Model predictive control of power plant superheater – comparison of multi model and nonlinear approaches Jaroslav Hlava, Jan Opálka

Tor Arne Johansen

Faculty of Mechatronics, Informatics and Interdisciplinary Studies Technical University of Liberec Liberec, Czech Republic {jaroslav.hlava; jan.opalka}@tul.cz

Department of Engineering Cybernetics Norwegian University of Science and Technology Trondheim Norway [email protected]

Abstract — Model predictive control of a three stage power plant superheater is investigated in this paper. This power plant subsystem consists of three heat exchangers and three attemperator sprays. Its dynamics is nonlinear and significantly load dependent. In the paper, the responses to large load demand changes (between 50 and 100% of boiler rated power) typical of today’s deregulated market conditions are considered. The ability of the predictive control system based on switched linear models and full nonlinear model to keep the steam temperature constant during such large transients is evaluated and the relative merits of the nonlinear vs. multi model approach are compared.

I.

INTRODUCTION

Control of fossil fuelled power plants is a very intricate subject. The number of controlled variables and control loops is high and many sources of nonlinearity and interactions are present. This paper focuses on one special but important control objective from this complex field: control of the superheated steam temperature. Tight control of this temperature is very important as it is directly related to the economic performance of a power plant. If the control is tight enough as to guarantee that the variability of the superheated steam temperature is sufficiently reduced, then the temperature setpoint can be increased. This increases the plant efficiency because efficiency is proportional to the superheated steam temperature. Smaller temperature fluctuations are also beneficial for the plant lifetime as they reduce the thermal stresses that could cause micro-cracks. The requirement on keeping the variations of the superheated steam temperature as small as possible is now both more important and more difficult to meet than before. Due to the market deregulation the demands on the load following capability of the power plants have been much raised. The superheated temperature variations must now be kept very small even during major and fast load changes.

This research was supported by the Technology Agency of the Czech Republic under contract No. TA02020109 “Predictive Control System for Stability Improvement and Higher Efficiency of Power Plants”

Tight superheated steam temperature control calls for application of advanced control techniques. In particular, model predictive control (MPC) should be considered. MPC has been well established in the process control field for more than one decade. On the other hand, MPC applications to power plant control remained surprisingly sporadic until quite recently. Despite this, several attempts to apply MPC to the superheater control can be found in the literature. In [1] a special kind of predictive controller (predictive adaptive multivariable multi-step adaptive regulator – MUSMAR) was applied to a simple superheater consisting of two heat exchangers and one attemperator spray. In [2] a classical form of predictive control: the dynamic matrix control algorithm was applied to a similar superheater system with one attemperator spray. In experiments described in [3], adaptive controller based on generalized predictive control was applied to improve the output temperature control of one superheater stage with attemperator. In our previous papers [4] and [5] we have reported an application of predictive control to a more complex superheater and reheater structures with several superheat/reheat stages and several spray attemperators. The predictive controller based on switched linear models was used to cope with the severe nonlinearity of the plant. All of the applications of the MPC to the superheater control mentioned above have two points in common. First, significant performance improvement was achieved by using MPC. Second, despite considerable nonlinearity of the superheater, control was based on linear models. The fact that superheater dynamics vary significantly with operating point was accounted for either by using on-line adaptation [1], [3] or model switching [4], [5]. In [2] the controller was based on one fixed linear model and hence only relatively small load changes (between 70 and 90%) could be considered. In all of these papers, nonlinear model predictive control (NMPC) was either not mentioned at all or it was dismissed offhand from the very beginning because of its notable problems such as non-convex optimization with the danger of sticking in local optima, complicated estimation of the unmeasurable states of the nonlinear model etc. However, the

research in the field of NMPC made a progress in the last decade as it can be seen from the research publications such as [6] and [7]. It also becomes evident that there are certain industrial applications where NMPC is necessary as the nonlinear effects are so dominant that a linear model is apriori unlikely to be adequate [8]. This motivated the research described in this paper. Its purpose is to compare the control performance of superheater controller based on switched linear MPC with the performance that can be obtained with full nonlinear MPC. The results described here were obtained within the framework of a research project whose purpose is to develop predictive controllers that could be used in several Czech steam power plants that are now being retrofitted. Thus, the starting point for the research was the plant retrofit design data together with the structure and parameters of the classical control system. Consequently, the control performance with classical control system is always taken as the reference point for all comparisons and the comparison is threefold: classical control system, multi model MPC and nonlinear MPC. II.

DESCRIPTION OF THE CONTROLLED PLANT

The structure of the superheater process under consideration is shown in Fig. 1. It includes three heat exchangers of the flue gas to steam type. Steam flowing in a number of parallel pipes is heated by hot flue gas. Output temperatures of the heat exchangers are controlled and they have specified setpoints. The most important temperature is the output from the last superheater stage – the main steam temperature. Temperatures measured after spray attemperators are auxiliary variables used for the purpose of cascade control. The figure includes also the structure of the classical control system. This system is based on independent PI cascades that are gain scheduled according to the power plant load. The control objective is the disturbance rejection. The main steam temperature must be kept constant despite load changes and other disturbances such as random fluctuations of the heat flow to heat exchangers, irregularities in functioning of the coal mills and variations of the boiler output steam temperatures. There are three actuators – attemperator spray valves V1 to V3. The control action can be just negative as the steam temperatures are decreased by cold water spray. Consequently, the higher are the values of the manipulated variables the smaller are the controlled temperatures. The valves are actuated using electrical motors with limited speed. This constrains the rate of change of valve positions.

Figure 1. Schematic of the superheater with the classical control system

The experiments described in this paper were performed in simulation. The basis of this simulation was a first principles model based on partial differential equations that were spatially discretized by dividing each heat exchanger into several control volumes. Model parameters were initially given by the boiler design data. In the second step we made use of the fact that some power plants with this superheater structure were already in operation and the model could be improved and fine-tuned by comparison with the measured data. The model does not describe just superheater but the whole supercritical once through boiler. Model details and its comparison with measured data are given in [5] and [9]. The main specifications of the boiler including the superheat stage were published by the boiler design company in [10]. For the purpose of this paper, the most important feature of the model is the fact that it has proved to be reliable with good agreement between model responses and measurements. III.

MULTI MODEL PREDICTIVE CONTROL

To design a multi model MPC it is first necessary to quantify the degree of the nonlinearity and to decide whether a multi model approach can provide benefits. Second, if the nonlinearity proves to be high, a suitable set of representative nominal models must be selected. The superheater subsystem has five inputs: three valve positions (manipulated variables), temperature of input steam and plant load (measured disturbances). The effect of plant load is not explicit in Fig. 1. However, plant load affects the heat flows Q1, Q2 and Q3 to the heat exchangers and its effect on superheater behavior is significant. The load can be estimated in real time from the values of several measured variables. Thus, it can be considered a measured disturbance. The number of possible combinations of values of these five variables may seem to be prohibitively high; requiring a big number of representative operating points. However, all output temperatures have specified setpoints and hence the steady state values of valve positions are uniquely determined. This significantly reduces the number of possible steady state operating points and it allows using plant load as the only scheduling variable. In quantifying the nonlinearity and selecting the nominal models, the method of model bank selection in multi-linear control proposed in [11] was followed. This method is based on the gap metric. The gap metric is a measure of the distance between two linear systems from the viewpoint of the closed loop control. If gap metric between two systems S1 and S2 is close to zero, these systems are similar in the sense that any controller stabilizing S1 is likely to stabilize S2, and the distance between the closed-loop systems will be small in the ∞-norm sense. However, if gap metric is close to one, these systems are unlikely to be stabilized by one linear controller. To evaluate the nonlinearity, linearized models were obtained for the load range from 50 to 100% with step 1% (tiny models in the terminology used in [11]). The load is expressed in the terms of the thermal power produced by the boiler and 50% of boiler rated power is just slightly above the minimum value at which electricity can be produced. Thus this load range actually covers the whole operating range of the plant during electricity production. In the second step, gap metric distances between the tiny models were computed.

Selected values with step 5% are shown in Table 1 (Mx is linearized model at x% of nominal load). It is evident that the submodels are far apart in the gap metric and the superheater is unlikely to be controlled by one linear controller. This is consistent with the fact that even the classical control system of the superheater uses gain scheduling. The parameters of the PI cascades are changed according to heuristic rules based on the rich experience of control system designers with the design and operation of power plant control systems. Finally the tiny models were divided into groups in such a way as to achieve maximum distance between the members of one group smaller then selected maximum γg. The gap metric can be used to specify guaranteed stability margins in linear robust control. Similar guarantees cannot be formulated in the case of multi model MPC and it cannot be said exactly that some value of γg is the most appropriate one. Usually the values of γg about 0.6 are used. In Table 2, the results of three choices close to this value are given (γg=0.5, γg=0.62 and γg=0.7). The table shows the resulting number of models and partitions of the load range and nominal models. The notation (a, b> % (c) means: in the load range above a (open interval) and less than or equal to b (closed interval) model at load level c is used as nominal. The nominal models in each range are selected as to minimize the maximum gap metric distance to the most distant tiny model in the respective range. All controllers in the multi model MPC use quadratic performance criterion. Using the notation from Fig. 1 (T1o,T2o, T3o output temperatures of the heat exchangers, rT1o, rT2o, rT3o their respective setpoints, V1, V2, V3 valve positions normalized to the range ), this criterion can be formulated as follows TABLE I.

J (k ) = +

∑ ∑ (ψ i ( rT ( k + p k )-Tio ( k + p k ) ) ) + N

3

p =1 i =1

N u −1 3

2

io

∑ ∑ ( λ j ΔV j ( k + p k ) )

2

p = 0 j =1

Increments of manipulated variables are weighted with uniform weight λ=20, T1o and T2o weights equal one, weight ψ3 on the main steam temperature T3o is varied between 1 and 10. Sampling time is 5 s, prediction horizon N=150 and control horizon Nu=20. Manipulated variables are constrained into the range ; their maximum rate of change is 1/30s-1. The controllers can be combined together either using hard switching or using various soft switching approaches. To avoid problems with selection of proper controller weighting and for computational reasons (just one controller running at a time) hard switching was used. To evaluate the performance of both multi model and nonlinear MPC, the response to the following load demand change was considered: ramp with the speed 10 MW/min from 50 % to 100 % of nominal load and back after 5 minutes time at 100% load. This relatively fast load change can be regarded as some kind of worst case scenario that can occur during wide range load following operation of the plant. In Figs 2 and 3 the responses of the main steam temperature to this load change are shown. The improvement over the classical control system is evident. If ψ3 is high (ψ3=10) even the performance of four model MPC is quite satisfactory. If ψ3 is smaller (ψ3=1), decreasing the number of models results in noticeable performance deterioration.

DISTANCES IN GAP METRIC BETWEEN MODELS AT DIFFERENT PLANT LOADS

M50 M55 M60 M65 M70 M75 M80 M85 M90 M95 M100 M50

0

0.58 0.88 0.98

M55 0.58

0

M60 0.88 0.53

1

1

1

1

1

0.53 0.82 0.92 0.95 0.97 0.98 0.98 0.98 0

M65 0.98 0.82 0.47

0.47 0.73 0.84 0.88 0.90 0.92 0.92 0

0.39 0.62 0.73 0.79 0.82 0.84

M70

1

0.92 0.73 0.39

M75

1

0.95 0.84 0.62 0.32

M80

1

0.97 0.88 0.73 0.52 0.26

M85

1

0.98 0.90 0.79 0.64 0.44 0.22

M90

1

0.98 0.92 0.82 0.70 0.56 0.39 0.19

M95

1

0.98 0.92 0.84 0.74 0.62 0.49 0.33 0.16

M100 1

1

0

0.32 0.52 0.64 0.70 0.74 0

0.26 0.44 0.56 0.62 0

0.22 0.39 0.49 0

0.19 0.33 0

0.99 0.93 0.85 0.76 0.66 0.56 0.43 0.29

0

0.15

0.99 0.93 0.85 0.76 0.66 0.56 0.43 0.29 0.15

TABLE II.

1

0.16

Figure 2.

Main steam temperature responses ψ3=1

Figure 3.

Main steam temperature responses ψ3=10

0

SELECTION OF NOMINAL MODELS AND LOAD RANGES

Model

γg =0.5 - 7 models

γg =0.62 - 5 models

γg =0.7 - 4 models

1 2 3 4 5 6 7

% (52) (54, 58> % (56) (58, 63> % (60) (63, 69> % (66) (69, 77> % (73) (77, 90> % (83) (90,100>% (95)

% (52) (55, 61> % (58) (61, 68> % (64) (68, 80> % (73) (88, 100> % (88)

% (53) (56, 63> % (59) (63, 74> % (68) (74, 100> % (84)

(1)

IV.

NONLINEAR MODEL PREDICTIVE CONTROL

A. The nonlinear MPC formulation Similarly as MPC based on multiple linear models, the nonlinear MPC (NMPC) uses a quadratic performance criterion. It can be formulated in the same way as (1) if output tracking is the control objective and one multivariable NMPC controller is used. However, a nonlinear state space model is now included in the optimization constraints. Hence NMPC is based on solving a nonlinear optimal control problem (OCP). The optimization constraints are now generally in the form x ( n + 1) = f ( x ( n ) , u ( n ) ) n ∈ k ; k + N

y ( n) = g ( x ( n) , u ( n))

umin ≤ u ( k + p | k ) ≤ umax ; Δumin ≤ Δu ( k + p | k ) ≤ Δumax

(2)

ymin ≤ y ( k + p | k ) ≤ ymax ; xmin ≤ x ( k + p | k ) ≤ xmax

where x are system states; u are manipulated variables (V1, V2, V3 in this particular case); y are output variables (T1o,T2o, T3o); f, g are functions of discrete time nonlinear state space model. In NMPC formulation (2) the disturbances are not involved in the nonlinear model. In the implementation of NMPC that is used in this paper, the disturbances are considered constant during the prediction horizon. Therefore disturbances are treated as parameters of the nonlinear model. The nonlinear OCP in NMPC is usually solved using the active set sequential quadratic programming methods (SQP) or interior-point methods (IPM) [7]. It is possible to find several software products for solving the optimization problem associated with NMPC like the ACADO toolkit [12] and the Yane framework [13]. The implementation of NMPC used in this paper is based on the ACADO toolkit. B. ACADO toolkit ACADO toolkit (Automatic Control and Dynamic Optimization) is a software environment written in C++. It contains a collection of algorithms which allow us to implement many control strategies like direct optimal control, MPC, NMPC and others. ACADO was originally designed for Linux operating system, but Windows is also supported. ACADO can be used independently or in connection with MATLAB. It is possible to implement the simulation of the closed loop NMPC in ACADO completely or the optimization can be done using ACADO and simulation of the controlled plant can be done using MATLAB. But in both cases it is necessary to implement the nonlinear model into the ACADO because it is a part of nonlinear OCP. The simulation model used in this paper that is outlined in section II and described in a greater detail in [9] is implemented in the MATLAB/Simulink environment. For this reason it was necessary to transform it into the ACADO environment using C++ code. MPC performance criterion includes increments of the manipulated variable. However, the OCP in ACADO is defined in values of the manipulated variable. For this reason, the original nonlinear model had to be modified by adding an integrator before the manipulated variable input. In this case it is possible to include a constraint on the value of manipulated variable and its rate limit into the OCP.

C. Distributed NMPC structure for superheated steam temperature control in once-through boiler MPC is a multivariable control strategy and the multi model MPC described in the previous section of this paper was implemented as one multivariable controller with three controlled variables, three manipulated variables and two measured disturbances. This structure allows coordination of manipulated variables and simple prioritization of control objectives by using an appropriate selection of weights (main steam temperature is more important than other temperatures). However, an attempt to implement nonlinear MPC in the form of one centralized multivariable controller was thwarted by several difficulties. The superheater model is of high order (46th order). This order results mainly from the lumped parameters approximation of the distributed parameters dynamics of the huge heat exchangers. There is some possibility of model reduction, however it is limited. As a result of this high order, the OCP of multivariable NMPC would have a high dimension with slow solution and questionable real time performance. Also the problem with non-convex OCP and possible trapping in local minima is exacerbated in this case. For these reasons it eventually turned out that distributed structure of NMPC using separate controller for each superheater stage is a better choice. This structure is shown in Fig. 4. The selection of this structure was also corroborated by the fact that superheater is not a general MIMO system as some couplings are zero (T1o and T2o are independent of V3 and T1o is independent of V2). The controllers in Fig. 4 work with relatively lower order models (1st and 2nd heat exchanger – 12th order models, output heat exchanger – 22nd order model) and they can run in parallel on different processors or cores. Hence the computations can be faster. E. Structure of one NMPC The structure of one NMPC is shown in Fig. 5. Each NMPC uses quadratic performance criterion J i (k ) = + αi

∑ ( ( rT ( k + p k )-Tio ( k + p k ) ) ) + N

p =1

N u −1

∑

p =0

2

io

( ΔVi ( k + p k ) )

2

(3)

where i=1,2,3 (HE1, HE2, HE3), αi is weighting coefficient, other symbols are the same as in (1). The constraints are in the form (2) where the nonlinear state space model is the model of individual heat exchanger, the constraints on manipulated variables and their increments are the same as in multi model case; output and state constraints are currently not used. In the

Figure 4.

Distributed structure of NMPC for superheater control

manipulated variable. For this reason a rather short horizon with move blocking had to be used: 1st move lasts 1 sampling period, 2nd move lasts 3 periods and 3rd move lasts to the end of prediction horizon i.e. 26 periods. The values of weights are αi=100, KKT tolerance is 10-5, integration accuracy 10-6. The nonlinear OCP was solved by SQP method. Gauss-Newton method was used for the Hessian matrix approximation. Discretization was done by multiple shooting method.

Figure 5. Structure of NMPC of one heat exchanger

second part of the criterion, it is possible to consider all values of p=0,..,Nu-1 or just some selected subset of values if some kind of move blocking strategy is used. The NMPC structure had to be kept as simple as possible to allow real time performance. For this reason, the use was made of the fact that the superheater model was stable and hence it was possible to use simple open-loop state observer. The difference of the outputs from the plant and observer is subtracted from the reference temperature in Fig. 5. This correction performs the implementation of simple disturbance observer. In this way an offset free regulation process is achieved [14] and possible errors in measuring and estimating the disturbances are compensated for. Similarly as in multi model MPC, each NMPC uses the information about plant load and input steam temperature to the respective heat exchanger as measured disturbances. In addition to it, the NMPCs for HE2 and HE3 use also the information about valve position(s) from the preceding exchanger(s) as measured disturbances as they have an influence on steam mass flow. The disturbances like input steam pressure and mass flow as well as the parameters of the cooling water and heat flows to the exchangers are calculated from the current load level using a look up table. The feedback is closed by feeding estimated states into the nonlinear OCP. F. Tuning of NMPC parametrs Appropriate tuning of NMPC is very important due to its influence on stability, control performance and computational demands of the algorithm. The following values of parameters were used in the simulation experiments described in this paper. Sampling time was 10 s. It is longer than the time used in multi model MPC (5 s) but its value was dictated by the real time computational requirements of the NMPC algorithm. The computation time for solving the OCP problem is slightly less than 10 s at fairly standard PC (double core CPU 2.4 GHz, 4 GB RAM). Hence the real time implementation of NMPC is possible in principle, if this sampling period is used. Prediction horizon was varied between 80 and 300 s (i.e. from 8 to 30 steps). This is very important parameter, because it has influence on stability [7]. Small values of prediction horizon may result in instability while too high values may result in slow responses with oscillations. The length of control horizon has very significant influence on computational demands of the NMPC algorithm as the computational complexity of the optimization problem depends directly on the number of free changes of the

G. Comparison of control performance The load change specified at the end of the section on multi model MPC (ramp 10 MW/min from 50% to 100% of nominal load and back after 5 minutes at 100% load) was used to test the performance of both nonlinear and multi model MPC (7 models, ψ3=10, ψ2=ψ1=1). The results are in Figs. 6, 7 and 8. The upper parts of these figures show the temperatures, the lower parts show the valve positions. The responses with the classical control system are also included for comparison. First, it is evident that both versions of MPC are much better than the classical control system. It must be emphasized that the classical control system was carefully designed by people with rich experience in designing power plant control systems. Responses with all controllers are identical in some intervals in Figs. 6 and 7 (T1o ca. 500-1700 s, T2o ca. 10001300 s). However, this is the feature of the superheater itself. The temperatures are too low even with zero spray and this cannot be compensated for by any controller. On the other hand, the control performances obtained with nonlinear and multi model MPC are very similar. The variations of the main steam temperature are almost negligible in both cases. The variations of other temperatures are also small (except for intervals where reaching the setpoint is impossible). The temperature responses can be changed by varying the weights. This works in predictable manner and if reasonable weights are used, the changes in control performances are definitely not radical. A more significant difference can be observed in valve positions, most importantly in V3. The valve positions are also quite similar in both cases. However, they are smooth functions of time in NMPC while they exhibit undesirable peaks in multi model MPC. Further analysis of the times when the controllers are switched reveals that these peaks arise due to controller switching. The switching itself is bumpless. The current value of the manipulated variable is sent to all inactive controllers and if any controller is switched to operation it calculates Δu with respect to this value. Also the state estimators of all inactive controllers are continuously running and updating their estimates to minimize the bumps during the controller switching. Despite all of this, the fact that new controller based on significantly different model is switched to operation results in relatively large initial changes of manipulated variable and hence the peaks appear. A possible remedy might be the replacement of hard switching with some soft switching mechanism. However, this implies increase in computational demands of multi model MPC because more than one controller must be running at a time if manipulated variable is to be calculated as weighted combination of the outputs of several controllers.

Figure 6. Response of T1o to load change

When evaluating the nonlinear MPC, it must first be said that the results were obtained with a very simplified controller. The whole problem was decomposed into three loosely coupled SISO loops. Simple open loop observer was used, the control horizon was extremely short and even the prediction horizon was at most one half of the plant settling time. Even with all of these simplifications, NMPC performance was similar to the performance of multi model controller that was multivariable with much longer horizons. Moreover, manipulated variables of NMPC changed smoothly without peaks. Even if the NMPC controller may not be ripe for practical applications at the moment, it has a significant potential for the future. Further performance improvements can be expected when these simplifications are gradually removed by continuing research including the latest improvements of ACADO toolbox described in [15]. REFERENCES [1]

[2]

[3] [4] Figure 7. Response of T2o to load change

[5]

[6] [7] [8]

[9] [10] Figure 8. Response of T3o (main steam temperature) to load change

V. CONCLUSIONS Finally, it is probably not possible to say unambiguously which MPC approach is better as the control performances of both are nearly identical. Multi model MPC based on several linear models has the clear advantage of relative simplicity and low computational demands. The time necessary for solving the optimization problem is a small fraction of the time required by the nonlinear optimization. For these reasons, field testing of superheater control that is planned for the next year will use multi model MPC. However, the switching mechanism will have to be changed as the switching artifacts are not acceptable.

[11] [12] [13] [14] [15]

J.M. Lemos, P.O. Shirley, R. Neves-Silva and B.A. Costa, “Adaptive Predictive Control of Superheated Steam and Economic Performance”, in Power Plant Applications of Advanced Control Techniques, Pal Szentanai, Ed. Verlag ProcessEng Engineering GmbH, pp. 17-42, 2010. A. Sánchez-López and G. A. Figueroa, “DMF and FLC for Steam Temperature Control”, in Power Plant Applications of Advanced Control Techniques, Pal Szentanai, Ed. Verlag ProcessEng Engineering GmbH, pp. 135-161, 2010. T. Moelbak, “Advanced control of superheater steam temperatures - an evaluation based on practical applications”, Control Engineering Practice, Vol. 7, Issue 1, Pages 1–10, January 1999. J. Hlava, L. Hubka and L. Tuma, “Modeling and predictive control of a nonlinear power plant reheater with switched dynamics”, in Proc. of 16th Int. Conference MMAR, pp. 284-289, Miedzyzdroje 2011. Hlava, J.; Hubka, L.; Tuma, L., "Multi model predictive control of a power plant heat exchanger network based on gap metric", in Proc. of the 16th International Conference on System Theory, Control and Computing (ICSTCC), pp.1-6, October 2012. T. A. Johansen, Selected Topics on Constrained and Nonlinear Control, chapter Introduction to Nonlinear Model Predictive Control and Moving Horizon Estimation, STU/NTNU, 2011 L. Grüne, J. Pannek. Nonlinear Model Predictive Control. London: Springer, 2011. G. C. Goodwin, M. G. Cea, M. M. Seron, D. Ferris, R. H. Middleton and B. Campos, “Opportunities and Challenges in the Application of Nonlinear MPC to Industrial Problems”, in Proceedings of the 4th IFAC Nonlinear Model Predictive Control Conference, Noordwijkerhout, NL, pp. 39-49, August 2012. L. Hubka, and O. Modrlak, "The practical possibilities of steam temperature dynamic models application", in Proceedings of the 13th International Carpathian Control Conference, pp.237-242, May 2012. M. Byrtus, “New boilers for low quality coal”, All for Power, vol. 4, pp. 24-28, 2009 (in Czech). J. Du, Ch. Song and P. Li P., “Application of gap metric to model bank determination in multilinear model approach,” Journal of Process Control, Vol. 19, pp. 231-240, 2009 B. Houska, H. J. Ferreau, M. Diehl. “ACADO Toolkit - An OpenSource Framework for Automatic Control and Dynamic Optimization”, Optimal Control Applications and Methods, 32 (3), pp. 298-312, 2011. The YANE [online], 2013 [10.3.2013] www: http://www.nonlinearmpc.com/ J. M. Maciejowski, Predictive Control with Constraint. England: Pearson Education Limited, 2002 M.Vukov et al., Auto-generated Algorithms for Nonlinear Model Predictive Control on Long and on Short Horizons , accepted for ECC 2013, Zürich Switzerland.