Application of hybrid modeling and control techniques to desalination plants

DESALINATION ELSEVIER

Desalination 152 (2002) 175-184

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Application of hybrid modeling and control techniques to desalination plants A. Gambier*, E. Badreddin Automation Laboratory, University of Mannheim, 68131 Mannheim, Germany Tel. +49 (621) 181-2740," Fax +49 (621) 181-2739; emails: gambier@,ti.uni-mannheim.de, badreddin@ti, uni-mannheim,de

Received 30 March 2002; accepted 15 April 2002

Abstract

One of the most recent and most intense efforts in control theory deals with handling systems whose behavior of interest is determined by interacting continuous and discrete dynamics. This approach can be applied not only to intrinsic hybrid processes but also to other systems as for example continuous processes with supervisory logic, multi-model control systems, switching control, etc. In this paper, hybrid systems are briefly introduced and possible applications to desalination plants are given by means of illustrative examples. Keywords: Hybrid systems; Hybrid control; Dynamic modeling; Desalination plants

1. Introduction

Many physical systems are hybrid in the sense that they have barriers or limitations. Inside the limitations they are modeled with differential equations. A natural way to model these systems is to use a mixture of differential equations and inequalities. Other systems have switches and *Corresponding author.

relays that can be naturally modeled as hybrid systems. These hybrid models appear in many areas. Typical examples are flight control, air traffic control, missile guidance, process control, robotics etc. Although modes can be strictly speaking not discrete, it can be advantageous to model systems in that way. An example o f this is when a nonlinear system is modeled with a set o f linear models each one covering a part of the state space. Desalination

Presented at the EuroMed 2002 conference on Desalination Strategies in South Mediterranean Countries: Cooperation between Mediterranean Countries of Europe and the Southern Rim of the Mediterranean. Sponsored by the European Desalination Society and Alexandria University Desalination Studies and Technology Center, Sharm El Sheikh, Egypt, May 4-6, 2002.

0011-9164/02/$- See front matter © 2002 Elsevier Science B.V. All rights reserved PII: S 0 0 1 1 - 9 1 6 4 ( 0 2 ) 0 1 0 6 0 - 3

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plants are also examples of complex large-scale systems that fit more than one of the above-described characteristics. Therefore, the application of hybrid modeling and control can be an interesting approach to improve plant performance, efficiency, safety and reliability. The first attempt to obtain a dynamical model of an MSF-process has already been reported in [1]. A second effort [2] applies empirical corrections for the evaporation rates. Simulations realized with this model by [3] showed significant deviations in the cooling water rate. Advanced models for MSF plants can be found in [4-9]. The most common approaches for controlling MSF plants are based on decentralized PID control loops. A survey on control of desalination plants can be found in [10]. Another approach was proposed by [11]. It consists in a parameter scheduling PID adaptive control for a subsystem defined by the most important six inputs and six outputs. Because brine temperature at the first-stage input is the most important variable in the plant, several efforts have been carried out to improve the control performance at this point by using advanced techniques. For example, in [12] fuzzy conti'ol was successfully applied, and [13] implemented an evolutionary PID fuzzy controller for the same purposes. Regarding reverse osmosis plants (RO), mass transfer models were reviewed by [14]. Moreover, [15] reported a simplified dynamic model for an industrial plant. In [16], an overview of process control of desalination plants is given and [17] presented advanced control techniques for RO plants. In [18], an approach based on DMC (Dynamic Matrix Control) was compared with standard PID control. Even though hybrid modeling and design techniques have developed in the last 10 years with good results particularly in the area of supervisory control, applications in the area of desalination have still not been reported in the literature. This paper gives a brief introduction to hybrid systems and examples how they can be applied to desalination plants.

2. Modeling hybrid systems There are many approaches to model hybrid systems. A common characteristic consists in that the state space S has both discrete and continuous variables, e.g. S c R" × Z ~. The equations can be linear or nonlinear and in general the discrete parts cannot be separated from the continuous parts. The models proposed by various researchers differ in definition of and restrictions on dynamic behavior. The difference between the models are on aspects as generality, allowance o f state jumps, dynamic restrictions, etc. Many models do not allow fast switching or sliding. A hybrid model, like all models, should be sufficiently complex to capture the rich behavior of the systems. However, it should also be easy enough to be analyzed, and formulated in such a way that simulation is possible. Different areas of control science have their proper model structures and there is no unified approach or any agreement on what is the most fruitful compromise between model generality and expressibility (for a review of different approaches see for example [19]. The most used approaches to model hybrid systems are hybrid automata [20] and Petri nets [21,22]. In this article, hybrid automata are used.

2.1. Hybrid automaton

The hybrid automaton is an extension of the traditional finite state machine. It can be defined by a three-couple (N, F, L). The network N is also called finite state machine or finite automaton. It is a directed graph characterized by the ordered pair (V,A). Visa set of vertices andA c Vx Visa set ofdir~ted arcs between vertices. Fig. 1 illusWates a network with three vertices V= {v~,v2,v3} and the arcs A = {(Vl,V2), (v2,v3), (v3,vl) }. A marked network is defined by N = (V,A,kt),where I~: Vg{0,1 }. ~t (vl) = 1 means that the vertex vj is marked, i.e. v t is the current state, kt can be described by a vector Ix, where the value of the i-th component is the marked vertex and the other components are equal to zero. The value that this vector takes

A. Gambler, E. Badreddin / Desalination 152 (2002) 175-184

at time t is denoted by I,t (t). Thus, kt represents the discrete state of the hybrid system. F is a finite Lipschitz continuous vector field. (1)

F={J],f2,...,f,}

where f : R" ---) R for i = 1, ... n. E a c h f maps the continuous state space R" back into itself and represents a continuous dynamical system. It generates state trajectories x: R ---) R" through the differential equation

xCt) = Z

u(t))

(2) (3)

x ( t o) = x o

The four-triple y = (Jc, x, x0, to) characterizes the continuous state of the hybrid system. L is the interface between N and F. It is represented as a mapping from the network's vertices and arcs of N in a prepositional logic P. The equations used for the labeling are of the form: • Invariant equations: e . g . f (x(t), u(t)) = 0 • Guard equations: they have the form g (x) > 0, where g: R" ~ R is a function defined over the continuous-state space R". The logical propositions labeling the network vertices and arcs can be defined in a variety of ways. In this contribution, it is assumed that networks arcs are labeled with guard equations and vertices with invariant equations This is shown in Fig. 1.

xE+x_~>0~(x,u)_~

Fig. 1. Example of a very simple hybrid automaton.

3. Example 1: brine heater model of a MSF plant

The brine heater is one of the most important subsystems in a MSF plant. It is the physical

177

decoupling interface between the electrical power subsystem and the desalination units. Damages in the tube bundle will produce damages in eleclrical unit (return of saline steam condensate). Fouled tubes introduce important changes in the plant performance. In [23], it was pointed out that the control of Top Brine Temperature (TBT) is decisive to reach the overall stability and economy of plant operation. The system also presents non-linear characteristics. For the brine heater, it is possible to define 22 variables and 19 equations, so that the degree of freedom is 3. Thus, tree control loops can be introduced in order to obtain an exactly specified equation system. A very important controlled and measured variable is the TBT on the heater output shown here as Tb. The TBT depends on the steam temperature ( T ) , the brine temperature (To), the brine flow rate (Fb) and the steam flow rate (F~), all at the heater input. T, depends on the temperature of the incoming steam, which is assumed constant, and on the water spray flow (Fd) (its control variable) from the desuperheater. F j is defined as control variable for Tb. Tb, is the temperature gained in the heat recovery section and therefore it is an output variable for this section and cannot be directly manipulated at this point. On the other hand, a minimum water level in the sump must be guaranteed in order to maintain the load of the condensate extraction pump constant. Hence, there is an additional controlled variable: the condensate level (/) in the sump, the control variable is Fco. Moreover, mass of steam (m), mass of brine (mb), specific enthalpy of steam (h) and condensate level (lc) can be selected as state variables. The equations for the brine heater are obtained from salt, mass and energy balance as well as from the thermodynamic properties of steam (IAPWS-IF97, [24] and the properties correlations for saltwater [25]. The model can be summarized as follows (see [26] for details): dm~ =Fs,(t)_Fci(t)

dt

(4)

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A. Gambier, E. Badreddin / Desalination 152 (2002) 175-184

1--~-[F~.,(t)[h,,(t)-h,(t)]+ F~,(t)[h~(t)-h~(t)]-Q~(,)]

if m~ > 0

ms

1--L[F~,(t)[h,(t)-h~(t)l-Q,(t)]

dt

if m~ > O a n d P > P ~ x

0

d~_ dt

if m,=O

1 [F~;(t)-F~(t)]

(8)

PcAo

F~; (t) ITs,(t) - Tc(t)] pcAol~

d~_ dt

if 0 < l c
l---!---[F~,(t)[T~,(t)T~ Qw(t) ~,~ (t)]- 1---C

if l~>lm~

peA& L

~

(9-12)

[Fc;(t)[T~,(t) - T~(t)]- Faom(t)ATm]

if 0
0

if /~=0

if Fb, (t) = F b(t)

dC~ = I 0

dt

~

dt

(5-7)

m s

1---~[Fd.,.Cb(t)-[F~;(t)+ Fd.,.(t)~7"~(t)]

if Fb, (t) ¢ F b(t)

(pcAolc

= & , ( t ) - F~(t)

(13,14)

(15)

if mb > 0

d~_ dt

(0]]

±Ira,,LR~,(,)[r~,(,)- r~(ol+ ~----[O.c.(') + Oc

if lc >/max

0

if m b = 0

(16-18)

A. Gambier, E. Badreddin / Desalination 152 (2002) 175-184 Q(t)

(19)

= % (t)Ah, AT,n

AT., =[rb,-

Tl,]/ln/Tsa' - T---b~

(20)

rso,-g,)

Fc(t ) = [ As2,mpA2i, P c ~ ' A ~ : 2 i , ~2[glc -(Pp,,,p - Psa,)/ Oc]

(21)

In order to simulate this model, a hybrid automaton can be implemented. That is, the model is driven by a finite state machine, which launches the correct equation when the corresponding conditions are satisfied. Table 1 shows some possible discrete states for the hybrid model. This hybrid automaton is shown in Fig. 2, where guards, jumps and states are shown. Note that the automaton also brings information for a higher supervision level, which can be used for fault handling purposes.

179

Table 1 Possible discrete states to define a hybrid model Discrete state

Mnemonic Equations

Normal operation NO High pressure HP No condensate NC No brine NB No steam NS No condensate--no brine N C N B No brine---no steam NBNS Salt in condensate SC High condensate level HCL

1,2,5,6,10,12,13 1,3,5,6,10,12,13 1,2,5,9,10,12,13 1,2,5,6,10,12,15 1,4,5,9,10,12,13 1,2,5,9,10,12,15 1,4,5,9,10,12,15 1,2,5,8,11,12,13 1,2,5,7,10,12,14

4. Example 2: top brine temperature supervisory control of a MSF plant Traditional control strategies for MSF plants arc based on fixed PID controllers. However, it can be shown that fixed PID controllers cannot bring satisfactory control performance for wide operating conditions. Therefore, in [ 12] a parameter scheduling adaptive scheme for six operating

Alarm signal:

Venting system is not working correctly

P~>P~x and

To5

'ondensate

To Salt in Condensa

e is brocken

Fig. 2. Automaton for the treatment of discrete events for the brine heater.

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A. Gambier, E Badreddin / Desalination 152 (2002) 175-184

points was proposed. Such strategy presents in general some difficulties as for example: detection of the operating point changes, controller switching method (or parameter switching for the same controller) bumpless parameter change and stability issues due to switching control. Here, these problems will be treated by introducing a hybrid automaton in the control system according to Fig. 3. reference

.~Ctmtro|lers~

Su~r

4.2. The supervisor The supervisor is responsible for detecting operating point changes and producing a bumpless switching when the parameters are changed. Its structure is schemed in Fig. 4. The first task is implemented by using the Min-switching strategy and the second one by using standard procedures for bumpless transfer [28].

4.3. Detection of operating point change: Minswitching strategy

H)brld

ContinuousModel~

I

Fig. 3. General hybrid control strategy.

4.1. Control law As control laws, the PID controller is used: u(t) =

g e K,,r(t) - y ( t ) + 1 fe('c)dr "l;g

with anti-windup mechanism (integration stop). The parameters for the PID controller can be tuned according to the Ziegler-Nichols rules.

- Tep(t)

The problem is to find a stable closed-loop control system for a continuous time process, several controllers and a logic system that commands the switches between these controllers. Malmborg [27] proposed a solution based on a set of controllers coupled to a set of Lyapunov functions. The key idea is to associate each linear model with a separated Lyapunov function and construct the logic-switching device in such a way that the composite system is stable. The switching strategy selects the controller corresponding to the Lyapunov function with the smallest value. This is known as the Min-switching strategy and it has been shown to be stable (see [28] for details of implementation).

Stateof nonlinearsystem

vj

[&2:l Hybrid Automaton

V3.1

;

]

r,)

&tl

s.p.)

Fig. 4. General scheme for the supervisor.

181

A, Gambier, E. Badreddin / Desalination 152 (2002) 175-184 4.4. Bumpless transfer

Because the controller is a dynamic system, a change in its parameters will result in changes of the control signal even if the input is kept constant. Methods to introduce bumpless switching differ in the form that they set e(t) and d(t) to zero. One important point here consists in deciding when the p a r a m e t e r s w i t c h i n g should be undertaken. One possibility is to do this when the new set point is reached. Another one consists in doing this when the trajectory crosses the switching surfaces. The fist case has the advantage that the switching is carried out when the steady state is reached (i.e. e(t) and ~(t) equal to zero) satisfying the condition for bumpless transfer. The drawback is the retard introduced between the change detection and the parameter switching. Fig. 5 shows the hybrid automaton for the supervisor.

Reverse osmosis desalination plants require that the feed seawater should be pretreated to minimize deposition of fine particles, mud and scaling compounds. This activity is normally carried out in tanks like it is shown in Fig. 6. The pump at the tank output is also very important in order to maintain the feed pressure at levels specified by the membrane manufacturer. Thus, a control level in the tank is necessary to avoid drainage and overflow, and to compensate flow rate variations due to changes in the chemical additives. The tank model comprises three different states corresponding to normal, empty and full operating conditions with the dynamic mass balance equation tacking a different form in each of them. The pump model has two states corresponding to operational (p = 1) and non-operational (p = 0) modes. The

Sj2> _ 0 and $41 > 0

$12< 0 and S41 > 0

e = 0 and &=

5. Example 3: level control of the pretreatment stage in a RO plant

e= 0 and k=0

S~z> 0 and $23 < 0

Sn2_<0 and $23 <

Oand ~=0 S12< 0 TM

,,=0 and ~=0

and S41 <~

e = 0 and S41 >and S~

$34 > 0 and $4~< 0

e = 0 and e = 0

e= 0 and e 0

Set CPi = Set Controller parameter set number i

Fig. 5. Hybrid automatonfor the supervisor.

~k~$23< 0 and ~ 534<0

34-<0 and 541 < 0

l

Sl2> 0 and \&3_>O

$34> 0 and S~ > 0

e= 0 and ~=0

$23>-"Oand $34< 0

Set Cond. = Set conditions for e = 0 and e = 0

182

A. Gambler; E. Badreddin / Desalination 152 (2002) 175-184

s.p.

pretreatment chemical

automated functions o f the plant. The introduction of hybrid modeling and control leads to more complex computer controlled systems and therefore to more complicated analysis and design tools, which are still being developed. The studies are just in an initial state and much effort should be undertaken in order to implement such systems.

seawat~

, to the ~embrane unit LP pump Fig. 6. Schematic diagram for the pre-treatment stage of a RO plant. transition depends on the control signal sent to the pump. The valve comprises one state for the normal operation and some states to describe the malfunctioning. Models for the system components are shown in Fig. 7. 6. S u m m a r y

In this contribution, the main idea o f hybrid systems has been presented. General properties and advantages o f hybrid modeling were highlighted and possible applications to desalination plants were illustrated be mean of simple examples. Simulation results showed that hybrid modeling and control can be promisingly for the desalination industry in order to extend and to improve the

I

2__d
IT

\_k

Fig. 7. Models for (a) the tank, (b) the pump and (c) the valve.

Symbols

A An, cx, C C F h l m P T t

--------------

p

--

Cross area, m e Heat transfer area, m 2 Heat transfer coefficient, W / ( m 2 ° C ) Salt concentration, g/kg Heat capacity, kJ/(kg.°C) Mass flow rate, kg/s Specific enthalpy, kJ/kg Level, m Mass, kg Pressure, MPa Heat flor rate, W Temperature, °C Time, s Density, kg/m 3

Subsripts

b c i s sat

- - Brine - - Condensate - - Input (e.g. Fbj is mass flow rate for brine input) - - Steam - - Saturated steam

A. Gambler, E. Badreddin / Desalination 152 (2002) 175-184

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[27] J. Malmborg, Analysis and design of hybrid control systems. PhD thesis, ISRN LUTFD2/TFRT-1050-SE, Department of Automatic Control, Lund Institute of Technology, 1998.

[28] A. Gambier, M. Fertig and E. Badreddin, Hybrid supervisory control of the brine heater for multi stage flash desalination plants. Accepted to be presented at 15th IFAC World Congress '02, Barcelona, Spain, 2002.