ε** to generate the nodal pressure reported in Table 2, assued as experiental pressure data at nodes where sensors are installed. Water distribution network siulation odel EPANET [7] was used to calculate nodal pressures, but other software can be eployed. Table 1. Pipe characteristics Pipe Diaeter () Length () Roughness ε** () Initial estiated roughness ε * () 1 250 200 0.60 0.5 2 150 400 0.30 0.3 3 200 300 0.10 0.7 4 300 190 0.70 0.4 5 80 210 0.40 0.5 6 200 300 1.50 0.7 7 150 160 0.30 0.3 8 300 200 1.00 0.7 9 200 180 0.10 0.4 10 250 140 0.10 0.7 11 200 360 0.60 0.3 12 150 200 0.40 0.5 13 150 340 0.90 0.4 14 80 180 0.20 0.4 15 150 180 0.05 0.3 16 200 345 0.60 0.5
Table 2. Nodal deand and pressure values Node Deand Pressure (l/s) () 1 20 86.63 2 30 67.36 3 20 67.09 4 10 67.10 5 20 70.27 6 30 67.83 7 10 67.81 8 10 79.01 9 20 72.96 10 10 68.07 11 30 67.26 The pipes are divided in four groups with the sae roughness coefficient, and the initial estiates of pipe roughness ε* reported in Table 1 are arbitrarily chosen. The reduction of the paraeter diension is a coon engineering practice for reducing the nuber of unknown paraeters in the WDS calibration. The influence of pipe groupings on the odel error and the odel prediction error was thoroughly analyzed by Mallick et al. [8]. For calibrating the network odel the approach proposed by Greco and Del Giudice [6] is adopted. Preliinary experients are developed by locating five pressure sensors at nodes 2, 3, 4, 6 and 11 and the corresponding easured pressure data of Table 2 are used for paraeter estiation. In the exaple, experiental variances are set to 0.01 for each pressure easureent y r,. The errors of experiental responses are assued to be independent fro each other. In this exaple, when one odel assues a relative probability saller than 2.5%, the odel is discarded [5]. Since soe authors argue strongly against the use of the odel probabilities in discriination procedure [9], in Eq. (3) is assued Z=0. Results of the preliinary experients are reported in Table 3. On the basis of the probability values, odel 2 can be eliinated, while between odels 1 and 3 the discriination is not possible and new experients ust be designed in order to discriinate between the. Moreover, Table 3 shows that the odel 1 has a larger probability to be the best odel, while the deterinant of the covariance atrix of paraeter estiates for odel 3 is saller than value obtained for odel 1. Table 3. Results after preliinary experients with five sensors Model ε 1 () ε 2 () ε 3 () ε 4 () P detv, 1 0.56 0.41 0.77 0.45 0.536 2.72 10-31 2 0.60 0.44 0.76 0.43 0.006 3.43 10-31 3 0.67 0.32 0.82 0.47 0.458 2.32 10-33 In order to select the optial location for the sixth pressure sensor, the Pareto front for the relative estiation function F E1 and F E2 and the relative discriination function F D is illustrated in Figure 2, by considering all the possible cases. The Pareto front shows that the two objectives are conflicting for both the odels and suggests that locating the additional sensor at node 5 leads to the ore precise roughness estiation, while the highest discriinant function is obtained if one install the sensor at node 8.
1 F Ei 0.8 0.6 0.4 node 7 node 10 node 5 node 9 node 1 node 8 0.2 0 0 0.2 0.4 0.6 0.8 1 F D Figure 2. Pareto front for the relative estiation function F E1 (red) and F E2 (black) and the relative discriination function F D for design of the sixth sensor. Table 4 shows the results obtained after inclusion of the sixth sensor at each of the potential nodes and the new paraeter estiation. In any of the cases one of two rival odels reaches a relative probability higher than 97.5% and the discriination is not attained. Table 4. Model probability after including additional sixth sensor node 7 node 10 node 5 node 9 node 1 node 8 P 1 0.40 0.65 0.35 0.10 0.90 0.15 P 2 0.60 0.35 0.65 0.90 0.10 0.85 On the basis of results reported in Figure 2, the sixth sensor is installed at node 8. Then, the procedure is repeated by considering all the reaining nodes for the installation of the seventh sensor. In Figure 3 is illustrated the Pareto front for the two objective functions with seven sensors, at all the possible positions. The analysis of the front shows that the optial solution is installing the additional sensor at node 5, where for odel 2 both objective functions are equal to 1. 1 F Ei 0.8 0.6 0.4 node 7 node 10 node 5 node 9 node 1 0.2 0 0 0.2 0.4 0.6 0.8 1 F D Figure 3. Pareto front for the relative estiation function F E1 (red) and F E2 (black) and the relative discriination function F D for design of the seventh sensor. The results obtained for seven sensors and after the new paraeter estiation are reported in Table 5 for all the nodes considered. When the seventh sensor is added at node 5, the odel probabilities indicate that odel 2 results not adequate and the odel 1 is selected as the best odel. Nowhere else the discriination is possible. Furtherore, after the sequential procedure, the final value of the deterinant of the posterior covariance atrix of paraeter estiates detṽ,1 is 1.20 10-34, a value significantly saller than that obtained initially for the selected odel and reported in Table 3.
Table 5. Model probability after including additional seventh sensor CONCLUSIONS node 7 node 10 node 5 node 9 node 1 P 1 0.15 0.08 0.98 0.18 0.06 P 2 0.85 0.92 0.02 0.82 0.94 This work presented an approach for discriinating rival odels in WDS analysis and siultaneously iproving the precise paraeter estiation during odel calibration. The proposed procedure was largely used in the field of the cheical engineering. As shown in the exaple, the procedure allows the discriination of the ost suitable odel aong those available during the odel building and perits to reduce significantly the uncertainty in the paraeter estiates, by selecting the best additional experients. Moreover, the analysis of the Pareto front shows when the two objectives are conflicting and helps the analyst in solving the saple design proble. REFERENCES [1] Alberton A. L., Schwaab M., Biscaia E. C. and Pinto J. C, Sequential experiental design based on ultiobjective optiization procedures, Cheical Engineering Science, Vol. 65, (2010), pp 5482-5494. [2] Schwaab M., Monteiro J. L. and Pinto J. C, Sequential experiental design for odel discriination taking into account the posterior covariance atrix of differences between odel predictions, Cheical Engineering Science, Vol. 63, (2008), pp 2408-2419. [3] Darvini G., Model discriination in water distribution systes, CCWI2013, Procedia Engineering, Vol. 70, (2014), pp 419-428, doi:10.1016/j.proeng.2014.02.047. [4] Bard Y., Nonlinear Paraeter Estiation, Acadeic Press, New York, (1974). [5] Schwaab M., Silva F. M., Queipo C. A., Barreto Jr. A. G., Nele M. and Pinto J. C., A new approach for sequential experiental design for odel discriination, Cheical Engineering Science, Vol. 61, (2006), pp 5791-5806. [6] Greco M. and Del Giudice G., New approach to water distribution network calibration, Journal of Hydraulic Engineering, Vol. 125, No. 8, (1999), pp 849-854. [7] Rossan L. A., EPANET 2 users anual, Water Supply and Water Resources Division, National Risk Manageent Research Laboratory, (2000). [8] Mallick K. N., Ahed I., Tickle K. S. and Lansey K. E., Deterining pipe groupings for water distribution networks, Journal of Water Resources Planning and Manageent, Vol. 128, (2002), pp 130-139. [9] Buzzi-Ferraris G. and Manenti F., Kinetic odels analysis, Cheical Engineering Science, Vol. 64, (2009), pp 1061-1074.