Partitioning for Large-scale Systems: A Sequential Distributed MPC Design *

Proceedings of the 20th World Congress Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of 20th Congress Proceedings of the the 20th World World Congress The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Available online at www.sciencedirect.com The International Federation of Control The International Federation of Automatic Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017

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Partitioning for Large-scale Systems: A Partitioning for Large-scale Systems: A  Partitioning for Large-scale Systems: A Sequential Distributed MPC Design  Sequential Distributed MPC Design  Sequential Distributed MPC Design

J. Barreiro-Gomez ∗,∗∗ C. Ocampo-Martinez ∗ N. Quijano ∗∗ ∗,∗∗ ∗,∗∗ C. Ocampo-Martinez ∗ ∗ ∗∗ J. Barreiro-Gomez Quijano ∗∗ ∗,∗∗ C. Ocampo-Martinez ∗ N. J. Barreiro-Gomez J. Barreiro-Gomez C. Ocampo-Martinez N. N. Quijano Quijano ∗∗ ∗ Automatic Control Department, Universitat Polit`ecnica de ∗ ∗ Automatic Control Department, Universitat Polit` ecnica de ∗ Automatic Control Department, Universitat Polit` cnica Catalunya, Institut de Rob` otica i Inform` atica Industrial Automatic Control Department, Universitat Polit`ee(CSIC-UPC), cnica de de Catalunya, Institut de Rob` o tica i Inform` a tica Industrial (CSIC-UPC), Catalunya, Rob` o tica ii Inform` tica Industrial (CSIC-UPC), Llorens de i Artigas, 4-6, 08028a Barcelona, Spain Catalunya, Institut Institut de Rob` o tica Inform` a tica Industrial (CSIC-UPC), Llorens Artigas, 4-6,cocampo}@ 08028 Barcelona, Barcelona, Spain Llorens iii Artigas, 4-6, 08028 Spain (e-mail: {jbarreiro, iri.upc.edu) Llorens Artigas, 4-6,cocampo}@ 08028 Barcelona, Spain (e-mail: {jbarreiro, iri.upc.edu) ∗∗ (e-mail: {jbarreiro, cocampo}@ iri.upc.edu) Departamento de Ingenier´ ıa El´ e ctrica y Electr´ o nica, Universidad de (e-mail: {jbarreiro, cocampo}@ iri.upc.edu) ∗∗ ∗∗ Departamento de Ingenier´ ıa El´ eectrica yy Electr´ o nica, Universidad de ∗∗ Departamento de Ingenier´ ıa El´ ctrica Electr´ o nica, Universidad los Andes, Carrera 1 No 18A-10, Bogot´ a , Colombia Departamento de Ingenier´ ıa El´ectrica y Electr´ oanica, Universidad de de los Andes, Carrera 1 No 18A-10, Bogot´ , Colombia los Andes, Carrera 1 No 18A-10, Bogot´ a ,, Colombia (e-mail: {j.barreiro135, nquijano}@ uniandes.edu.co) los Andes, Carrera 1 No 18A-10, Bogot´ a Colombia (e-mail: {j.barreiro135, {j.barreiro135, nquijano}@ uniandes.edu.co) uniandes.edu.co) (e-mail: (e-mail: {j.barreiro135, nquijano}@ nquijano}@ uniandes.edu.co) Abstract: Large-scale systems involve a high number of variables making challenging the design Abstract: Large-scale systems involve aa high number variables making challenging design Abstract: Large-scale systems involve of variables challenging the design of controllers because of information andof burden issues. the Normally, Abstract: Large-scale systems involveavailability a high high number number ofcomputational variables making making challenging the design of controllers because of information availability and computational burden issues. Normally, of controllers because of information availability and computational burden issues. Normally, the measurement of all the states in a large-scale system implies to have a big communication of controllers because of information availability and computational burden issues. Normally, the measurement of allbethe states in a large-scale system implies have a big communication the measurement of states in system implies to have communication network, which might quite expensive. On the other hand, the to treatment of large amount of the measurement of all allbethe the states in a a large-scale large-scale system implies to have aa big big communication network, which might quite expensive. On the other hand, the treatment of large amount network, which might might be quite quite expensive. expensive. On the the otherhigh hand, the treatment treatment of large large amount of of data to compute the appropriate control inputs implies computational costs. An alternative network, which be On other hand, the of amount of data to compute the appropriate control inputs implies high computational costs. An alternative data to compute the appropriate control inputs implies high computational costs. An alternative to mitigate the aforementioned issues isinputs to split the problem into severalcosts. sub-systems. Thus, data to compute the appropriate control implies high computational An alternative to mitigate thetasks aforementioned is to split the problem several sub-systems. Thus, to mitigate aforementioned issues is the into several sub-systems. Thus, computational may be splitissues and assigned to different local into controllers, letting to reduce the to mitigate the thetasks aforementioned issues is to to split split the problem problem into several letting sub-systems. Thus, computational may be split and assigned to different local controllers, to reduce the computational tasks may be split and assigned to different local controllers, letting to reduce the required time to compute the control inputs. Additionally, the partitioning of the system allows computational tasks may be split and inputs. assigned to different local controllers, letting to reduce the required time the partitioning of allows requireddesigners time to to compute compute the control control inputs. Additionally, Additionally, the partitioning of the the asystem system allows control to simplify the communication network.the This paper presents partitioning required time to compute the control inputs. Additionally, the partitioning of the system allows control designers to simplify the communication network. This paper partitioning control designers to the network. paper presents partitioning algorithm performed by considering an information-sharing graph thatpresents can be aaagenerated for control designers to simplify simplify the communication communication network. This This paper presents partitioning algorithm performed by considering an information-sharing graph that can be generated for algorithm performed by considering an information-sharing graph that can be generated for any control strategy and for any dynamical large-scale system. Finally, a distributed model algorithm performed by considering an information-sharing graph that can be generated for any control strategy and for any dynamical large-scale system. Finally, a distributed model any control strategy and for any dynamical large-scale system. Finally, a distributed model predictive control (DMPC) is designed for a large-scale system as an application of the proposed any control strategy and for any dynamical large-scale system. Finally, a distributed model predictive control control (DMPC) is is designed for for a large-scale system system as an an application of of the proposed proposed predictive (DMPC) partitioning method. predictive control (DMPC) is designed designed for aa large-scale large-scale system as as an application application of the the proposed partitioning method. partitioning method. partitioning © 2017, IFACmethod. (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Partitioning, large-scale systems, distributed model predictive control Keywords: Partitioning, Partitioning, large-scale systems, systems, distributed model model predictive control control Keywords: Keywords: Partitioning, large-scale large-scale systems, distributed distributed model predictive predictive control 1. INTRODUCTION types of systems, the problem of the thermal control for 1. INTRODUCTION INTRODUCTION types of systems, the problem the thermal control for 1. types of the problem of thermal control for buildings is studied applyingof partition clusters 1. INTRODUCTION types of systems, systems, theby problem of aathe the thermalinto control for buildings is studied by applying partition into clusters buildings is studied by applying a partition into clusters for decentralized control design ina Chandan and clusters Alleyne Large-scale systems are commonly associated to the con- buildings is studied by applying partition into for decentralized control design in Chandan and Alleyne Large-scale systems are commonly commonly associated to the the confor control in and Alleyne In Kleinberg et al.design (2014), partitioning Large-scale systems are to trol of a large number of states associated by manipulating also (2013). for decentralized decentralized control design in aaChandan Chandan and method Alleyne Large-scale systems are commonly associated to the conconIn Kleinberg et al. (2014), partitioning method trol of of aacontrol large number of this states by manipulating manipulating also (2013). (2013). In Kleinberg et al. (2014), a partitioning method is proposed based on capacitor reactive power domains for trol large number of states by also several inputs. In regard, controllers must (2013). In Kleinberg et al. (2014), a partitioning method trol of a large number of states by manipulating also is proposed based on capacitor reactive power domains for several tons control inputs. Inprocess this regard, regard, controllers must is proposed based on capacitor reactive power domains for the control of electric power distribution systems, and in several control inputs. In this controllers must merge of data and them in order to comis proposed based on capacitor reactive power domains for several control inputs. Inprocess this regard, controllers must the control of electric power distribution systems, and in merge tons of data and them in order to comthe control of electric power distribution systems, and in Nayeripour et al. (2016), power distribution networks are merge tons of data and process them in order to compute the appropriate control inputs and obtain a desired the control of electric power distribution systems, and in merge tons of data and process them inobtain order atodesired com- Nayeripour et al. (2016), power distribution networks are pute the appropriate control inputs and Nayeripour et al. (2016), power distribution networks are split into different areas by using a systematic approach pute the appropriate control inputs and obtain a desired performance for the closed-loop system. Another relevant Nayeripour et al. (2016), power distribution networks are pute the appropriate control inputs and obtain a desired into different areas by using aa systematic approach performance forlarge-scale the closed-loop closed-loop system. Anotherextended relevant split split into different areas by approach order control the voltage profile. On the other hand, performance for the system. Another relevant aspect is that systems are usually split intoto different areas by using using a systematic systematic approach performance forlarge-scale the closed-loop system. Anotherextended relevant in in order to control the voltage profile. On the other hand, aspect is that systems are usually in order to control the voltage profile. On the other a method to find an optimal decomposition structure of aspect is that large-scale systems are usually extended geographically throughout big areas, for which long comin order toto control theoptimal voltage decomposition profile. On the structure other hand, hand, aspect is that throughout large-scale big systems are usually extended a method find an of geographically areas, for which long coma method to find an optimal decomposition structure of distributed predictive controllers is presented in Xie et al. geographically throughout big areas, for which long communication links to transport measurements and control a method to find an optimal decomposition structure of geographically throughout big measurements areas, for whichand long com- distributed predictive controllers is presented in Xie et al. munication links to transport control distributed predictive controllers is presented in Xie et al. (2016) by using genetic algorithms. Moreover, the parmunication links to transport measurements and control signals are required, which might cause additional commudistributed predictive controllers is presented in Xie et al. munication links to transport measurements and control (2016) by using genetic algorithms. Moreover, theetparsignals are are required, which might might cause causeThe additional commu(2016) by using algorithms. Moreover, presented in Ocampo-Martinez al. signals required, which additional communication issues and economical partitioning of titioning (2016) byapproach using genetic genetic algorithms. Moreover, the theetparparsignals are required, which mightcosts. causeThe additional commutitioning approach presented in Ocampo-Martinez al. nication issues and economical costs. partitioning of titioning approach presented in Ocampo-Martinez et al. (2011) is devoted for the design of decentralized predictive nication issues and economical costs. The partitioning of large-scale systems appears as a plausible solution in order titioning approach presented in Ocampo-Martinez et al. nication issues and appears economical costs. Thesolution partitioning of (2011) is devoted for the design of decentralized predictive large-scale systems as a plausible in order (2011) is devoted for the design of decentralized predictive controllers. However, partitioning methodology is quite large-scale systems appears as a plausible solution in order to reduce the complexity of the control design, the costs (2011) is devoted for the design of decentralized predictive large-scale systems appearsofasthe a plausible solutionthe in order the partitioning quite to reduce reduce the the complexity control design, costs controllers. However, the methodology is general, i.e.,However, the partitioning can bemethodology implementedis prior to complexity of the design, thereduce costs controllers. associated to the communication issues and also to controllers. However, the partitioning partitioning methodology is quite quite to reduce the complexity of the control control design, the costs general, i.e., the partitioning can be implemented prior associated to the communication issues and also to reduce general, i.e., the partitioning can be implemented prior to defining the control strategy to apply. Other general associated to the communication issues and also to reduce the computational complexity. However, the partitioning i.e.,the thecontrol partitioning cantobe implemented prior associated to the communication issues and also to reduce general, to defining strategy apply. Other general the computational complexity. However, the partitioning to defining the control strategy to apply. Other general partitioning methodologies are presented in Ezhilarasi and the computational complexity. However, the partitioning task is quite challenging due to the existing dynamical couto defining the control strategy to apply. Other general the computational complexity. However, the partitioning partitioning methodologies are presented presented in Ezhilarasi Ezhilarasi and task is isamong quite challenging challenging due tothe thesystem, existingimposed dynamical cou- partitioning methodologies are in and Swarup (2012), where a harmony search algorithm is used, task quite due to the existing dynamical coupling elements within coupled partitioning methodologies are presented in Ezhilarasi and task is quite challenging due to the existing dynamical couSwarup (2012), where aaSalahshoor harmony search algorithm is used, pling among elements within the system, imposed coupled Swarup (2012), where harmony search algorithm is used, and in Kamelian and (2015), where the parpling among elements within the system, imposed coupled constraints, and objective-achievement warranties, among Swarup (2012), where a harmony search algorithm is used, pling among elements within the system, imposed coupled in Kamelian and Salahshoor (2015), where the parconstraints, and objective-achievement warranties, among and and in and Salahshoor (2015), the is obtained merging different sub-systems of constraints, objective-achievement among others. The and partitioning problem has warranties, gotten increasing and in Kamelian Kamelian andby Salahshoor (2015), where where the parparconstraints, and objective-achievement warranties, among titioning titioning is obtained by merging different sub-systems of others. The partitioning problem has gotten increasing titioning is obtained by merging different sub-systems of an initial grouping. There is still an significant interest others. The partitioning problem has gotten increasing importance in the automatic control community as systitioning is obtained by merging different sub-systems of others. The inpartitioning problem hascommunity gotten increasing an initial grouping. There is still an significant interest importance the automatic control as sysan initial grouping. There is still an significant interest in the development ofThere general partitioning procedures and importance the automatic as systems becomein challenging, andcontrol as thecommunity requirements and an initial grouping. is still an significant interest importance in the automatic control community as systhe development of general partitioning procedures and tems become become challenging, and become as the the requirements and in in the tems challenging, and as and desired closed-loop performance more strict. Many in the development development of of general general partitioning partitioning procedures procedures and and tems become challenging, and become as the requirements requirements and methodologies. methodologies. desired closed-loop performance more strict. Many methodologies. desired closed-loop performance become more strict. Many partitioning proposals focus on specific dynamical systems, methodologies. desired closed-loop performance become more strict. Many The main contribution of this paper is a partitioning partitioning proposals focusstrategy. on specific specific dynamicalparticular systems, The main contribution of this paper is a partitioning partitioning proposals focus on dynamical systems, or on a particular control Regarding partitioning proposals focusstrategy. on specific dynamicalparticular systems, approach The main contribution this paper a non-directedof information or on on aa particular particular control Regarding The mainof contribution ofgraph this representing paper is is aa partitioning partitioning or control strategy. Regarding particular approach of aa non-directed graph representing information orThis on awork particular particular  approach of non-directed graph representing information sharing inspired by the Kernighan-Lin algorithm (Gupta, has been control partiallystrategy. supportedRegarding by the project DEOCS approach of a non-directed graph representing information  sharing inspired by the Kernighan-Lin algorithm (Gupta, This work has been partially supported by the project DEOCS  sharing inspired by the Kernighan-Lin algorithm (Gupta, 1997), considering four different minimization objectives. (Ref. DPI2016-76493-C3-3-R). J. Barreiro-Gomez is partially supThis work has been partially supported by the project DEOCS  sharing inspired by the Kernighan-Lin algorithm (Gupta, ThisDPI2016-76493-C3-3-R). work has been partiallyJ.supported by the is project DEOCS 1997), considering four different minimization objectives. (Ref. Barreiro-Gomez partially supported by Colciencias and AGAUR. 1997), considering four different minimization objectives. (Ref. DPI2016-76493-C3-3-R). J. Barreiro-Gomez is partially sup1997), considering four different minimization objectives. (Ref. DPI2016-76493-C3-3-R). J. Barreiro-Gomez is partially supported by Colciencias and AGAUR.

ported ported by by Colciencias Colciencias and and AGAUR. AGAUR. Copyright © 2017 IFAC 9172Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 9172 Copyright © 2017 IFAC 9172 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 9172Control. 10.1016/j.ifacol.2017.08.1539

Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 J. Barreiro-Gomez et al. / IFAC PapersOnLine 50-1 (2017) 8838–8843

Most of the partitioning methods are performed based on a graph representation, i.e., algorithms consider graphs associated to the dynamics of the system. Differently, this paper proposes to generate a graph that describes the information dependence among variables taken into account in the control design. In this regard, the proposed methodology is general since the information-sharing graph may be generated for any control strategy and for any dynamical system. As an application to illustrate the advantages of the partitioning approach addressed by using an information representation instead of a dynamical-model representation, a large-scale water supply system is considered, and a model predictive control is designed. To this end, the information graph is computed in order to determine an appropriate partitioning by using the proposed algorithm. Notation: All column vectors are denoted by bold style and lower case, e.g., x. Matrices are denoted by bold upper case, e.g., A. In contrast, scalars are denoted by non-bold style, e.g., n. The sets are denoted by calligraphic upper case, e.g., A. The norm ||x|| of the vector x ∈ Rnx is √  defined as ||x|| = x x, and the cardinality of a set A is denoted by |A|. The vector 1 is the column vector with unitary entries and suitable dimensions, i.e., 1 = [1 . . . 1] , and the vector of null entries and suitable dimensions is denoted by 0. Real numbers are denoted by R, all the non-negative numbers are denoted by R≥0 , and all the strictly positive real numbers are denoted by R>0 . Similarly, the integer numbers, non-negative integer numbers, and the strictly positive integer numbers are denoted by Z, Z≥0 , and Z>0 , respectively. Throughout this document, both continuous and discrete time are treated. Therefore, k ∈ Z≥0 denotes the discrete time, whereas t is used to denote the continuous time. Regarding the discrete time notation for the MPC controller, xk+j|k denotes the prediction made at time k of the vector x for time k + j, where k, j ∈ Z≥0 , i.e., in the argument k + j|k, the first element k+j indicates discrete time for prediction, whereas the second element k indicates the current discrete time.

constant matrix representing all the relevance factors, i.e., cij = c(i, j). On the other hand, let K = {1, . . . , m} be the set of indices for the m ∈ Z>0 partitions of the graph G. The partitioning at time instant k is represented by the set Pk = {Vk :  ∈ K}, i.e., each partition of G at time instant k is an undirected connected  graph of the form Gk = (Vk , Ek ), for all  ∈ K, where ∈K Vk = ∅, and   ∈K Vk = V, for all k. Given a partition Pk , consider the function g : V → K that receives a node i ∈ V and returns the index  ∈ K that allows to identify to which partition the node i ∈ V belongs to, i.e., g(i) = { ∈ K : i ∈ Vk }. Finally, let Vk ∈ Rn×n be the time-varying matrix whose g(i) g(j) element vij,k = |Vk | − |Vk | if g(i) = g(j), and nodes i, and j are neighbors, i.e., j ∈ Ni . Remark 1. If Pk is an admissible partition, then the set { ∈ K : i ∈ Vk } is singleton, for all  ∈ K. ♦ 3. PROBLEM STATEMENT The m−partitioning problem consists in finding the optimal set of partitions denoted by P  such that the following objectives are minimized: (1) Links 1: minimize the amount of links connecting different partitions given by f1 , i.e., links (i, j) ∈ E such / Ek , for all  ∈ K, f1 (Pk ) =  that  (i, j) ∈ 1 ∈K i∈Vk j∈V\Vk aij . 2 (2) Nodes 2: minimize the difference   between the amount of nodes in the partitions Vk , for all  ∈ K, and the n average of total  nodes  n of the graph G given by m ,  1     i.e., m ∈K Vk − m . Notice that this objective may be expressed as the minimization of a function f2 depending matrix Vk , i.e., the time-varying   on f2 (Pk ) = ∈K i∈V  j∈V\V  vij,k . k k (3) Distance: minimize the distance among the nodes belonging  to the same partition, i.e., 1 ∈K i∈Vk j∈Vk \{i} dij . This objective may be 2 expressed conveniently considering the inverse distance of links that connecting  are   different partitions, i.e., f3 (Pk ) = 12 ∈K i∈V  j∈V\V  d−1 ij . k k (4) Relevance: minimize the information relevance connecting different partitions,   i.e.,  f4 (Pk ) = ∈K i∈V  j∈V\V  cij .

2. PRELIMINARIES Consider an undirected connected graph denoted by G = (V, E) representing the topology of an informationsharing network, which is determined depending on the information that a control strategy requires to compute the appropriate control inputs, where V = {1, . . . , n} is the set of n ∈ Z>0 nodes representing variables considered within the control strategy, and E ⊂ {(i, j) : i, j ∈ V} is the set of links of the graph representing possible information sharing among nodes. Furthermore, let A ∈ {0, 1}n×n be the adjacency matrix whose element aij = 1 if (i, j) ∈ E, and aij = 0, otherwise. The set of neighbors of a node i ∈ V is given by Ni = {j : (i, j) ∈ E}. The graph is undirected since it is assumed that links represent bidirectionalinformation channels. In the information-sharing network, it is defined a distance function among nodes from the set V whose mapping is given by d : V × V → R≥0 . Let D ∈ Rn×n ≥0 be the constant matrix representing distances, i.e., dij = d(i, j). Additionally, consider a function whose mapping is c : V × V → R≥0 , where c(i, j) determines how relevant the information shared between node j ∈ V and node i ∈ V is. Therefore, let C ∈ Rn×n be the ≥0

8839

k

k

The four aforementioned objectives are prioritized by setting the vector of weights denoted by ϕ ∈ R4≥0 . The optimal partitioning P  is obtained by solving the following optimization problem: 4  ϕj fj (Pk ), (1a) min P

s. t.

j=1



∈K

Vk = ∅,



∈K

Vk = V.

(1b)

4. PARTITIONING ALGORITHM In order to solve the optimization problem (1), consider the weighted graph G where Wk ∈ Rn×n is a time-varying weighting matrix, i.e., G = (V, E, Wk ). The elements of 1 2

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Presented as external balance in Ocampo-Martinez et al. (2011). Presented as internal balance in Ocampo-Martinez et al. (2011).

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Wk are denoted by wij,k ∈ R≥0 representing a cost associated to the link (i, j) ∈ E, where wij,k = ϕ1 aij + ϕ2 vij,k + ϕ3 d−1 ij + ϕ4 cij . Then, it is proposed to solve the m−partitioning problem (1) as follows:   wij,k , (2) min P

∈K i∈V  j∈V\V  k k

subject to constraints (1b). The set of nodes Vk of the subgraph Gk is composed of a set of internal nodes denoted by Vˇk , and a set of external nodes denoted by Vˆk , for all  ∈ K. The internal nodes from the set Vˇk only have connection to nodes that belong to the same partition. In contrast, the external nodes from the set Vˆk have connection to at least one node that belongs to a different partition. Formally, Vˇk = {i ∈ Vk : Ni ⊆ Vk }, for all  ∈ K, Vˆk = {i ∈ Vk : Ni ⊆ Vk }, for all  ∈ K, and Vk = Vˇk ∪ Vˆk , for all  ∈ K. Each external node i ∈ Vˆk , for all  ∈ K, represents a decision maker, which is able to select a partition from the set of available partitions defined as Qi,k = {g(j) : j ∈ Ni }\{g(i)}. Hence, each decision maker i ∈ {∪∈K Vˆk }, at time instant k, has associated an internal cost denoted by ˇ i , i.e., h  ˇ i (Pk ) = h wij,k , ∀ i ∈ Vˆ  , and  ∈ K, k

g(i)

j∈Vk

∩Ni

ˆ  , for each partition and an external benefit denoted by h i  from the set Qi,k , i.e.,  ˆ  (Pk ) = wij,k , ∀ i ∈ Vˆk , and  ∈ Qi,k . h i j∈Vk ∩Ni

Then, the best external benefit for the decision maker i ∈ Vˆk , for all  ∈ Qi,k , is obtained as follows: ˆ i (Pk ) = max h ˆ  (Pk ). h ∈Qi,k

i

Finally, as previously mentioned, the decision maker selects among the possible available partitions depending on a utility denoted by ui , i.e., if the decision maker has incentives to move from one partition to another one, then   ˆ i (Pk ) − h ˇ i (Pk ) , ∀ i ∈ Vˆ  , and  ∈ K. ui (Pk ) = max 0, h k

Further, consider an undirected graph G˜k = (V˜k , E˜k ) at time instant k –not necessarily connected– composed of all the decision makers –external nodes  of G–. Then, the set of nodes of G˜k is given by V˜k = ∈K Vˆk , and the set of links is given by E˜k ⊂ {(i, j) : i ∈ Vˆk , j ∈ Vˆkr ,  = r}. ˜ k ∈ {0, 1}|V˜k |×|V˜k | be the adjacency matrix of the Let A graph. Since G˜k is not necessarily a connected graph, it has q ∈ Z>0 components at time instant k, where the set of components of the graph is Ck = {1, . . . , q}. Each component is a graph denoted by G˜kp = (V˜kp , E˜kp ), with ˜ p , is connected and not necessarily adjacency matrix A k complete, for all p ∈ Ck . Furthermore, only the decision  maker with higher incentives winner decision maker in   the component p ∈ Ck denoted by ipk ∈ Wkp ⊆ V˜kp would make a decision to switch from its current partition to another one among the set of available partitions, i.e.,  (3) ipk ∈ arg maxp ui (Pk ) = Wkp , ∀ p ∈ Ck . ˜ i∈V k

Notice that (3) should be solved at each time instant  k ∈ Z≥0 . The best option for the decision maker ipk ∈ V˜kp , for all p ∈ Ck , to switch partition is  ˆ p  (Pk ). p ∈ argmax h (4) k

ik

∈Qip  ,k k

Hence, the partitioning is modified only if ui (Pk ) > , where  ∈ R>0 establishes a end-up condition. The updating is as follows: g ( ip  ) g ( ip  )  Vk+1k = Vk k \{ipk }, ∀ p ∈ Ck , (5a) p 

p 



k Vk+1 = Vkk ∪ {ipk }, ∀ p ∈ Ck .

(5b) {V01 , . . . , V0m }

Proposition 1. If the initial partitioning P0 = satisfies constraints (1b), then these constraints are satisfied by Pk = {Vk1 , . . . , Vkm } for all k ∈ Z≥0 under the partitioning update in (5). 5. FLOW-BASED DISTRIBUTION SYSTEMS

According to the algorithm for the m-partitioning problem, it is necessary to provide the number of partitions m, and the initial partition set P0 , i.e., V0 , for all  ∈ K. This section is devoted to present the procedure to determine these two elements in the context of flow-based distribution systems. Many engineering systems may be modeled as a flow-based distribution system as presented in Grosso (2015), e.g., water, energy or transportation systems. In general, flow-based distribution systems are composed of the following elements: (1) Storage: element that stores a resource and with both inflows and outflows. The set of storage elements is denoted by V¯st . (2) Actuator: element that manipulates the resource flow, having a unique inflow and outflow. The set of actuator elements is denoted by V¯ac . (3) Joint: element without storage capabilities with an associated mass-balance constraint, and with both inflows and outflows. The set of joint elements is denoted by V¯jo . (4) Sink: element that receives the resource from either a storage or joint element, e.g., resource demands in the system. The set of sink elements is denoted by V¯si . (5) Source: element providing the resource. The set of source elements is denoted by V¯so . (6) Flow: directed link (i, j) allowing flow from an element i ∈ {V¯st ∪ V¯ac ∪ V¯so ∪ V¯jo } (storage, actuator, source, or joint elements) to an element j ∈ {V¯st ∪ V¯ac ∪ V¯si ∪ V¯jo } (storage, sink, or joint elements). The ¯ set of flow elements is denoted by E. ¯ E) ¯ be a directed graph representing the Let G¯ = (V, flow-based distribution systems describing the possible direction of the flows, where V¯ = {V¯st ∪ V¯ac ∪ V¯si ∪ V¯so ∪ V¯jo } is the set of r ∈ Z>0 elements in the flow-based distribution system, i.e., storage, actuator, sink, source, and joint ¯ is elements. On the other hand, E¯ ⊂ {(i, j) : i, j ∈ V} ¯ ¯ the set of flows from the element i ∈ V to element j ∈ V. The introduced elements for a flow-based distribution system, and the representation of the system by a directed graph allow to identify some features of the system throughout indices an elements presented next.

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Definition 1. (Network resource-feeding index) A non¯ V¯so in the flow-based distribution source element i ∈ V\ system has a resource-feeding index denoted by θi ≤ |V¯so |, which gives information about the amount of source elements that may provide resource to the ith element, i.e., there are θi source elements that can feed the element ¯ V¯so . i ∈ V\ ♦ Definition 2. (Anchor elements) A non-source element i ∈ ¯ V¯so is anchor if there is only one source able to feed V\ it, i.e., the resource-feeding index of an anchor element is unitary. Let A ⊆ V¯ be the set of anchor elements within ¯ V¯so : θi = 1}. ♦ the flow-based network, i.e., A = {i ∈ V\ Definition 3. (Maximum resource-feeding index of the network) The maximum resource-feeding index is denoted by θ and it corresponds to θ = maxi∈V¯ θi . This index provides information about the non-source element in the system with more available source elements given by ♦ iθ ∈ arg maxi∈V¯ θi . Definition 4. (Maximum resource-feeding index per partition) Given a partition of the flow-based distribution system into m sub-systems, the maximum resource-feeding index per partition is denoted by θ = maxi∈V¯  θi , for all  ∈ K. Notice that source elements that do not belong to the current partition are also taken into account. Finally, the element with more available source elements is given by i ♦ ¯  θi , for all  ∈ K. θ ∈ maxi∈V Definition 5. (Resource-feeding co-relation index) The availability of resource at each partition  ∈ K is assessed with respect to the maximum resource index of the network given by the resource-feeding co-relation index denoted by ε ∈ [0, 1]. Formally, the resource-feeding co-relation index −1 is ε = θ (θ ) , for all  ∈ K. Notice that if ε = 1 for a partition  ∈ K, then the non-source element in the network with more available source elements belongs to the partition . ♦ Definition 6. (Available source elements for non-source elements) The available source elements that can provide resource to a non-source element is given by the set Ri ¯ V¯so . and with |Ri | = θi for all i ∈ V\ ♦ The number of partitions is determined by setting a desired minimum resource-feeding co-relation index (see Definition 5), i.e., partitions should satisfy that min∈K ε ≥ ε . In addition, further criteria to define the number of partitions may be included such that it is not desired that the elements i θ , for all  ∈ K, are not neighbors. Once the aforementioned elements i θ , for all  ∈ K are ¯ then those can be associated identified in the graph G, to the corresponding variables in the information-sharing graph G and the initial partition P0 can be determined by incorporation each node from the set V to the nearest m node from i1 (see Definition 3). θ , . . . , iθ 6. CASE STUDY: PARTITIONING ROLE IN THE DESIGN OF A NON-CENTRALIZED MPC Consider the Barcelona Water Supply Network (BWSN) whose flow-based dynamical model is given by xk+1 = Ad xk + Bd uk + Bl dk , (6a) 0 = Eu u k + E l d k , (6b) where x ∈ Rnx is the vector of system states corresponding to the tank volumes, u ∈ Rnu is the vector of control

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inputs corresponding to manipulated flows, and d ∈ Rnd is the vector of the time-varying disturbances corresponding to water demands with daily periodicity and constant mean value (Grosso, 2015; Ocampo-Martinez et al., 2011). The network must satisfy some mass-balance constraints described by (6b). Moreover, matrices Ad , Bd , Bl , Eu , and El are of suitable dimensions dictated by the flow network topology. A portion of the BWSN is presented in Figure 1(a), which is composed of nx = 17 tanks, 26 pumps and 35 valves, i.e., nu = 61, nine water sources, nd = 25 water demands, and 11 mass-balance nodes. The desired performance of the BWSN is determined by operational objectives. The first objective is the minimization of economical costs. The water costs are given by α 1 uk , where the vector α1 ∈ Rnu is constant, and the energy costs nu are given by α varies 2,k uk , where the vector α2 ∈ R with time. The second objective is to operate the system 2 smoothly to avoid possible damage, i.e., minimize ∆uk  , where ∆uk = uk − uk−1 , guaranteeing that the variations over the control inputs are smooth. Finally, it is desired that the volumes for all the tanks are higher than some predetermined constant volumes denoted by xs ∈ Rnx , i.e., constraint xk ≥ xs − ξ k should be satisfied where x ξ k ∈ Rn≥0 is an auxiliary variable. The third objective is to  2 minimize ξ k+j  . The same optimization problem as in (Grosso, 2015; Barreiro-Gomez et al., 2015) is considered for the design of the MPC controller. 6.1 Computing the information sharing graph The optimization problem behind the MPC contoller may be written as a quadratic programing problem of the form 1  Ω HΩk + h Ωk , 2 k s.t. GΩk = g,

min Ωk

where Ωk = [ω  k

...

(7a) (7b)

 ω k+Hp −1 ] , and each element of

 ξ π Ωk is ω k = [u k k k ] , with π k being the vector of slack variables for all the inequality constraints to make them equality constraints. Furthermore, the Lagrangian function corresponding to (7) is 1 HΩk + h Ωk + λ (GΩk − g) . L(Ωk , λ) = Ω 2 k Therefore, the Karush-Kuhn-Tucker conditions are obtained from ∇Ωk L(Ωk , λ ) = 0, and ∇λ L(Ωk , λ ) = 0 giving by the following equality:       Ωk −h H G = . (8) g λ G 0    Ψ

Notice that the matrix Ψ represents the information dependence among variables Ωk , and λ in order to solve in (8). In this regard, if there is a node per each variable in (8), then the adjacency matrix A of the graph G is given by aij = 1 if ψij = 0, and aij = 0, otherwise, defining the topology of the information-sharing network. 6.2 Anchor elements

In order to find the anchor elements in the BWSN, the ¯ E) ¯ is obtained by replacing each directed graph G¯ = (V,

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2

(a) (b) Fig. 1. (a) Aggregate model of the BWSN. (b) Information-sharing graph G presented in Section 6.1 of the BSWN. Links within the same partition with black color and links connecting different partitions with red color. The optimal physical partition is also presented (sub-systems 1 − 4 with colors green, blue, magenta, and red, respectively). Table 1. Resource-feeding network indices for the storage elements in the Barcelona WSN. Storage ¯st element i ∈ V x1 x2 x5 , x 8 x6 x14 , . . . , x16 x7 , x9 , . . . , x11 , x13 x3 , x4 , x12 , x17

Resource-feeding indices θi 1 1 1 2 2 5 7

Table 2. Costs comparison between the centralized and distributed MPC controllers.

Source element Ri s2 s1 s6 s1 , s6 s7 , s8 s1 , s 3 , s 6 , . . . , s 8 s1 , s 3 , s 4 , . . . , s 8

Day y 1 2 3 4 Total

element of the network, i.e., sources, actuators, tanks, sinks, and mass-balance joints, for nodes of the graph ¯ denoted by V¯ and each flow for a graph edge denoted by E. The network resource-feeding indices are shown in Table 1 (see Definition 1). Therefore the anchor storage elements from the i ∈ A are x1 , x2 , x5 , and x8 (see Definition 2). Moreover, these storage elements must belong to the same partition where their respective source elements are member, e.g., the tank x1 , and the source s2 should belong to the same partition. Finally, the maximum resourcefeeding index of the network is given by θ = 7. The appropriate number of partitions is determined based on the resource-feeding co-relation index, and further conditions may be added to this criterion. As an example, suppose that it is desired to have a resource-feeding co-relation index given by ε = 0.25 (i.e., it is desired that the partition with less available source elements has at least 25% of the maximum resource elements that can provide any element in the system θ ). Additionally, it is desired that the storage elements corresponding to the maximum resourcefeeding index per partition i θ , for all  ∈ K, do not have the same source elements Ri . Therefore, x3 , x4 , x12 , and x17 should belong to the same partition since they have a common set of available sources (first partition with resource-feeding co-relation index ε1 = 1). The same situation happens to x7 , x9 , x10 , x11 , and x13 (second partition with relation resource-feeding index ε2 = 0.7142); x14 , x15 , and x16 (third partition with resource-feeding corelation index ε3 = 0.2857); and x6 (fourth partition with resource-feeding co-relation index ε4 = 0.2857). 6.3 Partitioning procedure and results Once the appropriate number of partitions is determined by using a desired resource-feeding co-relation index ε ,

Energy Ce (y) 9.8133 8.5959 8.5959 8.5959 52.793

CMPC Water Cw (y) 5.8964 5.8829 5.8829 5.8829 35.311

Slew Rate C∆u (y) 0.23766 0.018947 0.018947 0.018947 0.33239

Energy Ce (y) 10.499 10.266 10.267 10.267 61.833

DMPC Water Cw (y) 2.6238 2.4903 2.489 2.4886 15.069

Slew Rate C∆u (y) 0.35532 0.010012 0.010074 0.010066 0.4056

Fig. 2. Hierarchical and information dependence among the m sub-systems.

Fig. 3. Evolution of five system states, and three control inputs for both the centralized and distributed MPC controllers. (a)-(c) show the evolution of x4 , x7 , and x12 , respectively. (d)-(f) show the evolution of u11 , u18 , and u23 , respectively. then the partitioning algorithm is run. In this paper, it is proposed to use the information-sharing graph G computed in Section 6.1, which is shown in Figure 1(b). The initial partition P0 considered within the algorithm is determined by using the nodes associated to the elements corresponding to the maximum partition resource-feeding indices i θ , for all  ∈ K. Therefore, the rest of nodes are incorporated to the closest and connected partition.

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The partitioning algorithm is performed with weights ϕ = [0.5 0.2 0.2 1] , and with the parameter  = 0 for the algorithm end-up condition. The optimal partitioning P  is the one presented in Figure 1(b). It is important to highlight that the total number of communication links in order to compute the optimal control input according to problem (8) is (1 A1)/2 = 361. Furthermore, the optimal partition P  has 13 links among partitions, which is the 3.6% of the total number of communication links, representing reduced communication dependence among different partitions, which is desired for the design of noncentralized controllers. With the optimal partition P  , the information-sharing graph is interpreted/translated into the physical system, obtaining the physical partition into m sub-systems presented in Figure 1(a) (the indices of the m sub-systems are given by the set K). With the m−partitioning, a local MPC controller is designed for each sub-system, identifying the information dependence among them as in Ocampo-Martinez et al. (2011). This procedure results in the sub-system dependence presented in Figure 2 (partitions from 1 to m correspond to colors green, blue, magenta, and red, respectively), where the terms µr are the information provided by sub-system r ∈ K to sub-system  ∈ K. The mentioned information among sub-systems is as follows:  ˜  ˜  ˜  ˜  ˜  u u u u u u µ12 = [˜ 40,k 47,k ] , 18,k 32,k 34,k 15,k µ13 = [˜ u 56,k

 ˜  u u 60,k ] , µ23 = [˜ 46,k

 ˜ 6,k , ˜  u 49,k ] , µ24 = u

˜ i,k = [ui,k . . . ui,k+Hp −1 ] ∈ RHp . Different where u from Ocampo-Martinez et al. (2011), there are not cycles in the hierarchical structure describing information dependence among sub-systems, therefore, it is not required to solve a constraint satisfaction problem (CSP). In order to evaluate the performance of the DMPC controller, a comparison with the performance of a centralized model predictive control (CMPC) is made. Both approaches are designed with the same prioritization weights, and simulations are made for four days. Table 2 shows the costs for each day denoted by y. The operational 24y  costs of energy Ce (y) = k=24y−24 α1 uk , the op24y  erational costs of water Cw (y) = k=24y−24 α2,k uk , and costs associated to the smooth operation (slew rate) 24y 2 C∆u (y) = k=24y−24 ∆uk  . It can be seen that, the control inputs computed with the distributed approach imply higher energy costs and also more variations over the control signals. In contrast, the costs associated to the water are lower with the distributed approach. Figure 3 shows the evolution of five system states and five control inputs for both the centralized, and distributed MPC controllers. It can be seen that the periodicity of the signals for both cases are the same. In general, the performance corresponding to the distributed MPC controller exhibits higher variations over the evolution of the system states. However, the computational tasks of the centralized MPC controller are divided and assigned to four different local MPC controllers and continue achieving the control objectives and satisfying all the constraints. 7. CONCLUSIONS AND FUTURE DIRECTION

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partitions, size of partitions, distance among elements, and importance of links has been presented in order to determine the appropriate partitions in a large-scale flowbased system. As one of the most relevant features of the proposed partitioning is that, it is a general methodology in the sense it can be implemented for any dynamical system, including flow-based network systems, and any control strategy since an information sharing graph is considered. As future work, motivated by the fact that some nominal conditions might vary along the time for a dynamical system, e.g., disturbances affecting the system under control, it is proposed to extend the partitioning procedure to the dynamical case. Thus, the large-scale system can be divided conveniently and permanently in order to enhance the performance of closed-loop systems operated by non-centralized controllers. REFERENCES Barreiro-Gomez, J., Ocampo-Martinez, C., and Quijano, N. (2015). Evolutionary-game-based dynamical tuning for multi-objective model predictive control. In S. Olaru, A. Grancharova, and F.L. Pereira (eds.), Developments in Model-Based Optimization and Control, 115–138. Springer Verlag. Chandan, V. and Alleyne, A. (2013). Optimal partitioning for the decentralized thermal control of buildings. IEEE Transactions on Control Systems Technology, 21(5), 1756–1770. Ezhilarasi, G.A. and Swarup, K. (2012). Network partitioning using harmony search and equivalencing for distributed computing. Journal of Parallel and Distributed Computing, 72(2012), 936–943. Grosso, J. (2015). Economic and Robust Operation of Generalised Flow-based Networks. Doctoral dissertation. Universidad Polit`ecnica de Catalunya. Automatic Control Department. Gupta, A. (1997). Fast and effective algorithms for graph partitioning and sparse-matrix ordering. IBM Journal of Research and Development, 41(1), 171–183. Kamelian, S. and Salahshoor, K. (2015). A novel graphbased partitioning algorithm for large-scale dynamical systems. International Journal of Systems Science, 46(2), 227–245. Kleinberg, M., Miu, K., Segal, N., Lehmann, H., and Figura, T. (2014). A partitioning method for distributed capacitor control of electric power distribution systems. IEEE Transactions on Power Systems, 29(2), 637–644. Nayeripour, M., Fallahzadeh-Abarghouei, H., Waffenschmidt, E., and Hasanvand, S. (2016). Coordinated online voltage management of distributed generationusing network partitioning. Electric Power Systems Research, 141(2016), 202–209. Ocampo-Martinez, C., Bovo, S., and Puig, V. (2011). Partitioning approach oriented to the decentralised predictive control of large-scale systems. Journal of Process Control, 21(2011), 775–786. Xie, L., Cai, X., Chen, J., and Su, H. (2016). GA based decomposition of large scale distributed model predictive control systems. Control Engineering Practice, 57(2016), 111–125.

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