paper moisture control-mpc

Model Predictive Midrange Control of the Moisture Content in Paper Production Ola Sltteke ABB Automation Technologies ABSTRACT Traditionally the moisture controller only manipulates the steam pressure set point of one dryer group (the lead group) and this makes it single-input-single-output control. From evaluations of a recently developed model, it is shown that by dividing the dryer section into two parts, each with its own lead group (two-input-single-output control), moisture variations in the final product can be reduced by midrange control.

The model is a nonlinear dynamic model for the dryer section. It is implemented in the object-oriented modeling language Modelica and is compared with measurements from a fine-paper machine.

INTRODUCTION The pulp and paper industry is a highly competitive and capital-intensive market that is under increasing cost pressure. Customers are demanding lower costs, better terms of delivery, and higher product quality. In order to meet these requirements, much effort is spent on process modeling [1]. The purpose of the models is varying. Some examples are (i) improved process understanding from experiments with what-if scenarios, (ii) identify the bottleneck in a process and suggest modifications, (iii) creating process simulators for operator training, or (iv) improved control system design.

Moisture is one of the most important quality parameters of the final paper product. It is important to keep this property well regulated, both at steady-state and at state-transitions. A good model of the dynamics of drying is essential for good moisture control. Mathematical modeling of cylinder drying started with the pioneer work by [2]. [3] gives an extensive review of drying models up to 1980 with some 130 references. Many drying models with different approximations and objectives have been proposed in the last decade. [4, 5, 6] present models based on non-linear steady-state relations. [7] develops a linear state-space model from physical relations. [8] develops a simplified dynamic model where the whole drying section is modeled as one or only a few large cylinder. Some physical properties are then adjusted to fit this assumption. [9] uses a combination of statistical multivariate models and physical models. [10, 11, 12, 13] use different kinds of black-box models. [14] introduces a high-order model, capable of describing moisture gradients and other properties inside the paper sheet. The model contains approximately 80.000 states, making simulations fairly tedious.

We will here introduce a dynamic physical model, built on heat and mass balances for steam, cylinder, and paper. The core of the model is based on work by [5, 15, 16]. The objective is control of the moisture at the reel-up and not to accurately describe micro-scale moisture variations inside the sheet. It is implemented in the object oriented modeling language Modelica [17, 18]. A similar attempt is found in [19]. Like any object-oriented programming language, Modelica provides the notions of classes, and instances, as fundamental abstractions. Properties like inheritance and abstract classes provide a structured approach to model structuring. Modelica also enables declarative programming, useful e.g. to express mathematical relations, as well as functional programming to express behavior interms of algorithms. The main advantages of Modelica are (i) it is built on a non-causal equation structure (ii) it permits mixing of physics with empirical models (iii) it is easy to go from simple models to high fidelity models by graphical editing (iv) it is easy to build and exchange model libraries and (v) it is suited for modeling in several engineering domains.

While the physical behavior of the process is formulated using partial differential equations (PDEs), numerical simulation requires the PDEs to be discretized in the spatial dimension(s). In this work, the paper process is discretized by partitioning the process into small control volumes where a mass and energy balance are defined for each volume. These control volumes are then put together so that the outflow of one becomes the inflow of the next. The precision of the model then depends on the size of the control volumes, where a finer discretization grid gives improved accuracy, but also increased computational complexity. In order to increase the clarity of the presentation,

the indices identifying each individual control volume has been dropped. In Fig. 3 and Fig. 4, however, the indices have been included to emphasize the discrete nature of the paper process model.

Many articles in the literature on the dryer section modeling introduce a model and show a few open loop simulation examples. This article takes the modeling effort one step further by using it to evaluate MPC control of a new process control structure.

THE MODEL

The Steam and Cylinder Process

Let qs (kg/s) be the mass flow rate of steam into the cylinder, qc be the condensation rate, qbt the blow through steam, and qw be the siphon flow rate. Also, let Vs and Vw (m3) be the volume of steam and water in the cylinder, and let s and w (kg/m3) be the densities. The mass balances for water and steam are then

( )( ) .

,

wcww

btcsss

qqVdtd

qqqVdtd

=

=

(1)

The energy balances for steam, water, and metal are

( )( )( ) ,

,

,)(

, pmmmp

mwwscwww

scsbtssss

QQTmCdtd

QhqhqVudtd

hqhqqVudtd

=

=

=

(2)

where Qm (W) is the power supplied from the water to the metal, Qp is the power supplied from the metal to the paper, hs (J/kg) is the steam enthalpy, hw is the water enthalpy, m (kg) the mass of the cylinder shell, Cp,m (J/kgK) the specific heat capacity of the shell, Tm (K) the mean temperature of the metal, us (J/kg) and uw are the specific internal energies of steam and water. The steam and water volumes add up to the total cylinder volume, V = Vs + Vw. The power flow to the metal is given by

( ),msscm TTAQ = (3) where sc (W/m2K) is the heat transfer coefficient from the steam-condensate interface to the centre of the cylinder shell, A (m2) is the inner cylinder area, and Ts the steam temperature. The power flow to the paper is

( ) ,pmcpp TTAQ = (4) where Tp is the paper temperature, cp the heat transfer coefficient from the cylinder shell to the paper, and (unit less) is the fraction of dryer surface covered by the paper web. Fig. 1a illustrates the temperature and heat flow given by (3) and (4). Since the steam flow to the cylinder cannot be manipulated directly, a valve model is also needed. From [20] we have

,)()( sshvvvs ppxfCq = (5)

where Cv (m2) is the valve conductance, xv is the position of the valve stem and the function fv (xv) is the valve characteristics, called valve trim. The valve stem varies from 0 (minimum valve opening) to 1 (maximum valve

opening). The supply pressure at the steam header is psh. We use equal percentage trim, since it is the most common characteristic in the process industry [21]. It is given by

.)( 1= vxvvv Rxf

Rv is a constant known as the rangeability since it is the ratio between the maximum and minimum valve opening. For simplicity, all steam within the cylinder cavity is assumed to be homogeneous with the same pressure and temperature. We also assume that the steam in the cylinder is saturated. This means that the enthalpy, density, and temperature are functions of the pressure only. Fitting polynomials to tabulated values for saturated steam gives

.1141ln43.52)(ln792.6)(ln3136.0

,10]26.64005048.0[

,10]5.748ln200)(ln77.18)(ln8842.0[

,10]1824ln260)(ln58.39)(ln887.2)(ln07402.0[

,5.124ln71.37)(ln388.3)(ln1723.0

23

3

323

3234

23

++=+=

+=+++=

++=

ppp

p

ppph

pppph

pppT

w

s

w

s

s

Equations (1)(5) are a crude nonlinear model for the steam-cylinder process in Fig. 1a. The system can be rewritten into a third order state equation (most steps are omitted here).

),()(

),()()(

,)()(

33

2221

1211

pmcpmsscm

msscwwsbtssshvvvw

btwsshvvvw

TTATTAdt

dTe

TTAhqhqhppxfCdt

dVedtdpe

qqppxfCdt

dVedtdpe

=

=+

=+

(6)

where

.,

,)()(,,)(

3322

211211

pssww

www

www

sws

swssw

ww

sw

mCehhe

Vp

hVp

VhphVV

pVVhee

pV

pVVe

==

++

+==

+=

model Cylinder equation Ms = rhos*Vs; der(Ms) = qs qc - qbt; % (eq.1) Mw = rhow*Vw; der(Mw) = qc - qw; Es = rhos*us*Vs; us = hs p/rhos; der(Es) = (qs qbt)*hs qc*hs; % (eq.2) Ew = rhow*uw*Vw; uw = hw p/rhow; der(Ew) = qc*hs qw*hw Qm; Em = m*Cp*Tm; der(Em) = Qm - Qp; V = Vs + Vw; Qm = alpha_sc*A*(Ts - Tm); % (eq.3) Qp = alpha_cp*A*eta*(Tm Tp); % (eq.4) end Cylinder;

Fig. 1a (left) A piece of the cross-section of a drying cylinder, showing the assumption on the temperature profile and energy flow. 1b (right) Code segment of the Modelica model of the steam cylinder.

In the rewritings of the energy balances above, the specific internal energy has been eliminated by the definitions us = hs p/s and uw = hw p/w. By solving for the derivatives in (6), the model can be directly implemented and simulated in e.g. Simulink.

By instead using Modelica the tedious and error prone procedure of transforming the system to explicit form is avoided and we let the simulation environment decide the state space realization. Since the transformation of equations is automated, it is also easier to change the model at a later stage. The model equations are put into the simulation environment in their original form, see Fig. 1b.

The paper web process

We will now expand the model to also include dynamics from the paper web. To describe the moisture in the paper we need a mass balance and to describe the paper temperature we need an energy balance. It is assumed that the temperature and moisture are spatially constant at a single cylinder due to the high machine speed, hence modeled as one control volume. Starting with the mass balance, we describe how much water is evaporating from the paper surface to the air. From [16] we get

,ln,

,

=

pvtot

avtot

p

wtotevap pp

ppRTKMpq

where qevap is the evaporation rate (kg/m2s), K is the mass transfer coefficient (m/s), Mw is the molecular weight of water (kg/mol), ptot the total pressure (Pa), pv,a the partial pressure for water vapor in the air (Pa), pv,p the partial pressure for the water vapor at the paper surface, R the gas constant (J/molK), and Tp the paper temperature (K). The partial pressure pv,a is given by the specific humidity of air, x (kg water vapor/kg dry air), and the total pressure,

.62.0, totav

px

xp +=

The vapor partial pressure at the paper surface is

,0, vpv pp =

where pv0 is the partial vapor pressure for free water. This is given by Antoines equation

,10 15.431690127.10

0

= pTvp

0 5 10 15 20 250

0.5

1.0

Moisture content (%)

Sorp

tion

isot

herm

0 5 10 15 20 250

500

1000

1500

Moisture content (%)

Hea

t of s

orpt

ion

(kJ/

kg) 30 oC

60 oC90 oC

30 oC60 oC90 oC

Fig. 2. Sorption isotherm, , and heat of sorption, Hs given by (7) and (8).

11 ievap

ixy qA

22 iixy ugvd

Paper i 1

Cylinder shell

Paper i Paper i +1

Cylinder shell

2ixy gvd

ievap

ixy qA

ievap

ixyqA

11 ++ ievap

ixy qA

11 iixy ugvd

1ixy gvd

iixy ugvd

ixy gvd

Fig. 3. The mass transport for water and fiber in the paper web. The shaded areas represent cylinder walls. When the paper is in the transition between two cylinders (the free draw), evaporation occurs from both paper surfaces.

As long as capillary transport can bring new water to the paper surface, the vapor partial pressure at the paper surface is equal to the partial pressure for free water. When the paper becomes more dry a correction factor called sorption isotherm, , is invoked which has a value between zero and one, see Fig. 2. The sorption isotherm of a paper web depends on its composition and temperature. It is not very well investigated when compared to other materials [22], but [23] gives an empirical expression for paper pulp, namely

),)273(10085.058.47exp(1 0585.1877.1 uTu p = (7)

where u is the moisture ratio (kg moisture/kg fiber). Also, let vx be the speed of the paper web (m/s), dy the width of the paper web (m), Axy the area of the dryer surface covered by paper (m2), and g the dry basis weight (kg/m2). Then the mass balance of moisture in the paper web can be written as

.)(

guvdqAugvddt

ugAdxyevapxyininxy

xy =

If there is a dryer felt between the web and air, a dryer reduction factor is inserted into the evaporation term to reduce qevap. A similar mass balance for moisture in the free draws can be derived from Fig. 3, which shows a schematic picture of the mass flows in the paper sheet. Analogously, the mass balance for fiber in the paper web is given by

.)( gvdgvdgAdtd

xyinxyxy =

Note that when the area of a paper element from the discretization equals the area of the paper covering one cylinder (for finer discretization grids several paper control volumes may be connected to the same cylinder), then Axy = A, cf. (4). To model the energy balance, we introduce

,1

,,, u

uCCC wpfiberppp +

+=

where Cp,p, Cp,fiber, and Cp,w is the specific heat capacity for the paper, fiber and water, respectively (J/kgK). As we can see, Cp,p is a weighted sum of the heat capacities of the parts. From [5] we have Cp,fiber = 1256 J/kgK.

Also, let Tp be the paper temperature and H be the amount of energy needed to evaporate the water from the paper surface. Analogously to the discussion on the mass balance, if the web is wet enough the required energy to evaporate the water is equal to the latent heat of vaporization for free water, Hvap (J/kgK). When the paper becomes more dry an extra amount of energy Hs (J/kgK) (the heat of sorption) is necessary besides the latent heat of vaporization for free water. The heat of sorption can be derived from the sorption isotherm by thermodynamic theory and this relation is known as the law of Clausius-Clapeyron

)(

)(1111

111

++i

aip

ixy

ipa

isvap

ievap

ixy

TTA

HHqA

22,

22 )1( + ipi ppiixy TCugvdPaper i 1

1ipQ

Cylinder shell

11,

11 )1( + ipi ppiixy TCugvdPaper i

)(

)(i

aip

ixy

ipa

isvap

ievap

ixy

TTA

HHqA

++

)(

)(i

aip

ixy

ipa

isvap

ievap

ixy

TTA

HHqA

++

ip

ipp

iixy TCugvd ,)1( +

Paper i +1

)(

)(1111

111

++++

+++

++i

aip

ixy

ipa

isvap

ievap

ixy

TTA

HHqA

1+ipQ

Cylinder shell

Fig. 4. The energy balance of the paper web. The shaded areas represent cylinder walls. When the paper is in the transition between two cylinders (the free draw), energy flow to ambient air occurs from both paper surfaces.

,)/1(

)(ln

=

pws Td

dM

RH

and by applying this on (7), we get

.110085.0 20585.1

wps M

RTuH = (8)

The amount of energy required to evaporate the water from the surface of the web is then given by

,svap HHH +=

where Hvap is equal to 2260 kJ/kg (at atmospheric pressure). Furthermore, let the energy transport due to convection between the paper surface and the air be

),( apxypaconv TTAQ =

where pa (W/m2K) is the heat transfer coefficient from paper to air and Ta (K) the ambient air temperature. Reference [22] investigated some sorption isotherms found in the literature. Many of those give a heat of sorption that goes to infinity as u goes to zero. This is physically unrealistic since the bond energy between the last fraction of water and a cellulose fiber must be finite. From [23], a finite heat of sorption at the origin which matches the hydrogen bond energy between waterfiber is given and is therefore found to be most appropriate.

In addition, since water is an incompressible medium there is no pressure volume work on the surroundings and we write the energy balance as a change in enthalpy. The energy balance of the paper web is thus modeled as

),()1()1())1((

,,,,

apxypapppxyevapxyinpppininxyppppxy TTATCugvdHqATCugvdQ

dtTCAugd +++=+

The energy balance for a free draw is similar and can be formulated using the schematic illustration of energy flows shown in Fig. 4. In addition, we let the heat transfer coefficient from the cylinder to the paper web depend on the moisture content in the web. From [5] the linear empirical relation

,955)0()( uu cpcp +=

is obtained, where cp(0) varies between 200 and 500 W/m2K. It is well known that cp depends on other things, e.g. the web tension, and surface smoothness of both paper and cylinder, but this is omitted here. If a dryer felt is between the cylinder and web, a dryer reduction factor is inserted to reduce the energy transfer.

STEADY STATE MODEL VALIDATION

The two partial models (cylinder model and paper model) in the previous section are validated separately in [15] and [16]. The combined model has been validated against steady-state data taken from a paper machine producing fine paper [24]. The machine is running at 708 m/min with a basis weight of 80 g/m2. The paper is over dried to a final moisture content of only 1.2 % (0.012 kg/kg), since this is a predryer which is immediately followed by a size press (a unit where starch is applied to the surface to obtain strength and water resistance) where a certain amount of rewetting occurs.

The moisture content is measured in 11 different positions, which can be seen in Fig. 5 together with the simulation result. The heat transfer coefficient sc is used as fitting parameter and 1100 W/m2K is found to give the best fit by visual inspection. The agreement between model and measurements is good. The model also captures the three zones in the drying process, the heating phase, the constant drying rate phase, and the falling drying rate phase [25]. In the heating phase (cyl. 15) most steam energy is used to heat the web and the evaporation rate is low. In the constant rate phase (cyl. 630), energy to the web is equal to the energy consumed for water vaporization. In the falling rate phase (cyl. 3137), drying rate begins to decrease due to the hygroscopic nature of the fibers, described in the previous section.

CONTROL OF MOISTURE BY MID-RANGING MPC

In the last decade, MPC (model predictive control) has found large attention in the process industry. It has been described as one of the most significant developments in process control [26] and the only methodology to handle constraints in a systematic way [27]. In this section, a new strategy to control the moisture in paper production is evaluated. It is implemented in a MPC structure and the analysis is done by simulations of the paper machine model described in the modeling section. A Matlab toolbox for MPC [28] (see www.control.lth.se/user /johan.akesson/mpctools) is linked to the Modelica environment. In this way, the advantages of the simple modeling technique in Modelica and the rich family of toolboxes in Matlab are used. The simulated machine consists of 60 cylinders, running at 1080 m/min (18 m/s), with inlet moisture content to the drying section of 60 % (1.5 kg moisture / kg fiber). Traditionally, the moisture is controlled by adjusting the pressure in the steam heated cylinders,

5 10 15 20 25 30 350

10

20

30

40

50

60

Cylinder number

Shee

t moi

stur

e (%

)

Fig. 5. Validation by comparing steady-state simulation to measurements.

in a single-input-single-output cascade loop structure, see Fig. 6. The inner loop is regulated by a PID-controller and the outer loop by some type of dead-time compensating model-based controller, e.g. an IMC [29] or a Dahlin controller [30].

The performance of the closed loop system in Fig. 6 is limited by the long transport dead-time in the drying section. By manipulating the steam pressure in the cylinders of the last part of the machine independently of the first part, a more effective moisture control system can be achieved, see Fig. 7. The objective is still to control the moisture in the sheet with the steam pressure in the cylinders but now the process has two inputs (and the same output as before) and this extra degree of freedom can be taken advantage of.

By identification of step responses on the high-order nonlinear physical model, a simple black-box model is achieved. In the Laplace transform, it is given by

,148

010.0148

098.02

51

14c

sc

s Ues

Ues

Y ++= (9)

where Y, Uc1, Uc2, are Laplace transforms of y, uc1, and uc2 respectively, y is the moisture content, uc1 is the steam pressure set point to the first 50 cylinders, and uc2 the set point to the last 10 cylinders. The PID-controllers in the inner loops are tuned according to a tuning method derived in [31]. The response from uc2 has a significantly shorter time delay but also smaller process gain. The advantage of the high gain from uc1 and fast dynamics from uc2 is utilized in a mid-ranging MPC structure.

The simplified model for the singe-loop case in Fig. 6, obtained correspondingly to (9), is given by

,148

11.0 13c

sUes

Y +=

The cost function being minimized in the mid-ranging MPC is

MBC Steam system Dryer

SetpointMoisture

Setpointsteam pressure

Steam pressure

PID-controller

1

Moisture

yr

uc

Fig. 6. The standard moisture control loop. A model based controller (MBC) is used to control the moisture in the paper sheet.

MPC Dryer

Setpointmoisture

Moisture

PID-controller

1

PID-controller

1

u1

u2Steam system Last part

Steam system First part

y

r1

Fig. 7. The proposed moisture control loop. A model predictive controller (MPC) is used to control the moisture in the paper sheet. The MPC used in the simulations, include both state estimation and error-free tracking.

,)()(

)()()|()(

)(3

0

2

2

149

0

2

22

1 ==

+++

++++=

i Rc

c

i Qcikuiku

ikuikrkikyikr

kJ (10)

where r1 is the set point for the moisture, and r2 the set point for uc2. Notation (k + i | k) denotes the i step-ahead prediction and is the difference operator. Since the MPC formulation is inherently discrete, the cost function is given as a summation. The idea of (10) is to let uc2 take care of the variations in paper moisture and let uc1 be positioned at a level where uc2 has an adequate control range in both directions in steady-state. In this way, the first part of the drying section serves as the base level of the drying while the last part controls the moisture. This is done by choosing appropriate weighting matrices, Q and R. For the simulations in this article they are chosen as

7.0

7.2

7.4

7.6

y (%

)

0 200 400 600 800 1000 1200430

440

450

Time (s)

u c (k

Pa)

Fig. 9. Simulation of single loop MPC. A set point change occurs at t = 100 s and a disturbance in inlet moisture content from 60.0 % to 60.5 % at t = 600 s.

7.0

7.2

7.4

7.6y

(%)

430

440

450

u c1 (

kPa)

0 200 400 600 800 1000 1200400

500

600

Time (s)

u c2 (

kPa)

Fig. 8. Simulation of mid-range MPC. There is a set point change at t = 100 s and a disturbance in inlet moisture content to the drying section from 60.0 % to 60.5 % at t = 600 s. The set point for u2 (which is r2) is 400 kPa.

.100

0100,

1000500

43

=

= RQ (11)

There is obviously a larger weight on deviations from r1 than from r2, since moisture control is the main objective. The weight on uc1 is larger than on uc2 making the controller to primarily use signal uc2 when acting on disturbances in moisture or on set point changes. The MPC settings in (11) are found from a combination of these rules of thumb and evaluation of performance by simulations. The sample time is 5 s, the prediction horizon is chosen to 50 and the control horizon is chosen to 4, see (10). The prediction horizon is set to approximately match five time constants of the open-loop system (to assure that the prediction sees the full response of a change in uc1 and uc2) and a small control horizon to limit the computational effort for the MPC. There is also a rate constraint for the control signals

.kPa50,kPa10 21 cc uu

The purpose of this is to avoid severe injections of disturbances into the steam system by allowing large variations in steam usage, which has negative effect on both steam production and other steam users.

Fig. 8 shows a simulation of the mid-range MPC and it clearly visualizes the thought of mid-ranging. The signal uc2 is used to quickly react to set point changes and disturbances while uc1 is used to push uc2 back to its set point in steady-state. Both during the set point change and disturbance, the rate constraint for uc2 is initially active. This reduces the performance of the controller slightly but is important for the steam consumption, as described above. Simulations show that leaving out the constraints gives a more aggressive use of uc2, but it is of course not free to use since there still is a cost from (10).

To make an evaluation of the proposed control structure, the mid-ranging MPC is compared to single-loop MPC (structure as in Fig. 6 but MBC is exchanged by MPC), see Fig. 9. The tuning is chosen so that the two control systems have the same maximum value of the sensitivity function, see Fig. 10. This implies that they, in some sense, have the same degree of robustness. However, due to the constraint handling, MPC is nonlinear and the comparison only serves as guidance. The cost function for the single-loop MPC is given by

[ ] ,)()|()()( 30

250

1

2 ==

++++=i

Rci

Q ikukikyikrkJ

and the chosen weights that give the sensitivity in Fig. 10, and constraint are

kPa10,1,10 == cuRQ

Fig. 10 also indicates the difference in performance between the two controllers, at least in the non-constrained case. The mid-range MPC has a limit frequency almost twice as large as the limit frequency of the single loop MPC. Since paper machines often are run from several hours to days with the same set point, disturbance rejection is important and it therefore makes sense to look at this property.

Fig. 11 shows the moisture response for the two different controllers together with the steam flow in the header. The difference in performance is apparent. The mid-range MPC has both better set point following and disturbance rejection. However, the transient steam consumption is twice as large for the mid-range MPC. This is the price being paid for the extra performance. Note that the change in steam consumption is rather abrupt but it should be remembered that in practice set points are ramped instead of changed in steps and the evaluation here is used for comparison only.

In Fig. 12, the single loop MPC has been tuned to give similar performance in disturbance rejection as the mid-range MPC, shown in Fig. 8 and Fig. 11. The output of the single loop MPC in this case is much more aggressive than it was in Fig. 9 and therefore the transient steam consumption is much larger. The aggressive steam consumption is

needed in order to match the performance of the mid-range MPC. The maximum value of the sensitivity is also larger, hence the control system is less robust, see Fig. 13.

The advantage with the proposed control structure is that, in general, it does not require any rebuild of the drying section. The cylinders are normally divided into different groups even though all groups follow the same set point. Therefore, it is simply a matter of changing the controller software and a majority of the main system vendors have the possibility to include a MPC package into their DCS system. However, it is an advantage if the flash steam from the last group is recirculated through a thermo compressor. If the flash steam is reused by another group it is important to let the steam pressure of the last group be constrained so that its pressure never falls below the steam pressure of the receiving group. This guarantees that the steam flows in the intended direction. A disadvantage with the proposed control structure can be that it is a bit more complicated to tune a mid-range MPC compared to e.g. the IMC or Dahlin controller used by many mills today.

6.8

7.0

7.2

7.4

7.6

Moi

stuur

e (%

)

0 200 400 600 800 1000 120015

16

17

18

19

20

Time (s)

Stea

m c

onsu

mpt

ion

(kg/

s)

Fig. 11. Comparison between mid-range MPC (solid) and single loop MPC (dotted), showing the moisture content and steam consumption.

10-4

10-3

10-2

10-10

0.5

1

1.5

Frequency (Hz)

Mag

nitu

de (a

bs)

Fig. 10. The sensitivity function for single loop MPC (dotted) and mid-range MPC (solid), using the linearized model (eq1). The maximum sensitivity is chosen to 1.25. The difference in limit frequency for the two systems is around a factor two.

CONCLUSIONS

A physical model, implemented in the object-oriented modeling language Modelica, for a drying section has been given. Components for steam cylinder, control valve, paper web, and different moisture controllers have been developed and collected in a model library. All equations are based on mass and energy balances, and algebraic constraints. By drag-and-drop features it is easy to build a simulation model of virtually any existing drying section. It is also easy to expand the model library with components for press and wire section.

The model has been validated against measurements on a real paper machine. By simply adjusting one parameter, the heat transfer coefficient for the condensate, a good fit is obtained.

6.8

7.0

7.2

7.4

7.6M

oistu

re (%

)

0 200 400 600 800 1000 120010

15

20

25

Time (s)

Stea

m c

onsu

mpt

ion

(kg/

s)

Fig. 12. Comparison between mid-range MPC (solid) and single loop MPC (dotted), showing moisture content and steam consumption when tuned to give similar performance in disturbance rejection.

10-4

10-3

10-2

10-10

0.5

1

1.5

Frequency (Hz)

Mag

nitu

de (a

bs)

Fig. 13. The sensitivity function for single loop MPC (dotted) and mid-range MPC (solid), when tuned to give similar performance.

The model is linked to a MPC toolbox in Matlab to evaluate a new strategy to control the moisture in the drying section. The strategy utilizes the possibility to divide the drying section into two parts. By controlling this multi-variable process with a mid-range structure, the performance of the closed loop system is greatly improved. The mid-ranging MPC is compared with single-loop MPC and the evaluation is done by comparing performance in terms of sensitivity function, disturbance rejection and steam consumption. An important advantage of the mid-ranging MPC is that it does not require any rebuild of the physical process.

ACKNOWLEDGEMENT The author acknowledges professor Karl Johan strm for valuable ideas and a large portion of inspiration during the process of developing the simulation model, and Johan kesson for the fruitful collaboration regarding the Modelica implementation. Both work at Lund University in Sweden.

REFERENCES

1. B. Foss, B. Lohmann, and W. Marquardt (1998), A field study of the industrial modeling process, Journal of Process Control, vol. 8, no. 5, pp. 325338.

2. A. Nissan and W. Kaye (1955), An analytical approach to the problem of drying thin fibrous sheets on multicylinder machines, Tappi Journal, vol. 38, no. 7, pp. 385398.

3. R. McConnell (1980), A literature review of drying research in the pulp and paper industry, Drying 80, Ed. A. Mujumdar, Hemisphere Publishing, NY, pp. 330337.

4. R. Ramesh (1991), Modeling of multicyliner drying of light-weight paper, PhD thesis, State University of New York, College of Environmental Science and Forestry.

5. B. Wilhelmsson (1995), An experimental and theoretical study of multi-cylinder paper drying, PhD thesis, Department of Chemical Engineering, Lund Institute of Technology.

6. S. Reardon, M. Davis, and P. Doe (2000), Computational modelling of paper drying machines, Tappi Journal, vol. 83, no. 9.

7. M. Berrada, S. Tarasiewicz, M. Elkadiri, and P. Radziszewski (1997), A state model for the drying paper in the paper product industry, IEEE Transactions on Industrial Electronics, vol. 44, no. 4, pp. 579586.

8. M. Rao, Q. Xia, and Y. Ying (1994), Modeling and advanced control for process industries: application to paper making processes, Springer-Verlag, New York.

9. P. Viitamki (2004), Hybrid modeling of paper machine grade changes, PhD thesis, Control Engineering Laboratory, Helsinki University of Technology.

10. S.-C. Chen (1995). Modelling of paper machines for control: theory and practice, Pulp and Paper Canada, vol. 96, no. 1, pp. 1721.

11. S. Menani, H. Koivo, T. Huhtelin, and R. Kuusisto (1998), Dynamic modelling of paper machine from grade change data, In Proceedings of Control Systems, Porvoo, Finland pp. 7986.

12. A. Skoglund, A. Brundin, and C.-F. Mandenius (2000), A multivariate process model for grade change in a paperboard machine, Nordic Pulp and Paper Research Journal, vol. 15, no. 3, pp. 183188.

13. X. Sun, K. Yi, and Y. Sun (2000), The modeling and control of basis weight and moisture content, Proceedings 3rd World Congress on Intelligent Control and Automation, Heifei, P. R. China, pp. 37243728.

14. M. Karlsson and S. Stenstrm (2005), Static and dynamic modelling of cardboard drying, Drying Technology, vol. 23, pp. 143163.

15. O. Sltteke, and K. J. strm (2005), Modeling of a steam heated rotating cylinder a grey-box approach, American Control Conference 2005, Portland, Oregon.

16. H. Persson (1998), Dynamic modelling and simulation of multi-cylinder paper dryers, Licentiate thesis, Department of Chemical Engineering, Lund Institute of Technology.

17. S. E. Mattsson, H. Elmqvist, (1997), Modelica - An international effort to design the next generation modeling language. 7th IFAC Symposium on Computer Aided Control Systems Design, Gent, Belgium.

18. P. Fritzon (2003), Principles of object-oriented modeling and simulation with Modelica 2.1, Wiley-IEEE Press.

19. J. Bergstrm and G. Dumont (1998), An object oriented framework for developing dynamic models of a paper machine, Dynamic Modeling Control Applications for Industry Workshop, IEEE Industry Applications, pp. 6369.

20. P. Thomas (1999), Simulation of industrial processes for control engineers, Butterworth Heinemann, London.

21. T. Hgglund (1991), Process control in practice, Studentlitteratur, Lund, Sweden.

22. M. Petterson, and S. Stenstrm (2000), Experimental evaluation of electric infrared dryers, Tappi Journal, vol. 83, no. 8.

23. P. Heikkil (1993), A study on the drying process of pigment coated paper webs, PhD thesis, Department of Chemical Engineering, bo Akademi, bo, Finland.

24. S. Stenstrm, B. Wilhelmsson, L. Nilsson, R. Krook, and R. Wimmerstedt (1994), Measurement of reference experimental drying data for the multi-cylinder paper dryer, Proceedings of the 9th International Drying Symposium (IDS94), Gold Coast, Australia, pp. 11791186.

25. M. Karlsson (2000), Paper making part 2, drying, Tappi Press.

26. F. J. Doyle (1999), Computational issues in the application of model-based control to the pulp and paper industry, AIChE Symposium Series No. 322, vol. 95, pp. 165172.

27. C. E. Garca, D. M. Prett, and M. Morari (1989), Model predictive control: theory and practice a survey, Automatica, vol. 25, no. 3, pp. 335348.

28. J. kesson (2003), Operator interaction and optimization in control systems, Licentiate thesis, Department of Automatic Control, Lund Institute of Technology.

29. M. Morari and E. Zafiriou (1986), Robust process control, Prentice-Hall, Englewood Cliffs, N. J.

30. E. B. Dahlin (1968), Designing and tuning digital controllers, Instruments and Control Systems, vol. 41, no.6, pp. 7783.

31. O. Sltteke, K. Forsman, T. Hgglund, and B. Wittenmark (2002). On identification and control tuning of cylinder dryers In Proceedings Control Systems 2002, pp. 298302, Stockholm, Sweden.

32. B. J. Allison and A. J. Isaksson (1998), Design and performance of mid-ranging controllers, Journal of Process Control, vol. 8, no. 56, pp. 469474.

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Model Predictive

Midrange Control of the

Moisture Content in

Paper Production

Ola Sltteke

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Outline

A physical model, implemented in the object-oriented

modeling language Modelica, for a drying section is

presented. It has been validated against a fine paper

machine.

The model is used to evaluate a new strategy to control

the moisture in the drying section (mid-ranging MPC)

with promising results.

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Steam pressure modeling

A graphical visualization of the temperature profile

and energy flow to the metal.

A first-principles model

Steam

Paper web

Dryer shellCondensate

Ts

Qm

Tm

Tp

Qp

p

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Steam pressure modeling

( )msm TTAQ =

Mass balances:

Energy balances:

Energy flow to metal:

( )

( ) wcww

btcsss

qqVdt

d

qqqVdt

d

=

=

( )

( )

( ) pmmp

mwwscwww

scsbtssss

QQTmCdt

d

QhqhqVudt

d

hqhqqVudt

d

=

=

=

)(


VVV ws =+

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Paper sheet modeling


evapxyqA

inxy guvd

Paper web

guvd xy

xv

)( svapevapxy HHqA +

inpppinxy TCugvd ,,)1( + pppxy TCugvd ,)1( +

pQ

Paper web

Mass balance moisture

Energy balance

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Implementation in Modelica

Pope

Moisture controller

Press

HeaderPres...

k={1000e3}P

i PID

F

PID

Header pressure

model Cylinder

equation

Ms = rhos*Vs;der(Ms) = qs qc - qbt;

Mw = rhow*Vw;

der(Mw) = qc - qw;

Es = rhos*us*Vs;us = hs p/rhos;

der(Es) = (qs qbt)*hs qc*hs;

Ew = rhow*uw*Vw;uw = hw p/rhow;

der(Ew) = qc*hs qw*hw Qm;

Em = m*Cp*Tm;

der(Em) = Qm - Qp;

V = Vs + Vw;

Qm = alpha_sc*Acyl*(Ts - Tm);

Qp = alpha_cp*Acyl*eta*(Tm - Tp);

end Cylinder;

Modeling of the drying section

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Calibration of the model


5 10 15 20 25 30 350

10

20

30

40

50

60

Cylinder number

S

h

e

e

t

m

o

i

s

t

u

r

e

(

%

)

Fine paper machine

Speed: 708 m/min

Basis weight: 80 g/m2

Heat transfer coefficient:

1100 W/(m2K)

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The last group is much faster


CsUe

sY 13

148

11.0

+=

MPC Steam system

Dryer

Setpoint

Moisture

Setpoint

steam pressureSteam

pressure

PID-

controller

Moisture

y

ucr

MPCDryer

Setpoint

moisture

Moisture

PID-

controller

PID-

controller

uc1

uc2

Steam system

Last part

Steam system

First part

y

r1

25

114

148

01.0

148

098.0C

sC

s Ues

Ues

Y

+

+=

Single loop moisture control

Proposed new control structure

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.small0

0large,

small0

0large

=

= RQ

MPCDryer

Setpoint

moisture

Moisture

PID-

controller

PID-

controller

uc1

uc2

Steam system

Last part

Steam system

First part

y

r1

==

+

++

++

++=

3

0

2

2

149

0

2

22

1

)(

)(

)()(

)|()()(

i Rc

c

i Qciku

iku

ikuikr

kikyikrkJ

25

114

148

01.0

148

098.0C

sC

s Ues

Ues

Y

+

+=

Mid-ranging control of the drying section

Mid-ranging MPC proposed by:Allison, B. J., and A. J. Isaksson (1998): Design and performance of mid-ranging controllers.

Journal of Process Control, 8(56), pp. 469474.

Tuning of the controller

Proposed new control structure

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uc2is faster that u

c1

Mid-ranging MPC

7.0

7.2

7.4

7.6y

(

%

)

430

440

450

u

c

1

(

k

P

a

)

0 200 400 600 800 1000 1200400

500

600

Time (s)

u

c

2

(

k

P

a

)

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6.8

7.0

7.2

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7.6

y

(

%

)

0 200 400 600 800 1000 1200430

440

450

Time (s)

u

c

(

k

P

a

)

10-4

10-3

10-2

10-1

0

0.5

1

1.5

Frequency (Hz)

M

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e

(

a

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s

)

Time (s)

0 200 400 600 800 1000 120014

16

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20

Time (s)

S

t

e

a

m

c

o

n

s

u

m

p

t

i

o

n

(

k

g

/

s

)

Single loop MPC

Mid-ranging MPC

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Single loop MPC

Mid-ranging MPC


6.8

7.0

7.2

7.4

7.6

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o

i

s

t

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e

(

%

)

0 200 400 600 800 1000 120010

15

20

25

Time (s)

S

t

e

a

m

c

o

n

s

u

m

p

t

i

o

n

(

k

g

/

s

)

10-4

10-3

10-2

10-1

0

0.5

1

1.5

Frequency (Hz)

M

a

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Conclusions

A simulation model based on first principles has

been given

A new strategy to control the moisture in the paper

machine has been proposed

The mid-ranging control has a large potential but

might also require an effective steam distribution

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