International Journal of Computers, Communications & Control Vol. II (2007), No. 3, pp. 241-251 Hybrid Control Accommodation for Water-asset Management of Hydraulic Systems Subjected to Large Operating Conditions Eric Duviella, Pascale Chiron, Philippe Charbonnaud Abstract: The Hybrid Control Accommodation (HCA) strategy was proposed to improve the water-asset management of hydraulic systems by resource allocation and setpoint assignment. Hydraulic system dynamics are taken into account during the setpoint assignment step which consists in controlling gates for large operating conditions. For hydraulic systems subjected to strong disturbances, transfer delays are variable, thus different operating modes must be considered. A multimodelling method, associated to a selection technique of transfer delay, allowing for the deter- mination of the number of models, is proposed. The simulation results on the first reach of the Neste canal show the effectiveness of the HCA strategy. Keywords: Hybrid control accommodation, resource allocation, setpoint assign- ment, multimodelling, water management. 1 Introduction Hydrographic networks naturally convey water quantities upstream to downstream. They are equipped with dams, catchment areas, channels, etc., and are instrumented for water resource management and sat- isfaction of human activity needs. Several control methods have been designed [9], and are still designed [8, 2], to supply hydraulic systems with water quantities corresponding to the management objectives and rejecting disturbances. Other techniques, such as LPV regulation [10] or supervised internal multi- model controller [3] were recently proposed in order to consider these systems for different operating points. These control methods are accurate for a local control. However, they are not designed to allo- cate water quantities in excess towards the catchments areas, and water quantities in lack amongst the users. An original proposition for the management of such situations consists in recalculating setpoints according to the resource value (continuous dynamics) and to the resource state (discrete events) of the hydrographic systems. The supervision and hybrid control accommodation strategy by resource alloca- tion and setpoint assignment proposed in [4] allows the water-asset management to take into account the management constraints. The strategy efficiency has been shown by simulation, in the case of an open-channel hydraulic system considering large operating conditions. The problem addressed in this paper deals with the water-asset management of a hydraulic system subjected to scenarii with large disturbances. In the second section, the supervision and hybrid control accommodation strategy, well adapted to consider large operating conditions, is presented. In the third section, a multimodelling method is proposed for determining the number of Operating Modes (OM) necessary to represent open-surface hydraulic system dynamics, accurately. For each OM, the transfer delay identification is done in the fourth section. Finally, the effectiveness of this proposed strategy is shown by simulation on the first reach of the Neste canal. 2 Hybrid control accommodation Hybrid Control Accommodation (HCA) strategy (see figure 1) allows the computation of new set- points for each gate G j of hydraulic systems according to data measured on the measurement points Mi (see figure 2). The setpoints are computed according to the resource state ei, and taking into account management objectives q j ob j, λ j and µ j weekly fixed by the Management Objective Generation module. Copyright © 2006-2007 by CCC Publications 242 Eric Duviella, Pascale Chiron, Philippe Charbonnaud Figure 1: HCA strategy scheme. Figure 2: Open-surface hydraulic system equipped with measurement points Mi and gates G j. Hybrid Control Accommodation for Water-asset Management of Hydraulic Systems Subjected to Large Operating Conditions 243 The resource state ei determination is carried out at the detection period Te by concurrent hybrid automata (see figure 3) designed for each measurement point Mi [4]. The five pertinent states retained correspond respectively to no-discrepancy state e0, two states where the discharge discrepancy is either positive (e+) or negative (e−) and constant c, and two states where the discharge discrepancy is either positive (e+) or negative (e−) and no constant ¬c. Transitions between states are defined as conditions on the measured discharge value and variation:    di : [|∆QMi| > thi] , ψi : [∆QMi < 0] , ωi : [∣∣Q̇Mi ∣∣ < dthi ] , (1) with ∆QMi = QMi − n ∑ j=ni q j ob j, where QMi is the measured discharge, q j ob j is the management objec- tive of the gate j, ni the index of the first gate downstream the ith measurement point Mi, n the number of gates, Q̇Mi the estimate derivative of QMi , thi and dthi respectively the detection and diagnosis thresholds. The discharge discrepancy allocation is carried out according to the resource state. When the state is e0, the gate setpoints correspond to their discharge objective q j ob j. When the state is e±∧c, the discharge discrepancy ∆QMi is allocated amongst gates, according to their weights λ j and µ j, by optimizing, for each measurement point, a cost function. Finally, when the state is e±∧¬c, the discharge discrepancy is allocated on gate each one on its turn [4]. At each detection date kTe, the discharge discrepancy allocation leads to the allocation vector qkMi : qkMi = [ 0 0 . . . 0 qkni q k ni+1 . . . qkn ]T , (2) Figure 3: Hybrid automata for the measurement point Mi. Then, to synchronize the control of the gate with the appearance of lacks or excess of the water due to the disturbances, the setpoints must be assigned at a date taking into account the transfer delays TMi, j between the measurement point Mi and the gate G j. It is computed according to the relation (3): TMi, j = TMi,ni + j ∑ r=ni+1 tr , (3) where tr is the time necessary for the water quantity to go from gate Gr−1 to gate Gr (see Figure 4). The delay tr depends on the physical characteristics of the hydraulic system and on the discharge value Qr. Thereafter, the allocation vector qkMi is associated with an allocation dates vector T arising 244 Eric Duviella, Pascale Chiron, Philippe Charbonnaud Figure 4: Time delays between Mi and G j. from the transfer delay values TMi, j. Finally, taking into account the allocation date, the setpoints are sent to gates periodically. The control period Tc is chosen as a multiple integer of Te. In the next section, a multimodelling method is proposed to identify the system dynamics for several OM. 3 Multimodelling steps The modelling of the free-surface hydraulic system dynamics is generally carried out starting from the diffusive wave equation (4) which is obtained by simplifying Saint Venant equations [1]. dQ dt + C dQ dx −D d2Q dx2 = 0 , (4) where C and D are respectively the celerity and diffusion coefficients. The diffusive wave equation can be linearized according to an operating discharge Qe [7], and the identified celerity and diffusion parameters, denoted Ce and De, are expressed as:    Ce = 1L2 ∂ J∂ Qe [ ∂ L ∂ x − ∂ JL∂ z ] , De = 1L ∂ J∂ Qe , (5) where L is the surface width, z is the discharge depth, J is the friction slope expressed with the Manning-Strickler relation as J = Q 2P 4 3 K2S 10 3 where K is the Strickler coefficient, P is the wetted perimeter and S the wetted surface. The open-channel systems dynamics can be modelled by transfer functions, as: F(s) = e −τ s 1+w1s+w2s2 , (6) where the coefficients w1, w2 and the pure delay τ are computed according to the coefficients Ce, De, to the open-channel system length X , and to the adimensional coefficient CL = 2CeX 9De ; if CL ≤ 49 , w2 = 0 and τ = 0, if 49 < CL ≤ 1, w2 = 0. This modelling method allows the identification of the free-surface hydraulic system dynamics with good accuracy only around an operating point. However, setpoint assignment must be done for hydraulic system subjected to large operating conditions. Thus, the hydraulic system dynamics must be identified for different OM. Based on the previous modelling method, the multimodelling approach consists in determining the number of OM and their corresponding operating points. The celerity coefficient C is a very relevant parameter of the hydraulic system dynamics. The model identified for each OM is available as soon as the error on the celerity coefficient is less than a predefined percentage ΠC. Thus, a validity domain is defined for each OM, and the number of OM which are nec- essary to identify the dynamics with a good accuracy, is determined. To limit the switching between two OM, an interval ∆C is shared by two successive OM validity domains. The selection of parameters ΠC and ∆C is carried out taking into account the system dynamics. The multimodelling steps are described Hybrid Control Accommodation for Water-asset Management of Hydraulic Systems Subjected to Large Operating Conditions 245 Table 1: Multimodelling algorithm. Input: Cmin, Cmax, ΠC, ∆C, Output: Cidr , Cin fr , Csupr , Cmed = Cmin+Cmax 2 , r = 1, For i: ⌊ ln Cmin Cmed ln 1+ΠC 1−ΠC ⌋ + 1 to ⌊ ln CmaxCmed ln 1+ΠC 1−ΠC ⌋ , Cidr = ( 1+ΠC 1−ΠC )i Cmed − sign(i)∆C1−sign(i)ΠC |i| ∑ j=1 ( 1+ΠC 1−ΠC )i−sign(i). j , Csupr = (1 + ΠC) ( 1+ΠC 1−ΠC )i Cmed −sign(i) (1+ΠC)∆C1−sign(i)ΠC |i| ∑ j=1 ( 1+ΠC 1−ΠC )i−sign(i). j , Cin fr = (1−ΠC) ( 1+ΠC 1−ΠC )i Cmed −sign(i) (1−ΠC)∆C1−sign(i)ΠC |i| ∑ j=1 ( 1+ΠC 1−ΠC )i−sign(i). j , r + +, EndFor by an algorithm (see Table 1), where the OM are determined starting with Cmed . This one is computed with the parameters Cmin and Cmax corresponding respectively to the minimum and maximum discharges of the system. The algorithm requires the definition of bxc which corresponds to the integer part of x, and sign(x) such as sign(x) = |x|x . Then, according to Cidr , the water elevation zidr of each r th OM is determined, with one millimeter accuracy, using the digital resolution of the relation (7) with the Newton method. Cid = √ JKS 5 3 P 2 3 L2 [ −12 ∂ L∂ z − L3P ( 2 ∂ P∂ z −5 PS ∂ S∂ z )] . (7) Then, the water elevation zidr value is used to compute the discharge Qidr with the relation (8). The same steps are used for parameters Cin fr and Csupr to obtain the domain validity boundaries of the OM: [Qin fr ; Qsupr [. Qidr = √ JKS 5 3 P 2 3 . (8) Finally, the celerity and diffusion coefficients Cidr and Didr (5), and the transfer function parame- ters w1, w2 and τ (6) are computed according to the discharge Qidr . These parameters allow for the computation and the selection of the transfer delays of each OM. This is described in the next section. 4 Transfer delay identification The hydraulic systems consist of several reaches, i.e. a part between two measurement points (see Figure 5.a), each reach being composed of Open-Channel Reach Section (OCRS), i.e. a part between two gates, or between a measurement point and a gate or between a gate and a measurement point. Thus, the reach dynamics are modelled with a concurrent Hybrid Automaton cHA (see Figure 5.b) composed of several HA defined for each OCRS. This representation is directly inspired from the concurrent hybrid automata proposed in [5, 6]. In the study case, the cHA is deterministic. The input of the first automaton HAoi of the first OCRS following Mi is the discharge measured on Mi. The input of the second automaton, HAoi+1 is the discharge downstream HAoi minus qni , the discharge setpoint assigned to the gate between this two OCRS. 246 Eric Duviella, Pascale Chiron, Philippe Charbonnaud Figure 5: (a) A canal reach and (b) its modelling by concurrent hybrid automaton (cHA). The OCRS dynamics (see Figure 6.a) are modelled using the multimodelling method described in the previous section, and are represented by a HA (see Figure 6.b). Each state corresponds to one OM identified as transfer function Fl given by the relation (6). The transition conditions are defined according to the upstream discharge Qo and the lth OM boundaries Qin fl and Qsupl . Figure 6: (a) OCRS with index o and (b) its modelling by Hybrid Automaton (HA). For each OM, the transfer delay tr is obtained from the step response of the corresponding transfer function. It is chosen as the time value for which ΠQ percent of step is reached. The percentage ΠQ can be tuned from simulation. In the next section, the HCA strategy is used to valorize water quantities of a hydraulic system subjected to large operating conditions. Hybrid Control Accommodation for Water-asset Management of Hydraulic Systems Subjected to Large Operating Conditions 247 Table 2: Geometrical characteristics of each OCRS. OCRS B [m] f X [m] J [rad] Qmin [m3/s] Qmax [m3/s] 1 5.73 0.79 204 7.10−4 0.8 10 2 5.09 0.96 702 7.10−4 0.8 9 3 5.21 0.95 562 6.10−4 0.7 7 4 3.72 0.94 1360 5.10−4 0.6 5 5 Simulation results The effectiveness of the proposed strategy is shown by simulation on the first reach of the Neste canal located in Gascogne, a french southwestern region. It is composed of one measurement point M1, three gates G1 to G3 and one output considered as a non-controlled gate G4 (see Figure 7). This canal reach is composed of four OCRS with trapezoidal profile characterized by the bottom width B, the average fruit of the banks f , the profile length X and the reach slope I (see Figure 8). The geometrical characteristics and operating conditions of each OCRS are detailed in Table 2. The fourth OCRS, between G3 and G4, is not modelled, because gate G4 is not locally controlled. In the case of a trapezoidal profile, celerity and diffusion coefficients C and D are expressed by the following relation: Figure 7: First reach of the Neste canal. { C = QL2 [ − f + L3 ( 2B Pz + 5L S − 2z )] D = Q2LJ , (9) where L = B + 2 f z, S = zB + f z2, P = B + 2z √ 1 + f 2. Figure 8: Geometrical characteristics of a trapezoidal profile. The multimodelling steps which are carried out with ΠC = 15 % and ∆C = 0.1, lead to the determi- nation of three OM for each OCRS. Their corresponding transfer function parameters are given in Table 3. The transfer delays tr are computed with a value ΠQ = 63 %. This value, chosen by simulation, leads to the best water quantity valorization. 248 Eric Duviella, Pascale Chiron, Philippe Charbonnaud Table 3: Multiple models of each OCRS. OCRS Q Qmin [m3/s] Qmax [m3/s] w1 w2 τ tr [s] 1.6 0.8 2.4 161 0 0 160 1 3.2 1.9 5.2 128 0 0 130 7.7 4.3 10 100 0 0 100 1.4 0.8 2.2 424 24331 145 575 2 3 1.7 4.9 364 0 89 450 7.4 4.1 9 355 0 0 355 1.2 0.7 1.9 377 0 129 505 3 2.4 1.5 4 357 0 29 385 5.8 3.2 7 320 0 0 320 The Neste canal is supplied with water corresponding to an objective discharge equal to 7 m3/s. The objective discharges and the weights of each gate are displayed in Table 4. The canal is subjected to strong withdrawals of 3 m3/s and 4.5 m3/s upstream to M1 (see Figure 9.a). This scenario was selected among several simulated scenarii, because it reveals accurately the inversion phenomenon of the dis- charge tendency. The HCA strategy goals consist in the allocation of the water resource amongst the gates according to their weights, and in the minimization of the discharge discrepancy at gate G4. The detection and control periods are selected as Te = 12 s and Tc = 120 s. In Figure 9, the simulation results are shown in continuous line when the transfer delay selection method is used, and in dashed line when the transfer delays are fixed to 100 s for the first OCRS, to 355 s for the second and to 320 s for the third. These transfer delay values are selected according to the operating point of each OCRS, i.e. 7 m3/s, 5 m3/s and 4 m3/s. They correspond to the third OM of each OCRS. 2 4 6 (a) Q M 1 [ m 3 /s ] E+ C E+ ~C E0 E− ~C E− C (b) 0 1 2 (c) q 1 [ m 3 /s ] 0.5 1 (d) q 2 [ m 3 /s ] 0 1 2 (e) q 3 [ m 3 /s ] 0 2 4 6 8 10 12 14 16 18 2 2.05 (f) q 4 [ m 3 /s ] Time[H] Figure 9: Discharge QM1 (continuous line) and objective discharge QM1ob j (dotted line) (a), resource states (b), setpoints assigned q j (continuous line when the delay selection method is used, dashed line in other case) and objective discharge q job j (dotted line) to G1 (c), G2 (d), G3 (e) and discharge at G4 ( f ). Hybrid Control Accommodation for Water-asset Management of Hydraulic Systems Subjected to Large Operating Conditions 249 Table 4: Gate objective discharges q job j, gate weights λ j and µ j, and minimum and maximum discharges q jmin and q jmax . Gate G1 G2 G3 G4 q job j [m3/s] 2 1 2 2 λ j 10 10 4 − µ j 10 4 10 − q jmin [m 3/s] 0.02 0.1 0.02 0.15 q jmax [m 3/s] 3.6 4.5 3.5 3 Table 5: Criteria computed when the transfer delay selection method is or not used. Criterion min(q4 [m3/s]) max(q4 [m3/s]) V [m3] With selection method 1.9778 2.0451 541 Without selection method 1.9763 2.0455 602 Figure 9 displays the discharges QM1 in (a), the corresponding discharge states in (b), the setpoint assigned at gate G1 in (c), G2 in (d), G3 in (e) and the resulting discharge at G4 in (f). The resource in lack measured on M1 is allocated on gates G1 and G3, as long as their setpoints are upper than their respectively minimum discharge characteristics q jmin , otherwise it is allocated on gate G2 (see Figures 9.c, 9.d and 9.e). Thus, the gates G1 and G3 are controlled between the 1st and 4th hours, and the gate G2 between the 4th and 7th hours. During all the simulation time, the discharge on G4 is close to the objective value of 2 m3/s (see Figure 9. f ). However, when the transfer delay selection method is not used, the setpoints are always assigned too early. Consequently at G4, the discharge is in excess when the water resource decreases on M1, and the discharge is in lack when the water resource increases on M1. The tendency of QM1 discharge is inverted at G4 the end of the canal reach. The use of the transfer delay selection method improves the performances of the HCA strategy and maintains the tendency. The maximum and minimum discharges reached at G4 and the water volume V which was not allocated, are displayed in Table 5. The maximum discharge discrepancy at G4 corre- sponds to 2.26 % of the objective discharge q4ob j when the transfer delay selection method is used and to 2.28 % in the other case. 6 Conclusion The HCA strategy is adapted to valorize the water resource by allocation and setpoint assignment of open-surface hydraulic system submitted to strong disturbances. A multimodelling method is proposed to identify by a determined number of linear models the open-surface hydraulic systems. Then, a transfer delay selection method is proposed to take into account variable transfer delays by the selection of the right setpoint assignment date according to the system dynamics and to the measured discharge. Finally, the HCA strategy performances are shown by simulation on the first reach of the Neste canal subjected to large operating conditions and to strong disturbances. The study of control stability proof of the HCA strategy is a future goal. 250 Eric Duviella, Pascale Chiron, Philippe Charbonnaud Bibliography [1] Chow, V. T., D. R. Maidment and L. W. Mays. Applied Hydrology. McGraw-Hill. New York, Paris, 1988. [2] Choy, S. and E. Weyer (2005). Reconfiguration schemes to mitigate faults in automated irrigation channels. CDC-ECC’05, 44 th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, 12-15 december, pp. 1875-1880, 2005. [3] Duviella, E., P. Charbonnaud, P. Chiron and F. Carrillo. Supervised internal multi-model control of a dam-gallery open-channel system. CDC-ECC’05, 44 th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, 12-15 december, pp. 1887-1892, 2005. [4] Duviella, E., P. Chiron, P. Charbonnaud and P. Hurand. Supervision and hybrid control accommoda- tion for water-asset management. Control Engineering Practice. Vol. 15, Issue 1, pp. 17-27, 2007. 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Escobet, P. Charbonnaud and E. Duviella. Identification and control of an open-flow canal using LPV models. CDC-ECC’05, 44 th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, 12-15 december, pp. 1893-1898, 2005. Eric Duviella Ecole des Mines de Douai Département Informatique et Automatique 941, rue Charles Bourseul, BP 10838, 59508 Douai Cedex, France Phone: 33.3.27.71.21.02 ; Fax : 33.3.27.71.29.80 E-mail: duviella@ensm-douai.fr Pascale Chiron Laboratoire Génie de Production Ecole Nationale d’Ingénieurs de Tarbes 47, Avenue d’Azereix, BP1629, 65016 Tarbes Cedex, France Phone: 33.5.62.44.27.69 ; Fax : 33.5.62.44.27.08 E-mail: pascale.chiron@enit.fr Philippe Charbonnaud Laboratoire Génie de Production Ecole Nationale d’Ingénieurs de Tarbes 47, Avenue d’Azereix, BP1629, 65016 Tarbes Cedex, France Hybrid Control Accommodation for Water-asset Management of Hydraulic Systems Subjected to Large Operating Conditions 251 Phone: 33.5.62.44.27.34 ; Fax : 33.5.62.44.27.08 E-mail: philippe.charbonnaud@enit.fr Received: January 29, 2007 Revised: May 22, 2007 Eric Duviella was born in France in 1978. In 2001, he re- ceived the Diploma in engineering from the National Engineering School, Tarbes, France. He received the Ph.D. degree from the In- stitut Polytechnique, Toulouse, France, in 2005. Since 2007, he has been an Assitant Professor in the Department Informatic and Automatic of the Ecole des Mines, Douai, France. He has pub- lished around 15 papers in journals or international conferences. His research interests include modelling, hybrid dynamical sys- tems, supervision, reactive control strategy. Pascale Chiron was born in France in 1961. She is assistant professor at the Ecole Nationale d’Ingénieurs de Tarbes, France since 2000. She has obtained his PhD in 1989 at Ecole Centrale in Nantes on ’matching and similarity criterion in medical imaging’. Now, she is involved in the theme ’Planning, Control, Supervision and Distributed Simulation’ in the ’Automated Production’ Group of Laboratoire de Génie de Production. Her domains of interest are modelling and simulation for system control. Philippe Charbonnaud was born in France in 1962. Since 2003, he is full Professor at the Ecole Nationale d’Ingénieurs de Tarbes. He is coordinator of the Spanish-French project GEPREDO deal- ing with the management of hydrographical networks and the pre- dictive management of the water resource. The main contribution lies in the development of a hybrid control method applied to dis- tributed systems. His main topic of interest concerns the real-time decision support systems, and more particularly supervision and control accommodation of distributed systems.