In this work, a novel distributed MPC algorithm, denoted D-SIORHC, is applied to upstream local control of a pilot water delivery canal. The D-SIORHC algorithm is based on MPC control agents that incorporate stability constraints and communicate only with their adjacent neighbors in order to achieve a coordinated action. Experimental results that show the effect of the parameters configuring the local controllers are presented

Cadete, Fernando

Igreja, José

Lemos, João

Rato, Luís

Rijo, Manuel

English

Repositório Científico da Universidade de Évora

10th International Conference on Hydroinformatics HIC 2012, Hamburg, GERMANY   EXPERIMENTAL APPLICATION TO A WATER DELIVERY CANAL OF A DISTRIBUTED MPC WITH STABILITY CONSTRAINTS   J. M. LEMOS (1), J. M. IGREJA (2), F. CADETE (3), M. RIJO (4), L. M. RATO (5) (1): INESC-ID, IST-Tech. Univ. of Lisbon, R. Alves Redol 9, 1000-029Lisboa, Portugal (2): INESC-ID, ISEL-Pol. Inst. Of Lisbon, Portugal  (3): INESC-ID, Portugal  (4): NuHCC, Univ. of Évora,Pólo da Mitra apartado 94, 7000-554 Évora,  Portugal  (5): Dep. Eng. Inform., Univ. Évora, R. Romão Ramalho 59, 7000-671 Évora, Portugal  In this work, a novel distributed MPC algorithm, denoted D-SIORHC, is applied to  upstream local control of a pilot water delivery canal. The D-SIORHC algorithm is based on MPC control agents that incorporate stability constraints and communicate only with their adjacent neighbors in order to achieve a coordinated action. Experimental results that show the effect of the parameters configuring the local controllers are presented.  INTRODUCTION   Model predictive control (MPC) is a tool for controller design that is generally recognized as a powerful mean to control various types of processes [1] and, in particular, water delivery canals. Indeed, since MPC is an optimization based design technique, it has features that are suitable for this type of application. In the first place, through an appropriate choice of the functional to optimize, MPC directly links the controller design with the objective of saving water. Furthermore, there are several ways to obtain decentralized versions of MPC, an important issue since water delivery canals are large, space distributed, systems. Some examples are provided in [2, 3, 4, 5].  In this article, a novel distributed MPC algorithm, denoted hereafter as D-SIORHC, is applied to local upstream water level control of a pilot water delivery canal. The D-SIORHC algorithm is based on local MPC control agents that communicate only with their neighbors in order to achieve a coordinated action. Each local control agent manipulates the canal gate to which it is associated such as to drive the corresponding local upstream level to the desired target value, but taking into account the control moves of their neighbors.  Local control agent MPC algorithms minimize a quadratic cost function using linear models, under the constraint that the plant state vanishes at a number of coincidence points locate in time close to the end of the prediction horizon [6]. Under appropriate conditions on the number of coincidence points, these constraints ensure stability of the local control loops, in case they are isolated [7].  When several local control agents are simultaneously connected to different plant inputs, they may yield competitive actions that destabilize the overall system. In order to prevent this undesirable effect, the actions of the different local control agents must be coordinated. For this sake, at the beginning of each sampling interval, a coordination procedure is run during which all control agents compute the value of the respective manipulated variables, using as accessible disturbances the previously computed value of the manipulated variable of their neighbors.  The contribution of this paper consists in the experimental illustration of  the effect of selecting different values for the parameters that configure the controller.  The paper is organized as follows. After this initial section that motivates the problem to consider and explains the approach followed in general terms, the water delivery canal used for experiments is described in section 2. The distributed MPC algorithm is described in section 3. The main results are presented in section 4, where the effect of different choices for the controller parameters  is shown experimentally. Finally, section 5 draws conclusions.  WATER DELIVERY CANAL DESCRIPTION  The water delivery canal used in the experiments is located in Évora, Portugal, and belongs to Núcleo de hidráulica e Controlo de Canais, a research unit of the University of Évora, Portugal [8]. Figure 1 shows some views of it.       Figure 1. Two of the gates of the pilot canal used in the experiments.  The structure of this experimental canal is shown schematically in Figure 2. The canal has four pools, each with a length of 35 m, separated by three undershoot gates (Figure 1 left), with the last pool ended by an overshot gate (Figure 1, right). The water is fed by gravity to the canal from a reservoir, with the inflow 0Q  imposed by a valve equipped with a flow controller of proportional type (denoted MONOVAR in Figure 2). The maximum value for 0Q  is 0.09 m3s-1. At the downstream section of each canal pool there is an offtake of the type orifice, used to simulate water consumption by farmers.  TankMonovar   valvePool 1 Pool 2 Pool 3Pool 4G1 G2G3G4M1 M2M3M4Q1 Q2 Q3 Q4Vo1 Vo2 Vo3 Vo4 y2 y3 y4y1Qo  Figure 2. A schematic view of the pilot canal used in the experiments.  Downstream of each pool there are water level sensors that provide measures iy , 4,,1i , ranging from 0 mm to 900 mm (canal bank). The opening of gate i  is denoted iu  and varies between 0 mm (gate closed) and 800 mm (gate fully open). In this work only the first three pools are used, and gate 4 is kept at a fixed position.  THE D-SIORHC ALGORITHM  The first step in designing a distributed controller consists in decomposing the plant in a number of subsystems and defining its interactions. The canal is decomposed in three subsystems, each one associated to a pool and a gate. To each subsystem, a local control agent (denoted controller 1, 2 and 3) computes the gate position using local information (pool water level measurement iy for controller i ) and information from their neighbor agents. So, the controller of pool 1 exchanges information about pool water level and gate position with pool 2, pool 2 does so with polls 1 and 3 and pool 3 with pool 2. It is remarked that the decomposition of the canal is purely conceptual and involves an approximation, while the decomposition of the control agents reflects the software structure that actually implements the controller.  The plant decomposition is implemented in practice by describing the overall system, when working around a fixed equilibrium point,  by a state-space model written as  )()()1( kBukaxkx                    (1)  )()( kCxky                      (2)  In (1, 2), k  is an integer number that denotes the discrete time, x  is the state of the overall plant,  Tuuuu 321  is the vector of manipulated variables,  Tyyyy 321 is the vector of observations and A , B and C  are matrices of parameters. To reflect the partitioning, matrices A  and B  are estimated imposing a priori the block structure  3332232221121100AAAAAAAA        332211000000BBBB                (3)  where ijA  is a matrix that reflects the influence of the state jx  on ix .  Each of the local controller agents computes its manipulated variable by minimizing a quadratic cost that takes into consideration not only the tracking error of the water level in the pool that it is controlling, but also the tracking error in adjacent pools. Hence, for pools 1, 2 and 3 the cost to be minimized are    NiikuikeikeJ121122211 )1()()(                 (4)   NiikuikeikeikeJ12222322212 )1()()()(               (5)   NiikuikeikeJ123123222 )1()()(                 (6)  where )()()( jyjrje iii  is the tracking error for the level in pool i  at time j , ir  being the reference to be tracked by pool level 1, the i are positive parameters that weight the penalty on the manipulated variable increments iu of  iu and N  is the prediction horizon. This minimization is done in a receding horizon sense under the stability constraint [6]  )()( jNkrjNky ii     for    Pj ,,1                 (7)  In a distributed framework, at the beginning of each sampling interval, a coordination recursive procedure is run a number of times that ensures convergence. In each cycle of the recursion, each iJ  is optimized with respect to iu , assuming knowledge of 1iu  and 1iu  (whenever defined) as computed in the previous iteration.  EXPERIMENTAL RESULTS  Several tests have been conducted in the pilot canal in order to study the effect of the parameters that configure the D-SIORHC controller. The results are reported in Figures 4 to 6. In all these experiments, after an initial period in which the setpoint of the level for reference 1 is made to vary, this setpoint is made constant and the valve off-take of pool 1 is open (at the instant corresponding to the vertical linea marked A) and the closed (at the 0 2000 4000 6000 8000 10000 12000500550600650700750800850Time [s]Water Level [mm]A By1y2y3 Figure 3. Experimental results with D-SIORHC. N=20, P=1, 10021   and 10003 .  0 2000 4000 6000 8000 10000 12000500550600650700750800850Time [s]Water Level [mm]A By1y2y3 Figure 4. Experimental results with D-SIORHC. N=25, P=1, 1021   and 10003 .     0 2000 4000 6000 8000 10000 12000500550600650700750800850Time [s]Water Level [mm]A By1y2y3 Figure 5. Experimental results with D-SIORHC. N=25, P=1, 10021   and 10003 .  0 2000 4000 6000 8000 10000 12000500550600650700750800850Time [s]Water Level [mm]A By1y2y3 Figure 6. Experimental results with D-SIORHC. N=25, P=1, 100021   and 10003 .  1  2  3  N  1D  [mm] 2D  [mm] 3D  [mm] 100 100 1000 20 10 8 6 10 10 1000 25 10 7 5 100 100 1000 25 9 7 6 1000 1000 1000 25 13 14 11  Table 1. Comparison of controller performances for different configurations.   instant corresponding to the vertical line marked B). The setpoints for the water levels of pools 2 and 3 are kept constant during the whose experiment. During all the experiments, gate 4 was kept at a constant position. Table 1 shows the mean deviation of the level with respect to the setpoint in pool i , denoted iD . This figure of merit is expressed in [mm] and is computed using all the data points for each experiment and hence it depends not only on the initial phase, in which the setpoint is varied, but also on the second phase, in which a disturbance is induced by opening the off-take of pool 1. The smaller iD , the better is the controller performance. Of-course the values of iD  would be different for other  sequences of manoeuvers.      In all the cases reported in this paper, the number of coincidence points was made P=1 and 10003 . In Figure 1, the prediction horizon is N=20. In the experiments reported in Figures 2, 3 and 4, the prediction horizon was N=25, greater than in Figure 1 and this led to an improvement in performance.  From Figure 2 to Figure 4, the values of  1  and 2  are selected with the increasing values of 10, 100 and 1000. As concluded from Table 1 and confirmed by the visual inspection of the figures, the better performance is obtained in this set of experiments for 10021   .   The SCADA system that interfaces the D-SIORHC controller to the canal imposes that a gate is moved with a dead band of 5 mm. This actuator nonlinearity causes a limitation in performance and, in particular, induces the oscillations observed in Figure 2, for 1021   . By observing Figure 3, it is concluded that the choice 10021   is less sensitive to these undesirable effects.   CONCLUSIONS  The effect of different values of the parameters that configure the distributed MPC D-SIORHC controller has been studied experimentally in a pilot water delivery canal.  With the two values of the prediction horizon considered, the largest yields the best result. This illustrates the advantage of using predictive control. It is remarked that, in other tests not reported in this paper, the performance degrades for N=30. A factor that contributes to this degradation is the loss of precision of predictive models when the predictive horizon grows, a fact observed in general in MPC algorithms. The number of coincidence points P is also an important variable. Increasing P with N=25 does not lead to better results, but other experiments (unreported here) show that the combination P=5 and N=25 leads to a better performance.  When selecting the weight in the penalty of the manipulated variables the best results are obtained with an intermediate value.   ACKNOWLEDGEMENT  This work was performed in the framework of project AQUANET – Decentralized and reconfigurable control for water delivery multipurpose canal systems, funded by FCT – Portugal under contract PTDC-EEA-CRO/102102/2008 and by INESC-ID PIDDAC pluri-annual funding.  REFERENCES  [1] Camacho E. F. and Bordons C., “Model predictive control”, 2nd ed., Springer, (2004).  [2] Georges D.., “Decentralized adaptive control for a water distribution system”, Proc. 3rd IEEE Conf. on Control Applications, where it took place, (1994), pp 1411-1416.   [3] Zhang Y. and Li S., “Networked model predictive control based on neighbourhood optimization for serially connected large-scale processes”, Journal of Process Control, Vol. 17, (2007), pp 37-50. [4] Negenborn R. R., Van Overloop P.-J., Kevicsky T. and De Schutter B., “Distributed model predictive control of irrigation canals”, Networks and Heterogeneous Media, Vol. 4, No. 2, (2009), pp. 359-380. [5] Maestre J M, Muñoz de la Peña D, Camacho E F and Alamo T. “Distributed model predictive control based on agent negotiation”. Journal of Process Control, Vol. 21, (2011),  pp. 685-697. [6] Mosca E., Lemos J. M. and Zhang J., “Stabilizing I/O receding horizon control”, Proc. 29th Conf. on Dec. and Control, Honolulu, Hawai, (1990), 2518-2523. [7] Mosca E., “Optimal, Predictive, and Adaptive Control”, Prentice Hall (1995). [8] http://canais.nuhcc.uevora.pt  

Experimental application to a water delivery canal of a distributed MPC with stability constraints

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Experimental application to a water delivery canal of a distributed MPC with stability constraints

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