Hierarchical MPC for Multiple Commodity ...

Chapter 33

Hierarchical MPC for Multiple Commodity Transportation Networks J. L. Nabais, R. R. Negenborn, R. B. Carmona-Benítez, L. F. Mendonça and M. A. Botto

Abstract Transportation networks are large scale complex systems spatially distributed whose objective is to deliver commodities at the agreed time and at the agreed location. These networks appear in different domain fields, such as communication, water distribution, traffic, logistics and transportation. A transportation network has at the macroscopic level storage capability (located in the nodes) and transport delay (along each connection) as main features. Operations management at transportation networks can be seen as a flow assignment problem. The problem dimension to solve grows exponentially with the number of existing commodities, nodes and connections. In this work we present a Hierarchical Model Predictive Control (H-MPC) architecture to determine flow assignments in transportation networks, while minimizing exogenous inputs effects. This approach has the capacity to keep track of commodity types while solving the flow assignment problem. A flow decomposition of the main system into subsystems is proposed to diminish the problem dimension to solve in each time step. Each subsystem is managed by a control J. L. Nabais(B) IDMEC, Department of Informatics and Systems Engineering, Setúbal School of Technology, Polytechnical Institute of Setúbal, Setúbal, Portugal e-mail: [email protected] R. R. Negenborn Transport Engineering and Logistics, Delft University of Technology, Delft, The Netherlands e-mail: [email protected] R. B. Carmona-Benítez School of Business and Economics, Universidad Anáhuac México Norte, Huixquilucan, México e-mail: [email protected] L. F. Mendonça Department of Marine Engineering, Escola Superior Naútica Infante D. Henrique, Paço d’Arcos, Portugal e-mail: [email protected] M. A. Botto IDMEC, Instituto Superior Técnico, Technical University of Lisbon, Lisboa, Portugal e-mail: [email protected] J. M. Maestre and R. R. Negenborn (eds.), Distributed Model Predictive Control 535 Made Easy, Intelligent Systems, Control and Automation: Science and Engineering 69, DOI: 10.1007/978-94-007-7006-5_33, © Springer Science+Business Media Dordrecht 2014

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agent. Control agents solve their problems in a hierarchical way, using a so-called push-pull flow perspective. Further problem dimension reduction is achieved using contracted projection sets. The framework proposed can be easily scaled to network topologies in which hundreds of commodities and connections are present.

33.1 Introduction In transportation networks (such as cargo transport [2, 4], postal networks, traffic networks [6], water distribution [8, 14], supply chains [10, 17]) the elementary objective is to deliver a certain commodity in the agreed quantity at the agreed time and at the agreed location [13]. The transport need can arise in two different forms: located downstream in the form of customer demand (clients in supply chains and water distribution) or located upstream as clients request to provide a service (deliver letters or containerized cargo). These two types of transport needs are disturbances to the network state and we consider them as exogenous inputs. The main control problem related to transportation networks can be categorized as a tracking control problem and stated as: find the optimal flows inside the network such that the exogenous inputs effects are eliminated and the network follows the desired state over time. In a water distribution application the optimal flows are assured through gate movements in order to keep water depths inside admissible levels in each canal, in cargo transportation the optimal flow is guaranteed by allocating transport capacity such that cargo is delivered at the final destination at the right time and with the exact quantity. The ability to access the stored volume per commodity type at each network node can be used to support operations management towards a more efficient, sustainable, cooperative and reliable transportation. Considering multiple commodities and network nodes a combinatorial issue arises. When realistic applications are considered this becomes a real problem in terms of computation time. Take an example from the freight transport: the Neuss Trimodal [5] terminal, recently added as a member of European Gateway Services.1 This hinterland intermodal container terminal situated at the Rhine river offers connections to the European hinterland through three transport modes: barge, train and truck. With 8 rail tracks it sustains 39 train connections weekly plus 7 inland shipping connections to Rotterdam and Antwerp ports using a quay of 230 m. Adding to these features all kind of container types (hazardous materials, reefer containers and other categories like size, weight and destination) much information has to be captured by the modeling framework. Information can be shared freely over the transportation network, restricted to some subnetworks or

1

European Gateway Services are a service provided by European Container Terminal (ECT) whose main objective is to create an integrated network of hinterland terminals cooperating to increase the ECT terminals throughput at the Rotterdam port. Neuss Trimodal has been a member of this network since 20 December 2011.

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confined to a single component as a consequence of the economical relations between the different parties present at the network. In case of vertical integration, when all parties belong to a same entity, information is usually shared freely. Using a central model to solve the flow assignment problem in a transportation network with multiple commodities is not an option because the problem dimension to be solved grows exponentially with the number of handled commodities, nodes and connections available. However, some connections can have no transport needs over some time (inactive connections), which means that the optimal solution is partially imposed, and it is also expected that the number of commodity types handled in an active connection (opposed to an inactive connection) is just a subset of all commodity types available at the network. We propose a Hierarchical Model Predictive Control (H-MPC) framework that is able to cope efficiently with this large-scale problem dimension by proposing explicitly measures to face the aspects mentioned above. The framework is based on the following: • The large-scale system is broken down into smaller subsystems using a decomposition inspired by flows. A subsystem can be related to an arc, path or cycle dependent on the specific network. • A control agent is assigned to each subsystem and formulates an optimization problem to solve the flow assignment. Control agents will only consider to solve problems related to active subsystems. • Subproblems will be simplified further by taking into account only the commodity types handled by the subsystem using contracted commodity sets. One important issue is the order by which the control agents should solve their problems. Following a flow perspective, control agents order is established in a so called push-pull flow perspective [15] based on the exogenous inputs location. If the exogenous input is located downstream, a pull flow perspective is applied and therefore the control agents responsible to move commodities to that downstream node are set to a higher priority. If the exogenous inputs are located upstream, a push flow perspective is applied and control agent responsible to move commodities from the source nodes get a higher priority. A simultaneous push-pull flow perspective is possible. Adding more connections to the network has as a consequence the addition of more control agents. The original problem remains solvable in a reasonable time even for large scale networks with hundreds of commodities, nodes and connections. In Sect. 33.2 the transportation network model is described to cope with different network topologies. The optimization problem is formulated for the whole network in Sect. 33.3. The H-MPC framework is presented in Sect. 33.3.2 based on a model decomposition inspired by flows and using contraction projected sets. In Sect. 33.4 numerical results are presented, in which the H-MPC architecture is compared with the MPC architecture.

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33.2 Multiple Commodity Network Model 33.2.1 Motivation A transportation network can be represented by a graph G = (V, E) where nodes V represent centers or intersections and arcs E represent the existing connections between nodes [1]. The assumptions made in this work are intended to produce a general framework able to describe different transportation network topologies that can be found in different research fields such as: traffic networks, water distribution, supply chain, cargo and passenger transportation, postal networks. At a macroscopic level, transportation networks exhibits two major phenomena: storage capability in well-defined areas (where commodities can be produced, manufactured or simply stored) and the transport delay which is the time necessary to transport commodities between different locations. To distinguish these two phenomena inside the transportation network we define two components: Center Node: is a network node with a significant storage capacity where commodities can be stored temporarily before moving to another network node. The center node degree is always equal of bigger than one. If the center node indegree is zero the node is categorized as a source (upstream) node, if the center node outdegree is zero the node is categorized as a destination (downstream) node. When center nodes have simultaneously the indegree and outdegree bigger than one (see Fig. 33.1a) they are categorized as store nodes; Connection: is a path between two center nodes and is used to model the transport phenomena. It is composed of a succession of nodes with an indegree and outdegree equal to one, which means that there is only one arc arriving and another arc departing from each node. Connection i is composed of n ci nodes and n ci + 1 arcs as indicated in Fig. 33.1b. Connections are modeled using a push-pull flow perspective: pushing commodities from the connection upstream node and delivering it to the connection downstream node.

(a)

(b) node j

arc 1 arc 2 .. arc j .

node i

arc j + 1 arc j + 2 .. . arc deg(i)

node j + 1

connection i node 1

arc 1 upstream

node 2

arc 2

…

arc 3

node nci arc nci arc nci + 1 downstream

Fig. 33.1 Elementary components in a transportation network (deg(i) stands for node degree). a Center node of store type and b connection (path) between two center nodes ( j and j + 1)

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All transportation networks are generally composed of center nodes and connections. The complexity of the network model is determined by the following parameters: • • • •

n t : number of commodity types considered; n c : number of connections existing in the network; n ci : number of nodes belonging exclusively to connection i; n n : number of center nodes in the network that are further divided into source (upstream) nodes n un , destination (downstream) nodes n dn and store nodes n sn . The store nodes can be distributed through several levels based on similar characteristics, such as distribution centers, consolidation centers; • n l : number of levels present in the transportation network, including the source (upstream) and destination (downstream) levels.

For illustration purposes consider the transportation network indicated in Fig. 33.2. nc n ci = 52 nodes associated exclusively The network is composed of 61 nodes ( i=1 to connections and n n = 9 center nodes shared by several connections: three source nodes, four store nodes and two destination nodes) and n c = 16 connections with 68 transport flows. This network is divided into four levels (n l = 4), including the source and the destination levels which are level 1 and level 4 respectively. The network topology is generic including: connections between nodes on nonadjacent levels (connections from node 53 to node 58 using the walk 53–1–2–3–4–5–58 and between nodes 55 and node 59 using the walk 55–18–19–20–21–22–59) and cycles between level two and level three (for example cycle formed by nodes 56–26–27–28–59– 44–45–46–56). Center nodes can have multiple connections arriving and departing while connections can share limited infrastructure resources to guarantee the desired flows between nodes. Transportation networks are therefore complex systems with coupled dynamics and coupled constraints.

Fig. 33.2 Example of a transportation network

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33.2.2 Network Model Framework A generic framework to model different transportation network is intended but adaptation can be required to accommodate modeling assumptions made for each case scenario. The total number of nodes inside the network is associated with the network topology and is given by, nc  n ci . (33.1) ny = nn + i=1

For each node in the transportation network a state-space vector x¯ j (k) is defined, and these are merged to form the state-space vector x of the complete network, ⎡

⎤ x 1j (k) ⎢ x 2 (k) ⎥ ⎢ j ⎥ x¯ j (k) = ⎢ . ⎥ , j = 1, . . . , n y , ⎣ .. ⎦ nt x j (k)

⎡

x¯ 1 (k) ⎢ x¯ 2 (k) ⎢ x(k) = ⎢ . ⎣ ..

⎤ ⎥ ⎥ ⎥, ⎦

(33.2)

x¯ n y (k)

where x tj (k) is the volume of commodities of type t at node j at time step k. The state-space dimension x is given by n t n y corresponding to the number of commodity types handled and the number of nodes existing in the network. The state-space vector contains information about the quantity per commodity type not only at the center nodes, with significant storage capacity, but also at connection nodes. The total volume per commodity type inside the network is always accessible through the state-space vector. The model for the network dynamics can be represented in a state-space form as, x(k + 1) = Ax(k) + Bu u(k) + Bd d(k), y(k) = Cx(k), x(k) ≥ 0, u(k) ≥ 0, y(k) ≤ ymax , Puu u(k) ≤ umax ,

(33.3) (33.4) (33.5) (33.6) (33.7) (33.8)

x(k) ≥ Pxu u(k), x(k) ∈ X ,

(33.9) (33.10)

u(k) ∈ U,

(33.11)

where u is the control action vector with length n u = n t (n y − n n + n c ), d is an exogenous input vector assumed known with dimension n t (n un + n dn ), y is the current volume at all nodes with dimension n y , ymax are the maximum node storage capacities, umax the available infrastructure resources according to the network structural layout, A, Bu , Bd and C are the state-space matrices, Pxu is the projection from the

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control action set U into the state-space set X and Puu is the projection matrix from the control action set U into the infrastructure resource capacity set Umax . The transportation network state of x at the next time step, k + 1, is determined using (33.3) as a function of the current network state x plus the contribution due to the control action u and the corresponding exogenous inputs d capturing the external requests on the transportation network. The control action u is the flow of commodities between nodes and is imposed through the available infrastructure resources. Inequalities (33.5)–(33.9) are necessary in this framework for imposing the network structural layout and assumptions made: Nonnegativity of States and Control Actions: negative storage and negative control actions (flows) are not physically possible, imposed by (33.5) and (33.6); Storage Capacity: each network node has to respect its own storage capacity, this is represented by (33.7); Maximum Control Actions: the network structural layout in terms of available hardware in quantity and type used to guarantee the desired flows is represented by (33.8); Feasible Control Actions: not all control actions that satisfy (33.5) and (33.6) are feasible. The control action has to respect the existence of commodity type in the related network node. Inequality (33.9) imposes this relation.

33.2.3 Network Model Decomposition Taking into account that real transportation networks may serve tens of center nodes and handle hundreds of commodity types it is critical to alleviate the computational burden when considering the sparse central model (33.3)–(33.11) to support operations management. Using a node/arc numbering in a push-flow perspective (from the sources towards the destinations) as indicated in Fig. 33.2, it is possible to obtain a highly structured model without the need for further mathematical manipulations [16]. A connection is by definition the path between two center nodes. The interaction of a single connection into the set of center nodes is done solely at two nodes. To take advantage of the model structure we define a subsystem i as the node collection related to a connection i plus the associated source and destination nodes [11]. The state-space vector xi for subsystem i will be composed of the corresponding x¯ j state-space vectors, ⎡

⎤ x¯ n Ci −n ci +1 (k) ⎢ x¯ n C −n c +2 (k) ⎥ ⎢ i i ⎥ ⎢ .. ⎥ i ⎢. ⎥ 

 ⎢ ⎥ xi (k) = ⎢ x¯ n −1 (k) nc j , 1 ≤ i ≤ nc, ⎥ , n Ci = ⎢ Ci ⎥ j=1 ⎢ x¯ (k) ⎥ ⎢ n Ci ⎥ ⎣ x¯ in (k) ⎦ i x¯ iout (k)

(33.12)

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with length n t (n ci + 2) belonging to state-space set Xi where x¯ iin and x¯ iout are the statespace vectors related to the source and destination nodes for connection i respectively. The control action vector ui for subsystem i is given by the corresponding u¯ j statespace vectors, ⎤ u¯ n Ui −n ci (k) ⎢ u¯ n U −n c +1 (k) ⎥ i i i ⎥ ⎢ 

 ⎥ ⎢ .. ui (k) = ⎢ . nc j + 1 , ⎥ , n Ui = ⎥ ⎢ j=1 ⎣ u¯ n U −1 (k) ⎦ ⎡

1 ≤ i ≤ nc,

(33.13)

i

u¯ n Ui (k)

with length n t (n ci + 1) belonging to control action set Ui . The state-space model for subsystem i is given by, xi (k + 1) = Ai xi (k) + Bui ui (k) + Bdi di (k) +

nc 

Bui, j u j (k)

(33.14)

j=1, j=i

yi (k) = Ci xi (k),

(33.15)

where yi is the quantity per commodity type at subsystem i nodes, di is the exogenous input vector associated with subsystem i, Ai , Bui , Bui, j Bdi and Ci are the state-space matrices for subsystem i. The last term in (33.14) is critical to assure information exchange between control agents, in particular regarding their future behavior, to avoid that two or more control agents respond to the same demand. The complete subsystem i model is obtained including constraints of nonnegativity of states and control actions, storage capacity, maximum control actions, and feasible control actions to the state space model (33.14)–(33.15).

33.3 Hierarchical Network-Flow Control In this section we present the Hierarchical Model Predictive Control architecture (H-MPC). First, operations management for the transportation network is formulated accordingly to a central MPC approach [3, 9]. After, using a push-pull flow perspective, the large-scale system is decomposed into subsystems to each of which a control agent is assigned. Control agents are associated to a network level if they are delivering commodities to the center nodes located at that level for a pull flow perspective or if they are taking commodities from the center nodes located at that level in a push flow perspective. The problem dimension to be solved at each time step is diminished by considering only control agents related to active connections over the prediction horizon. Further reduction is achieved using the contracted and global commodity sets to formulate the optimization problem considering only the number of commodity types affected over the prediction horizon .

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33.3.1 Centralized MPC Formulation The cost function is defined in accordance to the application domain and is generally a function of the network states, control actions and desired states over the prediction horizon Np , Np −1

J (˜xk , u˜ k , x˜ ref ) =



f (x(k + 1 + l), u(k + l), xref (k + 1 + l)) ,

(33.16)

l=0

where x˜ k is the vector composed of the state-space vectors for each time step

T over the prediction horizon xT (k + 1) , . . . , xT (k + Np ) , u˜ k is the vector composed by the control action vectors for each time step over the prediction horizon

T T u (k) , . . . , uT (k + Np − 1) , xref is the state-space reference vector and x˜ ref is the vector composed of the state-space reference vectors for each time step over

T T (k + N ) T . The weights to be used the prediction horizon xref (k + 1) , . . . , xref p in the objective function (33.16) are considered time varying to allow changing flow priorities according to the different behaviors desired for the transportation network over time. The MPC problem for the transportation network can be formulated as: min J (˜xk , u˜ k , x˜ ref ) u˜ k

(33.17)

subject to x(k + 1 + l) = Ax(k + l) + Bu u(k + l) + Bd d(k + l),

(33.18)

y(k + l) = Cx(k + l), l = 0, . . . , Np − 1, x(k + 1 + l) ≥ 0,

(33.19) (33.20)

u(k + l) ≥ 0, y(k + l) ≤ ymax ,

(33.21) (33.22)

Puu u(k + l) ≤ umax , x(k + l) ≥ Pxu u(k + l), Pdx x(k + 1 + l) ≤ dd (k + l),

(33.23) (33.24) (33.25)

where ymax is the maximum storage capacity by center node, umax the maximum available infrastructure resources according to the network structural layout, dd is the vector responsible to introduce the exogenous inputs, Pdx is the projection matrix from the state-space set into the disturbance set and Puu is the projection matrix from the control action set into the infrastructure resource set Umax . Constraint (33.25) is included in the MPC problem formulation to introduce the network exogenous inputs prediction.

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33.3.2 Hierarchical Formulation 33.3.2.1 MPC Formulation for One Control Agent The cost function of a control agent is defined in accordance to the application field and is generally a function of the states, control actions and desired states of the subsystem the agent controls over the prediction horizon Np ,



Ji x˜ k,i , u˜ k,i , x˜ ref,i =

Np −1



 f xi (k + 1 + l), ui (k + l), xref,i (k + 1 + l)

l=0

(33.26) where x˜ k,i is the vector composed of the state-space vectors for each time step

T over the prediction horizon xiT (k + 1) , . . . , xiT (k + Np ) for control agent i, u˜ k,i is the vector composed by the control action vectors for each time step over the

T prediction horizon uT (k) , . . . , uT (k + Np − 1) for control agent i, xref,i is the state-space reference vector for control agent i and x˜ ref,i is the vector composed of the state-space reference vectors for each time step over the prediction horizon

T T (k + N ) T xref,i (k + 1) , . . . , xref,i . Transportation networks are large scale sysp tems spatially distributed therefore it is common to have connections with rather different features, in particular the transport delay. For larger transport delays the optimization problem requires a larger prediction horizon in order commodities have enough time to reach the destination node such that this effect is reflected in the cost function. For smaller transport delays smaller prediction horizons can be used at the cost of some performance decrease. The MPC formulation for control agent i can be stated as:

 min Ji x˜ k,i , u˜ k,i , x˜ ref,i

(33.27)

u˜ k,i

subject to xi (k + 1 + l) = Ai xi (k + l) + Bui ui (k + l) + · · · + Bdi di (k + l) +

nc 

Bui, j u j (k + l) (33.28)

j=1, j=i

yi (k + l) = Ci xi (k + l), l = 0, . . . , Np − 1,

(33.29)

xi (k + 1 + l) ≥ 0, ui (k + l) ≥ 0,

(33.30) (33.31)

yi (k + l) ≤ ymax,i , Puu,i ui (k + l) ≤ umax,i ,

(33.32) (33.33)

xi (k + l) ≥ Pxu,i ui (k + l), Pdx,i xi (k + 1 + l) ≤ ddi (k + l),

(33.34) (33.35)

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where ymax,i is the maximum capacity for control agent i nodes, umax,i the available infrastructure resources according to the network structural layout for agent i, ddi is the vector responsible to introduce the exogenous inputs for control agent i, Puu,i is the projection matrix from the control action set Ui into the infrastructure resource set for control agent i, Pxu,i is the projection from the control action set Ui into the state-space set Xi and Pdx,i is the projection matrix from the state-space set Xi into the exogenous input set of control agent i.

33.3.2.2 Projections Between Contracted and Global Commodity Sets Considering the network model as a collection of subsystems reduces the optimization problem dimension to be solved at each time step. It is not expected that each connection in the network is transporting simultaneously all commodity types, therefore further reduction of the problem dimension to be solved in each time step can be made if we consider only the handled commodities over the prediction horizon. Define the following sets: • T := {1, . . . , n t } is the set of all commodity types handled in the transportation network with cardinality |T | = n t ; • Ti (k) = {1, . . . , n iac (k)} is the set of the commodity types handled by subsystem i over the prediction horizon at time instant k with cardinality |Ti | = n iac . It is important to note that the cardinality of Ti is made time varying to allow different commodity flows over time. The following relation between sets can be derived, (33.36) Ti (k) ⊂ T . The model (33.14)–(33.15) can be written for a new state-space variable xic and a new control action uic whose dimensions are a subset of the network commodity set T by eliminating from the state-space vector xi and from the control action vector ui all variables related to commodity types that are not included in the contracted commodity set Ti and for this reason are not expected to change over the prediction horizon. The original state-space representation can be recovered using, 

xi (k) = Pcx,i (k)xic (k) ui (k) = Pcu,i (k)uic (k)

(33.37)

where Pcx,i and Pcu,i are time varying projection matrices between the contracted commodity set Ti into the global commodity set T for the state-space and control action vectors respectively. This procedure allows to look for the optimal solution regarding only significant control actions. The control actions associated to the eliminated variables are zero by default.

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33.3.2.3 Hierarchical Framework The order in which the control agents solve their problems at each time step can be fixed over time or depend on the current network state. For setting the control agents order at each time step it is important to recall the transportation network type. Depending on where the network exogenous inputs occur the control agents order is set in a push flow perspective (for upstream exogenous inputs) or in a pull flow perspective (for downstream exogenous inputs). In either case the control agents order is set by groups corresponding each group to a network level and only after the control agents order for a level is determined the order for the control agents associated to the next level is determined. At the beginning of each time step all control agents update their state using the information about exogenous inputs. After, control agents determine in parallel their expected workload over the prediction horizon to follow the network desired state, Np    F xiF (k + 1 + l) − xref (k + 1 + l) , i = 1, . . . , n l , ci (k) = αi (k)

(33.38)

l=0

where αi is a time varying penalty term to account commodities transport costs using F are the state-space and reference vector for the connection i over time, xiF and xref edge node of connection i (upstream center node for a push flow perspective and downstream center node for a pull flow perspective). Each control agent shares its workload information ci , for the current time step at the network, with the central coordinator that sets the order o(k) in which the control agents should solve their problems. After analyzing all network levels the complete order o(k) = o1 . . . on c with 1 ≤ oi ≤ n c such that, co1 (k) > · · · > con1 (k) · · · co l (k) > · · · > conc (k), n −n +1 c       c c first level last level

(33.39)

where n 1c is the number of connections associated to the first network level to be solved and nlc is the number of connections associated to the last network level to be solved. The central coordinator is responsible to set the total amount of infrastructure resources that are available θ 0 = umax and the current prediction for future decisions set P 0 = {u˜ k−1,o1 , . . . , u˜ k−1,onc }. The control agent to start (o1 ) has all infrastructure resources available. After the initial configuration the iterations are executed in which each control agent oi (i = 1, . . . , n c ), one after another, performs the following tasks (see Fig. 33.3): • the maximum admissible resource for control agent oi is determined as the minimum between the subsystem maximum infrastructure resource umax,oi and the infrastructure resources not yet assigned,

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0,

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o, x o1

Agento1

o1

o2

Agento2

o2

..

.

onc −1 onc −1

Agentonc uopt,o1

uopt,o2

...

uopt,onc

uopt Fig. 33.3 H-MPC schematics at a time step



umax,oi = min Pmax,oi θ oi−1 ; umax,oi ,

(33.40)

where Pmax,oi is the projection matrix from the global infrastructure resource set Umax to the maximum infrastructure resource set Umax,oi for subsystem oi ; • in case the workload coi is zero the optimal control action uopt,oi is zero by default. In case the workload coi is nonzero the optimal control action uopt,oi is found solving the MPC problem (33.27)–(33.35) taking into account the contracted commodity set (33.36). The control agent state-space and control action vectors are recovered using (33.37); • the available resources to the next control agent oi+1 are updated: θ oi+1 = θ oi − Pmu,oi (k)uopt,oi (k)

(33.41)

where Pmu,oi (k) is the projection matrix from agent oi infrastructure resources set Uoi to the control action set Umax ; • the predictions for future decisions are updated and denoted by P oi+1 replacing the control agent initial prediction u˜ k−1,oi by the new optimal sequence found u˜ opt,oi . The procedure to follow is represented schematically in Fig. 33.3 and is presented in Algorithm 33.1.

33.4 Application Results The presented H-MPC architecture has been used as a tool to support the resource allocation in a hinterland intermodal container terminal [11] serving three transport modalities: barge, train and truck. In this example the terminal operator perspective is considered and all parties present at the terminal are assumed cooperative in sharing

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Algorithm 33.1 Hierarchical Model Predictive Control for Transportation Network 1: repeat 2: control agents determine in parallel the expected workload using (33.39) 3: control agents determine their contracted set Toi (k) and projections matrices 4: central coordinator updates the control agents order as in (33.40) 5: central coordinator initialize the infrastructure resource and future decision predictions set 6: for i = 1 → n c do 7: update the admissible resources for control agent using (33.41) 8: solve optimization problem (33.27)–(33.36) for agent oi 9: recover the global commodity set 10: the optimal control action uopt,oi is the first component of u˜ opt,oi 11: update the future decision predictions set using u˜ opt,oi 12: end for 13: apply the optimal solution uopt to the transportation network 14: update time step k 15: until simulation time is reached

information. The container terminal is integrated in a network composed of 4 terminals and faces an average weekly throughput of 16,800 TEU (twenty-foot equivalent units). The terminal allows the berth of two barges simultaneously (Barge A and Barge B), three connections per berth are made on a daily basis. The quay cranes allow a maximum handling capacity of 90 TEU/h, for Barge A this full capacity can be used while for Barge B a maximum rate of 45 TEU/h is possible. As a consequence, Barge A and Barge B will be competing for the same resource at the quay. For the train modality there are two rail tracks (Train A and Train B) that serve four trains each in a single day. The maximum handling capacity for each rail track is 40 TEU/h but the train gate only offers a maximum capacity of 40 TEU/h. In this case both train tracks will be competing for the gate handling capacity. Although geographically confined to the port area, this network has simultaneously two types of exogenous inputs in the form of request of containers to unload and containers to load to different transports available at the terminal, see Fig. 33.4. The request is presented as the number of containers of each type considered in the network. The containers to unload represent a push of containers towards the Central Yard and the containers to load are pulled from the Central Yard. Considering that each transport available at the terminal has simultaneously an outflow and inflow we use both flows to define a network path passing through the common node (Central Yard). This network path (linking unload and load areas for each available transport at the terminal) will be used to decompose the system into subsystems. In this network, the source and destination nodes are associated exclusively to a single path therefore they were categorized as connection nodes. This is a simple example that shows the need for small adjusts when applying the presented framework. The considered terminal structural layout is translated into the network graph presented in Fig. 33.4. In this graph there are n ci = 5 exclusive nodes per connection, the containers are categorized into n t = 5 different classes (four destinations A, B, C, D, plus empty containers) and the number of connections that can be served simultaneously at the terminal is n c = 5. The Central Yard is a common node to

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Fig. 33.4 Intermodal container terminal network

all connections and is responsible for the dynamic coupling. A total of 26 nodes are present at the terminal. For this setup the terminal is described by 130 states using the central model (33.3)–(33.11), or by 30 states per subsystem if the decomposed model (33.14)–(33.15) is used. More details about the terminal handling resources and weekly schedules at the terminal are available in [12].

33.4.1 Simulation Setup The H-MPC architecture and the centralized MPC approach uses the same type of cost function and weights in the optimization problem to allow a fair comparison. It is possible to assign different weights to different terminal nodes, container classes and connections depending on their role in the container terminal dynamics and the desired behavior. The weight assigned to the Import Area at the Central Yard is zero as it acts as a warehouse for containers between delivery and pick up times. The weights at the Load Area are taken negative, such that containers are pulled from the Import Area. The main criterion to assign weights is related to the connection priority according to the volume of containers to handle: the higher the volume the higher the priority. A time step of 1 h is considered. A prediction horizon of Np = 3 steps is used. To guarantee pulling containers from the Import Area to the Load Area the weights assigned should respect the relation

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− (q3+i + q4+i ) >



q5+i i = 0, . . . , n c − 1,

(33.42)

j=1

where q3 , q4 and q5 are the weights associated to containers located at the Export Area, Export Shake Hands and Load Area for the first connection respectively. The simulation is performed using MatLab R2009b on a personal computer with a processor Intel(R) Core(TM) i7 at 1.60 GHz with 8 Gb RAM memory in a 64bit Operating System. The optimization problem is solved at each time step of the simulation using the MPT v2.6.3 toolbox [7] with the CDD Criss–Cross solver for linear programming problems.

33.4.2 Tested Scenario The scenario presents 1 week. Different criteria to establish the order in which the control agents should be solved

in the H-MPC approach were tested; case H-MPC1 the call size p = 1 1 1 1 1 ; case H-MPC2 benefiting from sustainable transport modalities p = 2 2 1 1 0.5 and case H-MPC3 inverting the order considered 1 1 1.5 1.5 2 . Control strategies are compared using in the MPC strategy p = two criteria: 1) the sum of the cost function over the entire simulation and 2) the computation time. In Fig. 33.5 it is clear that both strategies lead to almost the same terminal behavior over time. This similarity can be confirmed by the cost function performance indicated in Table 33.1. Both strategies achieve a similar performance with a slightly better score for the centralized approach. Interesting to note that all H-MPC strategies tested achieved similar performance. In terms of computation time, the H-MPC approach outperforms the MPC strategy, Table 33.1.

(b)

(a) 60

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x C6 x C10 xH 6 xH 10

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Fig. 33.5 Quantity of containers for the tested scenario (C stands for centralized MPC architecture, H stands for H-MPC architecture). a Barge B and b Import Area at the Central Yard

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Table 33.1 Control strategies comparison Strategy

Max (s)

Mean (s)

Stdv (s)

Cost function performance

H-MPC1 H-MPC2 H-MPC3 MPC

4.71 8.28 7.39 367.83

2.66 2.84 2.83 118.16

1.14 1.26 1.21 67.18

−4.660 × 105 −4.660 × 105 −4.660 × 105 −4.766 × 105

Fig. 33.6 Quantity per container class at the Central Yard for the H-MPC framework

3000

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x 1i x 2i x 3i x 4i x 5i 20

40

60 80 time step k

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Figure 33.6 shows the volume per container class at the Import Area in the Central Yard over the simulation. This model ability is partially responsible for the large problem dimension. However, when looking to the total volume at the terminal it is almost constant (around 9,000 TEU, Fig. 33.5b). The model complexity is the price to pay to have more information regarding the state of the terminal which is a key element for the transportation network.

33.5 Conclusions and Future Research This work presents a Hierarchical Model Predictive Control (H-MPC) approach for operations management of large-scale transportation networks with multiple commodity types. The large-scale system is broken down into smaller subsystems, in a flow perspective, to which a control agent is assigned. Based on the network application domain, agents solve their problems in a push-pull flow perspective. The computation burden of considering a sparse central model to support operations management is avoided and a solution is obtained in reasonable time. Further dimension reduction is achieved using contracted and global commodity sets. Future research will focus on decomposition methods such that the network components or parties are captured into a subsystem. The cooperation mechanism

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between control agents when multiple parties are present in the transportation network will also be addressed. Extending the approach to more application domains is currently under investigation. Acknowledgments This work was supported by the Portuguese Government, through Fundação para a Ciência e a Tecnologia, under the project PTDC/EEACRO/102102/2008 - AQUANET, through IDMEC under LAETA and by the VENI project “Intelligent multi-agent control for flexible coordination of transport hubs” (project 11210) of the Dutch Technology Foundation STW, a subdivision of the Netherlands Organisation for Scientific Research (NWO).

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