The proposed methodology for the integration of scheduling and control consists of two main steps. Firstly, a control scheme is deployed for the process at hand. Secondly, the process with its developed control scheme is then used to derive an approximate model, based on which a scheduling formulation is derived according to the principles in [183].
Process Control Strategy (PCS)
The process control strategy follows step-by-step the principles described in Chapter 2. The key point in this step comes after the formulation of the mp-MPC controllers. More specifically, the ‘high fidelity’ model with the verified controller scheme is treated as a single entity. The degrees of freedom of this entity are therefore the operational set-points of the process rather than the actual degrees of freedom of the ‘high fidelity’ model. This statement can be mathematically depicted in Equations 7.2 and 7.3.
‘High fidelity’ model d dtx(t) = f(x(t), uc(t), Y (t), d(t)) y= g(x(t), uc(t), Y (t), d(t)) (7.2) Approximate model d dtx(t) = f(x(t), uc,T, Y(t), d(t)) y= g(x(t), uc,T, Y(t), d(t)) uc,T = Ki· ◊T + ri, ’◊T œ CRi ◊T = [xT; uc,T≠1; dT; yT; yTSP] (7.3)
where x, d and y are the states, measured disturbances and outputs of the system re- spectively. uc are the optimal control actions and Y denotes any binary variables within the
formulation, here not a decision variable. In Eq. 7.2 the degrees of freedom of the system are the control variables. They are the only variables that the user can manipulate in order to affect the output of the system y. Note that we assume that the disturbances d are manipu- lated by external factors that cannot be affected. Via the formulation of a multi-parametric MPC and the inclusion of the optimal actions within the formulation the degrees of freedom to the system are altered. In Eq. 7.3 the control variables are a linear function of the states, the previous control variables, the disturbances, the outputs and the output set-points, in other words the parametric vector. It is clear that apart from the output set-points the rest of the parameters are dependent on the operation of the system. The subscript T in the Eq. 7.3 is used to denote the piecewise constant nature of the control variable within a single time step of the receding horizon optimization and is in line with the application of MPC. It becomes clear that the degrees of freedom of Eq. 7.3 are the output set-points. It is also clear that a step change in the output set-points does not result in an immediate and exact change in the system output due to constraints posed in the control problem (e.g. the mismatch until the set-point is reached in Figures 4.10 and 4.11).
The financially optimal operation of a system in the form of Eq. 7.3 is determined by two factors. Firstly, the conditions under which the system operates that result in an optimal profit, given the disturbances that appear in the system and secondly, the ability of the system to reach and maintain the aforementioned conditions efficiently. In other words, its profitable operation is determined by (i) a reactive scheduling formulation under uncertainty and (ii) the ability of the schedule to account and interact with the control scheme. Next, we describe the formulation of such a scheduling formulation that takes into account the ‘high fidelity’ model with its control scheme.
Process Scheduling Strategy (PSS)
The solution of multi-parametric scheduling problems has been previously discussed in [299] and [360]. In [183] the procedure for deriving and solving scheduling problems via multi- parametric programming, using a state-space model representation [336] and a mp-MILP reformulation is presented. The models used for scheduling in [183] and [336] are based on a stochastic model approach as they do not consider information regarding the process at hand or its dynamics.
Utilizing such an approach without including process information, for an integrated scheduling and control application, will result into a schedule that is not consistent with the process and will therefore create a mismatch between the ‘high fidelity’ mathematical model and the state-space model representation used in the schedule. In this work we pro- pose a scheduling formulation via a state-space model that is based on the closed-loop control
behavior of the ‘high fidelity’ model. Our approach is presented in Figure 7.1.
Approximate Model
with control system dynamics
Multi-Parametric Programming
Control aware scheduling formulation
(mpMILP) Process
‘High Fidelity’ Dynamic Model
with control
Figure 7.1: The PAROC based framework for the derivation of scheduling mpMILPs, based on the ‘high fidelity’ model with control
The procedure presented here is based on the PAROC framework and is described in the following steps:
PSS 1: ‘High Fidelity’ Model with control scheme – The ‘high fidelity’ model
with the control scheme is the basis upon which the scheduling strategy is designed. It is the result of the application of the PAROC framework as described in the Chapter 2. Its form is similar to Eq. 7.3.
PSS 2: Approximate model – Following an approximation step a linear state space
model is derived. On the contrary to the approximate models derived in Chapters 2 and 4, the approximate model in this case takes into account the control system dynamics. More specifically, in Chapters 2 and 4 the set of data used to identify the approximate models consist of step or impulse changes in the control variables and disturbances (uc and d of
Eq. 7.2) and tracking of the outputs (y of Eq. 7.2). In this case though, the degrees of freedom of the ‘high fidelity’ model have changed due to the inclusion of the control scheme. Therefore, the step or impulse changes of the control variables are replaced with step or impulse changes of the output set-points. An identified approximate state-space model is created with (i) linear dynamics in discrete time, (ii) discretization step of several orders of magnitude larger than the time step of the control and (iii) awareness of the process dynamics. Note that different control schemes for a single process could require multiple state-space approximate models in order to fully describe the systems operation.
PSS 3: mp-MILP and mp-QP reformulation and solution – The approximate
state-space model is used to formulate a MILP problem that corresponds to the economic scheduling of the process at hand. The multi-parametric version of this procedure is described in detail in [183]. It features an economic linear objective function that minimizes cost of operation (or maximizes revenue), linear equality constraints that arise from the state space model and polytopic constraints on the states, inputs, outputs and disturbances of the system. Commonly, binary variables are used to denote switching between different operating conditions. For instance, consider a power generation system with the ability to (i) buy power from a central grid in order to cover demand shortfalls and (ii) sell surplus power to the grid. This mutually exclusive operations are binary decision variables in the context of a scheduling formulation. Furthermore, in the case of multiple state-space models corresponding to different control schemes, binary variables decide which control scheme is applied. The schedule MILP is solved multi-parametrically using the POP•toolbox [258].R
The parameters in this case are the disturbances introduced to the system and the system operation (commonly denoted by the states variables of the state-space model).
The approximate model derived in PSS 2 involves a mismatch against the ‘high fidelity’ process model. A common characteristic of the PAROC framework when dealing with con- trol problems, which is hedged against via explicitly taking into account that mismatch. Part of this characteristic behavior of the MPC is attributed to its quadratic programming formulation and the system feedback1. In the case of the schedule though, the formulation
is linear and based on economics rather than process performance. Therefore, the mismatch is not at this point handled. This translates into the solution of the scheduling problem to include a mismatch. This is stochastically depicted in Figure 7.2.
Figure 7.2: Conceptual set-point mismatch due to the approximation step
Furthermore, the scheduling formulation is based on a state-space model with larger discretization time. Therefore, a single step in the scheduling receding horizon algorithm
1Note that in the case of linear MPC the absolute value formulation of the objective function serves in
requires a number of steps in the control receding horizon algorithm. During a single step in the scheduling algorithm there is no feedback from the ‘high fidelity’ model with control to the scheduling level. Therefore, the use of a receding horizon optimization formulation that will (i) bridge the time-scale gap between schedule and control and (ii) take into account the process model mismatch needs to be derived. The proposed methodology is presented in Figure 7.3 and described below. We formulate a QP that minimizes the mismatch between the schedule predicted output and the ‘high fidelity’ model with the control scheme. This surrogate model recalculates the optimal scheduling action and the control set-points at the control time interval level. Note that in the case of multiple modes of operation as a result of an optimal schedule (e.g. in process with multiple products) multiple surrogate models need to be derived. Both formulations are solved explicitly.
Approximate Model
with control system dynamics
Multi-Parametric Programming
Control aware surrogate for corrective
actions (mpQP) Process
‘High Fidelity’ Dynamic Model
with control
Figure 7.3: The PAROC based framework for the derivation of surrogate QPs based on the ‘high fidelity’ model with control
Together, the scheduling mp-MILP formulation and the surrogate mp-QP formulation represent the control aware schedule of the process (Figure 7.4).
Note that, as described in Chapter 1, the unification of design, scheduling and control can be expressed by a large scale MIDO problem, both the mismatch and the different time scales considered is a mathematical artifact inherent to the procedures used to make the problem approachable via the available optimization techniques.
PSS 4: Closed-Loop Validation – The closed loop validation of the scheme involves all
three stages and formulations. In a similar manner as in the PAROC framework the receding horizon optimization policies are tested against the original ‘high fidelity’ model using the
mpMILP
Control aware Scheduling
Process
‘High Fidelity’ Dynamic Model
with control
mpQP Approximate Model
with controlled system dynamics
Control aware
Scheduler
Control aware
Surrogate
Figure 7.4: The PAROC based framework for the derivation of control aware schedules with mismatch treatment
same computational tools. An overall depiction of the closed loop formulation is shown in Figure 7.5. In the figure, SP stands for set-point, u stands for optimization variable and y for system output (and it denotes the feedback). OP is the operating policy that the solution at each stage dictates for the lower stage. Furthermore, MM is the process mismatch and the subscripts T s and T c correspond to the scheduling and control time intervals and denote the frequency of the information exchange between the different stages.
Schedule (mpMILP & mpQP) Control Schemes (mpMPC) High-Fidelity Model Demand Scenario Measurements Optimal Action Control Setpoints & Feedback Discrete Decisions