Coalitional model predictive control of an irrigation canal

Coalitional model predictive control of an irrigation canal⋆ Filiberto Felea,∗, Jos´e M. Maestrea, Mehdi Hashemy Shahdanyb, David Mu˜noz de la Pe˜naa, Eduardo F. Camachoa aDepartment of Systems Engineering and Automation, University of Seville, 41092 Seville, Spain bDepartment of Water Management, Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands Abstract We present a hierarchical control scheme for large-scale systems whose components can exchange information through a data network. The main goal of the supervisory layer is to find the best compromise between control performance and communicational costs by actively modifying the network topology. The actions taken at the supervisory layer alter the control agents’ knowledge of the complete system, and the set of agents with which they can communicate. Each group of linked subsystems, or coalition, is independently controlled based on a decentralized model predictive control (MPC) scheme, managed at the bottom layer. Hard constraints on the inputs are imposed, while soft constraints on the states are considered to avoid feasibility issues. The performance of the proposed control scheme is validated on a model of the Dez irrigation canal, implemented on the accurate simulator for water systems SOBEK. Finally, the results are compared with those obtained using a centralized MPC controller. Keywords: Model predictive control, coalitional control, hierarchical control, irrigation canal 1. Introduction The progress made in data networks technology and the derived decreasing implementation costs — to which wireless networks have hugely contributed — have enabled the application of advanced control techniques in systems where the high cost of communication was a substantial obstacle. In particular, large scale systems related with public infrastructures are now fully within the scope of real-time control engineering, fostering the potential of a positive impact on the fundamental services provided countrywide [2]. ⋆©2014. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/. The final version can be found at https://doi.org/10.1016/j.jprocont.2014.02.005 (please cite as [1]). Financial support from the HYCON2 EU-project within the ICT-FP7, and MEC-Spain DPI2008-05818 and FPI grant is gratefully acknowledged. ∗Corresponding author Email address: [email protected] (Filiberto Fele) Preprint submitted to Elsevier October 7, 2023 This paper deals with water management in irrigation canals, a demanding task which entails finding the right trade-off among different sectors in direct competition (agricultural, municipal, and industrial). Since irrigated agriculture constitutes the largest consumer of freshwater resources, the modernization of canal operational management could drastically improve water conservation efficiency and supply flexibility. Moving in this direction, several advanced control strategies have been proposed over the last decades (see, e.g., the survey [3] and references therein). In [4], an optimal quadratic criteria is used to adjust the parameters of downstream level feedback controllers. Different classes of controllers are considered, ranging from PI controllers at each gate to a centralized controller. The improvement derived from the communication of control actions among neighboring pools is also investigated. In [5], the effectiveness of model predictive control in water systems is studied and compared to classical feedback and feedforward strategies. Among several challenging aspects regarding irrigation systems, geographical distance is one of the most interesting. Water networks are generally very disperse, and often different parts of the system are owned by independent entities, expectedly unwilling to coordinate their control actions unless strictly necessary. Moreover, permanent communication between the various parts of the network can be impractical. Consequently, the use of a traditional centralized control approach is hampered, even when the water network is owned and managed by a single entity. Considering all these factors, distributed control schemes can provide solutions able to satisfy the different actors involved. Thus, irrigation canals have become a popular benchmark to assess the performance of hierarchical and distributed control schemes. In [6], a thorough classification of these is given. A survey of centralized and distributed MPC schemes for water systems is provided in [7]. In this same work, the potential of the application of a distributed MPC scheme — based on an augmented Lagrangian formulation — is investigated with a simulation study on an irrigation canal. Also based on an augmented Lagrangian formulation, the work of [8] describes the decomposition of the receding-horizon optimal control problem for heterogeneous irrigation systems, aimed to reduce the computational complexity and to conform to the system topology. The performance objective considered in [8] accounts for the costs of water pumping and water losses, and the profits from power generation. A two-layer control scheme is proposed in [9], where the top layer follows a risk management strategy to cope with unexpected changes in the demand, failures or additional maintenance costs, and the bottom layer optimizes the values of water flows by means of a distributed MPC technique. A section of an irrigation canal located in Spain is considered as case study. The idea behind these solutions is to partition the centralized problem among a given number of local controllers or agents (see, e.g., [10]). Depending on the degree of dynamic interaction between the subsystems, the controllers are categorized in the literature either as distributed or as decentralized. In the first class the agents need to communicate to coordinate their operations [11,12,13,14]. By contrast, in the second class the limited degree of interaction allows the agents to tackle their control tasks with no need of communication [15,16]. Between these two classes lie the coalitional control schemes. In these, the control strategy is adapted to the varying coupling conditions between different parts of the system, promoting cooperation among the control agents most concerned at any given time. The formation of groups of cooperative agents based on the 2 active coupling constraints is considered in [17]. The work of [18] describes a hierarchical framework where information among the agents is exchanged at each time step within clusters of strongly dynamically coupled subsystems, while a slower communication rate is required between different clusters. In [19], the complexity of the model predictive control problem of the Barcelona drinking water network is reduced by means of a partitioning algorithm, in order to control in a hierarchical-distributed manner the resulting subnetworks. In [20] a flexible hierarchical MPC scheme is proposed for a hydro-power valley, where the priority of the agents in optimizing their control actions can be rearranged according to the different operational conditions. The present work focuses on how the interaction between the subsystems varies with time. Consider a large-scale system with an associated network, in which a number of control agents communicate in order to derive their knowledge of the overall system. Here, a two-layer hierarchical control strategy that manipulates the network topology with regard to both the current state of the system and the communicational cost is proposed. Those data links that do not yield a significant improvement of the control performance, compared with their relative cost of use, are disconnected. This feature is interesting, e.g., for communication infrastructures based on battery-powered wireless communication devices. Thus, any agent will be able to communicate only with those agents whose cooperation is most relevant, and the overall system will be partitioned into coalitions working in a decentralized fashion. The properties of a multi-agent control scheme based on this idea are discussed in [21], where the time-variant relevance of the communication within a set of dynamically coupled, unconstrained linear systems is analyzed using tools from cooperative game theory [22]. The research was extended towards input constrained systems in a preliminary version of this work [23], employing a model predictive control strategy [24,25] at the bottom layer. A local Luenberger observer was used within each coalition in order to estimate the dynamic influence caused by external subsystems. In this article, constraints on both states and inputs are considered, and local Kalman filters are used to estimate the dynamic couplings between different coalitions, viewed as perturbations. Information about any measurable disturbance can be now employed at the bottom layer. The proposed control scheme is validated on a detailed model of a 45 km section of the Dez irrigation canal, implemented on the SOBEK hydrodynamic simulator. For comparison, the results are shown along with those obtained using a centralized MPC controller. The paper is organized as follows. In §2, a formulation of the control problem is presented. The new distributed control algorithm is introduced in §3. The performance of the controller is finally validated in §4, employing a model of a section of the Dez irrigation canal as case study. 2. Problem formulation 2.1. System model On the basis of the work of [5], where the implementation of model-based control techniques on water systems has been examined, a discrete-time linear approximation of the dynamics of the irrigation canal, namely the ID model [26], is adopted here. According to this model, each reach is characterized as a 3 transport delay in series with an integrator.1A minimal order representation of the dynamics of the canal follows, which is suitable for the application to the control scheme presented in this article, where the computational burden is a major issue. The canal is partitioned into its essential components, i.e., pairs composed by a gate (the actuator) and its downstream reach, forming a set V={1, . . . , N} of subsystems. The dynamics of any subsystem i∈ V are described by the linear model: xi(k+ 1) = Aiixi(k) + Biiui(k) + Eipi(k) + Giwi(k) (1a) wi(k) = X j∈Ni Aijxj(k) + Bijuj(k) (1b) where xi∈Rniand ui∈Rmiare the state and input vectors respectively, pi∈ Rliis a measurable perturbation due to the offtake flow, and wi∈Rridescribes the influence on xiof the neighbors’ states and inputs. In (1b), xj∈Rnjand uj∈Rmjare the state and input vectors of each neighbor j∈ Niof subsystem i. The neighborhood set Niis defined as: Ni={j∈ V|Aij =0∨Bij =0, j =i}(2) i.e., it is the set composed by any subsystem j=iwhose state and/or input produce some effect on the dynamics of subsystem i. Remark 1. Even if model (1)is tailored to the case study, it is indeed a general formulation fitting a wide variety of large-scale systems. The state vector xi(k)≡[qi(k−1), . . . , qi(k−di), ei(k)]⊺ gathers information about the flow qialong the reach and the water level error eiwith respect to a desired value. Notice that an augmented representation is used in order to take into account the flow transport delay di. The input ui(k)≡∆qi(k) is the variation of the flow entering the reach i, controlled at its upstream gate. For each subsystem, the measure of the water level error ei(k) in the backwater section of the reach is available to its control agent; the rest of the state variables (water flow in different sections of the reach) are observable. 2.2. Exchange of information All the control agents can communicate through a data network whose topology is described by means of the undirected graph G= (V,Λ), where to each subsystem in Vis assigned a node. Let L ⊆ V × V be the set of edges corresponding to the existing communication links between the agents. Each link ℓij ={i, j}={j, i}=ℓji ∈ L can be either enabled or disabled. Then the network topology Λ(k)⊆ L is defined as the set of links enabled at a given time, i.e., ℓij ∈Λ(k) if and only if it is enabled at time k. Each active link has a cost 1A more detailed description of the model and the parameters of the canal is given in §4. 4 cℓ>0 per time of use. This cost can vary, e.g., as a function of the available bandwidth. For simplicity, a constant and unique value of cℓis considered in the remainder for all the links.2 Definition 1. Any two agents are said to be connected if and only if there exists a path between them in G= (V,Λ). Assumption 1. Any two agents can communicate if and only if they are connected. From Definition 1and Assumption 1it follows that any given network topology induces a partition of the whole agent set Vinto disjoint communication components [27]. As agents within the same communication component will benefit from cooperation — i.e., sharing information in order to aggregate their control tasks — we will refer to such components as coalitions, and the partition resulting by a given network topology Λ(k) will be denoted as V/Λ = {C1,C2,...,C|V/Λ|}, where |·|represents the cardinality of the set. To ease the notation, let us define the set of indices VΛ={1,...,|V/Λ|}. Any partition V/Λ originates a set of coalitions satisfying the following conditions [28]: (i) Ci=∅,∀i∈ VΛ; (ii) |V/Λ| S i=1 Ci=V; (iii) Ci∩ Cj=∅,∀i, j ∈ VΛ,i=j. The number of coalitions |V/Λ|produced by any topology Λ pertains to the interval [1, N], whose extremes represent the centralized control case (all the N subsystems connected3) and the case where each subsystem forms a coalition on its own (all links disabled). 2.3. Coalition dynamics Thus, to describe the dynamics of each coalition Ci∈ V/Λ, the following extension of (1a) holds: ξi(k+ 1) = Ξiiξi(k)+Υiiυi(k)+Φiρi(k)+Ψiωi(k) (3) where ξiand υiare respectively the state and input vectors of coalition Ci, composed by stacking the vectors of all the subsystems in the coalition: ξi={xs}s∈Ci, υi={us}s∈Ci, i ∈ VΛ Similarly, ρi={ps}s∈Ciis the vector grouping the offtake flows. As an extension of (1b), ωiexpresses the influence of the neighbor coalitions’ states and inputs on ξi: ωi(k) = X j∈NCi Ξijξj(k)+Υijυj(k) (4) 2The value of cℓis a given parameter in the described framework; also, it is assumed as constant during the prediction horizon. With these premises, the complexity of the topology selection problem is not related to differences in the costs of use of the links. On the other hand, the solutions of the problem will depend on the costs. 3Notice that, according to Assumption 1, this condition does not necessarily require all the links to be active. 5 where the set of neighbors NCiis an extension of (2), indexing any coalition Cj,j=iwhose state and/or inputs produce some effect on the dynamics of (any subsystem inside) coalition Ci. Matrices Ξii, Ξij, Υii, Υij, Φiand Ψiare composed accordingly. 2.4. Control objective In the application considered here, the control objective is to regulate the water level error of all the reaches to zero while minimizing a cost that depends on the state and input trajectories. An additional term in the cost function will take into account the use of network resources. In order to meet the objective, the offset caused by the offtake flows is canceled by steering each coalition’s state to a suitable setpoint ¯ ξi,¯υi. By imposing the steady state condition for all the coalitions, and setting the water level errors to zero, the following system of equations is obtained: I−Ξ−Υ Γ0 ¯ ξ ¯ υ=Φρ(k) 0(5) where ξ≡ {ξi}i∈VΛand υ≡ {υi}i∈VΛ, i.e., the global state and input vectors permuted according to the partition V/Λ. Similarly, Ξ∈Rn×nand Υ∈Rn×m are the permuted global state and input matrices. The offtake vector ρ≡ {ρi}i∈VΛand Φare composed likewise. The matrix Γ∈RN×nis determined such that Γξis the stacked vector of the water level errors of all the coalitions, i.e.: Γξ={es}s∈Ci,∀i∈ VΛ Denoting the shifted state and input of coalition Cias ζi=ξi−¯ ξiand νi=υi−¯υi, respectively, the cost function can be divided into a term Jsrepresenting the optimal performance objective, and a term Jnexpressing the network-related cost: Js,i = Np−1 X t=0 (ζ⊺ i(t|k)Qiζi(t|k) + ν⊺ i(t|k)Riνi(t|k)) + ζ⊺ i(Np|k)Piζi(Np|k) (6a) Jn,j =Np cℓ 2nℓ,j (Λ) (6b) where the notation α(t|k) corresponds to the value of αpredicted at time k+t, based on the knowledge at time k, and Qi≥0, Ri>0 and Pi=P⊺ i>0 are constant weighting matrices.4In (6b), nℓ,j(Λ) is the number of active links directly connecting agent jto other agents according to the network topology Λ. Note that each agent shares the cost of a link with the agent located at the other side of that link. The overall control problem can be posed as the following receding-horizon optimization: min υ,ΛX i∈VΛ Js,i(ξi(k), υi,Λ) + X j∈V Jn,j(Λ) (7) 4Matrix Qionly weights deviations of the water level. 6 s.t. ξi(t+ 1|k)=Ξiiξi(t|k)+Υiiυi(t|k)+Φiρi(k)+Ψiˆωi(k) ξi(t|k)∈ Xi,∀t∈[0, Np] υi(t|k)∈ Ui,∀t∈[0, Np−1] ξi(0|k) = ξi(k) Λ⊆ L Remark 2. In general, as the planning of the offtakes is made in advance, the perturbation ρi(k+t)can be known for the entire prediction horizon. In this work, however, forecasts are not considered, and a constant value of the offtake flow, corresponding to the current measure, is maintained along the horizon. Remark 3. Notice that (4)cannot be used directly, as coalition Cihas no knowledge of the states and inputs of external subsystems. An estimate ˆωiof the perturbation they cause on ξiis thus performed at each time step k, and its value is assumed constant along the prediction horizon. Details are given in §3.2. Problem (7) constitutes a dynamic programming optimization with mixed integer variables, which is generally not practical to solve. Notice that the integer variables do not explicitly appear in (7). Since any topology corresponds to a partition of the global system, the composition of the resulting coalitions’ state and input vectors and matrices will implicitly depend on Λ. The choice of the network topology is made within a discrete set whose size, in the general case, grows exponentially with the number of links. In the remainder, we formulate a hierarchical multi-agent control algorithm which provides a suboptimal, yet less computationally expensive solution. 3. The control algorithm The hierarchical multi-agent strategy proposed in this article is based upon an approximation of problem (7), with the intent of reducing the computational requirements when dealing with large-scale systems. The resulting control problem will provide a suboptimal solution to (7), and it will be split in two layers: a top layer that will take charge of the choice of the network mode, and a bottom layer which will handle the estimation and the real-time control tasks. A conceptual diagram of the proposed coalitional MPC strategy is shown in Figure 1. 3.1. Top layer The discrete part of (7), related to the choice of the optimal network topology, is separated and assigned to the top layer. As this constitutes the most computationally demanding part of the problem, its solution is computed on a coarser time scale (with respect to the sample time required for the control of the system). Several network topologies are compared at the top layer, and the most appropriate is selected for the following interval TΛ. Let L+= 7 Figure 1: Functional diagram of the coalitional MPC. {Λ1,Λ2,...,Λ|L+|}be the set of possible network topologies to be evaluated. Then, let us define the function J:Rn× L 7→ Ras follows [29]: J(ξ,Λ) = X i∈VΛ ζ⊺ iPiζi+cℓ|Λ|TΛ,Λ∈L+(9) where Pi=P⊺ i>0, |Λ|is the number of enabled links and cℓis the cost of use of one link, considered over the interval TΛ. It is not pragmatic to see L+as the set containing every possible configuration of links. Because the number of all possible topologies grows exponentially with the number of links, the set L+should be defined as a reasonably sized set of relevant topologies for the system to be controlled. The composition of L+could either be static or evolving depending, for example, on the current state of the system, on the network constraints, or on the willingness to cooperate among the agents. Of all the configurations considered at a given moment, the one giving the minimum value of (9), denoted as Λ∗∈L+, will be applied during the next interval TΛ. As a consequence to the choice of any given topology Λ, the set of agents is partitioned into a specific set of coalitions {C1,C2,...,C|V/Λ|}. To attain the optimal performance objective (6a), a feedback gain Kfor the whole system is computed at the top layer.5In conformity with the system partition V/Λ, K will be composed of a set of decentralized feedback gains, each one associated to a coalition, i.e., K=diag{K1, . . . , K|V/Λ|}. Let P>0 be the block matrix having {P1, . . . , P|V/Λ|}on its diagonal, and consider the Lyapunov function V(ξ) = ξ⊺Pξ, where ξ≡ {ξi}i∈VΛis the global state vector, permuted according to the partition V/Λ. For V(ξ) to constitute an upper bound on the infinitehorizon performance objective, the constraints of the following LMI problem have to be satisfied (see for example [30]): max K,PTrP−1(10) s.t. P=P⊺>0 (Ξ+ΥK)⊺P(Ξ+ΥK)−P≤ −Q−K⊺RK 5In the global coalition case, the feedback law Kwill coincide with the LQR gain. 8 where Ξand Υare respectively the global state and input matrices, composed to match ξand υ≡ {υi}i∈VΛ. Similarly, Q=diag{Q1, . . . , Q|V/Λ∗|} ≥ 0 and R=diag{R1, . . . , R|V/Λ∗|}>0 are the global weighting matrices. By the solution of (10), a set of feedback control laws υi=Kiξiwhich minimize V(ξ) and a set of matrices Pi,i∈ VΛis obtained. These matrices are then used to compute the value of (9) and find its minimizer Λ∗. Remark 4. Notice that the evaluation of different network topologies is independent and can be executed in parallel on a multi-processor platform. Also, the set of control laws associated with any network topology could be stored and reused whenever the same topology is considered again, without the need of solving more than once the relative LMI problem. 3.2. Bottom layer At the bottom layer, the control is decentralized into the coalitions arising from the partition V/Λ∗. With the term decentralized we designate the complete absence of communication among different coalitions; agents within a coalition share their information at each sample time k. As a consequence, the term ωi related with the water demand in neighboring coalitions cannot be computed through (4). Every coalition gets an estimate ˆωiby means of a local Kalman filter, based on the available measures of water level errors and current offtake flows. Given the slow dynamics of the system and the steady nature of the offtake flows, transient dynamics are neglected. Specifically, ˆωiis viewed as a constant integrating disturbance, included in the model defining a suitable augmented state vector. Remark 5. In general, inter-pool transient dynamics are not negligible. However, (i) the system examined in the case study is inherently stable, and (ii) the presence of local flow controllers at each gate is considered. If the response of these local control loops is sufficiently fast, the interaction between two adjacent pools reduces to a one-way perturbation [26]. Then, implementing a standard offset-free scheme for regulation [31], the agents steer their subsystems to an appropriate setpoint in order to compensate for both the estimated disturbance and the offset caused by the measurable offtake flows. Notice that any offset due to mismatches between the linear model and the actual system will be also included in ˆωi. The Kalman filter also serves as an observer for the values of water flows. For any coalition Ci, the setpoint {¯ ξi,¯υi}is obtained by the solution of the linear system: I−Ξii −Υii Γi0 ¯ ξi ¯υi=Φiρi(k)+Ψiˆωi(k) 0(11) where Γiis an output matrix defined such that Γiξi={es}s∈Ci, i.e., the vector composed by the water level deviations of all the subsystems in coalition Ci. Note that, because of the estimation of ˆωi, the setpoint computed through (11) is expected to change at each time step. Moreover, when the topology is changed, the Kalman filters structure changes as well, according to the composition of the new set of coalitions. To avoid undesired drifts on the computed value of the setpoint, a good initial guess of the state and the covariance matrix is 9 The plots in Figure 4refer to Scenario 1. In the upper plot, the evolution of water level errors is shown. It can be seen how the sudden decrease in the offtake flows at time k= 72 causes the water levels to rise above the desired setpoint, reaching in some pools a peak of 0.5 m. In the bottom plot of Figure 4, blue lines indicate active links. The agents in charge of the first four upstream reaches act jointly when the disturbance occurs. This coordination allow to minimize the perturbation on reach 3 (and consequently on reach 2 and 1) due to the recovery maneuver of the agent in charge of reach 4. During the transient response of the local flow controllers, the sudden variation of the offtake flows produces an immediate perturbation in downstream direction. Therefore, in order to improve the performance of the overall system, the agents are organized into bigger coalitions while the effect of the offtakes’ variation is perceived. As the disturbance is rejected, the data links are disabled and most of the agents continue to control their reaches in a decentralized way. It can be seen that a coalition among upstream agents is profitable and thus does not eventually disappear. One reason to this might be the fact that, in the system under study, an accurate decentralized estimation of neighbor’s state is not possible, and water levels at upstream reaches tend to deviate from the setpoint as soon as their agents stop to communicate. This may suggest that in an optimal partition of the overall system the upstream agents would be bound into the same coalition. Figure 5shows the inflows qi(k) to the reaches. Starting from time k= 72, the flows are reduced in response to the decreased offtakes in order to bring the water levels back to their setpoints. The results relative to Scenario 2 are shown in Figure 6. Before the offtakes variation at time k= 72, the network topology changes to a decentralized configuration while the water levels are kept at their setpoints. When the decrease in the offtakes occurs, the system reacts by reducing the inflows (Fig. 7), while data links are enabled in order to coordinate the operations along the canal. At time k= 144 the offtake flows are restored, which is matched with an increase in the input flows. As the error in the water levels is attenuated, downstream data links are disabled. Notice in the last part of the simulation the persistence of the coalition among the upstream agents, and also the coalition formed by agents 5 and 6, since the water level in reach 5 does not converge to its setpoint. It can be seen in Figures 5and 7that the constraints on the water flows are satisfied in both scenarios. For comparison, the performance of a centralized MPC controller in the same scenarios is shown in Figures 8and 9. The centralized control law is computed as in (14), using the same parameters of Table 2(except for the cost of active links, which is set to zero). The centralized controller can coordinate the response of the entire canal to provide a faster reaction to the disturbance, which yields about 30% reduction in the level error peaks.7As expected, due to the absence of network topology switching with centralized control, a smoother response of the system is obtained. Notice that, even without taking into account any offtake forecast, the results achieved with the proposed distributed control scheme are within the admitted range of canal operation. Table 3displays a comparison 7An exception to this is the water level in reach 13, as no further downstream reaches can be employed to improve its response (the water discharge at the downstream gate in reach 13 cannot be manipulated by the controller). 16 Coal. (·103) Centr. (·103) Scn. 1 cℓ= 0 5.62 2.06 cℓ= 0.6 10.44 14.06 Scn. 2 cℓ= 0 9.36 3.89 cℓ= 0.6 14.80 15.89 Table 3: Comparison of the average costs of the proposed coalitional scheme and centralized MPC for the two scenarios. n◦dec. var. n◦coal. Scn. 1 5.6 8.2 Scn. 2 6.3 7.6 Centr. MPC 39 1 Table 4: Average number of decision variables for the decentralized MPC controllers at the bottom layer, relative to both scenarios. of the average performance costs of the proposed coalitional scheme and the centralized MPC, for both scenarios. In Figure 10 the accumulated costs relative to Scenario 1 are represented, evidencing the impact of communication-related costs in a centralized framework. The average number of decision variables for the decentralized MPC problems solved by the coalitions at the bottom layer is shown in Table 4, along with the average number of coalitions. 5. Conclusions We have presented a hierarchical control scheme aimed to large-scale networked systems, that dynamically adapts the information flow between agents as a function of the state of the system and the cost of use of the network resources. The proposed scheme promotes cooperation between different agents whenever it results in a sensible performance improvement. The controller have been tested within a case study of the control of water levels in the Dez irrigation canal, using the accurate hydrodynamic simulation environment SOBEK. Although suboptimal w.r.t. a centralized MPC control, the performance of the proposed strategy lies within the admitted range of operation of the canal. On the other hand, the need of constant exchange of information through the entire system is obviated. Also, a sensible reduction of the computational requirements can be attained in high-dimensional systems. Regarding the network topology optimization, (i) the evaluation of different topologies is independent and can be executed in parallel on a multi-processor platform, and (ii) the control laws resulting from the evaluation of any network topology could be stored and reused. Another advantage of the proposed control strategy over a centralized controller is the possibility to explicitly manage any link failure by switching to a different network configuration, improving the robustness to network-related issues. Given the availability of a communication infrastructure (e.g., point-to-point wireless network) across the canal, both layers of the proposed scheme can be distributed among the local control agents (Fig. 11). The network topology optimization task can be shared, e.g., performing the distributed synthesis of the global feedback law with the method presented in [40,41]. Alternatively, 17 0 50 100 150 200 250 −0.2 0 0.2 0.4 0.6 Water level errors [m] Time steps Reach 1 Reach 2 Reach 3 Reach 4 Reach 5 Reach 6 Reach 7 Reach 8 Reach 9 Reach 10 Reach 11 Reach 12 Reach 13 0 50 100 150 200 250 1 2 3 4 5 6 7 8 9 10 11 12 13 Time steps Agents Figure 4: Scenario 1: At k= 72 reaches 4, 9, 10 and 13 undergo a step decrease in the offtake flows. The upper plot shows how the controller regulates the water level errors along the canal. Each blue line in the bottom plot represents an active data link between two control agents. Starting from a centralized configuration, links are deactivated one at a time until the variation in the offtakes is sensed. Then links are enabled to form coalitions among the most concerned control agents, until the disturbance is eventually rejected. 0 50 100 150 200 250 0 20 40 60 80 100 Input Flows [m3/s] Time steps Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Gate 9 Gate 10 Gate 11 Gate 12 Gate 13 Figure 5: Scenario 1: Water flows in input to each reach. 18 0 50 100 150 200 250 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Water level errors [m] Time steps Reach 1 Reach 2 Reach 3 Reach 4 Reach 5 Reach 6 Reach 7 Reach 8 Reach 9 Reach 10 Reach 11 Reach 12 Reach 13 0 50 100 150 200 250 1 2 3 4 5 6 7 8 9 10 11 12 13 Time steps Agents Figure 6: Scenario 2: The offtake flows in reaches 4, 9, 10 and 13 undergo a step decrease at k= 72, and are restored at k= 144. The upper plot shows the water level errors in all the reaches. The use of data links between the control agents is represented by the blue lines in the bottom plot. With the water levels at their setpoints, the network changes toward a decentralized topology. In reaction to the offtake change, the control agents are organized into coalitions to improve the overall response. 0 50 100 150 200 250 0 20 40 60 80 100 120 140 Input Flows [m3/s] Time steps Gate 1 Gate 2 Gate 3 Gate 4 Gate 5 Gate 6 Gate 7 Gate 8 Gate 9 Gate 10 Gate 11 Gate 12 Gate 13 Figure 7: Scenario 2: Water flows in input to each reach. 19 0 50 100 150 200 250 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Water level errors [m] Time steps Reach 1 Reach 2 Reach 3 Reach 4 Reach 5 Reach 6 Reach 7 Reach 8 Reach 9 Reach 10 Reach 11 Reach 12 Reach 13 Figure 8: Scenario 1: Water level errors with centralized MPC controller. 0 50 100 150 200 250 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 Water level errors [m] Time steps Reach 1 Reach 2 Reach 3 Reach 4 Reach 5 Reach 6 Reach 7 Reach 8 Reach 9 Reach 10 Reach 11 Reach 12 Reach 13 Figure 9: Scenario 2: Water level errors with centralized MPC controller. 0 50 100 150 200 250 102 103 104 105 106 107 CC MPC CC w/ net MPC w/ net Figure 10: Scenario 1: Accumulated costs comparison between the proposed coalitional control strategy and a centralized MPC, w/ and w/o network costs. 20 Figure 11: Scheme of a possible implementation of the proposed control strategy. one of the agents can assume the role of supervisor, and carry out the topology optimization as well as its bottom layer control tasks. Both solutions will require data exchange among all the agents before each topology switch. Within members of a given coalition, a higher rate of communication is required for the decentralized MPC at the bottom layer. Future work shall focus on how the switching network topology influence the global state estimation. In the proposed strategy, deviations of the water levels w.r.t. to their initial values have been penalized through the cost functions. Constraints on the water level have not been considered to avoid feasibility issues (mainly caused by the drifts in the state estimate occurring at each topology switch). Moreover, due to the variance in the estimate, a robust design of the controller and the observer is in order for the correct handling of state constraints. For these same reasons, soft constraints have been chosen for the water flows. 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