State Estimation in Water Distribution Networks through a New Successive Linear Approximation

TO APPEAR IN THE 58TH CONFERENCE ON DECISION AND CONTR OL, NICE, FRANCE, DECEMBER 11–13, 2019 1 State Estimation in W ater Distrib ution Networks thr ough a New Successiv e Linear Appr oximation Shen W ang ˚ , Ahmad F . T aha ˚ , Lina Sela ; , Nikolaos Gatsis ˚ , and Marcio H. Giacomoni : Abstract —State estimation (SE) of water distribution networks (WDNs) is difficult to solve due to nonlinearity/nonconv exity of water flow models, uncertainties fr om parameters and demands, lack of redundancy of measurements, and inaccurate flow and pressur e measurements. This paper proposes a new , scalable successive linear approximation to solve the SE problem in WDNs. The approach amounts to solving either a sequence of linear or quadratic programs—depending on the operators’ objectives. The proposed successive linear approximation offers a seamless way of dealing with valve/pump model nonconv exities, is different than a first order T aylor series linearization, and can be incorporated into with rob ust uncertainty modeling. T wo simple test-cases are adopted to illustrate the effectiveness of proposed approach using head measurements at select nodes. I . I N T R O D U C TI O N A N D P A P E R C O N T R I B U T I O N S W ater distrib ution networks (WDNs) are designed to de- liv er water to v arious residential and business consumers with sufficient pressure and flow [ 1 ]. The calculation of flows and heads/pressures in WDNs can be obtained by the principles of conservation of mass and energy . The former implies the continuity of flow at nodes, and the latter states that energy difference stored in a component equals the energy increases minus ener gy losses, such as, frictional and minor losses [ 2 ]. The challenging part of monitoring WDNs is that pipes are usually buried underground and are inaccessible [ 3 ]. Hence, it is impossible to monitor the flow in all pipes and the head at all nodes ev en with modern supervisory control and data acquisition (SCAD A) systems, let alone enable continuous monitoring of WDNs, which is limited in practice due to high in vestment, operations and maintenance costs [ 4 ]. A practical approach to gain a network-wide observability , while addressing the aforementioned limitations, is to use state estimation (SE), which can determine the unknown v ariables of a system based on a set of local measurements and a hydraulic network model [ 5 ]. Usually , the set of measurements consists of heads at key nodes and the flows through key links. Ho wev er , the SE problem is dif ficult due to uncertainty from pipe roughness coefficients, demands, and measurement errors [ 6 ]. One way to reduce uncertainty is by introducing redundancy of observations, which significantly impro ves the ˚ Department of Electrical and Computer Engineering, The Univ ersity of T exas at San Antonio. : Department of Civil and En vironmental Engineering, The University of T exas at San Antonio, ; Department of Civil, Architec- tural and En vironmental Engineering, Cockrell School of Engineering, The Univ ersity of T exas at Austin. Emails: mvy292@my .utsa.edu, { ahmad.taha, nikolaos.gatsis, marcio.giacomoni } @utsa.edu, linasela@utexas.edu. This ma- terial is based upon work supported by the National Science Foundation under Grant CMMI-DCSD-1728629. performance of the SE procedure. The degree of redundancy is achie ved by combining actual measurements (e.g., heads and flows) with the pseudo-measurements (e.g., demands); howe ver , due to limited measurement availability , the application of SE algorithms to WDNs is an ongoing research [ 3 ]. In WDNs, the SE problem is predominantly cast as an in verse problem to determine unknown system conditions with an objectiv e, e.g., weighted least-squares (WLS), to minimize the mismatch between measurements and hydraulic model estimations [ 7 ]. The authors in [ 8 ] discuss a way to obtain the solution from over -determined measurements. The study [ 9 ] produces solutions that are consistent with available SCAD A data by adjusting estimated demands based on WLS method. The authors in [ 10 ] use Monte Carlo simulation (MCS) to ev aluate the effect of variable demands on pressure and water quality , and their w ork is e xtended by [ 11 ]. In order to ov ercome the computational time of MCS, a new approximate method for uncertainty analysis is proposed in [ 12 ]. The authors in [ 13 ] propose a SE in the presence of control devices with switching behavior , such as pressure reducing valv es after a minor modification of existing WLS solvers. An approach combining regression-trees with genetic algorithms to fit demands to the observations was proposed in [ 14 ]. In [ 6 ], the authors solv e the real-time SE problem using interval lineariza- tion of the nonlinear flo w equations and successi vely tightening the interval bounds. In summary , the SE problem results in nonlinear and nonconv ex system of equations, which exhibit serious scalability issues when applied to realistic WDNs. This paper proposes a new , scalable successiv e linear ap- proximation to solve the SE problem in WDNs. The approach amounts to solving either a sequence of linear or quadratic programs (LP/QP), depending on the operator’ s objecti ves. The proposed successive linear approximation of fers a seamless way of dealing with valv e/pump model nonconv exities, is different than a first order T aylor series linearization, and can be easily incorporated into uncertainty modeling. The paper’ s contributions can be summarized as follows: ‚ The classical, highly nonlinear and noncon vex state estimation problem is conv erted into successi ve conv ex LP/QP problems via geometric pr ogramming (GP) approximation [ 15 ]. ‚ A new optimization technique is introduced to solve the noncon vex SE problem through a tractable computational al- gorithm for a general WDN topology . The proposed research builds on our recent work on pump control of WDNs using GP [ 16 ], but offers a different approach through LP formulation, in comparison with our prior work. The paper organization is giv en next. Section II describes SE TO APPEAR IN THE 58TH CONFERENCE ON DECISION AND CONTR OL, NICE, FRANCE, DECEMBER 11–13, 2019 2 T ab. I: Hydraulic models of pipes and pumps and their con verted models (time index k is ignored for each v ariable for simplicity). Pipes Pumps Original Hydraulic Model ∆ h P ij “ h i ´ h j “ R ij q ij | q ij | µ ´ 1 (1) ∆ h M ij “ h i ´ h j “ ´ s 2 ij p h 0 ´ r p q ij s ´ 1 ij q ν q (2) GP F orm ˆ h i ˆ h ´ 1 j r p C P ij s ´ 1 ˆ q ´ 1 ij “ 1 (3) ˆ h i ˆ h ´ 1 j r p C M 1 s ´ 1 r ˆ q ij s ´ p C M 2 “ 1 (4) Linear F orm ∆ h P ij “ h i ´ h j “ C P ij ` q ij (5) ∆ h M ij “ h i ´ h j “ C M 1 ` C M 2 q ij (6) formulation. In Section III , the proposed optimization-based SE technique is introduced, con version of noncon vex SE into LP/QP is giv en, and two test-cases are used to illustrate the effecti veness of our approach in Section IV . Finally , Section V presents the paper’ s limitations and future research directions. I I . M O D E L I N G A N D S TA T E E S T I M A T I O N O F W D N S WDN is modeled by a directed graph p V , E q . Set V defines nodes and is partitioned as V “ J Ť T Ť R where J , T , and R stand for the collection of junctions, tanks, and reservoirs. Let E Ď V ˆ V be the set of links, and define the partition E “ P Ť M Ť W , where P , M , and W stand for the collection of pipes, pumps, and v alves. For the i th node, set N i collects its neighboring nodes and is partitioned as N i “ N in i Ť N out i , where N in i and N out i stand for the collection of inflow and outflow nodes. According to the principles of conservation of mass and energy , we present the modeling in WDNs next. A. Modeling WDNs In this section, we introduce the modeling of WDNs. 1) T anks and Reservoirs: The water hydraulic dynamics in the i th tank can be e xpressed by a discrete-time dif ference equation [ 16 ] h TK i p k ` 1 q “ h TK i p k q` ∆ t A TK i ¨ ˝ ÿ j P N in i q j i p k q´ ÿ j P N out i q ij p k q ˛ ‚ . (7) where h TK i , A TK i respectiv ely stand for the head, cross-sectional area of the i th tank, and ∆ t is sampling time; q j i p k q , j P N in i is inflow , while q ij p k q , j P N out i is outflow of the j th neighbor . W e assume that reserv oirs hav e infinite w ater supply and the head of the i th reservoir is fixed as h R set i [ 17 ], [ 18 ] which is perfectly accurate. This also can be viewed as an operational constraint ( 9a ). 2) J unctions and Pipes: Junctions are the points where water flow mer ges or splits. The expression of mass conservation of the i th junction can be written as ÿ j P N in i q j i p k q ´ ÿ j P N out i q ij p k q “ d i p k q , (8) where d i p k q stands for end-user demand that is extracted from node i at time k . The major head loss of a pipe from node i to j is due to friction and is determined by ( 1 ) from T ab . I , where R is the pipe resistance coef ficient and µ is the constant flow exponent, both are determined by the corresponding formula, Hazen- W illiams, Darcy-W eisbach, or Chezy-Manning. The approach we proposed considers any of the three formulae [ 18 ]. Minor head losses are not considered in this paper, but can be easily modeled through equiv alent pipe length. 3) Head Gain in Pumps: A head increase/gain can be generated by a pump between suction node i and deliv ery node j . Generally , the head gain can be expressed as ( 2 ), where h 0 , r , and ν are the pump curve coefficients; q ij is the flo w through a pump; s ij P r 0 , s max ij s is the relativ e speed of the pump, we assume that the speed is fixed and can be expressed as s ij “ s max ij “ 1 . Notice that head gain h M ij is always negativ e, and can be viewed as an operational constraint ( 9b ). For all the operational limitations of head at each junction and flow though each pipe, we list them as ( 9c ). Hence, the compact constraints are Constrain ts : h R i p k q “ h R set i (9a) h M ij p k q ď 0 (9b) h min i ď h i p k q ď h max j , q min ij ď q ij p k q ď q max ij . (9c) B. State Estimation F ormulation Classical state estimation problems are typically presented as y “ g p ξ q `  , (10) where ξ is the unknown variable, vector y includes all measured quantities, the g p ξ q is the model of system including nonlinear functions, and the  represents error between true model and measured values via sensors. As we mentioned in Section I , it is impossible to measure flows and pressures in the entire WDN, except for key locations. Hence, ξ can be a vector collecting all unknown variables and defined as ξ p k q fi ! h J p k q , h R p k q , h TK p k q , q P p k q , q M p k q ) . (11) where h J , h R , and h TK collects the heads at junctions, reser - voirs, and tanks; q P and q M collects the flow through pipes and pumps. The y can be treated as the vector collecting se veral measured key heads in the scenario of WDNs (sensors are assumed av ailable to measure head). The overall WDN-SE problem can now be written as WDN-SE : min ξ p k q f p  q “ ř T k “ 1  J p k q W p k q  p k q s . t . h TK p k ` 1 q “ h TK p k q ` E TK q P p k q (12a) d p k q “ E q „ q P p k q q M p k q  (12b) Constrain ts ( 9 ) (12c) where k is time-index; T is time-horizon; ξ p k q for all k “ 1 , . . . , T is the optimization variable that includes unmeasured heads h and flo ws q ; n e represents the number of measure- ments;  p k q P R n e ˆ 1 is the error to be minimized; f p  q is a WLS objecti ve function and  p k q “ E h „ ∆ h P p k q “ Φ P p q P p k qq ∆ h M p k q “ Φ M p q M p k qq  loooooooooooooooooomoooooooooooooooooon g p ξ q ´ ∆ r h p k q lo omo on y , TO APPEAR IN THE 58TH CONFERENCE ON DECISION AND CONTR OL, NICE, FRANCE, DECEMBER 11–13, 2019 3 where Φ P p¨ ¨ ¨q and Φ M p¨ ¨ ¨q collect the nonlinear head loss ( 1 ) of all pipes and the nonlinear head gain ( 2 ) of all pumps. The residual  is reminiscent of the model in ( 10 ) and captures the error between the true model and differences ∆ r h p k q between head measurements, while matrix E h is related to the position of sensors. The weight matrix at time k is giv en by a diagonal matrix W p k q : smaller diagonal elements in W p k q imply more accurate measurements. In practice, sensors are usually fixed in key nodes, and accuracy of sensors can also be assumed as fixed. Hence, W p k q is assumed to be a constant matrix. The objecti ve function is thus designed to minimize the weighted error , and we refer to W as accuracy matrix . The constraints in WDN-SE are discussed next. Equation ( 12a ) collects the dynamics ( 7 ) of all tanks in the network, and E TK is formed by the coef ficients of flo ws in ( 7 ). In fact, this constraint can be added and remov ed according to the situation; e.g., this constraint can be removed when performing single period analysis. Equation ( 12b ) collects ( 8 ) at all nodes, where matrix E q is defined by WDNs topology , and vector d p k q collects water demands at all junctions. W e do not consider demand uncertainty , and thus d p k q is assumed known and perfectly accurate. Constraint ( 12c ) includes the linear constraints presented in Section II-A ; i.e., the heads of reservoirs are usually fixed and equal to their elev ation. Hence, we assume that the measurement of h R is v ery accurate. Notice that any head difference can be expressed as the linear combination of nonlinear models of pipes and pumps using E h . This key observ ation is thoroughly illustrated in Fig. 3(a) in Section IV . W e note the following about the WDN-SE problem. ‚ T wo scenarios exist in SE problem in WDNs [ 8 ]: Scenario 1 is described by having sufficient measurements, e.g., all states can be determined if heads at tanks and reserv oirs are kno wn, see the blue line in Fig. 3(a) . In fact, the SE in this scenario has equal number of variables and equations. Scenario 2 refers to the case with o ver-determined equations, e.g., additional sets of head are measured at sev eral ke y nodes besides the head at tanks and reservoirs, see the red line in Fig. 3(a) . Numerical tests are given in Section IV for both scenarios. ‚ WDN-SE Problem ( 12 ) is nonconv ex due to the nonlinearity and nonconv exity of head loss/gain models Φ P p¨ ¨ ¨q and Φ M p¨ ¨ ¨q of pipes and pumps—the noncon vexity shows up in the objecti ve function, rather than the constraints. The only optimization vari- able in WDN-SE is ξ p k q , and other v ariables such as ∆ h P p k q are expressions of v ectors included in ξ ( 11 ). Finally , certain variables in ξ p k q , notably the measured heads at reservoirs and tanks, are considered to be known. ‚ While WDN-SE pertains to SE given a batch of measurements for k “ 1 , . . . , T and then reconstructs the estimates ξ p k q for that time-period, a simple windo wing algorithm can yield near real-time state estimates. I I I . N E W L I N E A R A P P R OX I M A T I O N O F W D N - S E In this section, we propose a new method inspired by geometric programming to con vert the nonlinear head loss model ( 1 ) and head gain model ( 2 ) into GP constraints which can also be rewritten as linear constraints. A basic introduction to GP is gi ven at first, and then a ne w optimization technique related with GP is proposed for ensuing sections. A. Geometric Pr ogram and A New Optimization T echnique A geometric program [ 19 ] is a type of optimization problem can be expressed as GP: min x ą 0 f 0 p x q s . t . f i p x q ď 1 , i “ 1 , ¨ ¨ ¨ , m (13) g i p x q “ 1 , i “ 1 , ¨ ¨ ¨ , p, where x is an entry-wise positiv e optimization variable, f i p x q are posynomial functions and g i p x q are monomials. One main requirement of the GP formulation is the positive- ness of the decision v ariables, which limits some decision vari- ables and physical constraints in our setting, e.g., flows in pipes and headloss equation. T o ov ercome this modeling limitation we are inspired by linear programming (LP) techniques. In the simplex method [ 20 ], for example, the free v ariables are split into a positive and ne gative part, both being nonnegati ve. In our case, we introduce an exponential function f p x q “ b x to con vert a nonpositiv e variable to a positiv e one, since f p x q is always positiv e. Using this technique, we can con vert some problems with negativ e feasible regions into a new problem with a positive feasible region, and then solve it by using modern optimization solvers. This technique has been successfully applied to solve the control of WDNs in our recent work [ 16 ]. The SE problem here is similar to the control problem of WDNs; howe ver , in the current paper , we conv ert the SE problem ( 12 ) into an LP or QP problem instead of a GP , which provides more elegant—and computationally more efficient—solutions. B. Con version of Energy Balance Equations Based on the newly introduced optimization technique in Section III-A , we first con vert the nonlinear hydraulic model of WDNs into its GP form and then into its LP form. Here, we con vert the head at the i th node h i and the flo w q ij into positiv e values ˆ h i and ˆ q ij through exponential functions, ˆ h i fi b h i , ˆ q ij fi b q ij , where b “ 1 ` δ is a constant base and δ is a small positive number . Notice that we only need to con vert the Φ P p¨ ¨ ¨q and Φ M p¨ ¨ ¨q into linear form because others are already linear . T ab . I show detailed conv ersions of all physical models that are all discussed in the following sections. Next, we con vert the pipe model, and let ˆ h P ij be the GP form of head loss of a pipe, which is obtained by exponentiating both sides of ( 1 ) as follo ws ˆ h i ˆ h ´ 1 j “ b q ij p R ij | q ij | µ ´ 1 ´ 1 q ˆ q ij “ p C P ij p q ij q ˆ q ij , where p C P ij p q ij q “ b q ij p R ij | q ij | µ ´ 1 ´ 1 q is a function of q ij . Hence, the head loss constraint for each pipe can be written as a monomial equality constraint, which is expressed as ( 3 ), if a an estimate of p C P ij p q ij q is known. In order to make it linear , we can execute the log function on both sides of ( 3 ) and obtain ( 5 ) where p C P ij p q ij q turns into C P ij p q ij q “ q ij p R ij | q ij | µ ´ 1 ´ 1 q . W e note that the expres sion above is linear with respect to C P ij p q ij q if q ij is kno wn, hence we dev elop a method to find q ij by sequentially updating q ij and C P ij p q ij q . The technique is introduced here. At first, we can make an initial guess denoted TO APPEAR IN THE 58TH CONFERENCE ON DECISION AND CONTR OL, NICE, FRANCE, DECEMBER 11–13, 2019 4 by x q ij y 0 for the 0 th iteration ( x C P ij y 0 can be obtained if x q ij y 0 is known), thus, for the n th iteration, the corresponding values are denoted by x q ij y n and x C P ij y n . If the flow rates are close to each other between two successiv e iterations, we can approximate x C P ij y n using x C P ij y n ´ 1 , that is x C P ij y n « x C P ij y n ´ 1 . Then, for each iteration n , we ha ve x C P ij y n “ x q ij y n ´ 1 ´ R |x q ij y n ´ 1 | µ ´ 1 ´ 1 ¯ , and it can be approximated by a constant given the flow value x q ij y n ´ 1 from the previous iteration. Similarly , the ne w variables ˆ q ij “ b q ij and ˆ s ij “ b s ij for p i, j q P M are introduced for pumps. Let ˆ h M ij be the GP form of head increase of a pump ˆ h i ˆ h ´ 1 j “ b ´ s 2 ij h 0 p b q ij q rq ν ´ 1 ij s 2 ´ ν ij “ p C M 1 p ˆ q ij q p C M 2 , where p C M 1 “ b ´ s 2 ij h 0 and p C M 2 “ r q ν ´ 1 ij s 2 ´ ν ij . Hence, the approximating equation for the pump head increase becomes the monomial equality constraint ( 4 ) in T ab. I . After executing log function on both sides of ( 4 ), the equation ( 6 ) can be obtained, which is a linear constraint. And at the same time, the parameters p C M 1 and p C M 2 become C M 1 and C M 2 , that is C M 1 “ ´ s 2 ij h 0 , C M 2 “ r q ν ´ 1 ij s 2 ´ ν ij . Parameters C M 1 are fix ed, while C M 2 follow a similar iterative process as C P ij . That is, starting with an initial guess for the flow rates and relativ e speeds, the constraints are approximated at e very iteration via constraints abiding by the linear form, as listed in T ab. I . This process continues until a termination criterion is met. The details are further discussed in Algorithm 1 . C. LP/QP F ormulation of SE After the con version of pipe and pump model constraints, we can e xpress the con verted problem as LP/QP-SE : min ξ p k q f p  q (14a) s . t . ( 12a ) ´ ( 12c ) (14b) ∆ h P p k q “ q P p k q ` C P p k q (14c) ∆ h M p k q “ C M 1 p k q ` C M 2 p k q q M p k q (14d) where constraints and variables remain the same as in ( 12 ) except that constraints ( 14c ) and ( 14d ) are no w linear and viewed as constraints. The parameters C P is a R n p ˆ 1 vector collecting the C P ij for each pipe. Similarly , the C M 1 and C M 2 are a R n m ˆ n m diagonal matrices collecting C M 1 and C M 2 for each pump. W e note the following: (i) LP/QP-SE ( 14 ) is only an ap- proximation of WDN-SE ( 12 ) at a specific point (the flow through pipes and pumps q ij ), in other words, the noncon vex WDN-SE can be approximated by successiv e conv ex LP/QP- SE . (ii) The con verted model is linear but it is not the equiv alent to the first order T aylor series linearization. W e present the geometric meaning of the con version we applied via a concrete example in Section IV -A1 . (iii) LP/QP-SE can be e xpressed as either an LP or QP depending on the objective function. When f p  q is modeled through the absolute weighted error , i.e., ř T k “ 1 ř n e i “ 1 w i p k q|  i p k q| , the problem can be written as an LP . When it is based on WLS, then it becomes a standard QP . Algorithm 1: Successive approximation of WDN-SE . Input: WDN topology , x ξ y 0 , demand t d p k qu T k “ 1 , measurements of head r h , the accuracy matrix W Output: The estimated state v alue t ξ SE p k qu T k “ 1 1 Set ξ sav e : “ x ξ y 0 , n “ 1 , step “ 4 , a “ 3 2 while error ě threshold OR n ď maxIter do 3 Obtain x C P ij y n , x C M 1 y n , and x C M 2 y n from x ξ y n ´ 1 4 Generate constraints and form it as L P / Q P- S E ( 14 ) 5 Solve ( 14 ) and obtain x ξ y n 6 if mo d p n, step q “ 0 then 7 ∆ ξ “ x ξ y n ´ x ξ y n ´ 2 8 x ξ y n “ x ξ y n ` a ∆ ξ 9 end if 10 Calculate error : “ norm px ξ y n ´ ξ sav e q 11 Update ξ sav e “ x ξ y n and n “ n ` 1 12 end while 13 Set ξ SE “ x ξ y n h 2 AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKewGIR4DXjzGRx6QLGF20psMmZ1dZmaFsOQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2AkmN3O/84RK81g+mmmCfkRHkoecUWOlh/GgNihX3Kq7AFknXk4qkKM5KH/1hzFLI5SGCap1z3MT42dUGc4Ezkr9VGNC2YSOsGeppBFqP1ucOiMXVhmSMFa2pCEL9fdERiOtp1FgOyNqxnrVm4v/eb3UhNd+xmWSGpRsuShMBTExmf9NhlwhM2JqCWWK21sJG1NFmbHplGwI3urL66Rdq3pu1bu7qjTu8ziKcAbncAke1KEBt9CEFjAYwTO8wpsjnBfn3flYthacfOYU/sD5/AH5lY2h AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKewGIR4DXjzGRx6QLGF20psMmZ1dZmaFsOQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2AkmN3O/84RK81g+mmmCfkRHkoecUWOlh/GgNihX3Kq7AFknXk4qkKM5KH/1hzFLI5SGCap1z3MT42dUGc4Ezkr9VGNC2YSOsGeppBFqP1ucOiMXVhmSMFa2pCEL9fdERiOtp1FgOyNqxnrVm4v/eb3UhNd+xmWSGpRsuShMBTExmf9NhlwhM2JqCWWK21sJG1NFmbHplGwI3urL66Rdq3pu1bu7qjTu8ziKcAbncAke1KEBt9CEFjAYwTO8wpsjnBfn3flYthacfOYU/sD5/AH5lY2h AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKewGIR4DXjzGRx6QLGF20psMmZ1dZmaFsOQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2AkmN3O/84RK81g+mmmCfkRHkoecUWOlh/GgNihX3Kq7AFknXk4qkKM5KH/1hzFLI5SGCap1z3MT42dUGc4Ezkr9VGNC2YSOsGeppBFqP1ucOiMXVhmSMFa2pCEL9fdERiOtp1FgOyNqxnrVm4v/eb3UhNd+xmWSGpRsuShMBTExmf9NhlwhM2JqCWWK21sJG1NFmbHplGwI3urL66Rdq3pu1bu7qjTu8ziKcAbncAke1KEBt9CEFjAYwTO8wpsjnBfn3flYthacfOYU/sD5/AH5lY2h AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKewGIR4DXjzGRx6QLGF20psMmZ1dZmaFsOQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2AkmN3O/84RK81g+mmmCfkRHkoecUWOlh/GgNihX3Kq7AFknXk4qkKM5KH/1hzFLI5SGCap1z3MT42dUGc4Ezkr9VGNC2YSOsGeppBFqP1ucOiMXVhmSMFa2pCEL9fdERiOtp1FgOyNqxnrVm4v/eb3UhNd+xmWSGpRsuShMBTExmf9NhlwhM2JqCWWK21sJG1NFmbHplGwI3urL66Rdq3pu1bu7qjTu8ziKcAbncAke1KEBt9CEFjAYwTO8wpsjnBfn3flYthacfOYU/sD5/AH5lY2h h 3 AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKexqQI8BLx7jIw9IljA76U2GzM4uM7NCWPIJXjwo4tUv8ubfOEn2oIkFDUVVN91dQSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfjm5nffkKleSwfzSRBP6JDyUPOqLHSw6h/2S9X3Ko7B1klXk4qkKPRL3/1BjFLI5SGCap113MT42dUGc4ETku9VGNC2ZgOsWuppBFqP5ufOiVnVhmQMFa2pCFz9fdERiOtJ1FgOyNqRnrZm4n/ed3UhNd+xmWSGpRssShMBTExmf1NBlwhM2JiCWWK21sJG1FFmbHplGwI3vLLq6R1UfXcqndXq9Tv8ziKcAKncA4eXEEdbqEBTWAwhGd4hTdHOC/Ou/OxaC04+cwx/IHz+QP7GY2i AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKexqQI8BLx7jIw9IljA76U2GzM4uM7NCWPIJXjwo4tUv8ubfOEn2oIkFDUVVN91dQSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfjm5nffkKleSwfzSRBP6JDyUPOqLHSw6h/2S9X3Ko7B1klXk4qkKPRL3/1BjFLI5SGCap113MT42dUGc4ETku9VGNC2ZgOsWuppBFqP5ufOiVnVhmQMFa2pCFz9fdERiOtJ1FgOyNqRnrZm4n/ed3UhNd+xmWSGpRssShMBTExmf1NBlwhM2JiCWWK21sJG1FFmbHplGwI3vLLq6R1UfXcqndXq9Tv8ziKcAKncA4eXEEdbqEBTWAwhGd4hTdHOC/Ou/OxaC04+cwx/IHz+QP7GY2i AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKexqQI8BLx7jIw9IljA76U2GzM4uM7NCWPIJXjwo4tUv8ubfOEn2oIkFDUVVN91dQSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfjm5nffkKleSwfzSRBP6JDyUPOqLHSw6h/2S9X3Ko7B1klXk4qkKPRL3/1BjFLI5SGCap113MT42dUGc4ETku9VGNC2ZgOsWuppBFqP5ufOiVnVhmQMFa2pCFz9fdERiOtJ1FgOyNqRnrZm4n/ed3UhNd+xmWSGpRssShMBTExmf1NBlwhM2JiCWWK21sJG1FFmbHplGwI3vLLq6R1UfXcqndXq9Tv8ziKcAKncA4eXEEdbqEBTWAwhGd4hTdHOC/Ou/OxaC04+cwx/IHz+QP7GY2i AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKexqQI8BLx7jIw9IljA76U2GzM4uM7NCWPIJXjwo4tUv8ubfOEn2oIkFDUVVN91dQSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfjm5nffkKleSwfzSRBP6JDyUPOqLHSw6h/2S9X3Ko7B1klXk4qkKPRL3/1BjFLI5SGCap113MT42dUGc4ETku9VGNC2ZgOsWuppBFqP5ufOiVnVhmQMFa2pCFz9fdERiOtJ1FgOyNqRnrZm4n/ed3UhNd+xmWSGpRssShMBTExmf1NBlwhM2JiCWWK21sJG1FFmbHplGwI3vLLq6R1UfXcqndXq9Tv8ziKcAKncA4eXEEdbqEBTWAwhGd4hTdHOC/Ou/OxaC04+cwx/IHz+QP7GY2i h 4 AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKexKIB4DXjzGRx6QLGF20psMmZ1dZmaFsOQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2AkmN3O/84RK81g+mmmCfkRHkoecUWOlh/GgNihX3Kq7AFknXk4qkKM5KH/1hzFLI5SGCap1z3MT42dUGc4Ezkr9VGNC2YSOsGeppBFqP1ucOiMXVhmSMFa2pCEL9fdERiOtp1FgOyNqxnrVm4v/eb3UhNd+xmWSGpRsuShMBTExmf9NhlwhM2JqCWWK21sJG1NFmbHplGwI3urL66R9VfXcqndXqzTu8ziKcAbncAke1KEBt9CEFjAYwTO8wpsjnBfn3flYthacfOYU/sD5/AH8nY2j AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKexKIB4DXjzGRx6QLGF20psMmZ1dZmaFsOQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2AkmN3O/84RK81g+mmmCfkRHkoecUWOlh/GgNihX3Kq7AFknXk4qkKM5KH/1hzFLI5SGCap1z3MT42dUGc4Ezkr9VGNC2YSOsGeppBFqP1ucOiMXVhmSMFa2pCEL9fdERiOtp1FgOyNqxnrVm4v/eb3UhNd+xmWSGpRsuShMBTExmf9NhlwhM2JqCWWK21sJG1NFmbHplGwI3urL66R9VfXcqndXqzTu8ziKcAbncAke1KEBt9CEFjAYwTO8wpsjnBfn3flYthacfOYU/sD5/AH8nY2j AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKexKIB4DXjzGRx6QLGF20psMmZ1dZmaFsOQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2AkmN3O/84RK81g+mmmCfkRHkoecUWOlh/GgNihX3Kq7AFknXk4qkKM5KH/1hzFLI5SGCap1z3MT42dUGc4Ezkr9VGNC2YSOsGeppBFqP1ucOiMXVhmSMFa2pCEL9fdERiOtp1FgOyNqxnrVm4v/eb3UhNd+xmWSGpRsuShMBTExmf9NhlwhM2JqCWWK21sJG1NFmbHplGwI3urL66R9VfXcqndXqzTu8ziKcAbncAke1KEBt9CEFjAYwTO8wpsjnBfn3flYthacfOYU/sD5/AH8nY2j AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKexKIB4DXjzGRx6QLGF20psMmZ1dZmaFsOQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2AkmN3O/84RK81g+mmmCfkRHkoecUWOlh/GgNihX3Kq7AFknXk4qkKM5KH/1hzFLI5SGCap1z3MT42dUGc4Ezkr9VGNC2YSOsGeppBFqP1ucOiMXVhmSMFa2pCEL9fdERiOtp1FgOyNqxnrVm4v/eb3UhNd+xmWSGpRsuShMBTExmf9NhlwhM2JqCWWK21sJG1NFmbHplGwI3urL66R9VfXcqndXqzTu8ziKcAbncAke1KEBt9CEFjAYwTO8wpsjnBfn3flYthacfOYU/sD5/AH8nY2j q 23 AAAB7XicbVBNSwMxEJ34WetX1aOXYBE8ld0q6LHgxWMV+wHtUrJpto3NJmuSFcrS/+DFgyJe/T/e/Dem7R609cHA470ZZuaFieDGet43WlldW9/YLGwVt3d29/ZLB4dNo1JNWYMqoXQ7JIYJLlnDcitYO9GMxKFgrXB0PfVbT0wbruS9HScsiMlA8ohTYp3UfOxl1fNJr1T2Kt4MeJn4OSlDjnqv9NXtK5rGTFoqiDEd30tskBFtORVsUuymhiWEjsiAdRyVJGYmyGbXTvCpU/o4UtqVtHim/p7ISGzMOA5dZ0zs0Cx6U/E/r5Pa6CrIuExSyySdL4pSga3C09dxn2tGrRg7Qqjm7lZMh0QTal1ARReCv/jyMmlWK75X8W8vyrW7PI4CHMMJnIEPl1CDG6hDAyg8wDO8whtS6AW9o4956wrKZ47gD9DnDz/ajvM= AAAB7XicbVBNSwMxEJ34WetX1aOXYBE8ld0q6LHgxWMV+wHtUrJpto3NJmuSFcrS/+DFgyJe/T/e/Dem7R609cHA470ZZuaFieDGet43WlldW9/YLGwVt3d29/ZLB4dNo1JNWYMqoXQ7JIYJLlnDcitYO9GMxKFgrXB0PfVbT0wbruS9HScsiMlA8ohTYp3UfOxl1fNJr1T2Kt4MeJn4OSlDjnqv9NXtK5rGTFoqiDEd30tskBFtORVsUuymhiWEjsiAdRyVJGYmyGbXTvCpU/o4UtqVtHim/p7ISGzMOA5dZ0zs0Cx6U/E/r5Pa6CrIuExSyySdL4pSga3C09dxn2tGrRg7Qqjm7lZMh0QTal1ARReCv/jyMmlWK75X8W8vyrW7PI4CHMMJnIEPl1CDG6hDAyg8wDO8whtS6AW9o4956wrKZ47gD9DnDz/ajvM= AAAB7XicbVBNSwMxEJ34WetX1aOXYBE8ld0q6LHgxWMV+wHtUrJpto3NJmuSFcrS/+DFgyJe/T/e/Dem7R609cHA470ZZuaFieDGet43WlldW9/YLGwVt3d29/ZLB4dNo1JNWYMqoXQ7JIYJLlnDcitYO9GMxKFgrXB0PfVbT0wbruS9HScsiMlA8ohTYp3UfOxl1fNJr1T2Kt4MeJn4OSlDjnqv9NXtK5rGTFoqiDEd30tskBFtORVsUuymhiWEjsiAdRyVJGYmyGbXTvCpU/o4UtqVtHim/p7ISGzMOA5dZ0zs0Cx6U/E/r5Pa6CrIuExSyySdL4pSga3C09dxn2tGrRg7Qqjm7lZMh0QTal1ARReCv/jyMmlWK75X8W8vyrW7PI4CHMMJnIEPl1CDG6hDAyg8wDO8whtS6AW9o4956wrKZ47gD9DnDz/ajvM= AAAB7XicbVBNSwMxEJ34WetX1aOXYBE8ld0q6LHgxWMV+wHtUrJpto3NJmuSFcrS/+DFgyJe/T/e/Dem7R609cHA470ZZuaFieDGet43WlldW9/YLGwVt3d29/ZLB4dNo1JNWYMqoXQ7JIYJLlnDcitYO9GMxKFgrXB0PfVbT0wbruS9HScsiMlA8ohTYp3UfOxl1fNJr1T2Kt4MeJn4OSlDjnqv9NXtK5rGTFoqiDEd30tskBFtORVsUuymhiWEjsiAdRyVJGYmyGbXTvCpU/o4UtqVtHim/p7ISGzMOA5dZ0zs0Cx6U/E/r5Pa6CrIuExSyySdL4pSga3C09dxn2tGrRg7Qqjm7lZMh0QTal1ARReCv/jyMmlWK75X8W8vyrW7PI4CHMMJnIEPl1CDG6hDAyg8wDO8whtS6AW9o4956wrKZ47gD9DnDz/ajvM= q 34 AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hV0N6DHgxWMU84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqqb7q4o4cxY3//2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCKK92OsKGcSdqwzHLaTjTFIuK0FY2up37riWrDlLy344SGAg8kixnB1knNx152UZ30SmW/4s+AlkmQkzLkqPdKX92+Iqmg0hKOjekEfmLDDGvLCKeTYjc1NMFkhAe046jEgpowm107QadO6aNYaVfSopn6eyLDwpixiFynwHZoFr2p+J/XSW18FWZMJqmlkswXxSlHVqHp66jPNCWWjx3BRDN3KyJDrDGxLqCiCyFYfHmZNM8rgV8Jbqvl2l0eRwGO4QTOIIBLqMEN1KEBBB7gGV7hzVPei/fufcxbV7x85gj+wPv8AULljvU= AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hV0N6DHgxWMU84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqqb7q4o4cxY3//2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCKK92OsKGcSdqwzHLaTjTFIuK0FY2up37riWrDlLy344SGAg8kixnB1knNx152UZ30SmW/4s+AlkmQkzLkqPdKX92+Iqmg0hKOjekEfmLDDGvLCKeTYjc1NMFkhAe046jEgpowm107QadO6aNYaVfSopn6eyLDwpixiFynwHZoFr2p+J/XSW18FWZMJqmlkswXxSlHVqHp66jPNCWWjx3BRDN3KyJDrDGxLqCiCyFYfHmZNM8rgV8Jbqvl2l0eRwGO4QTOIIBLqMEN1KEBBB7gGV7hzVPei/fufcxbV7x85gj+wPv8AULljvU= AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hV0N6DHgxWMU84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqqb7q4o4cxY3//2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCKK92OsKGcSdqwzHLaTjTFIuK0FY2up37riWrDlLy344SGAg8kixnB1knNx152UZ30SmW/4s+AlkmQkzLkqPdKX92+Iqmg0hKOjekEfmLDDGvLCKeTYjc1NMFkhAe046jEgpowm107QadO6aNYaVfSopn6eyLDwpixiFynwHZoFr2p+J/XSW18FWZMJqmlkswXxSlHVqHp66jPNCWWjx3BRDN3KyJDrDGxLqCiCyFYfHmZNM8rgV8Jbqvl2l0eRwGO4QTOIIBLqMEN1KEBBB7gGV7hzVPei/fufcxbV7x85gj+wPv8AULljvU= AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hV0N6DHgxWMU84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqqb7q4o4cxY3//2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCKK92OsKGcSdqwzHLaTjTFIuK0FY2up37riWrDlLy344SGAg8kixnB1knNx152UZ30SmW/4s+AlkmQkzLkqPdKX92+Iqmg0hKOjekEfmLDDGvLCKeTYjc1NMFkhAe046jEgpowm107QadO6aNYaVfSopn6eyLDwpixiFynwHZoFr2p+J/XSW18FWZMJqmlkswXxSlHVqHp66jPNCWWjx3BRDN3KyJDrDGxLqCiCyFYfHmZNM8rgV8Jbqvl2l0eRwGO4QTOIIBLqMEN1KEBBB7gGV7hzVPei/fufcxbV7x85gj+wPv8AULljvU= (a) 3-node network. (b) Visualization of equation ( 15 ) of 3-node network. Fig. 1: 3-node network and visualization of its equations. D. Iterative Algorithm In order to solve all the unkno wn variables, our algorithm needs to kno w the basic information at first, e.g., the topology of tested network to form the matrices E q , demand d , measure- ments of head r h , and the accuracy matrix W . Notice that all variables are collected in ξ by ( 11 ) and the notation x ξ y n in Algorithm 1 stands for the n th iteration value ξ . For the 0 th iteration, we initialize all flo w x q P y 0 and x q M y 0 in x ξ y 0 with historical average flows. In fact, this algorithm still works by initializing all the flow in x ξ y 0 with random number . Howe ver , the con vergence is relativ ely slow . All initial statuses of pumps, tanks, and reservoirs are initialized with the value set in “.inp” source file which is a standard input file used by EP ANET , e.g., the initial status (open or close) and speed (if open) of pumps, and the initial head value x h R y 0 and x h TK y 0 of tanks and reservoirs. The parameters x C P ij y 1 , x C M 1 y 1 , and x C M 2 y 1 are then calculated by initialized values according to Section III-B , and all constraints are automatically generated for dif ferent WDNs topologies. After solving ( 14 ) and obtaining the current solution x ξ y n , and defining the iteration error as the Euclidean distance be- tween two consecutive iterations, we set x ξ y n as the sav ed v alue for error calculation in next iteration by assigning ξ sav e “ x ξ y n . The iteration continues until the error is less than a predefined threshold ( threshold ) or a maximum number of iterations ( maxIter ) is reached, and the final solution is ξ SE . The bottle- neck of this algorithm is solving a scalable LP/QP successi vely TO APPEAR IN THE 58TH CONFERENCE ON DECISION AND CONTR OL, NICE, FRANCE, DECEMBER 11–13, 2019 5 T ab. II: Formulations of 3-node network (time index k is ignored). Original f ormulation min k Φ P 23 p q 23 q ` Φ P 34 p q 34 q ´ ∆ r h 24 k (15) s . t . q 23 ´ q 34 “ d 3 SE f ormulation QP-SE f ormulation min  J  (16) s . t . 0 “ ” 1 ´ 1 ı « q 23 q 34 ff ` d 3 ∆ h P “ « Φ P 23 p q 23 q Φ P 34 p q 34 q ff min  J  (17) s . t . 0 “ ” 1 ´ 1 ı « q 23 q 34 ff ` d 3 ∆ h P “ « q 23 q 34 ff ` « C P 23 C P 34 ff which should not cause a lar ge computational burden, e ven if tens of iterations are required to con verge. W e note that Steps 7 and 8 are used to accelerate the computational times, since the direction of the search is known, and the acceleration parameter a in Step 8 is needed to be adjusted according to the specific WDN. This will be inv estigated in future work. W e finally note that Algorithm 1 does not show the windowing process of performing real-time SE, as the algorithm only sho ws batch state estimation. Ho wev er , a moving horizon windo w can be implemented within Algorithm 1 thereby allowing real-time state estimation. I V . C A S E S T U D I E S W e present two simulation examples to illustrate the appli- cability of our approach. The first 3-node network is used to illustrate the geometric meaning of proposed method, and then we test the 8-node network to illustrate that our approach can handle looped topology . All numerical tests are simulated using EP ANET Matlab T oolkit [ 21 ] on Ubuntu 16.04.4 L TS with an Intel(R) Xeon(R) CPU E5-1620 v3 @ 3.50 GHz. CVX [ 22 ] is used to solve the optimization problem. W e set the base b “ 1 . 001 when conv erting the variables. All case studies are performed for T “ 1 time-horizon; the head unit is ft ; and the flow unit is GPM . All codes, parameters, and tested networks are a vailable in [ 23 ]. Fig. 2: (Left) 3D visualization of ( 15 ); (Right) Iteration process of solving scenario 1 (sufficient measurements) for 3-node network. A. Thr ee-node Network The 3-node network comprised of 3 junctions and 2 pipes is shown in Fig. 1(a) , and no demand at Junctions 2 and 4. T ab. III: Results of 3-node and 8-node network. (a) Results for the 3-node network. Sufficient measur ements scenario Over-determined measur ements scenario V ariables q 23 q 34 q 23 q 34 Fmincon with GlobalSear ch 238.607 38.607 234.690 34.690 Algorithm 1 238.538 38.528 235.007 35.007 (b) Results for the 8-node network. h TK 8 h 3 h 5 q 46 T rue value: ξ EP ANET 834.00 875.89 863.49 82.50 Measur ement: ˜ ξ 834.60 875.64 — — Estimation from Algorithm 1 p ξ SE q Case 1 834.56 876.40 864.01 82.44 Case 2 833.89 875.81 863.36 83.81 1) Sufficient measur ements scenario: Suppose that we mea- sure head dif ference ∆ r h 24 between Junction 2 and 4, and estimates of the flows q 23 and q 34 are sought. According to ( 10 ), the classical SE is presented as ( 15 ) in T ab . II . In fact, it can be visualized as Fig. 1(b) where the red surface is nonlinear Φ P 23 p q 23 q ` Φ P 34 p q 34 q , the blue surface is linear conservation of mass constraint,and the gray surface is measured head difference ∆ r h 24 . The solution is in the intersection of these three surfaces. In order to see the feasible set, we can view 3D plot from top (ignore ∆ r h 24 dimension). Notice that the feasible set can be viewed as the intersection of red and gray surfaces, and it is highly noncon vex in 2D. After con version, the corresponding QP form of SE ( 16 ) is presented as ( 17 ) in T ab. II , where  “ “ 1 1 ‰ ∆ h P ´ ∆ r h 24 . Iteration process is presented in Fig. 2 , and intersection of blue and red surface in the left plot are approximated by the intersec- tion of blue surface and multi-green surfaces in the right plot. As we mentioned, this is a new type of linear approximation but not the first order T aylor series of the nonlinear function, we can notice this from the green surfaces. 2) Over-determined measur ements scenario: Suppose that we measure two head differences ∆ r h 23 and ∆ r h 24 and that the ∆ r h 23 is ten times more accurate than ∆ r h 24 . Therefore, the weight in the objecti ve function must be updated. Hence, the SE problem can simply be presented as min k Φ P 23 p q 23 q´ ∆ r h 23 k ` 0 . 1 k Φ P 23 p q 23 q` Φ P 34 p q 34 q ´ ∆ r h 24 k s . t . q 23 ´ q 34 “ d 3 (18) W ith such changes, the objectiv e function in ( 17 ) becomes  J diag p 1 , 0 . 1 q  and the corresponding expression is  “ „ 1 0 1 1  ∆ h P ´ « ∆ r h 23 ∆ r h 24 ff , and notice that it is still a QP . In order to pro ve the ef fectiveness of our approach, our results is compared with solutions from other solv ers. On one hand, the nonlinear optimization prob- lems ( 15 ) and ( 18 ) can be solved optimally via fmincon with GlobalSearch option in Matlab, on the other hand, it can be solved via Algorithm 1 . The final results and comparisons are listed in T ab . III(a) . W e can see that the proposed algorithm yields similar solutions to fmincon and if measurement ∆ r h 23 is more reliable than ∆ r h 24 , then final results change accord- ingly . TO APPEAR IN THE 58TH CONFERENCE ON DECISION AND CONTR OL, NICE, FRANCE, DECEMBER 11–13, 2019 6  M 12 AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokRdCFi4IbN0IF+4Amhsl00g6dmYSZiVBCNv6KGxeKuPUz3Pk3TtostPXAhcM593LvPWHCqNKO821VVlbX1jeqm7Wt7Z3dPXv/oKviVGLSwTGLZT9EijAqSEdTzUg/kQTxkJFeOLku/N4jkYrG4l5PE+JzNBI0ohhpIwX2kdce0yBzm/lD5nGkx5Jnt3leC+y603BmgMvELUkdlGgH9pc3jHHKidCYIaUGrpNoP0NSU8xIXvNSRRKEJ2hEBoYKxInys9kDOTw1yhBGsTQlNJypvycyxJWa8tB0FjeqRa8Q//MGqY4u/YyKJNVE4PmiKGVQx7BIAw6pJFizqSEIS2puhXiMJMLaZFaE4C6+vEy6zYbrNNy783rrqoyjCo7BCTgDLrgALXAD2qADMMjBM3gFb9aT9WK9Wx/z1opVzhyCP7A+fwB56ZZM AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokRdCFi4IbN0IF+4Amhsl00g6dmYSZiVBCNv6KGxeKuPUz3Pk3TtostPXAhcM593LvPWHCqNKO821VVlbX1jeqm7Wt7Z3dPXv/oKviVGLSwTGLZT9EijAqSEdTzUg/kQTxkJFeOLku/N4jkYrG4l5PE+JzNBI0ohhpIwX2kdce0yBzm/lD5nGkx5Jnt3leC+y603BmgMvELUkdlGgH9pc3jHHKidCYIaUGrpNoP0NSU8xIXvNSRRKEJ2hEBoYKxInys9kDOTw1yhBGsTQlNJypvycyxJWa8tB0FjeqRa8Q//MGqY4u/YyKJNVE4PmiKGVQx7BIAw6pJFizqSEIS2puhXiMJMLaZFaE4C6+vEy6zYbrNNy783rrqoyjCo7BCTgDLrgALXAD2qADMMjBM3gFb9aT9WK9Wx/z1opVzhyCP7A+fwB56ZZM AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokRdCFi4IbN0IF+4Amhsl00g6dmYSZiVBCNv6KGxeKuPUz3Pk3TtostPXAhcM593LvPWHCqNKO821VVlbX1jeqm7Wt7Z3dPXv/oKviVGLSwTGLZT9EijAqSEdTzUg/kQTxkJFeOLku/N4jkYrG4l5PE+JzNBI0ohhpIwX2kdce0yBzm/lD5nGkx5Jnt3leC+y603BmgMvELUkdlGgH9pc3jHHKidCYIaUGrpNoP0NSU8xIXvNSRRKEJ2hEBoYKxInys9kDOTw1yhBGsTQlNJypvycyxJWa8tB0FjeqRa8Q//MGqY4u/YyKJNVE4PmiKGVQx7BIAw6pJFizqSEIS2puhXiMJMLaZFaE4C6+vEy6zYbrNNy783rrqoyjCo7BCTgDLrgALXAD2qADMMjBM3gFb9aT9WK9Wx/z1opVzhyCP7A+fwB56ZZM AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokRdCFi4IbN0IF+4Amhsl00g6dmYSZiVBCNv6KGxeKuPUz3Pk3TtostPXAhcM593LvPWHCqNKO821VVlbX1jeqm7Wt7Z3dPXv/oKviVGLSwTGLZT9EijAqSEdTzUg/kQTxkJFeOLku/N4jkYrG4l5PE+JzNBI0ohhpIwX2kdce0yBzm/lD5nGkx5Jnt3leC+y603BmgMvELUkdlGgH9pc3jHHKidCYIaUGrpNoP0NSU8xIXvNSRRKEJ2hEBoYKxInys9kDOTw1yhBGsTQlNJypvycyxJWa8tB0FjeqRa8Q//MGqY4u/YyKJNVE4PmiKGVQx7BIAw6pJFizqSEIS2puhXiMJMLaZFaE4C6+vEy6zYbrNNy783rrqoyjCo7BCTgDLrgALXAD2qADMMjBM3gFb9aT9WK9Wx/z1opVzhyCP7A+fwB56ZZM  P 23 AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokVdCFi4IblxHsA5oYJtNpO3RmEmYmQgnZ+CtuXCji1s9w5984abPQ1gMXDufcy733RAmjSjvOt1VZWV1b36hu1ra2d3b37P2DjopTiUkbxyyWvQgpwqggbU01I71EEsQjRrrR5Kbwu49EKhqLez1NSMDRSNAhxUgbKbSPfG9Mw6x5nj9kPkd6LHnm5XkttOtOw5kBLhO3JHVQwgvtL38Q45QToTFDSvVdJ9FBhqSmmJG85qeKJAhP0Ij0DRWIExVkswdyeGqUARzG0pTQcKb+nsgQV2rKI9NZ3KgWvUL8z+unengVZFQkqSYCzxcNUwZ1DIs04IBKgjWbGoKwpOZWiMdIIqxNZkUI7uLLy6TTbLhOw727qLeuyziq4BicgDPggkvQArfAA22AQQ6ewSt4s56sF+vd+pi3Vqxy5hD8gfX5A4GlllE= AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokVdCFi4IblxHsA5oYJtNpO3RmEmYmQgnZ+CtuXCji1s9w5984abPQ1gMXDufcy733RAmjSjvOt1VZWV1b36hu1ra2d3b37P2DjopTiUkbxyyWvQgpwqggbU01I71EEsQjRrrR5Kbwu49EKhqLez1NSMDRSNAhxUgbKbSPfG9Mw6x5nj9kPkd6LHnm5XkttOtOw5kBLhO3JHVQwgvtL38Q45QToTFDSvVdJ9FBhqSmmJG85qeKJAhP0Ij0DRWIExVkswdyeGqUARzG0pTQcKb+nsgQV2rKI9NZ3KgWvUL8z+unengVZFQkqSYCzxcNUwZ1DIs04IBKgjWbGoKwpOZWiMdIIqxNZkUI7uLLy6TTbLhOw727qLeuyziq4BicgDPggkvQArfAA22AQQ6ewSt4s56sF+vd+pi3Vqxy5hD8gfX5A4GlllE= AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokVdCFi4IblxHsA5oYJtNpO3RmEmYmQgnZ+CtuXCji1s9w5984abPQ1gMXDufcy733RAmjSjvOt1VZWV1b36hu1ra2d3b37P2DjopTiUkbxyyWvQgpwqggbU01I71EEsQjRrrR5Kbwu49EKhqLez1NSMDRSNAhxUgbKbSPfG9Mw6x5nj9kPkd6LHnm5XkttOtOw5kBLhO3JHVQwgvtL38Q45QToTFDSvVdJ9FBhqSmmJG85qeKJAhP0Ij0DRWIExVkswdyeGqUARzG0pTQcKb+nsgQV2rKI9NZ3KgWvUL8z+unengVZFQkqSYCzxcNUwZ1DIs04IBKgjWbGoKwpOZWiMdIIqxNZkUI7uLLy6TTbLhOw727qLeuyziq4BicgDPggkvQArfAA22AQQ6ewSt4s56sF+vd+pi3Vqxy5hD8gfX5A4GlllE= AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokVdCFi4IblxHsA5oYJtNpO3RmEmYmQgnZ+CtuXCji1s9w5984abPQ1gMXDufcy733RAmjSjvOt1VZWV1b36hu1ra2d3b37P2DjopTiUkbxyyWvQgpwqggbU01I71EEsQjRrrR5Kbwu49EKhqLez1NSMDRSNAhxUgbKbSPfG9Mw6x5nj9kPkd6LHnm5XkttOtOw5kBLhO3JHVQwgvtL38Q45QToTFDSvVdJ9FBhqSmmJG85qeKJAhP0Ij0DRWIExVkswdyeGqUARzG0pTQcKb+nsgQV2rKI9NZ3KgWvUL8z+unengVZFQkqSYCzxcNUwZ1DIs04IBKgjWbGoKwpOZWiMdIIqxNZkUI7uLLy6TTbLhOw727qLeuyziq4BicgDPggkvQArfAA22AQQ6ewSt4s56sF+vd+pi3Vqxy5hD8gfX5A4GlllE=  P 37 AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokKtSFi4IblxHsA5oYJtNpO3RmEmYmQgnZ+CtuXCji1s9w5984abPQ1gMXDufcy733RAmjSjvOt1VZWV1b36hu1ra2d3b37P2DjopTiUkbxyyWvQgpwqggbU01I71EEsQjRrrR5Kbwu49EKhqLez1NSMDRSNAhxUgbKbSPfG9Mw+yimT9kPkd6LHnm5XkttOtOw5kBLhO3JHVQwgvtL38Q45QToTFDSvVdJ9FBhqSmmJG85qeKJAhP0Ij0DRWIExVkswdyeGqUARzG0pTQcKb+nsgQV2rKI9NZ3KgWvUL8z+unengVZFQkqSYCzxcNUwZ1DIs04IBKgjWbGoKwpOZWiMdIIqxNZkUI7uLLy6Rz3nCdhnt3WW9dl3FUwTE4AWfABU3QArfAA22AQQ6ewSt4s56sF+vd+pi3Vqxy5hD8gfX5A4mFllY= AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokKtSFi4IblxHsA5oYJtNpO3RmEmYmQgnZ+CtuXCji1s9w5984abPQ1gMXDufcy733RAmjSjvOt1VZWV1b36hu1ra2d3b37P2DjopTiUkbxyyWvQgpwqggbU01I71EEsQjRrrR5Kbwu49EKhqLez1NSMDRSNAhxUgbKbSPfG9Mw+yimT9kPkd6LHnm5XkttOtOw5kBLhO3JHVQwgvtL38Q45QToTFDSvVdJ9FBhqSmmJG85qeKJAhP0Ij0DRWIExVkswdyeGqUARzG0pTQcKb+nsgQV2rKI9NZ3KgWvUL8z+unengVZFQkqSYCzxcNUwZ1DIs04IBKgjWbGoKwpOZWiMdIIqxNZkUI7uLLy6Rz3nCdhnt3WW9dl3FUwTE4AWfABU3QArfAA22AQQ6ewSt4s56sF+vd+pi3Vqxy5hD8gfX5A4mFllY= AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokKtSFi4IblxHsA5oYJtNpO3RmEmYmQgnZ+CtuXCji1s9w5984abPQ1gMXDufcy733RAmjSjvOt1VZWV1b36hu1ra2d3b37P2DjopTiUkbxyyWvQgpwqggbU01I71EEsQjRrrR5Kbwu49EKhqLez1NSMDRSNAhxUgbKbSPfG9Mw+yimT9kPkd6LHnm5XkttOtOw5kBLhO3JHVQwgvtL38Q45QToTFDSvVdJ9FBhqSmmJG85qeKJAhP0Ij0DRWIExVkswdyeGqUARzG0pTQcKb+nsgQV2rKI9NZ3KgWvUL8z+unengVZFQkqSYCzxcNUwZ1DIs04IBKgjWbGoKwpOZWiMdIIqxNZkUI7uLLy6Rz3nCdhnt3WW9dl3FUwTE4AWfABU3QArfAA22AQQ6ewSt4s56sF+vd+pi3Vqxy5hD8gfX5A4mFllY= AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokKtSFi4IblxHsA5oYJtNpO3RmEmYmQgnZ+CtuXCji1s9w5984abPQ1gMXDufcy733RAmjSjvOt1VZWV1b36hu1ra2d3b37P2DjopTiUkbxyyWvQgpwqggbU01I71EEsQjRrrR5Kbwu49EKhqLez1NSMDRSNAhxUgbKbSPfG9Mw+yimT9kPkd6LHnm5XkttOtOw5kBLhO3JHVQwgvtL38Q45QToTFDSvVdJ9FBhqSmmJG85qeKJAhP0Ij0DRWIExVkswdyeGqUARzG0pTQcKb+nsgQV2rKI9NZ3KgWvUL8z+unengVZFQkqSYCzxcNUwZ1DIs04IBKgjWbGoKwpOZWiMdIIqxNZkUI7uLLy6Rz3nCdhnt3WW9dl3FUwTE4AWfABU3QArfAA22AQQ6ewSt4s56sF+vd+pi3Vqxy5hD8gfX5A4mFllY=  P 78 AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokIrQLFwU3LiPYBzQxTKbTdujMJMxMhBKy8VfcuFDErZ/hzr9x0mahrQcuHM65l3vviRJGlXacb6uytr6xuVXdru3s7u0f2IdHXRWnEpMOjlks+xFShFFBOppqRvqJJIhHjPSi6U3h9x6JVDQW93qWkICjsaAjipE2Umif+N6EhlmzlT9kPkd6Innm5XkttOtOw5kDrhK3JHVQwgvtL38Y45QToTFDSg1cJ9FBhqSmmJG85qeKJAhP0ZgMDBWIExVk8wdyeG6UIRzF0pTQcK7+nsgQV2rGI9NZ3KiWvUL8zxuketQKMiqSVBOBF4tGKYM6hkUacEglwZrNDEFYUnMrxBMkEdYmsyIEd/nlVdK9bLhOw727qrevyziq4BScgQvggiZog1vggQ7AIAfP4BW8WU/Wi/VufSxaK1Y5cwz+wPr8AZFolls= AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokIrQLFwU3LiPYBzQxTKbTdujMJMxMhBKy8VfcuFDErZ/hzr9x0mahrQcuHM65l3vviRJGlXacb6uytr6xuVXdru3s7u0f2IdHXRWnEpMOjlks+xFShFFBOppqRvqJJIhHjPSi6U3h9x6JVDQW93qWkICjsaAjipE2Umif+N6EhlmzlT9kPkd6Innm5XkttOtOw5kDrhK3JHVQwgvtL38Y45QToTFDSg1cJ9FBhqSmmJG85qeKJAhP0ZgMDBWIExVk8wdyeG6UIRzF0pTQcK7+nsgQV2rGI9NZ3KiWvUL8zxuketQKMiqSVBOBF4tGKYM6hkUacEglwZrNDEFYUnMrxBMkEdYmsyIEd/nlVdK9bLhOw727qrevyziq4BScgQvggiZog1vggQ7AIAfP4BW8WU/Wi/VufSxaK1Y5cwz+wPr8AZFolls= AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokIrQLFwU3LiPYBzQxTKbTdujMJMxMhBKy8VfcuFDErZ/hzr9x0mahrQcuHM65l3vviRJGlXacb6uytr6xuVXdru3s7u0f2IdHXRWnEpMOjlks+xFShFFBOppqRvqJJIhHjPSi6U3h9x6JVDQW93qWkICjsaAjipE2Umif+N6EhlmzlT9kPkd6Innm5XkttOtOw5kDrhK3JHVQwgvtL38Y45QToTFDSg1cJ9FBhqSmmJG85qeKJAhP0ZgMDBWIExVk8wdyeG6UIRzF0pTQcK7+nsgQV2rGI9NZ3KiWvUL8zxuketQKMiqSVBOBF4tGKYM6hkUacEglwZrNDEFYUnMrxBMkEdYmsyIEd/nlVdK9bLhOw727qrevyziq4BScgQvggiZog1vggQ7AIAfP4BW8WU/Wi/VufSxaK1Y5cwz+wPr8AZFolls= AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokIrQLFwU3LiPYBzQxTKbTdujMJMxMhBKy8VfcuFDErZ/hzr9x0mahrQcuHM65l3vviRJGlXacb6uytr6xuVXdru3s7u0f2IdHXRWnEpMOjlks+xFShFFBOppqRvqJJIhHjPSi6U3h9x6JVDQW93qWkICjsaAjipE2Umif+N6EhlmzlT9kPkd6Innm5XkttOtOw5kDrhK3JHVQwgvtL38Y45QToTFDSg1cJ9FBhqSmmJG85qeKJAhP0ZgMDBWIExVk8wdyeG6UIRzF0pTQcK7+nsgQV2rGI9NZ3KiWvUL8zxuketQKMiqSVBOBF4tGKYM6hkUacEglwZrNDEFYUnMrxBMkEdYmsyIEd/nlVdK9bLhOw727qrevyziq4BScgQvggiZog1vggQ7AIAfP4BW8WU/Wi/VufSxaK1Y5cwz+wPr8AZFolls= 5 AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ZMUvHhswX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2M72Z++wmV5rF8MJME/YgOJQ85o8ZKjat+ueJW3TnIKvFyUoEc9X75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NApObPKgISxsiUNmau/JzIaaT2JAtsZUTPSy95M/M/rpia88TMuk9SgZItFYSqIicnsazLgCpkRE0soU9zeStiIKsqMzaZkQ/CWX14lrYuq51a9xmWldpvHUYQTOIVz8OAaanAPdWgCA4RneIU359F5cd6dj0VrwclnjuEPnM8ff3mMtQ== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ZMUvHhswX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2M72Z++wmV5rF8MJME/YgOJQ85o8ZKjat+ueJW3TnIKvFyUoEc9X75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NApObPKgISxsiUNmau/JzIaaT2JAtsZUTPSy95M/M/rpia88TMuk9SgZItFYSqIicnsazLgCpkRE0soU9zeStiIKsqMzaZkQ/CWX14lrYuq51a9xmWldpvHUYQTOIVz8OAaanAPdWgCA4RneIU359F5cd6dj0VrwclnjuEPnM8ff3mMtQ== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ZMUvHhswX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2M72Z++wmV5rF8MJME/YgOJQ85o8ZKjat+ueJW3TnIKvFyUoEc9X75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NApObPKgISxsiUNmau/JzIaaT2JAtsZUTPSy95M/M/rpia88TMuk9SgZItFYSqIicnsazLgCpkRE0soU9zeStiIKsqMzaZkQ/CWX14lrYuq51a9xmWldpvHUYQTOIVz8OAaanAPdWgCA4RneIU359F5cd6dj0VrwclnjuEPnM8ff3mMtQ== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ZMUvHhswX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2M72Z++wmV5rF8MJME/YgOJQ85o8ZKjat+ueJW3TnIKvFyUoEc9X75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NApObPKgISxsiUNmau/JzIaaT2JAtsZUTPSy95M/M/rpia88TMuk9SgZItFYSqIicnsazLgCpkRE0soU9zeStiIKsqMzaZkQ/CWX14lrYuq51a9xmWldpvHUYQTOIVz8OAaanAPdWgCA4RneIU359F5cd6dj0VrwclnjuEPnM8ff3mMtQ== 6 AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1JMUvHhswX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2M72Z++wmV5rF8MJME/YgOJQ85o8ZKjat+ueJW3TnIKvFyUoEc9X75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NApObPKgISxsiUNmau/JzIaaT2JAtsZUTPSy95M/M/rpia88TMuk9SgZItFYSqIicnsazLgCpkRE0soU9zeStiIKsqMzaZkQ/CWX14lrYuq51a9xmWldpvHUYQTOIVz8OAaanAPdWgCA4RneIU359F5cd6dj0VrwclnjuEPnM8fgP2Mtg== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1JMUvHhswX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2M72Z++wmV5rF8MJME/YgOJQ85o8ZKjat+ueJW3TnIKvFyUoEc9X75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NApObPKgISxsiUNmau/JzIaaT2JAtsZUTPSy95M/M/rpia88TMuk9SgZItFYSqIicnsazLgCpkRE0soU9zeStiIKsqMzaZkQ/CWX14lrYuq51a9xmWldpvHUYQTOIVz8OAaanAPdWgCA4RneIU359F5cd6dj0VrwclnjuEPnM8fgP2Mtg== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1JMUvHhswX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2M72Z++wmV5rF8MJME/YgOJQ85o8ZKjat+ueJW3TnIKvFyUoEc9X75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NApObPKgISxsiUNmau/JzIaaT2JAtsZUTPSy95M/M/rpia88TMuk9SgZItFYSqIicnsazLgCpkRE0soU9zeStiIKsqMzaZkQ/CWX14lrYuq51a9xmWldpvHUYQTOIVz8OAaanAPdWgCA4RneIU359F5cd6dj0VrwclnjuEPnM8fgP2Mtg== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1JMUvHhswX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2M72Z++wmV5rF8MJME/YgOJQ85o8ZKjat+ueJW3TnIKvFyUoEc9X75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NApObPKgISxsiUNmau/JzIaaT2JAtsZUTPSy95M/M/rpia88TMuk9SgZItFYSqIicnsazLgCpkRE0soU9zeStiIKsqMzaZkQ/CWX14lrYuq51a9xmWldpvHUYQTOIVz8OAaanAPdWgCA4RneIU359F5cd6dj0VrwclnjuEPnM8fgP2Mtg== 4 AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoCcpePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2VmrVBueJW3QXIOvFyUoEcjUH5qz+MWRqhNExQrXuemxg/o8pwJnBW6qcaE8omdIQ9SyWNUPvZ4tAZubDKkISxsiUNWai/JzIaaT2NAtsZUTPWq95c/M/rpSa88TMuk9SgZMtFYSqIicn8azLkCpkRU0soU9zeStiYKsqMzaZkQ/BWX14n7auq51a9Zq1Sv83jKMIZnMMleHANdbiHBrSAAcIzvMKb8+i8OO/Ox7K14OQzp/AHzucPffWMtA== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoCcpePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2VmrVBueJW3QXIOvFyUoEcjUH5qz+MWRqhNExQrXuemxg/o8pwJnBW6qcaE8omdIQ9SyWNUPvZ4tAZubDKkISxsiUNWai/JzIaaT2NAtsZUTPWq95c/M/rpSa88TMuk9SgZMtFYSqIicn8azLkCpkRU0soU9zeStiYKsqMzaZkQ/BWX14n7auq51a9Zq1Sv83jKMIZnMMleHANdbiHBrSAAcIzvMKb8+i8OO/Ox7K14OQzp/AHzucPffWMtA== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoCcpePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2VmrVBueJW3QXIOvFyUoEcjUH5qz+MWRqhNExQrXuemxg/o8pwJnBW6qcaE8omdIQ9SyWNUPvZ4tAZubDKkISxsiUNWai/JzIaaT2NAtsZUTPWq95c/M/rpSa88TMuk9SgZMtFYSqIicn8azLkCpkRU0soU9zeStiYKsqMzaZkQ/BWX14n7auq51a9Zq1Sv83jKMIZnMMleHANdbiHBrSAAcIzvMKb8+i8OO/Ox7K14OQzp/AHzucPffWMtA== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoCcpePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2VmrVBueJW3QXIOvFyUoEcjUH5qz+MWRqhNExQrXuemxg/o8pwJnBW6qcaE8omdIQ9SyWNUPvZ4tAZubDKkISxsiUNWai/JzIaaT2NAtsZUTPWq95c/M/rpSa88TMuk9SgZMtFYSqIicn8azLkCpkRU0soU9zeStiYKsqMzaZkQ/BWX14n7auq51a9Zq1Sv83jKMIZnMMleHANdbiHBrSAAcIzvMKb8+i8OO/Ox7K14OQzp/AHzucPffWMtA== 7 AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqCcpePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2VmrVBueJW3QXIOvFyUoEcjUH5qz+MWRqhNExQrXuemxg/o8pwJnBW6qcaE8omdIQ9SyWNUPvZ4tAZubDKkISxsiUNWai/JzIaaT2NAtsZUTPWq95c/M/rpSa88TMuk9SgZMtFYSqIicn8azLkCpkRU0soU9zeStiYKsqMzaZkQ/BWX14n7auq51a95nWlfpvHUYQzOIdL8KAGdbiHBrSAAcIzvMKb8+i8OO/Ox7K14OQzp/AHzucPgoGMtw== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqCcpePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2VmrVBueJW3QXIOvFyUoEcjUH5qz+MWRqhNExQrXuemxg/o8pwJnBW6qcaE8omdIQ9SyWNUPvZ4tAZubDKkISxsiUNWai/JzIaaT2NAtsZUTPWq95c/M/rpSa88TMuk9SgZMtFYSqIicn8azLkCpkRU0soU9zeStiYKsqMzaZkQ/BWX14n7auq51a95nWlfpvHUYQzOIdL8KAGdbiHBrSAAcIzvMKb8+i8OO/Ox7K14OQzp/AHzucPgoGMtw== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqCcpePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2VmrVBueJW3QXIOvFyUoEcjUH5qz+MWRqhNExQrXuemxg/o8pwJnBW6qcaE8omdIQ9SyWNUPvZ4tAZubDKkISxsiUNWai/JzIaaT2NAtsZUTPWq95c/M/rpSa88TMuk9SgZMtFYSqIicn8azLkCpkRU0soU9zeStiYKsqMzaZkQ/BWX14n7auq51a95nWlfpvHUYQzOIdL8KAGdbiHBrSAAcIzvMKb8+i8OO/Ox7K14OQzp/AHzucPgoGMtw== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqCcpePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2VmrVBueJW3QXIOvFyUoEcjUH5qz+MWRqhNExQrXuemxg/o8pwJnBW6qcaE8omdIQ9SyWNUPvZ4tAZubDKkISxsiUNWai/JzIaaT2NAtsZUTPWq95c/M/rpSa88TMuk9SgZMtFYSqIicn8azLkCpkRU0soU9zeStiYKsqMzaZkQ/BWX14n7auq51a95nWlfpvHUYQzOIdL8KAGdbiHBrSAAcIzvMKb8+i8OO/Ox7K14OQzp/AHzucPgoGMtw== 2 AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoCcpePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2VmrVBueJW3QXIOvFyUoEcjUH5qz+MWRqhNExQrXuemxg/o8pwJnBW6qcaE8omdIQ9SyWNUPvZ4tAZubDKkISxsiUNWai/JzIaaT2NAtsZUTPWq95c/M/rpSa88TMuk9SgZMtFYSqIicn8azLkCpkRU0soU9zeStiYKsqMzaZkQ/BWX14n7VrVc6te86pSv83jKMIZnMMleHANdbiHBrSAAcIzvMKb8+i8OO/Ox7K14OQzp/AHzucPeu2Msg== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoCcpePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2VmrVBueJW3QXIOvFyUoEcjUH5qz+MWRqhNExQrXuemxg/o8pwJnBW6qcaE8omdIQ9SyWNUPvZ4tAZubDKkISxsiUNWai/JzIaaT2NAtsZUTPWq95c/M/rpSa88TMuk9SgZMtFYSqIicn8azLkCpkRU0soU9zeStiYKsqMzaZkQ/BWX14n7VrVc6te86pSv83jKMIZnMMleHANdbiHBrSAAcIzvMKb8+i8OO/Ox7K14OQzp/AHzucPeu2Msg== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoCcpePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2VmrVBueJW3QXIOvFyUoEcjUH5qz+MWRqhNExQrXuemxg/o8pwJnBW6qcaE8omdIQ9SyWNUPvZ4tAZubDKkISxsiUNWai/JzIaaT2NAtsZUTPWq95c/M/rpSa88TMuk9SgZMtFYSqIicn8azLkCpkRU0soU9zeStiYKsqMzaZkQ/BWX14n7VrVc6te86pSv83jKMIZnMMleHANdbiHBrSAAcIzvMKb8+i8OO/Ox7K14OQzp/AHzucPeu2Msg== AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoCcpePHYgv2ANpTNdtKu3WzC7kYoob/AiwdFvPqTvPlv3LY5aOuDgcd7M8zMCxLBtXHdb6ewsbm1vVPcLe3tHxwelY9P2jpOFcMWi0WsugHVKLjEluFGYDdRSKNAYCeY3M39zhMqzWP5YKYJ+hEdSR5yRo2VmrVBueJW3QXIOvFyUoEcjUH5qz+MWRqhNExQrXuemxg/o8pwJnBW6qcaE8omdIQ9SyWNUPvZ4tAZubDKkISxsiUNWai/JzIaaT2NAtsZUTPWq95c/M/rpSa88TMuk9SgZMtFYSqIicn8azLkCpkRU0soU9zeStiYKsqMzaZkQ/BWX14n7VrVc6te86pSv83jKMIZnMMleHANdbiHBrSAAcIzvMKb8+i8OO/Ox7K14OQzp/AHzucPeu2Msg== 3 AAAB6HicdVDLSsNAFJ3UV62vqks3g0VwFSYxtHUjBTcuW7APaEOZTCft2MkkzEyEEvoFblwo4tZPcuffOGkrqOiBC4dz7uXee4KEM6UR+rAKa+sbm1vF7dLO7t7+QfnwqKPiVBLaJjGPZS/AinImaFszzWkvkRRHAafdYHqd+917KhWLxa2eJdSP8FiwkBGsjdS6GJYryL6sV12vCpGNUM1xnZy4Ne/Cg45RclTACs1h+X0wikkaUaEJx0r1HZRoP8NSM8LpvDRIFU0wmeIx7RsqcESVny0OncMzo4xgGEtTQsOF+n0iw5FSsygwnRHWE/Xby8W/vH6qw7qfMZGkmgqyXBSmHOoY5l/DEZOUaD4zBBPJzK2QTLDERJtsSiaEr0/h/6Tj2g6ynZZXaVyt4iiCE3AKzoEDaqABbkATtAEBFDyAJ/Bs3VmP1ov1umwtWKuZY/AD1tsn3ICM9Q== AAAB6HicdVDLSsNAFJ3UV62vqks3g0VwFSYxtHUjBTcuW7APaEOZTCft2MkkzEyEEvoFblwo4tZPcuffOGkrqOiBC4dz7uXee4KEM6UR+rAKa+sbm1vF7dLO7t7+QfnwqKPiVBLaJjGPZS/AinImaFszzWkvkRRHAafdYHqd+917KhWLxa2eJdSP8FiwkBGsjdS6GJYryL6sV12vCpGNUM1xnZy4Ne/Cg45RclTACs1h+X0wikkaUaEJx0r1HZRoP8NSM8LpvDRIFU0wmeIx7RsqcESVny0OncMzo4xgGEtTQsOF+n0iw5FSsygwnRHWE/Xby8W/vH6qw7qfMZGkmgqyXBSmHOoY5l/DEZOUaD4zBBPJzK2QTLDERJtsSiaEr0/h/6Tj2g6ynZZXaVyt4iiCE3AKzoEDaqABbkATtAEBFDyAJ/Bs3VmP1ov1umwtWKuZY/AD1tsn3ICM9Q== AAAB6HicdVDLSsNAFJ3UV62vqks3g0VwFSYxtHUjBTcuW7APaEOZTCft2MkkzEyEEvoFblwo4tZPcuffOGkrqOiBC4dz7uXee4KEM6UR+rAKa+sbm1vF7dLO7t7+QfnwqKPiVBLaJjGPZS/AinImaFszzWkvkRRHAafdYHqd+917KhWLxa2eJdSP8FiwkBGsjdS6GJYryL6sV12vCpGNUM1xnZy4Ne/Cg45RclTACs1h+X0wikkaUaEJx0r1HZRoP8NSM8LpvDRIFU0wmeIx7RsqcESVny0OncMzo4xgGEtTQsOF+n0iw5FSsygwnRHWE/Xby8W/vH6qw7qfMZGkmgqyXBSmHOoY5l/DEZOUaD4zBBPJzK2QTLDERJtsSiaEr0/h/6Tj2g6ynZZXaVyt4iiCE3AKzoEDaqABbkATtAEBFDyAJ/Bs3VmP1ov1umwtWKuZY/AD1tsn3ICM9Q== AAAB6HicdVDLSsNAFJ3UV62vqks3g0VwFSYxtHUjBTcuW7APaEOZTCft2MkkzEyEEvoFblwo4tZPcuffOGkrqOiBC4dz7uXee4KEM6UR+rAKa+sbm1vF7dLO7t7+QfnwqKPiVBLaJjGPZS/AinImaFszzWkvkRRHAafdYHqd+917KhWLxa2eJdSP8FiwkBGsjdS6GJYryL6sV12vCpGNUM1xnZy4Ne/Cg45RclTACs1h+X0wikkaUaEJx0r1HZRoP8NSM8LpvDRIFU0wmeIx7RsqcESVny0OncMzo4xgGEtTQsOF+n0iw5FSsygwnRHWE/Xby8W/vH6qw7qfMZGkmgqyXBSmHOoY5l/DEZOUaD4zBBPJzK2QTLDERJtsSiaEr0/h/6Tj2g6ynZZXaVyt4iiCE3AKzoEDaqABbkATtAEBFDyAJ/Bs3VmP1ov1umwtWKuZY/AD1tsn3ICM9Q== 8 AAAB6HicdVDLSgNBEJyNrxhfUY9eBoPgKcwEMZuLBLx4TMA8IFnC7KQ3GTP7YGZWCCFf4MWDIl79JG/+jbNJBBUtaCiquunu8hMptCHkw8mtrW9sbuW3Czu7e/sHxcOjto5TxaHFYxmrrs80SBFBywgjoZsoYKEvoeNPrjO/cw9Kizi6NdMEvJCNIhEIzoyVmu6gWCJlQgilFGeEVi+JJbWaW6EuppllUUIrNAbF9/4w5mkIkeGSad2jJDHejCkjuIR5oZ9qSBifsBH0LI1YCNqbLQ6d4zOrDHEQK1uRwQv1+8SMhVpPQ992hsyM9W8vE//yeqkJXG8moiQ1EPHloiCV2MQ4+xoPhQJu5NQSxpWwt2I+ZopxY7Mp2BC+PsX/k3alTEmZNi9K9atVHHl0gk7ROaKoiuroBjVQC3EE6AE9oWfnznl0XpzXZWvOWc0cox9w3j4B0PuM7g== AAAB6HicdVDLSgNBEJyNrxhfUY9eBoPgKcwEMZuLBLx4TMA8IFnC7KQ3GTP7YGZWCCFf4MWDIl79JG/+jbNJBBUtaCiquunu8hMptCHkw8mtrW9sbuW3Czu7e/sHxcOjto5TxaHFYxmrrs80SBFBywgjoZsoYKEvoeNPrjO/cw9Kizi6NdMEvJCNIhEIzoyVmu6gWCJlQgilFGeEVi+JJbWaW6EuppllUUIrNAbF9/4w5mkIkeGSad2jJDHejCkjuIR5oZ9qSBifsBH0LI1YCNqbLQ6d4zOrDHEQK1uRwQv1+8SMhVpPQ992hsyM9W8vE//yeqkJXG8moiQ1EPHloiCV2MQ4+xoPhQJu5NQSxpWwt2I+ZopxY7Mp2BC+PsX/k3alTEmZNi9K9atVHHl0gk7ROaKoiuroBjVQC3EE6AE9oWfnznl0XpzXZWvOWc0cox9w3j4B0PuM7g== AAAB6HicdVDLSgNBEJyNrxhfUY9eBoPgKcwEMZuLBLx4TMA8IFnC7KQ3GTP7YGZWCCFf4MWDIl79JG/+jbNJBBUtaCiquunu8hMptCHkw8mtrW9sbuW3Czu7e/sHxcOjto5TxaHFYxmrrs80SBFBywgjoZsoYKEvoeNPrjO/cw9Kizi6NdMEvJCNIhEIzoyVmu6gWCJlQgilFGeEVi+JJbWaW6EuppllUUIrNAbF9/4w5mkIkeGSad2jJDHejCkjuIR5oZ9qSBifsBH0LI1YCNqbLQ6d4zOrDHEQK1uRwQv1+8SMhVpPQ992hsyM9W8vE//yeqkJXG8moiQ1EPHloiCV2MQ4+xoPhQJu5NQSxpWwt2I+ZopxY7Mp2BC+PsX/k3alTEmZNi9K9atVHHl0gk7ROaKoiuroBjVQC3EE6AE9oWfnznl0XpzXZWvOWc0cox9w3j4B0PuM7g== AAAB6HicdVDLSgNBEJyNrxhfUY9eBoPgKcwEMZuLBLx4TMA8IFnC7KQ3GTP7YGZWCCFf4MWDIl79JG/+jbNJBBUtaCiquunu8hMptCHkw8mtrW9sbuW3Czu7e/sHxcOjto5TxaHFYxmrrs80SBFBywgjoZsoYKEvoeNPrjO/cw9Kizi6NdMEvJCNIhEIzoyVmu6gWCJlQgilFGeEVi+JJbWaW6EuppllUUIrNAbF9/4w5mkIkeGSad2jJDHejCkjuIR5oZ9qSBifsBH0LI1YCNqbLQ6d4zOrDHEQK1uRwQv1+8SMhVpPQ992hsyM9W8vE//yeqkJXG8moiQ1EPHloiCV2MQ4+xoPhQJu5NQSxpWwt2I+ZopxY7Mp2BC+PsX/k3alTEmZNi9K9atVHHl0gk7ROaKoiuroBjVQC3EE6AE9oWfnznl0XpzXZWvOWc0cox9w3j4B0PuM7g== 1 AAAB6HicdVDLSgNBEOyNrxhfUY9eBoPgKcwEMclFAl48JmAekCxhdjKbjJl9MDMrhCVf4MWDIl79JG/+jbNJBBUtaCiquunu8mIptMH4w8mtrW9sbuW3Czu7e/sHxcOjjo4SxXibRTJSPY9qLkXI20YYyXux4jTwJO960+vM795zpUUU3ppZzN2AjkPhC0aNlVpkWCzhMsaYEIIyQqqX2JJ6vVYhNUQyy6IEKzSHxffBKGJJwEPDJNW6T3Bs3JQqI5jk88Ig0TymbErHvG9pSAOu3XRx6BydWWWE/EjZCg1aqN8nUhpoPQs82xlQM9G/vUz8y+snxq+5qQjjxPCQLRf5iUQmQtnXaCQUZ0bOLKFMCXsrYhOqKDM2m4IN4etT9D/pVMoEl0nrotS4WsWRhxM4hXMgUIUG3EAT2sCAwwM8wbNz5zw6L87rsjXnrGaO4Qect0/GX4zn AAAB6HicdVDLSgNBEOyNrxhfUY9eBoPgKcwEMclFAl48JmAekCxhdjKbjJl9MDMrhCVf4MWDIl79JG/+jbNJBBUtaCiquunu8mIptMH4w8mtrW9sbuW3Czu7e/sHxcOjjo4SxXibRTJSPY9qLkXI20YYyXux4jTwJO960+vM795zpUUU3ppZzN2AjkPhC0aNlVpkWCzhMsaYEIIyQqqX2JJ6vVYhNUQyy6IEKzSHxffBKGJJwEPDJNW6T3Bs3JQqI5jk88Ig0TymbErHvG9pSAOu3XRx6BydWWWE/EjZCg1aqN8nUhpoPQs82xlQM9G/vUz8y+snxq+5qQjjxPCQLRf5iUQmQtnXaCQUZ0bOLKFMCXsrYhOqKDM2m4IN4etT9D/pVMoEl0nrotS4WsWRhxM4hXMgUIUG3EAT2sCAwwM8wbNz5zw6L87rsjXnrGaO4Qect0/GX4zn AAAB6HicdVDLSgNBEOyNrxhfUY9eBoPgKcwEMclFAl48JmAekCxhdjKbjJl9MDMrhCVf4MWDIl79JG/+jbNJBBUtaCiquunu8mIptMH4w8mtrW9sbuW3Czu7e/sHxcOjjo4SxXibRTJSPY9qLkXI20YYyXux4jTwJO960+vM795zpUUU3ppZzN2AjkPhC0aNlVpkWCzhMsaYEIIyQqqX2JJ6vVYhNUQyy6IEKzSHxffBKGJJwEPDJNW6T3Bs3JQqI5jk88Ig0TymbErHvG9pSAOu3XRx6BydWWWE/EjZCg1aqN8nUhpoPQs82xlQM9G/vUz8y+snxq+5qQjjxPCQLRf5iUQmQtnXaCQUZ0bOLKFMCXsrYhOqKDM2m4IN4etT9D/pVMoEl0nrotS4WsWRhxM4hXMgUIUG3EAT2sCAwwM8wbNz5zw6L87rsjXnrGaO4Qect0/GX4zn AAAB6HicdVDLSgNBEOyNrxhfUY9eBoPgKcwEMclFAl48JmAekCxhdjKbjJl9MDMrhCVf4MWDIl79JG/+jbNJBBUtaCiquunu8mIptMH4w8mtrW9sbuW3Czu7e/sHxcOjjo4SxXibRTJSPY9qLkXI20YYyXux4jTwJO960+vM795zpUUU3ppZzN2AjkPhC0aNlVpkWCzhMsaYEIIyQqqX2JJ6vVYhNUQyy6IEKzSHxffBKGJJwEPDJNW6T3Bs3JQqI5jk88Ig0TymbErHvG9pSAOu3XRx6BydWWWE/EjZCg1aqN8nUhpoPQs82xlQM9G/vUz8y+snxq+5qQjjxPCQLRf5iUQmQtnXaCQUZ0bOLKFMCXsrYhOqKDM2m4IN4etT9D/pVMoEl0nrotS4WsWRhxM4hXMgUIUG3EAT2sCAwwM8wbNz5zw6L87rsjXnrGaO4Qect0/GX4zn (a) 8-node network topology and over -determined mea- surements scenario (additional measurement h 3 ), blue line is head dif ference ∆ r h 18 , and red line is ∆ r h 13 . (b) Error between solution from EP ANET and our approach in sufficient measurements scenario. Fig. 3: 8-node network and its results under multi-scenarios. B. 8-node Network The 8-node network from EP ANET Users Manual [ 18 ] is a looped network, and labels for various components are all shown in Fig. 3(a) . If the head at reservoir and tank, h R 1 and h TK 8 , are known, then it satisfies the suf ficient scenario which means the solution can be determined with just these two measure- ments. The solution ξ SE solved by Algorithm 1 and ξ EP ANET from EP ANET are given in T ab . III(b) when h TK 8 “ 834 ft and h R 1 “ 700 ft , and final error k ξ SE ´ ξ EP ANET k presented in Fig. 3(b) reaches 0.1 which illustrates the effecti veness of proposed method. As we mentioned in Section II-B , we assume that the mea- surements of heads at tanks and reservoirs are very accurate. W e measure one more head at Junction 3 ( h 3 ) thereby defining the ov er-determined scenario. There are two cases based on which measurement is more trustful. F or Case 1, if we postulate that h TK 8 “ 834 . 60 ft is more accurate and setting the accuracy matrix as W “ diag p 1 , 0 . 1 q , we see the resulting h TK 8 is very close to the measured value in T ab. III(b) , while h 3 is far from its measurement since it is considered less accurate. For Case 2, if h 3 is considered more accurate, h 3 is close to its measured value 875 . 64 ft . Besides that, the two estimated variables h 5 and q 46 are shown, and we can see that both are close to the v alues provided by the EP ANET software, but v ary slightly between Cases 1 and 2. V . P A P E R ’ S L I M I TA T I O N S A N D F U T U R E W O R K The paper’ s limitations include the lack of uncertainty quantification from nodal water demands, leaks in pipes, and pipe roughness parameters. The proposed approach can handle ellipsoidal and cardinal-polyhedral uncertainty by updating the constraints through the successive linear approximation, gi ven historical sets of demands and network parameters. T o this end, future work will focus on deri ving a robust, yet still scalable state estimation routine that addresses uncertainty stemming from the aforementioned sources, in addition to modeling various types of valves in the SE problem, and thoroughly comparing our approach to the state-of-the-art in the literature. V I . A C KN O W L E D G M E N T S W e gratefully ackno wledge the constructive comments from the conference editor and the revie wers. W e also acknowledge the financial support from National Science Foundation through Grants 1728629 and 1917164. R E F E R E N C E S [1] W . Zhao, T . H. Beach, and Y . Rezgui, “Optimization of potable water distribution and waste water collection networks: A systematic revie w and future research directions. ” IEEE T rans. Systems, Man, and Cyber- netics: Systems , vol. 46, no. 5, pp. 659–681, 2016. [2] V . Puig, C. Ocampo-Martinez, R. P ´ erez, G. Cembrano, J. Quevedo, and T . Escobet, Real-Time Monitoring and Operational Contr ol of Drinking- W ater Systems . Springer , 2017. [3] K. S. Tshehla, Y . Hamam, and A. M. Abu-Mahfouz, “State estimation in water distrib ution network: A revie w , ” in Industrial Informatics (INDIN), 2017 IEEE 15th International Confer ence on . IEEE, 2017, pp. 1247– 1252. [4] A. Aisopou, I. Stoianov , and N. J. Graham, “In-pipe water quality monitoring in water supply systems under steady and unsteady state flow conditions: A quantitativ e assessment, ” W ater r esearch , vol. 46, no. 1, pp. 235–246, 2012. [5] W . Cheng, T . Y u, and G. Xu, “Real-time model of a large-scale water distribution system, ” Pr ocedia Engineering , vol. 89, pp. 457–466, 2014. [6] S. Vrachimis, D. Eliades, and M. Polycarpou, “Real-time hydraulic interval state estimation for water transport networks: A case study , ” Drinking W ater Engineering and Science , vol. 11, pp. 19–24, 03 2018. [7] A. Bargiela, “On-line monitoring of water distribution networks, ” Ph.D. dissertation, Durham Univ ersity , 1984. [8] R. Powell, “State estimation in water networks: a tutorial, ” Measur ement and control , vol. 32, no. 4, pp. 101–104, 1999. [9] J. Da vidson and F .-C. Bouchart, “ Adjusting nodal demands in SCAD A constrained real-time water distribution network models, ” Journal of Hydraulic Engineering , vol. 132, no. 1, pp. 102–110, 2006. [10] B. D. Barkdoll and H. Didigam, “Effect of user demand on water quality and hydraulics of distrib ution systems, ” in W orld W ater & Envir onmental Resour ces Congress 2003 , 2003, pp. 1–10. [11] M. Pasha and K. Lansey , “ Analysis of uncertainty on water distribution hydraulics and water quality , ” in Impacts of Global Climate Change , 2005, pp. 1–12. [12] D. Kang, M. Pasha, and K. Lansey , “ Approximate methods for uncer- tainty analysis of water distribution systems, ” Urban W ater Journal , vol. 6, no. 3, pp. 233–249, 2009. [13] F . Fusco and E. Arandia, “State estimation for water distribution networks in the presence of control devices with switching beha vior, ” Pr ocedia Engineering , vol. 186, pp. 592–600, 2017. [14] A. Preis, A. J. Whittle, A. Ostfeld, and L. Perelman, “Efficient hy- draulic state estimation technique using reduced models of urban water networks, ” Journal of W ater Resour ces Planning and Management , vol. 137, no. 4, pp. 343–351, 2011. [15] R. Duffin, E. Peterson, and C. Zener, Geometric progr amming - theory and application . New Y ork: Wile y , 1967. [16] S. W ang, A. F . T aha, N. Gatsis, and M. Giacomoni, “Geometric programming-based control for nonlinear , dae- constrained water distribution netw orks, ” in 2019 American Contr ol Confer ence, Philadelphia, US , July 2019, pp. 1470–1475. https://arxiv .org/pdf/1902.06026.pdf [17] A. S. Zamzam, E. Dall’Anese, C. Zhao, J. A. T aylor, and N. Sidiropou- los, “Optimal water-power flo w problem: Formulation and distributed optimal solution, ” IEEE T ransactions on Contr ol of Network Systems , 2018. [18] L. A. Rossman et al. , “Epanet 2: users manual, ” 2000. [19] S. Boyd, S.-J. Kim, L. V andenberghe, and A. Hassibi, “ A tutorial on geometric programming, ” Optimization and engineering , vol. 8, no. 1, p. 67, 2007. [20] I. J. Lustig, R. E. Marsten, and D. F . Shanno, “Interior point methods for linear programming: Computational state of the art, ” ORSA J ournal on Computing , vol. 6, no. 1, pp. 1–14, 1994. [21] D. El ´ ıades and M. Kyriakou, “Epanet matlab toolkit, ” University of Cyprus, Republic of Cyprus , 2009. [22] I. CVX Research, “CVX: Matlab software for disciplined conv ex programming, version 2.0, ” http://cvxr .com/cvx , Aug. 2012. [23] “Shenwang9202/state estimation, ” accessed 15 March 2019. https: //github .com/ShenW ang9202/StateEstimation