16 Industrial controls

16 Industrial controls C. de Prada1 , D. Sarabia1 , C. Sonntag2 , and S. Engell2 1

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Universidad de Valladolid, Dpt. Ingenier´ıa de Sistemas y Autom´ atica, Facultad de Ciencias, c/ Real de Burgos s/n, 47011, Valladolid, Spain, [email protected] Technische Universit¨ at Dortmund, Fakult¨ at Bio- und Chemieingenieurwesen, Lehrstuhl f¨ ur Systemdynamik und Prozessf¨ uhrung, 44221 Dortmund, Germany, c.sonntag|[email protected]

16.1 Model-predictive control of hybrid systems: application to industrial case studies 16.1.1 Introduction The aim of this section is to show that engineered model-predictive control techniques can successfully solve industrial control problems that involve continuous and discrete inputs and switched dynamics. Two case studies will be discussed: The first one is the control of a supermarket refrigeration system. It was provided by Danfoss, a leading manufacturer of refrigeration, air conditioning and heating systems as well as of control systems. The system is composed of a set of display cases each with an evaporator system and a compressor rack. Here, all decision elements are discrete (on/off), and the goal is to maintain the temperature of the goods in the display cases in a certain range with minimum energy consumption and wear of the compressors. The second case study concerns a multi-stage set of evaporators and was provided by Bayer Technology Services. The plant performs an industrial purification process where volatile components are removed from a liquid stream that contains the desired product. The control task is the optimization of the start-up procedure, i.e., to generate a control strategy that drives the evaporators from a shut-down (cold and empty) state to nominal operation in minimal time and with minimal resource consumption. In general, hybrid systems have very different dynamic characteristics, and the range of control goals and of decision variables involved is large so that no single control technique will cover all cases. The case studies considered here are realistic examples of systems with complex dynamics and switched inputs that cannot be handled straightforwardly by techniques for switched linear (piecewise affine) systems described by ODE models. While the solutions presented are problem-specific, they provide examples of a general approach to

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Authors Suppressed Due to Excessive Length

engineered solutions to complex hybrid control problems, based upon hierarchical structures and a suitable parameterization of the degrees of freedom in the finite-horizon optimization problems. This section is structured as follows: For each case study, first the process is described and the goals of feedback control are formulated. Then, hybrid MPC control approaches are presented that result in optimization problems that can be solved in realistic computation times and provide good performance, as demonstrated by simulations. The section ends with some conclusions. 16.1.2 Control of a supermarket refrigeration system Description of the supermarket refrigeration system Display cases for refrigerated goods are present in all supermarkets and food stores. A schematic representation of the process, with two display cases, is depicted in Fig. 16.1, and the cross section of a display case with its evaporation system is shown in Fig. 16.2. The temperatures of the goods in the display cases must be kept within a specified range in order to preserve them for consumption. The temperatures of the air inside the display cases are used as an indirect measurement of the temperatures of the goods. The compressor rack creates a low pressure in the suction manifold and in the evaporators which increases the rate of evaporation of the refrigerant, resulting in a transfer of heat from the walls of the display cases to the refrigerant, which is the basic mechanism for cooling the air around the refrigerated goods. The compressed refrigerant is condensed and sent back to the inlets of the evaporators. The overall system is subject to significant disturbances, e.g. opening of the display cases by the customers and for refill, changes in the amount of goods placed in the display cases, and the transitions from day to night where the ambient temperature varies. In order to control the temperatures in the display cases, the inlet valves of the display cases and the state of the compressors (on/off) are used as actuators.

Fig. 16.1. Simplified layout of a supermarket refrigeration system.

16 Industrial controls

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Fig. 16.2. Cross section of a refrigerated display case.

The process dynamics is continuous in nature. The hybrid character of the control problem results from the fact that the inlet valves of the evaporators are on/off valves and that the compressors can only be switched on or off, generating an inherent oscillatory behavior of the overall system. The process is a representative of the class of continuous or quasi-continuous processes operated with alternating on/off actions. A full dynamic model of the system is given in (Larsen et al., 2007a,b). The number of dynamic equations depends on the number of display cases that are modeled (four differential equations per display case, one of which is of variable structure). An additional differential equation is used to model the pressure in the suction manifold. Thus, for a system with two display cases, the model comprises 9 nonlinear differential equations, and with five display cases, the model consists of 21 differential equations. The control goal is to operate the system such that the temperatures of the air in the display cases are kept within a prescribed range (2 - 5 ◦ C) with minimum energy consumption and a minimum number of switching of the manipulated variables, in spite of disturbances and the transitions from day to night. Moreover, the pressure in the suction manifold must not exceed a prescribed value. The conventional control structure for a supermarket refrigeration system employs decentralized hysteresis temperature controllers in each evaporator, manipulating the on/off inlet valves, and PI-control of the suction pressure, with a quantifier on the controller output to determine the number of compressors in operation. This type of control, while cheap and reliable, gives rise to a phenomenon of synchronization in the operation of the display cases, i.e. the switching of the inlet valves occurs in a synchronized fashion, leading to frequent switching of the compressors which increases the wear of the com-

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pressors and creates peaks in the energy demand. This is the main problem that motivated research in more sophisticated control structures. NMPC formulations for the supermarket refrigeration system Improving the conventional control by using a model of the hybrid system and optimization criteria for computing the control actions in a natural way leads to a finite-horizon optimization or MPC formulation. A direct solution of the hybrid problem was obtained in (Larsen et al., 2005) using a MLD formulation in which binary variables represent the control actions at every sampling time, and changes in the model structure are translated into a set of linear constraints involving binary variables. If used with the nonlinear continuous process dynamics, this approach leads to a MINLP optimization problem that grows quickly with the considered look-ahead horizon to a size where solutions become too time-consuming. We have therefore investigated two different engineered NMPC formulations (Sarabia et al., 2007; Sonntag et al., 2007, 2008a). The ”embedded logic” approach (Sarabia et al., 2007) is based upon two key elements: •

•

An internal model is formulated in continuous time that integrates the first-principles-based model of the continuous dynamics, the logic of operation, and the binary actions. This model incorporates all discontinuities and possibly a variable structure using the formalism of the simulation environment. Using state-of-the-art simulation technology ((ECO, 2008), see also chapter 11 of this handbook), both, the values of the cost function over the prediction horizon and the values of the constraint residuals can be computed as a function of a given control policy over this horizon. A parameterization of the binary decision variables in terms of times of the occurrence of events is used. This parameterization is illustrated in Fig. 16.3 for the case of an evaporator inlet on/off valve. The time pattern of the valve is included in the simulation and the optimizer decides on i i , Tof the duration of the pulses, Ton f , which are real variables. In addition,

1 On/off valve

0 Toff1

Ton1

Pulse 1

Toff2

Ton2

Toff3

Ton3 Prediction time

Real decision variables: Times Toffi and Toni i = 1, 2, 3, …

Fig. 16.3. Sequence of pulses of an on/off evaporator valve.


HM PC CONTROLLER

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Tkon Tkoff

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J

Constraints Non-linear Optimizer min J

Tkon Tkoff

Process + Converter u i (t) PI Controller continuous to on/off signals

y(t)

Fig. 16.4. Hybrid control implementation.

and extending the concept of control horizon of standard MPC, the pattern corresponding to the last pulse is repeated till the end of the prediction horizon, saving decision variables and imposing a certain regularity in the future behavior of the system.This parameterization has two advantages: the number of decision variables is usually smaller than in a formulation with binary variables for each sampling interval, and on the other hand, as all decision variables are continuous, the associated optimization problem can be formulated as one of NLP type, which can be solved more efficiently than a MINLP. A similar policy can be applied to the compressor on/off actions but, for the sake of simplicity, a PI-control of the suction pressure, with a quantifier on the controller output has been maintained for the compressors in the simulation presented. The NLP optimization problem is solved in every sampling period using a sequential approach as depicted in Fig. 16.4. In the second approach (Sonntag et al., 2007, 2008a), the same general idea is pursued, but the distribution of the decision variables is different. In this approach, a hierarchical scheme is employed: The temperatures in the display cases are regulated by low-level controllers that open the inlet valves when a temperature threshold is reached. The durations of the openings and the switching strategy of the compressors are determined on a higher layer by optimization over a finite look-ahead horizon (see Fig. 16.5). A branch-andbound technique is employed where patterns of possible compressor switching in the near future are investigated one-by-one, thus leading to a sequence of optimization problems with purely continuous degrees of freedom. If a pattern with no or few switches results in meeting the constraints for optimized inlet valve switching times, the search is terminated, otherwise it is continued for a larger number of switches. The optimization horizon is divided into two parts. In the more remote future, the compressor switching is approximated by a real number of active compressors while in the near future, only integer values of the number of active compressors are considered (see Fig. 16.6).

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High-level NMPC control Continuous and discrete controller parameters

Low-level process control Discrete inputs

Plant (a)

(b)

Compressor capacity

Fig. 16.5. A scheme of the hierarchical control approach (a) and the operation of the low-level temperature controllers (b).

100 % 75 % 50 %

uc tc ,1 t s ,1

25 % 0% Discrete

Continuous

tp

t

Fig. 16.6. Optimization of the switching for four compressors, shown exemplarily for a single compressor switching. tp is the prediction horizon, ts,1 and tc,1 are the time points at which the compressors are switched, and uc is a fictitious continuous compressor capacity.

This enables a larger look-ahead horizon for the optimization. An additional low-level system monitors the system and switches the compressors directly if fast and large changes in the external disturbances occur. Simulation results for the supermarket refrigeration system Results obtained with the first approach (Sarabia et al., 2007) for a system with two display cases and two compressors are shown in Fig. 16.7. The hybrid controller was tested in simulations using a cost function that penalized the deviation of the temperatures and pressure from their desired ranges. Fig. 16.7 shows the simulation of a four-hour experiment where the amount of goods in the display cases changed and a day/night transition took place after two hours. The sampling time was 1 minute and the controller could operate in real time. On the right, the time evolution of the settings of two valves and


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two compressors can be seen, while the pressure, the air temperatures, and the goods temperatures appear on the left from top to bottom, showing very good behavior and no synchronization. The second approach (Sonntag et al., 2007, 2008a) was applied to a system with five display cases and three compressors (with relative capacities of 30 %, 30 %, and 40 %). The results are shown in Fig. 16.8. The controller

u d isp , 1 a nd u d isp ,2 (o n/off)

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Fig. 16.7. Evolution of the main variables for a system with two display cases and two compressors using the first approach, before and after a day/night transient.

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Fig. 16.8. Evolution of the main variables for a system with five display cases and three compressors using the second approach, before and after a day/night transient. The lower left figure shows an enlargement of the upper left figure.

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and the simulation model were implemented in Matlab, and the NLP problems were solved using the TOMLAB-based solver NPSOL (Holmstrm et al., 2007). The computation time per iteration of the high-level optimizer was set to 200 seconds. The controller is capable of keeping all process variables within the bounds and desynchronizes the air temperatures, which are completely synchronized initially, very quickly. Furthermore, the low-level pressure controller detects the drastic change in the external disturbances at the transition between day- and night operation and switches off all compressors almost instantaneously. 16.1.3 Start-up of a multiple-effect evaporator system The evaporator system The multi-stage evaporator case study deals with the start-up of a triple-effect evaporation system. Fig. 16.9 shows a simplified flowsheet of the evaporation process. It consists of three similar evaporation stages which are used to concentrate a feed stream (F1 ) that contains a non-volatile organic component A (the product), as well as an alcoholic solvent and water. A stream of cold feed (F1 ) is injected into the first evaporator E1. The pumps (C) together with valves V2 and V3 are used to transfer liquid from one evaporator to the next (by controlling the flows F2 and F3 ). In normal operation, the heat supply to the first two stages is provided by condensation of the hot vapor from the following evaporator in a heat exchanger. The last evaporator (E3) is heated by fresh hot steam controlled by the valve VV 2 . The valves VV S,1 and VV S,2 can be used to switch between the supply of fresh steam to E1 and E2 and vapor from the following evaporator in the start-up phase. Once the liquid mixture in an evaporator starts to boil, the volatile components (water, alcohol) evaporate and can be used as heating vapor for the preceding stages.

Fig. 16.9. The multi-stage evaporation system.


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In order to establish the desired operation, the pressure profile must increase from the left to the right (i.e. P1 < P2 < P3 ), since the boiling temperature of the mixture rises with an increase of pressure. The pressure in the first evaporator, P1, can be controlled by adjusting the flow rate FC of the cooling water to the condenser CON. The final product is continuously drained from E3 if the concentration of the product A meets the specification. The hybrid nature of the process is due to the fact that the process is operated using both continuous and discrete inputs: the regulation valves and the two on/off valves, VV S,1 and VV S,2 . During transients, the process exhibits different continuous dynamics (evaporating, non-evaporating), leading to a variable-structure model. Note that the structure of the model changes also if the on/off valves are switched. The control goals are to drive the process from an empty and cold initial state to a final state where the levels are in the range of 60 − 64 % and the output concentration of A in E3 is in the range kg 0.8 − 0.84 kg , while respecting constraints on the levels, the pressures, and the temperatures, and while minimizing the consumption of fresh steam and the duration of the start-up operation. A non-linear model of the process based on first principles, as well as its description as a hybrid automaton, is given in (Sonntag and Stursberg, 2005). Computation of an optimal start-up policy The start-up of a single evaporator was optimized in (Sonntag et al., 2008b) using a finite-horizon formulation with branch-and-bound search over the discrete variables and embedded nonlinear optimization of the continuous variables. The combinatorial complexity of the resulting problem is such that only relatively short horizons can be considered, necessitating an engineering of the cost functions based on knowledge on the global shape of the optimal trajectories. The approach taken in the optimization of the start-up of the three-stage system by (De Prada et al., 2007) takes the logic steps that are used in the start-up operation according to the physical nature of the system into account, as well as the final control structure that is required for regulation of the process once the normal regime has been reached. A key observation is that the control structure for steady-state operation supports the start-up operation and that the switching of the on/off valves VV S,1 and VV S,2 can be imposed when a certain pressure is reached. If this logic and continuous control structure is implemented, the number of degrees of freedom of the optimization is largely reduced. Consequently, for the optimization, logic and basic controllers are embedded in the hybrid non-linear model of the process, following the ”logic-embedded” approach mentioned above. The remaining degrees of freedom are only continuous variables. Thus, the optimal start-up can be solved as a NLP problem using a sequential method. The implementation requires a two-layer architecture as shown in Fig. 16.10. The lower layer includes PIcontrollers and logic controllers that act on some process variables, while the

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Process model with Basic control and logic

Basic control and logic

Optimizer

Process Fig. 16.10. Optimization architecture for start-up of the evaporators.

upper layer optimizes a reduced set of key variables. The normal operation of the evaporation system requires, at least, level control of all evaporators, pressure control in the vapor space of evaporator E1, and concentration control of product A in evaporator E3. Several alternative schematics are possible. One of them is displayed in Fig. 16.11, where the condenser is considered as a pressure boundary condition. Notice that, if implemented in the initial phase, these loops will provide the right valve openings as required for the start-up: While the levels are below their set points, the valve at the liquid input of one evaporator will be open allowing the liquid to flow through the system, and when the level reaches the target, the controller keeps the level around the set point. In the same way, the output concentration controller will keep the liquid output valve of evaporator E3 closed until the product concentration

s

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Fig. 16.11. Basic control loops of the evaporation system.


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reaches its target, and then, it will maintain the concentration around the set point. The only degree of freedom that is left in the system is the fresh steam supply to evaporator E3, which can be manipulated in order to optimize the transition of the station from the empty and cold state to normal operation. The set point of the vapor pressure in evaporator E1, or the switching pressure levels of valves VV S,1 and VV S,2 , could be considered as additional degrees of freedom, but the sensibility of the solution to these variables is quite small. The optimization problem was formulated as a dynamic optimal control problem in the NMPC framework where two performance objectives combined in a cost function with the appropriate weighting. The first one refers to minimizing the start-up time tf , that is, the time needed to reach the target region, and the second one is to minimize the amount of fresh steam required for the start-up. The manipulated variable, the fresh steam valve opening that governs the steam flow Fv to the evaporator E3, is designed for minimizing the start-up cost taking into consideration the non-linear dynamic model of the plant and its basic control system as well as constraints on the inputs and the outputs of the process using a suitable control vector parameterization technique and a sequential approach for computing the cost function. A problem with the hybrid control approach presented in this section is associated with the estimation of the gradients for the NLP optimization. The estimation of the gradients, or equivalently of the sensitivities with respect to the decision variables, can often be performed in spite of the fact that each time a discontinuity takes place along the prediction horizon, the sensitivities have to be reset to a new value. In problems where the discontinuities are due to the on/off nature of the manipulated variables and where the order and the number of states and of switches does not change along the prediction horizon, as in the displays cases example, the computation of the gradients can be done safely. In contrast, when the number of switches of the on/off variables is not fixed or the number of states may change, we can expect difficulties with single-shooting NLP algorithms due to the discontinuities in the gradients. Therefore, in the evaporator example, a parameterization of the decision variable (flow of fresh steam) was used that adapts the intervals where this variable is constant to the periods between events created by the underlying controllers during start-up. Simulation results for the start-up of the evaporator system Results of the simulation are shown in Fig. 16.12 together with the main constraints, showing good behavior. The optimal start-up takes 3 hours, and all variables are maintained within their range as can be seen in the graphs. The sampling time is 2 minutes. For the computation of the cost function to be minimized, which depends on the time needed to reach normal operation conditions, the simulation stops when the target is reached. In case it is not

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Pressures

Valve opening. Supply steam to E3

Temperatures

Concentrations E3

Fig. 16.12. Evolution of the main variables in the optimal start-up of the evaporator system.

reached within the prediction horizon length of eight hours, a penalty term is applied. 16.1.4 Conclusions Two case studies have been considered in this section, a supermarket refrigeration system, an example of a continuous process operated with sequences of on/off actions, and a multi-stage evaporation system for which the optimal start-up has been examined. By using physical insight, the logic-embedded approach and special parameterizations, hybrid decision problems that give rise to large dynamic mix-integer optimization problems if a brute-force approach is used were converted into simpler ones that involve no or only few discrete decision variables. The approach described here has been shown to lead to practical solutions to several small- to medium-sized important industrial hybrid control problems. The ideas developed need further research to obtain a general framework and to reach the stage of maturity of other advanced strategies, in particular continuous linear and nonlinear MPC.

References

Ecosimpro user manual. EA Internacional, 2008. http://www.ecosimpro.com. C. De Prada, S. Cristea, and J. J. Rosano. Optimal start-up of an evaporation station. In Proc. 8th International Symposium on Dynamics and Control of Process Systems (DYCOPS), pages 115–120, 2007. K. Holmstrm, A. O. Gran, and M. M. Edvall. User’s guide for TOMLAB, 2007. http://www.tomopt.com. L. F. S. Larsen, T. Geyer, and M. Morari. Hybrid MPC in supermarket refrigeration systems. In Proc. 16th IFAC World Congress, 2005. Th-E12TO/5. L. F. S. Larsen, R. Izadi-Zamanabadi, and R. Wisniewski. Supermarket refrigeration system - benchmark for hybrid system control. In Proc. European Control Conference (ECC), pages 113–120, 2007a. L. F. S. Larsen, R. I. Zamanabadi, R. Wisniewski, and C. Sonntag. Supermarket refrigeration systems - a benchmark for the optimal control of hybrid systems. Technical report for the HYCON NoE, 2007b. http://tinyurl.com/23nrkc. D. Sarabia, F. Capraro, L. F. S. Larsen, and C. de Prada. Hybrid control of a supermarket refrigeration system. In Proc. American Control Conference (ACC), pages 4178–4185, 2007. C. Sonntag and O. Stursberg. Optimally controlled start-up of a multi-stage evaporation system. Technical report for the HYCON Network of Excellence, Technische Universitt Dortmund, 2005. http://tinyurl.com/36xsrn. C. Sonntag, A. Devanathan, S. Engell, and O. Stursberg. Hybrid nonlinear model-predictive control of a supermarket refrigeration system. In Proc. IEEE Multi-Conference on Systems and Control (MSC/CCA), pages 1432– 1437, 2007. C. Sonntag, A. Devanathan, and S. Engell. Hybrid NMPC of a supermarket refrigeration system using sequential optimization. 2008a. Accepted for: 17th IFAC World Congress, Seoul, Korea.

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References

C. Sonntag, W. Su, O. Stursberg, and S. Engell. Optimized start-up control of an industrial-scale evaporation system with hybrid dynamics. Control Engineering Practice, 16(8):976–990, 2008b.