IFAC PapersOnLine 51-20 (2018) 295–300
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10.1016/j.ifacol.2018.11.028
© 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Economic Nonlinear Model Predictive Control for Flexible Operation of
Air Separation Units
Adrian Caspari, Johannes M. M. Faust, Pascal Sch¨afer, Adel Mhamdi, Alexander Mitsos∗
AVT Process Systems Engineering, RWTH Aachen University, 52056 Aachen, Germany
∗e-mail: [email protected], [email protected]
Abstract: The integration of renewables into the portfolio of energy sources implies that dynamic operation of energy intensive processes, such as air separation, may give an economic advantage. Dynamic process operation can be achieved by applying economic nonlinear model predictive control (eNMPC). In this work, we present an in-silico case study of dynamic operation of an air separation process under fluctuating electricity prices. Using a day-ahead electricity price profile, an offline dynamic optimization (DO) problem is solved, which is used as an initial guess to start a fast update method deployed by the eNMPC. The eNMPC uses the same objective and constraints as the offline DO. However, the prediction horizon is shorter and current states and disturbances are taken into account. A first-principle air separation process model implemented in Modelica is used in all optimization problems. All optimization problems are solved using the DO framework DyOS. The process is flexibly operated by the application of the eNMPC, which leads to near-optimal economic process behavior during operation. This work demonstrates the contribution of model based process control to the integration of renewable energy sources in the supply chain of the process industry.
Keywords: economic model predictive control, renewable energy sources, nonlinear model predictive control, optimal control problem, demand side management, air separation
1. INTRODUCTION
The major renewable energy sources, wind and solar power, are naturally fluctuating and so is the electricity demand. Thus, the penetration of photovoltaic and wind turbines results in fluctuating electricity prices. Operators of energy intensive processes can obtain an economic bene- fit if they operate dynamically; this also allows for a higher penetration of renewable energy sources.
The intelligent management of the electricity demand is referred to as demand side management (DSM) (cf. the re- view of Zhang and Grossmann (2016)). An early contribu- tion is the work of Daryanian et al. (1989) presenting a con- ceptual framework for DSM focusing on a general problem formulation for industrial processes. Ghobeity and Mitsos (2010) apply DSM to a seawater reverse osmosis, assuming pseudo steady-state behavior. In other cases, the process dynamics have to be considered, and can be exploited ex- plicitly for DSM. This can be achieved online, during oper- ation by economic nonlinear model predictive control (eN- MPC) (Gr¨une and Pannek (2011); Amrit et al. (2013)).
eNMPC integrates the two traditionally separated fields of process scheduling and control (Pistikopoulos and Di- angelakis (2016)) by using an economic objective to be
The authors gratefully acknowledge the financial support of the Kopernikus project SynErgie by the Federal Ministry of Education and Research (BMBF) and the project supervision by the project management organization Projekttr¨ager J¨ulich (PtJ).
optimized during process operation with respect to the model equations and additional constraints. The resulting large-scale dynamic optimization (DO) problems have to be solved online in real time over a long time horizon in order to determine the optimal production schedule and the process inputs on the controller level at the same time.
The average performance and stability of eNMPC has been shown, e.g., in the work of Angeli et al. (2012). They show that the average eNMPC performance can not be worse than the best steady-state operation. In contrast to real time scheduling and control approaches such as presented by Pattison et al. (2017), eNMPC directly accounts for the economics inside the controller formulation. A major difficulty in eNMPC for large-scale processes is that large nonlinear dynamic process models have to be optimized in real time. Thus, fast update methods have been suggested enabling approximate solutions based on an initial input trajectory (cf. Wolf and Marquardt (2016)). Sensitivity- based update methods update the initial guess by tracking the solution of the optimality conditions. In sub-optimal update methods, introduced by the real time iteration concept of Diehl et al. (2002), the SQP iterations within the DO algorithm are restricted to a small number. One of the possibilities for selecting the initial guess is the the solution of an offline DO. This allows for an efficient fast update and near optimal solution since the offline solution is expected to be similar to the optimal solution during operation.
Copyright © 2018 IFAC 331
Economic Nonlinear Model Predictive Control for Flexible Operation of
Air Separation Units
Adrian Caspari, Johannes M. M. Faust, Pascal Sch¨afer, Adel Mhamdi, Alexander Mitsos∗
AVT Process Systems Engineering, RWTH Aachen University, 52056 Aachen, Germany
∗e-mail: [email protected], [email protected]
Abstract: The integration of renewables into the portfolio of energy sources implies that dynamic operation of energy intensive processes, such as air separation, may give an economic advantage. Dynamic process operation can be achieved by applying economic nonlinear model predictive control (eNMPC). In this work, we present an in-silico case study of dynamic operation of an air separation process under fluctuating electricity prices. Using a day-ahead electricity price profile, an offline dynamic optimization (DO) problem is solved, which is used as an initial guess to start a fast update method deployed by the eNMPC. The eNMPC uses the same objective and constraints as the offline DO. However, the prediction horizon is shorter and current states and disturbances are taken into account. A first-principle air separation process model implemented in Modelica is used in all optimization problems. All optimization problems are solved using the DO framework DyOS. The process is flexibly operated by the application of the eNMPC, which leads to near-optimal economic process behavior during operation. This work demonstrates the contribution of model based process control to the integration of renewable energy sources in the supply chain of the process industry.
Keywords: economic model predictive control, renewable energy sources, nonlinear model predictive control, optimal control problem, demand side management, air separation
1. INTRODUCTION
The major renewable energy sources, wind and solar power, are naturally fluctuating and so is the electricity demand. Thus, the penetration of photovoltaic and wind turbines results in fluctuating electricity prices. Operators of energy intensive processes can obtain an economic bene- fit if they operate dynamically; this also allows for a higher penetration of renewable energy sources.
The intelligent management of the electricity demand is referred to as demand side management (DSM) (cf. the re- view of Zhang and Grossmann (2016)). An early contribu- tion is the work of Daryanian et al. (1989) presenting a con- ceptual framework for DSM focusing on a general problem formulation for industrial processes. Ghobeity and Mitsos (2010) apply DSM to a seawater reverse osmosis, assuming pseudo steady-state behavior. In other cases, the process dynamics have to be considered, and can be exploited ex- plicitly for DSM. This can be achieved online, during oper- ation by economic nonlinear model predictive control (eN- MPC) (Gr¨une and Pannek (2011); Amrit et al. (2013)).
eNMPC integrates the two traditionally separated fields of process scheduling and control (Pistikopoulos and Di- angelakis (2016)) by using an economic objective to be
The authors gratefully acknowledge the financial support of the Kopernikus project SynErgie by the Federal Ministry of Education and Research (BMBF) and the project supervision by the project management organization Projekttr¨ager J¨ulich (PtJ).
optimized during process operation with respect to the model equations and additional constraints. The resulting large-scale dynamic optimization (DO) problems have to be solved online in real time over a long time horizon in order to determine the optimal production schedule and the process inputs on the controller level at the same time.
The average performance and stability of eNMPC has been shown, e.g., in the work of Angeli et al. (2012). They show that the average eNMPC performance can not be worse than the best steady-state operation. In contrast to real time scheduling and control approaches such as presented by Pattison et al. (2017), eNMPC directly accounts for the economics inside the controller formulation. A major difficulty in eNMPC for large-scale processes is that large nonlinear dynamic process models have to be optimized in real time. Thus, fast update methods have been suggested enabling approximate solutions based on an initial input trajectory (cf. Wolf and Marquardt (2016)). Sensitivity- based update methods update the initial guess by tracking the solution of the optimality conditions. In sub-optimal update methods, introduced by the real time iteration concept of Diehl et al. (2002), the SQP iterations within the DO algorithm are restricted to a small number. One of the possibilities for selecting the initial guess is the the solution of an offline DO. This allows for an efficient fast update and near optimal solution since the offline solution is expected to be similar to the optimal solution during operation.
Copyright © 2018 IFAC 331
Economic Nonlinear Model Predictive Control for Flexible Operation of
Air Separation Units
Adrian Caspari, Johannes M. M. Faust, Pascal Sch¨afer, Adel Mhamdi, Alexander Mitsos∗
AVT Process Systems Engineering, RWTH Aachen University, 52056 Aachen, Germany
∗e-mail: [email protected], [email protected]
Abstract: The integration of renewables into the portfolio of energy sources implies that dynamic operation of energy intensive processes, such as air separation, may give an economic advantage. Dynamic process operation can be achieved by applying economic nonlinear model predictive control (eNMPC). In this work, we present an in-silico case study of dynamic operation of an air separation process under fluctuating electricity prices. Using a day-ahead electricity price profile, an offline dynamic optimization (DO) problem is solved, which is used as an initial guess to start a fast update method deployed by the eNMPC. The eNMPC uses the same objective and constraints as the offline DO. However, the prediction horizon is shorter and current states and disturbances are taken into account. A first-principle air separation process model implemented in Modelica is used in all optimization problems. All optimization problems are solved using the DO framework DyOS. The process is flexibly operated by the application of the eNMPC, which leads to near-optimal economic process behavior during operation. This work demonstrates the contribution of model based process control to the integration of renewable energy sources in the supply chain of the process industry.
Keywords: economic model predictive control, renewable energy sources, nonlinear model predictive control, optimal control problem, demand side management, air separation
1. INTRODUCTION
The major renewable energy sources, wind and solar power, are naturally fluctuating and so is the electricity demand. Thus, the penetration of photovoltaic and wind turbines results in fluctuating electricity prices. Operators of energy intensive processes can obtain an economic bene- fit if they operate dynamically; this also allows for a higher penetration of renewable energy sources.
The intelligent management of the electricity demand is referred to as demand side management (DSM) (cf. the re- view of Zhang and Grossmann (2016)). An early contribu- tion is the work of Daryanian et al. (1989) presenting a con- ceptual framework for DSM focusing on a general problem formulation for industrial processes. Ghobeity and Mitsos (2010) apply DSM to a seawater reverse osmosis, assuming pseudo steady-state behavior. In other cases, the process dynamics have to be considered, and can be exploited ex- plicitly for DSM. This can be achieved online, during oper- ation by economic nonlinear model predictive control (eN- MPC) (Gr¨une and Pannek (2011); Amrit et al. (2013)).
eNMPC integrates the two traditionally separated fields of process scheduling and control (Pistikopoulos and Di- angelakis (2016)) by using an economic objective to be
The authors gratefully acknowledge the financial support of the Kopernikus project SynErgie by the Federal Ministry of Education and Research (BMBF) and the project supervision by the project management organization Projekttr¨ager J¨ulich (PtJ).
optimized during process operation with respect to the model equations and additional constraints. The resulting large-scale dynamic optimization (DO) problems have to be solved online in real time over a long time horizon in order to determine the optimal production schedule and the process inputs on the controller level at the same time.
The average performance and stability of eNMPC has been shown, e.g., in the work of Angeli et al. (2012). They show that the average eNMPC performance can not be worse than the best steady-state operation. In contrast to real time scheduling and control approaches such as presented by Pattison et al. (2017), eNMPC directly accounts for the economics inside the controller formulation. A major difficulty in eNMPC for large-scale processes is that large nonlinear dynamic process models have to be optimized in real time. Thus, fast update methods have been suggested enabling approximate solutions based on an initial input trajectory (cf. Wolf and Marquardt (2016)). Sensitivity- based update methods update the initial guess by tracking the solution of the optimality conditions. In sub-optimal update methods, introduced by the real time iteration concept of Diehl et al. (2002), the SQP iterations within the DO algorithm are restricted to a small number. One of the possibilities for selecting the initial guess is the the solution of an offline DO. This allows for an efficient fast update and near optimal solution since the offline solution is expected to be similar to the optimal solution during operation.
Copyright © 2018 IFAC 331
Economic Nonlinear Model Predictive Control for Flexible Operation of
Air Separation Units
Adrian Caspari, Johannes M. M. Faust, Pascal Sch¨afer, Adel Mhamdi, Alexander Mitsos∗
AVT Process Systems Engineering, RWTH Aachen University, 52056 Aachen, Germany
∗e-mail: [email protected], [email protected]
Abstract: The integration of renewables into the portfolio of energy sources implies that dynamic operation of energy intensive processes, such as air separation, may give an economic advantage. Dynamic process operation can be achieved by applying economic nonlinear model predictive control (eNMPC). In this work, we present an in-silico case study of dynamic operation of an air separation process under fluctuating electricity prices. Using a day-ahead electricity price profile, an offline dynamic optimization (DO) problem is solved, which is used as an initial guess to start a fast update method deployed by the eNMPC. The eNMPC uses the same objective and constraints as the offline DO. However, the prediction horizon is shorter and current states and disturbances are taken into account. A first-principle air separation process model implemented in Modelica is used in all optimization problems. All optimization problems are solved using the DO framework DyOS. The process is flexibly operated by the application of the eNMPC, which leads to near-optimal economic process behavior during operation. This work demonstrates the contribution of model based process control to the integration of renewable energy sources in the supply chain of the process industry.
Keywords: economic model predictive control, renewable energy sources, nonlinear model predictive control, optimal control problem, demand side management, air separation
1. INTRODUCTION
The major renewable energy sources, wind and solar power, are naturally fluctuating and so is the electricity demand. Thus, the penetration of photovoltaic and wind turbines results in fluctuating electricity prices. Operators of energy intensive processes can obtain an economic bene- fit if they operate dynamically; this also allows for a higher penetration of renewable energy sources.
The intelligent management of the electricity demand is referred to as demand side management (DSM) (cf. the re- view of Zhang and Grossmann (2016)). An early contribu- tion is the work of Daryanian et al. (1989) presenting a con- ceptual framework for DSM focusing on a general problem formulation for industrial processes. Ghobeity and Mitsos (2010) apply DSM to a seawater reverse osmosis, assuming pseudo steady-state behavior. In other cases, the process dynamics have to be considered, and can be exploited ex- plicitly for DSM. This can be achieved online, during oper- ation by economic nonlinear model predictive control (eN- MPC) (Gr¨une and Pannek (2011); Amrit et al. (2013)).
eNMPC integrates the two traditionally separated fields of process scheduling and control (Pistikopoulos and Di- angelakis (2016)) by using an economic objective to be
The authors gratefully acknowledge the financial support of the Kopernikus project SynErgie by the Federal Ministry of Education and Research (BMBF) and the project supervision by the project management organization Projekttr¨ager J¨ulich (PtJ).
optimized during process operation with respect to the model equations and additional constraints. The resulting large-scale dynamic optimization (DO) problems have to be solved online in real time over a long time horizon in order to determine the optimal production schedule and the process inputs on the controller level at the same time.
The average performance and stability of eNMPC has been shown, e.g., in the work of Angeli et al. (2012). They show that the average eNMPC performance can not be worse than the best steady-state operation. In contrast to real time scheduling and control approaches such as presented by Pattison et al. (2017), eNMPC directly accounts for the economics inside the controller formulation. A major difficulty in eNMPC for large-scale processes is that large nonlinear dynamic process models have to be optimized in real time. Thus, fast update methods have been suggested enabling approximate solutions based on an initial input trajectory (cf. Wolf and Marquardt (2016)). Sensitivity- based update methods update the initial guess by tracking the solution of the optimality conditions. In sub-optimal update methods, introduced by the real time iteration concept of Diehl et al. (2002), the SQP iterations within the DO algorithm are restricted to a small number. One of the possibilities for selecting the initial guess is the the solution of an offline DO. This allows for an efficient fast update and near optimal solution since the offline solution is expected to be similar to the optimal solution during operation.
Copyright © 2018 IFAC 331
Economic Nonlinear Model Predictive Control for Flexible Operation of
Air Separation Units
Adrian Caspari, Johannes M. M. Faust, Pascal Sch¨afer, Adel Mhamdi, Alexander Mitsos∗
AVT Process Systems Engineering, RWTH Aachen University, 52056 Aachen, Germany
∗e-mail: [email protected], [email protected]
Abstract: The integration of renewables into the portfolio of energy sources implies that dynamic operation of energy intensive processes, such as air separation, may give an economic advantage. Dynamic process operation can be achieved by applying economic nonlinear model predictive control (eNMPC). In this work, we present an in-silico case study of dynamic operation of an air separation process under fluctuating electricity prices. Using a day-ahead electricity price profile, an offline dynamic optimization (DO) problem is solved, which is used as an initial guess to start a fast update method deployed by the eNMPC. The eNMPC uses the same objective and constraints as the offline DO. However, the prediction horizon is shorter and current states and disturbances are taken into account. A first-principle air separation process model implemented in Modelica is used in all optimization problems. All optimization problems are solved using the DO framework DyOS. The process is flexibly operated by the application of the eNMPC, which leads to near-optimal economic process behavior during operation. This work demonstrates the contribution of model based process control to the integration of renewable energy sources in the supply chain of the process industry.
Keywords: economic model predictive control, renewable energy sources, nonlinear model predictive control, optimal control problem, demand side management, air separation
1. INTRODUCTION
The major renewable energy sources, wind and solar power, are naturally fluctuating and so is the electricity demand. Thus, the penetration of photovoltaic and wind turbines results in fluctuating electricity prices. Operators of energy intensive processes can obtain an economic bene- fit if they operate dynamically; this also allows for a higher penetration of renewable energy sources.
The intelligent management of the electricity demand is referred to as demand side management (DSM) (cf. the re- view of Zhang and Grossmann (2016)). An early contribu- tion is the work of Daryanian et al. (1989) presenting a con- ceptual framework for DSM focusing on a general problem formulation for industrial processes. Ghobeity and Mitsos (2010) apply DSM to a seawater reverse osmosis, assuming pseudo steady-state behavior. In other cases, the process dynamics have to be considered, and can be exploited ex- plicitly for DSM. This can be achieved online, during oper- ation by economic nonlinear model predictive control (eN- MPC) (Gr¨une and Pannek (2011); Amrit et al. (2013)).
eNMPC integrates the two traditionally separated fields of process scheduling and control (Pistikopoulos and Di- angelakis (2016)) by using an economic objective to be
The authors gratefully acknowledge the financial support of the Kopernikus project SynErgie by the Federal Ministry of Education and Research (BMBF) and the project supervision by the project management organization Projekttr¨ager J¨ulich (PtJ).
optimized during process operation with respect to the model equations and additional constraints. The resulting large-scale dynamic optimization (DO) problems have to be solved online in real time over a long time horizon in order to determine the optimal production schedule and the process inputs on the controller level at the same time.
The average performance and stability of eNMPC has been shown, e.g., in the work of Angeli et al. (2012). They show that the average eNMPC performance can not be worse than the best steady-state operation. In contrast to real time scheduling and control approaches such as presented by Pattison et al. (2017), eNMPC directly accounts for the economics inside the controller formulation. A major difficulty in eNMPC for large-scale processes is that large nonlinear dynamic process models have to be optimized in real time. Thus, fast update methods have been suggested enabling approximate solutions based on an initial input trajectory (cf. Wolf and Marquardt (2016)). Sensitivity- based update methods update the initial guess by tracking the solution of the optimality conditions. In sub-optimal update methods, introduced by the real time iteration concept of Diehl et al. (2002), the SQP iterations within the DO algorithm are restricted to a small number. One of the possibilities for selecting the initial guess is the the solution of an offline DO. This allows for an efficient fast update and near optimal solution since the offline solution is expected to be similar to the optimal solution during operation.
Copyright © 2018 IFAC 331
Economic Nonlinear Model Predictive Control for Flexible Operation of
Air Separation Units
Adrian Caspari, Johannes M. M. Faust, Pascal Sch¨afer, Adel Mhamdi, Alexander Mitsos∗
AVT Process Systems Engineering, RWTH Aachen University, 52056 Aachen, Germany
∗e-mail: [email protected], [email protected]
Abstract: The integration of renewables into the portfolio of energy sources implies that dynamic operation of energy intensive processes, such as air separation, may give an economic advantage. Dynamic process operation can be achieved by applying economic nonlinear model predictive control (eNMPC). In this work, we present an in-silico case study of dynamic operation of an air separation process under fluctuating electricity prices. Using a day-ahead electricity price profile, an offline dynamic optimization (DO) problem is solved, which is used as an initial guess to start a fast update method deployed by the eNMPC. The eNMPC uses the same objective and constraints as the offline DO. However, the prediction horizon is shorter and current states and disturbances are taken into account. A first-principle air separation process model implemented in Modelica is used in all optimization problems. All optimization problems are solved using the DO framework DyOS. The process is flexibly operated by the application of the eNMPC, which leads to near-optimal economic process behavior during operation. This work demonstrates the contribution of model based process control to the integration of renewable energy sources in the supply chain of the process industry.
Keywords: economic model predictive control, renewable energy sources, nonlinear model predictive control, optimal control problem, demand side management, air separation
1. INTRODUCTION
The major renewable energy sources, wind and solar power, are naturally fluctuating and so is the electricity demand. Thus, the penetration of photovoltaic and wind turbines results in fluctuating electricity prices. Operators of energy intensive processes can obtain an economic bene- fit if they operate dynamically; this also allows for a higher penetration of renewable energy sources.
The intelligent management of the electricity demand is referred to as demand side management (DSM) (cf. the re- view of Zhang and Grossmann (2016)). An early contribu- tion is the work of Daryanian et al. (1989) presenting a con- ceptual framework for DSM focusing on a general problem formulation for industrial processes. Ghobeity and Mitsos (2010) apply DSM to a seawater reverse osmosis, assuming pseudo steady-state behavior. In other cases, the process dynamics have to be considered, and can be exploited ex- plicitly for DSM. This can be achieved online, during oper- ation by economic nonlinear model predictive control (eN- MPC) (Gr¨une and Pannek (2011); Amrit et al. (2013)).
eNMPC integrates the two traditionally separated fields of process scheduling and control (Pistikopoulos and Di- angelakis (2016)) by using an economic objective to be
The authors gratefully acknowledge the financial support of the Kopernikus project SynErgie by the Federal Ministry of Education and Research (BMBF) and the project supervision by the project management organization Projekttr¨ager J¨ulich (PtJ).
optimized during process operation with respect to the model equations and additional constraints. The resulting large-scale dynamic optimization (DO) problems have to be solved online in real time over a long time horizon in order to determine the optimal production schedule and the process inputs on the controller level at the same time.
The average performance and stability of eNMPC has been shown, e.g., in the work of Angeli et al. (2012). They show that the average eNMPC performance can not be worse than the best steady-state operation. In contrast to real time scheduling and control approaches such as presented by Pattison et al. (2017), eNMPC directly accounts for the economics inside the controller formulation. A major difficulty in eNMPC for large-scale processes is that large nonlinear dynamic process models have to be optimized in real time. Thus, fast update methods have been suggested enabling approximate solutions based on an initial input trajectory (cf. Wolf and Marquardt (2016)). Sensitivity- based update methods update the initial guess by tracking the solution of the optimality conditions. In sub-optimal update methods, introduced by the real time iteration concept of Diehl et al. (2002), the SQP iterations within the DO algorithm are restricted to a small number. One of the possibilities for selecting the initial guess is the the solution of an offline DO. This allows for an efficient fast update and near optimal solution since the offline solution is expected to be similar to the optimal solution during operation.
Copyright © 2018 IFAC 331
Optimal operation of air separation processes (ASU) is challenging since it requires the fast DO of a large-scale nonlinear process model under strong restrictions on the product purity and other constraints. Additionally, ASU is important as a large electricity consumer; it accounted for 2.5 % of the total industrial electricity consumption in the U.S. in 20101. Chen et al. (2010) showed a NMPC method applied to the isolated upper column of an ASU.
They used a tracking controller based on full discretization and a reduced order dynamic column model. Huang and Biegler (2012) presented a framework of eNMPC using an electricity price prediction model and the known electricity price from the day ahead auction. In a real time case study, they presented closed-loop simulation results of an ASU. They used their earlier introduced advanced-step NMPC (Huang et al. (2009)), that applies a sensitivity- based fast update using simultaneous collocation. Patti- son et al. (2017) presented a real time scheduling and control strategy of closed-loop scheduling on a moving horizon based on dynamic models, which they applied to an ASU. They used data-driven models for the closed-loop process representation, which they further optimized on- line. However, their framework requires the entire closed- loop process represented as a model to be optimized. A linearizing tracking controller was used by Sch¨afer et al.
(2018) to improve the agility of an ASU. To the authors’
best knowledge, eNMPC with a fast update method based on a single-shooting algorithm was not applied yet to a complete ASU.
In this work, an eNMPC scheme based on a sub-optimal fast update method is applied to an ASU. The eNMPC computation is initialized by the solution of an offline DO problem. A closed-loop simulation over a time horizon of one day is performed based on the known electricity price as from the German day-ahead auction, and assuming a perturbed product demand profile. This is a realistic assumption in cases where the process is directly coupled to a downstream process and the exact product demand is not known much time before but rather becomes known in the course of the operation. Different eNMPC settings are evaluated. The closed-loop eNMPC process operation scheme is compared to the standard steady-state process operation and to the optimal operation under known dis- turbances. The eNMPC improves the process economics and achieves near-optimal process operation despite the presence of disturbances. The optimization problems are solved using the DO framework DyOS2. In this work, full state feedback for the eNMPC is used. Plant-model mismatch is not accounted for and it is assumed that the eNMPC considers the disturbances as soon as they are within the prediction horizon. Further, the process is directly controlled by the eNMPC, no base-layer control is assumed.
The article is structured as follows. In Section 2 the overall process operation scheme is introduced. The ASU case study is discussed in Section 3, before the conclusions are drawn in Section 4.
1 United States Energy Information Administration Database.
Available at: https://www.eia.gov. Accessed April 2018.
2 RWTH Aachen University, Aachener Verfahrenstechnik - Process Systems Engineering (AVT.SVT), url: http://permalink.avt.rwth- aachen.de/?id=295232. Accessed April 2018.
electricity market, demand forecast
Offline Dynamic Optimization
eNMPC
Process disturbances
initial guess
process
control feedback
Fig. 1. eNMPC scheme for flexible process operation.
2. ENMPC STRATEGY
The process operation scheme depicted in Figure 1 is applied in this work using a moving horizon, where a DO problem is repeatedly solved online using the current process states and a process model (Binder et al. (2001)).
A DO problem on a finite time horizonT = [t0, tf] of the following form has to be solved online:
minu tf
t0
L(x(t),y(t),u(t),p(t))dt (1a) s.t.x(t) =˙ f(x(t),y(t),u(t),p(t)), ∀t∈ T (1b) 0=g(x(t),y(t),u(t),p(t)), ∀t∈ T (1c) 0=h(x(t0),y(t0),p(t0)) (1d) 0≥c( ˙x(t),x(t),y(t),u(t),p(t)), t∈ T (1e) wheref:Rnx×Rny×Rnu×Rnp→Rnxandg:Rnx×Rny× Rnu ×Rnp → Rny define the semi-explicit differential- algebraic system of index 1, whileh:Rnx×Rny ×Rnu× Rnp →Rnx indicates the initial conditions and c:Rnx× Rnx×Rny×Rnu×Rnp→Rncare the constraints. The DO problem (1) aims at finding optimal control trajectories u: T →Rnu, differential and algebraic state trajectories x : T → Rnx and y : T → Rny, respectively, for given parameter valuesp:T →Rnpincluding the disturbances, in the sense of a local minimum of the objective function that is the integral of L: Rnx ×Rny ×Rnu ×Rnp → R. cmaybe used, e.g., to constrain rates of changes of input variables.
Problem (1) may be too hard to be solved online in real time to optimality. In this work, a suboptimal fast update is applied within the eNMPC. The solution of an offline DO is used to initialize the eNMPC. Larger disturbances would require to solve the DO again during operation to improve the initial guesses for the eNMPC (cf. the process operation architecture of Kadam and Marquardt (2004)).
Within the offline DO step, (1) is solved over a time horizon of one day assuming unperturbed profiles for the disturbance trajectoriesp(t). In this case study, p(t) the product demand and electricity price of the day ahead as shown in Figures 3 and 4. The computed solution of the DO is used as an initial guess for the eNMPC,
liq
Liquifier Evaporator
Rectification Column Heat
Exchanger Zone 1
Heat Exchanger
Zone 2 Turbine
Compressor
mac
Nitrogen Product
Waste
Storage Tank
Integrated Reboiler and
Condenster
Drain
top
drain
Feed Air
tank
Fig. 2. ASU flowsheet. The large bold symbols mark the control variables for the DO and the eNMPC.
which updates it by performing limited SQP iterations on problem (1) over the time of the prediction horizon.
The control horizon of the eNMPC is the same as its prediction horizon. The eNMPC uses exactly the same problem formulation (1) as the offline DO except that the time horizon is different and the initial conditions are set to the current process state following the moving horizon paradigm. The assumptions for the DO are the same for each day, such that the eNMPC always starts from the same initial guess extracted for the respective prediction horizon.
The updated input trajectory for one prediction horizon resulting from the eNMPC is directly sent to the process.
The process is then operated using the input trajectories for one sampling time, before the current process state is fed back to the eNMPC as new initial state of the process model from which the eNMPC problem is solved again.
The process model (cf. Section 3) is implemented in Modelica3 and exported as FMU4, which is used as the model for all optimizations as well as for the process simulations. The DO problems are solved in the software framework DyOS using adaptive single-shooting (Schlegel et al. (2005)). The closed-loop simulation is implemented in MATLAB 2017b5 using DyOS via a MEX subroutine.
SNOPT (Gill et al. (2005)) is applied as NLP solver and LIMEX (Schlegel et al. (2004)) as DAE integrator.
A Windows 7 desktop computer equipped with an Intel Core(TM) i3-6100 processor running at 3.7 GHz and 8 GB RAM is used for all computations of this work.
3. AIR SEPARATION APPLICATION
The configuration of the cryogenic ASU, depicted in Figure 2, is similar to the one presented by Pattison et al. (2017), where a detailed process description can be found. The main energy consumption of the process is the feed air compression and the liquefaction of the product nitrogen.
Air is compressed from its ambient state to 10 bar and cooled down by heat-integration and expansion. It is separated into its components in a rectification column at
3 https://www.modelica.org/. Accessed April 2018
4 FMI standard (2014). Functional mock-up interface for model exchange and co-simulation. url: http://fmi-standard.org. Accessed April 2018.
5 The MathWorks, Inc.
0 4 8 12 16 20 24
20 30 40 50
time [h] cel[e/MWh]
Fig. 3. Electricity price of the day ahead auction adapted from6. This is a typical, non-extreme electricity price profile.
0 4 8 12 16 20 24
155 165
Time [h] Fproduct [mol/s]
Fig. 4. Product demand assumed in the dynamic optimiza- tion. The solid line shows the molar nitrogen product stream as assumed in the dynamic optimization. The dashed line shows the perturbed molar product stream .
a pressure of 6.6 bar comprising of an integrated reboiler and condenser. Nitrogen leaves the process as product. 3.1 Process Model
The process model comprises of 127 differential and 2940 algebraic states, 5 input variables, and 1032 constants. It is similar to the one used in Pattison et al. (2017). The most important differences are summarized in the following. Thermodynamics: The vapor liquid equilibrium is calcu- lated using the isofugacity condition. The pure compo- nents’ vapor pressures are computed using the extended Antoine equation. The ideal gas law is used for the vapor phase. The liquid phase activity coefficient is calculated using the NRTL model. All parameter values for the ther- modynamic models are retrieved from Aspen Plus version 8.87. Using NRTL is expected to be more accurate than the Margules model used by Pattison et al. (2017). Rectification Column: The rectification column is de- scribed by a stage-by-stage model. The column stages are modeled with dynamic MESH-equations using the following assumptions: (i) negligible vapor hold up, (ii) fast temperature dynamics, (iii) a constant pressure difference between neighboring stages and (iv) a linear relation for stage hydraulics. The column is modeled by 40 equilibrium stages.
Heat exchanger: The heat exchanger is modeled by a spatially distributed model discretized using steady-state energy and mass balance equations. The heat exchanger
6 Fraunhofer ISE. Electricity production and spot prices in Germany in May 2017. url: https://www.energy- charts.de/price.htm?year=2017&auction=1h&month=5. Accessed April 2018.
7 Aspen Technology, Inc.
liq
Liquifier Evaporator
Rectification Column Heat
Exchanger Zone 1
Heat Exchanger
Zone 2 Turbine
Compressor
mac
Nitrogen Product
Waste
Storage Tank
Integrated Reboiler and
Condenster
Drain
top
drain
Feed Air
tank
Fig. 2. ASU flowsheet. The large bold symbols mark the control variables for the DO and the eNMPC.
which updates it by performing limited SQP iterations on problem (1) over the time of the prediction horizon.
The control horizon of the eNMPC is the same as its prediction horizon. The eNMPC uses exactly the same problem formulation (1) as the offline DO except that the time horizon is different and the initial conditions are set to the current process state following the moving horizon paradigm. The assumptions for the DO are the same for each day, such that the eNMPC always starts from the same initial guess extracted for the respective prediction horizon.
The updated input trajectory for one prediction horizon resulting from the eNMPC is directly sent to the process.
The process is then operated using the input trajectories for one sampling time, before the current process state is fed back to the eNMPC as new initial state of the process model from which the eNMPC problem is solved again.
The process model (cf. Section 3) is implemented in Modelica3 and exported as FMU4, which is used as the model for all optimizations as well as for the process simulations. The DO problems are solved in the software framework DyOS using adaptive single-shooting (Schlegel et al. (2005)). The closed-loop simulation is implemented in MATLAB 2017b5 using DyOS via a MEX subroutine.
SNOPT (Gill et al. (2005)) is applied as NLP solver and LIMEX (Schlegel et al. (2004)) as DAE integrator.
A Windows 7 desktop computer equipped with an Intel Core(TM) i3-6100 processor running at 3.7 GHz and 8 GB RAM is used for all computations of this work.
3. AIR SEPARATION APPLICATION
The configuration of the cryogenic ASU, depicted in Figure 2, is similar to the one presented by Pattison et al. (2017), where a detailed process description can be found. The main energy consumption of the process is the feed air compression and the liquefaction of the product nitrogen.
Air is compressed from its ambient state to 10 bar and cooled down by heat-integration and expansion. It is separated into its components in a rectification column at
3 https://www.modelica.org/. Accessed April 2018
4 FMI standard (2014). Functional mock-up interface for model exchange and co-simulation. url: http://fmi-standard.org. Accessed April 2018.
5 The MathWorks, Inc.
0 4 8 12 16 20 24
20 30 40 50
time [h]
cel[e/MWh]
Fig. 3. Electricity price of the day ahead auction adapted from6. This is a typical, non-extreme electricity price profile.
0 4 8 12 16 20 24
155 165
Time [h]
Fproduct [mol/s]
Fig. 4. Product demand assumed in the dynamic optimiza- tion. The solid line shows the molar nitrogen product stream as assumed in the dynamic optimization.
The dashed line shows the perturbed molar product stream .
a pressure of 6.6 bar comprising of an integrated reboiler and condenser. Nitrogen leaves the process as product.
3.1 Process Model
The process model comprises of 127 differential and 2940 algebraic states, 5 input variables, and 1032 constants. It is similar to the one used in Pattison et al. (2017). The most important differences are summarized in the following.
Thermodynamics: The vapor liquid equilibrium is calcu- lated using the isofugacity condition. The pure compo- nents’ vapor pressures are computed using the extended Antoine equation. The ideal gas law is used for the vapor phase. The liquid phase activity coefficient is calculated using the NRTL model. All parameter values for the ther- modynamic models are retrieved from Aspen Plus version 8.87. Using NRTL is expected to be more accurate than the Margules model used by Pattison et al. (2017).
Rectification Column: The rectification column is de- scribed by a stage-by-stage model. The column stages are modeled with dynamic MESH-equations using the following assumptions: (i) negligible vapor hold up, (ii) fast temperature dynamics, (iii) a constant pressure difference between neighboring stages and (iv) a linear relation for stage hydraulics. The column is modeled by 40 equilibrium stages.
Heat exchanger: The heat exchanger is modeled by a spatially distributed model discretized using steady-state energy and mass balance equations. The heat exchanger
6 Fraunhofer ISE. Electricity production and spot prices in Germany in May 2017. url: https://www.energy- charts.de/price.htm?year=2017&auction=1h&month=5. Accessed April 2018.
7 Aspen Technology, Inc.
10 20 30 40 50
Impurity[10−6mol/mol]
eNMPC offline DO Bounds Benchmark optimal disturbed
2.5 2.6 2.7 2.8 2.9
TankHoldup[106mol]
0 4 8 12 16 20 24
0 5 10 15
Time [h]
Ftank,out[mol/s]
Fig. 5. Trajectories of selected constrained states as a result of the offline DO, the closed-loop simulation (eNMPC), the benchmark simulation, and the op- timal operation under known uncertainties (optimal disturbed). Benchmark is a constant production rate.
The eNMPC uses 3 SQP iterations, a prediction hori- zon of 8 hours, and a sampling time of 15 min.
is separated into two segments corresponding to the ther- modynamic state, two phase and one phase respectively, of the streams, each comprising of 5 finite volumes.
Utilities: The turbine and compressor are modeled with an isentropic efficiency. A maximum electrical efficiency of 0.9 at the steady-state operating point is used. The liquefier is modeled as an equilibrium stage. The heat is withdrawn assuming a refrigeration cycle with an efficiency of 40%.
The evaporator is modeled as an equilibrium stage where the heat is supplied from the environment.
3.2 Operation Scenarios
The operating cost of the process is taken as the objective in (1) , i.e.,L=cel×(Pcomp+Pliq−Ptur), wherecel is the the electricity price and Pcomp/tur/liq is the electric- ity demand/supply of the compressor, turbine, and the liquifier, respectively. The streamsFmac,Fliq,Ftop,Ftank, and Fdrain are the control variables u in problem (1).
Accordingly, the product stream is given as p(t) and the eNMPC has to determine how much of the product stream is withdrawn from the tank and how much is coming directly from the rectification column. Temperatures, tem-
perature differences inside the heat exchanger, the product impurity, and the storage tank level are constrained. The impurity is path-constrained by 5·10−5 mol/mol as the upper bound. The temperatures of the streams leaving the heat exchanger are constrained according to the vapor and dew points of these streams to guarantee the desired phases. The temperature differences are constrained to be positive in order to guarantee a physically meaningful solution. The reboiler holdup is constrained to a minimum of 105 mol due to the fact, that a bath type condenser is used, which has to be fully submerged in liquid. The drain and the storage tank level are constrained to a minimum of zero and the storage tank level is additionally endpoint- constrained, such that its content at the end is at least that of the beginning. The tank level is endpoint-constrained to be larger than 279·104 mol, which corresponds to the product amount of 5 hours constant production.
The offline DO is solved using 5 grid refinement steps for the adaptive control vector parametrization starting with an initial grid of 8 equidistant piecewise-constant elements.
The eNMPC uses a sampling time of 15 minutes and a prediction horizon of 8 hours. Each eNMPC iteration uses the control variable profile obtained by the DO for the respective time horizon. The number of SQP iterations is restricted to 3. The controller performance is compared to an ideal eNMPC, i.e., an eNMPC with a maximum of 250 SQP iteration, i.e., literally without restriction of SQP iterations. The absolute and relative integrator tolerance is set to 10−4. The NLP solver uses 10−4 as optimality tolerance and feasibility tolerance. As apposed to full discretization, the shooting algorithm, applied in this work, always guarantees the model equations are solved in the given integrator tolerance even if the SQP iterations are restricted. Different prediction horizons have been investigated. In this work, only the results for the 8 hour prediction horizon are presented due to space limitations.
As a benchmark scenario, a typical steady-state opera- tion is considered. Due to an increased product demand, product is withdrawn from the storage tank. The tank is consequently less filled with product. This difference is considered by adding the average cost for producing the storage tank holdup difference, i.e., the cost of producing exactly the amount of product required to fill the tank up to its nominal level using the average electricity price of the current day. The same correction is applied for a deviating tank hold up of the eNMPC-controlled process.
In addition, the closed-loop simulation is compared to the optimal process operation under known disturbances.
Therefore, the problem (1) is solved over a horizon of one day using the disturbed input profile shown in Figure 4.
3.3 Closed-loop Simulations
The closed-loop process results of the scenarios are pre- sented in this subsection; closed-loop refers to repeatedly solving the eNMPC problem on a moving horizon.
The most interesting trajectories are shown in Figures 5 and 6. The results indicate a satisfying and near-optimal process operation through the eNMPC strategy. The tra- jectories of the eNMPC-controlled process are similar to the optimal trajectories under known disturbances. As Figure 5 indicates, the constraints are satisfied, except for an intermediate violation of the impurity constraint, which
Table 1. Cost correction and economic evaluation of the benchmark process operation, the optimal solution under known disturbances, and the eNMPC-controlled closed-loop process simulation for 3 and 250 SQP iterations using a prediction horizon of 8 hours and a sampling time of 15 min. Tank holdup differences are considered by correcting the operation cost, thus it is a posteriory accounted for missing product in the tank. The economic improvement relates
to the benchmark process operation.
Benchmark Optimal eNMPC 3 SQP eNMPC 250 SQP
operating cost [e] 3249 3256 3276 3256
product [103mol] 13536 13536 13536 13536
tank holdup difference [103mol] 288 0 37.466 26.870
cost correction [e] 69 0 9 6
corrected cost [e] 3318 3256 3285 3262
economic improvement [%] 0.0 1.9 1.0 1.7
is assumed to be negligible.
The benchmark operation satisfies the constraints, how- ever, the intended tank holdup level is not reached, due to the disturbance of an increased product demand. The eNMPC-controlled process seems to converge to the de- sired tank holdup, however does not reach it exactly.
Therefore the operating cost is corrected as explained before. The results of this correction are shown in Table 1.
The eNMPC strategy leads to an economical improvement of about 1 % compared to the benchmark.
The control variable trajectories are shown in Figure 6.
A comparison with Figure 3 reveals that the process feed streamFmacis high at low electricity prices and vice versa.
During peak production, the storage tank is filled as the Ftank profile indicates. As it can be seen from Table 1, there is a trade off between the 3 SQP iterations restricted and the ideal eNMPC in economic improvement and com- putational burden, i.e., the ideal eNMPC is closer to the optimal solution, however, it is not online applicable due to high CPU times. The ideal eNMPC is closer to the optimal case than the 3 SQP eNMPC and achieves an economic improvement of 1.7 % with respect to the benchmark.
More SQP iterations lead to a larger deviation of the solution from its initial guess. There is thus an otpimal restriction of SQP iterations balancing the computational effort and economic improvement. It can further be seen that the tank level is closer to its nominal value for the ideal eNMPC than for the 3 SQP eNMPC which results in lower additional cost due to the cost correction. The operational cost of the ideal eNMPC case is the same as for the optimal case; the economical losses are only due to the tank level difference cost correction.
In addition, it should be noted that the tank level end- point constraint limits the flexibility of the process under eNMPC control, since it forces the eNMPC to operate the process such that the tank is at least at its nominal level after each prediction horizon. However, by using this constraint an extensive unloading of the tank during one control horizon is limited, which could be disadvantegous in later control cycles. It is worth further studying this effect as well as possible reformulations of the eNMPC op- timization problem to account for the endpoint constraint.
In summary, the appliedbottom-upstrategy based on eN- MPC that uses a sub-optimal fast update method leads to an efficient, near-optimal process operation. The economic advantage is expected to be even stronger, since a produc- tion rate dependent electrical efficiency is assumed in the model, which has its maximum at the constant operation point of the benchmark. This means, the higher the devia-
tion of the process state from the benchmark operation is, the smaller is the electrical efficiency that directly influ- ences the operational cost. In addition, it can be expected that larger disturbances and more extreme electricity price profiles will further increase the advantage of the eNMPC strategy compared to the benchmark operation, since the eNMPC can exploit the prices differences more extensively while satisfying the operation constraints.
The solution time for the offline DO is less than 8 hours. The solution time for the eNMPC using 3 SQP iterations is less than 120 CPU seconds. This is less than the sampling time of the eNMPC and can be used online. The solution time for the ideal eNMPC is much longer than the eNMPC sampling time and can thus not be regarded to be real time applicable. The solution times motivate further algorithmic improvements to accelerate the optimization and obtain online tractability of the method even for larger process models than used here.
4. CONCLUSION
A process operation strategy based on an eNMPC using a sub-optimal fast update method, that updates the solution of an offline DO during process operation, is applied. Both the offline DO and the eNMPC layer use the same first- principle model. A closed-loop scenario is simulated in- silico assuming a product demand disturbance during op- eration in contrast to the constant demand profile assumed for the DO. The eNMPC successfully controls the process and achieves near optimal process behavior. Based on an in-silico closed-loop case study, the trade-off between computational time and economical performance for dif- ferent SQP iteration number restrictions of the eNMPC is illustrated. The solution times of the offline DO and the eNMPC would allow online applicability, however they also motivate further algorithmic improvements. The so- lution time may also be reduced by using reduced models, however this also may lead the model unacceptable accu- racy. The delays resulting from the solution times of the optimization problems will be addressed in the future. The comparison of the eNMPC operation with and without base-layer control is also an interesting task of further work as well as studying the influence of different initial trajectories of the eNMPC.
ACKNOWLEDGEMENTS
The authors thank Andreas Peschel, Florian Schliebitz, and Gerhard Zapp from Linde AG and Falco C. Jung from AVT, RWTH Aachen for fruitful discussions.
Table 1. Cost correction and economic evaluation of the benchmark process operation, the optimal solution under known disturbances, and the eNMPC-controlled closed-loop process simulation for 3 and 250 SQP iterations using a prediction horizon of 8 hours and a sampling time of 15 min. Tank holdup differences are considered by correcting the operation cost, thus it is a posteriory accounted for missing product in the tank. The economic improvement relates
to the benchmark process operation.
Benchmark Optimal eNMPC 3 SQP eNMPC 250 SQP
operating cost [e] 3249 3256 3276 3256
product [103mol] 13536 13536 13536 13536
tank holdup difference [103mol] 288 0 37.466 26.870
cost correction [e] 69 0 9 6
corrected cost [e] 3318 3256 3285 3262
economic improvement [%] 0.0 1.9 1.0 1.7
is assumed to be negligible.
The benchmark operation satisfies the constraints, how- ever, the intended tank holdup level is not reached, due to the disturbance of an increased product demand. The eNMPC-controlled process seems to converge to the de- sired tank holdup, however does not reach it exactly.
Therefore the operating cost is corrected as explained before. The results of this correction are shown in Table 1.
The eNMPC strategy leads to an economical improvement of about 1 % compared to the benchmark.
The control variable trajectories are shown in Figure 6.
A comparison with Figure 3 reveals that the process feed streamFmacis high at low electricity prices and vice versa.
During peak production, the storage tank is filled as the Ftank profile indicates. As it can be seen from Table 1, there is a trade off between the 3 SQP iterations restricted and the ideal eNMPC in economic improvement and com- putational burden, i.e., the ideal eNMPC is closer to the optimal solution, however, it is not online applicable due to high CPU times. The ideal eNMPC is closer to the optimal case than the 3 SQP eNMPC and achieves an economic improvement of 1.7 % with respect to the benchmark.
More SQP iterations lead to a larger deviation of the solution from its initial guess. There is thus an otpimal restriction of SQP iterations balancing the computational effort and economic improvement. It can further be seen that the tank level is closer to its nominal value for the ideal eNMPC than for the 3 SQP eNMPC which results in lower additional cost due to the cost correction. The operational cost of the ideal eNMPC case is the same as for the optimal case; the economical losses are only due to the tank level difference cost correction.
In addition, it should be noted that the tank level end- point constraint limits the flexibility of the process under eNMPC control, since it forces the eNMPC to operate the process such that the tank is at least at its nominal level after each prediction horizon. However, by using this constraint an extensive unloading of the tank during one control horizon is limited, which could be disadvantegous in later control cycles. It is worth further studying this effect as well as possible reformulations of the eNMPC op- timization problem to account for the endpoint constraint.
In summary, the appliedbottom-upstrategy based on eN- MPC that uses a sub-optimal fast update method leads to an efficient, near-optimal process operation. The economic advantage is expected to be even stronger, since a produc- tion rate dependent electrical efficiency is assumed in the model, which has its maximum at the constant operation point of the benchmark. This means, the higher the devia-
tion of the process state from the benchmark operation is, the smaller is the electrical efficiency that directly influ- ences the operational cost. In addition, it can be expected that larger disturbances and more extreme electricity price profiles will further increase the advantage of the eNMPC strategy compared to the benchmark operation, since the eNMPC can exploit the prices differences more extensively while satisfying the operation constraints.
The solution time for the offline DO is less than 8 hours.
The solution time for the eNMPC using 3 SQP iterations is less than 120 CPU seconds. This is less than the sampling time of the eNMPC and can be used online. The solution time for the ideal eNMPC is much longer than the eNMPC sampling time and can thus not be regarded to be real time applicable. The solution times motivate further algorithmic improvements to accelerate the optimization and obtain online tractability of the method even for larger process models than used here.
4. CONCLUSION
A process operation strategy based on an eNMPC using a sub-optimal fast update method, that updates the solution of an offline DO during process operation, is applied. Both the offline DO and the eNMPC layer use the same first- principle model. A closed-loop scenario is simulated in- silico assuming a product demand disturbance during op- eration in contrast to the constant demand profile assumed for the DO. The eNMPC successfully controls the process and achieves near optimal process behavior. Based on an in-silico closed-loop case study, the trade-off between computational time and economical performance for dif- ferent SQP iteration number restrictions of the eNMPC is illustrated. The solution times of the offline DO and the eNMPC would allow online applicability, however they also motivate further algorithmic improvements. The so- lution time may also be reduced by using reduced models, however this also may lead the model unacceptable accu- racy. The delays resulting from the solution times of the optimization problems will be addressed in the future. The comparison of the eNMPC operation with and without base-layer control is also an interesting task of further work as well as studying the influence of different initial trajectories of the eNMPC.
ACKNOWLEDGEMENTS
The authors thank Andreas Peschel, Florian Schliebitz, and Gerhard Zapp from Linde AG and Falco C. Jung from AVT, RWTH Aachen for fruitful discussions.
320 330 340 350
Fmac[mol/s]
145 150 155 160 165
Ftop[mol/s]
0 5 10
Ftank[mol/s]
35 40 45
Fliq[mol/s]
0 4 8 12 16 20 24
0 10 20 30 40 50
Time [h]
Fdrain[mol/s]
eNMPC offline DO Benchmark optimal disturbed Bounds
Fig. 6. Trajectories of selected control variables as a re- sult of the offline DO, the closed-loop simulation (eNMPC), the benchmark simulation, and the op- timal operation under known uncertainties (optimal disturbed). Benchmark is a constant production rate.
The eNMPC performs 3 SQP iterations, uses a pre- diction horizon of 8 hours and a sampling time of 15 min.
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