that this model can be approximated using Genetic Algorithm optimization of a fuzzy .... the occurrence of external events, an internal state transition transpires ...
M&l.
Comput. Modelling Vol. 23, No. 11/12, pp. 215-228, 1996 Copyright@1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/96 $15.00 -t 0.00
Pergamon
SO8957177(96)00074-X
DEVS Approximation of Infiltration Using Genetic Algorithm Optimization of a Fuzzy System B. P. ZEIGLER AND Y. MOON Department of Electrical and Computer Engineering University of Arizona, Tucson, AZ 85721, U.S.A. Qece.arizona.edu v. L. LOPES School of Renewable Natural Resources University of Arizona, Tucson, AZ 85721, U.S.A. vlopescDsrnr.a.rizona.edu J. KIM Department of Electrical and Computer Engineering University of Arizona, Tucson, AZ 85721, U.S.A. jwkimBece.arizona.edu Abstract-The infiltration process is generally described by a nonlinear differential equation, which can be solved by iteration methods such as a Newton-Raphson method. In this paper we propose a Discrete Event System Specification (DEVS) model for Green-Ampt infiltration. We show that this model can be approximated using Genetic Algorithm optimization of a fuzzy system. The fuzzy approximation is shown to be more accurate than the Taylor series approximation recently proposed. Keywords-Discrete equation.
event system specification,
Fuzzy systems, Genetic algorithms,
Green-Ampt
1. INTRODUCTION Modeling
rainfall
decades. small
runoff
One scheme cells which
enough, equations Discrete efficiency
Although
for such which event the
In this
efficiency, paper
source has
we adopt
a continuous
the
model
most
a model approach described
of them
that
in hydrology
a large
If the
the
into many
area of each
cell is small
form
models
of nonlinear
advantages models
such
differential
System
Specification
fully takes
advantage
(DEVS) of discrete
a DEVS
model
equation
This research was supported by NSF HPCC Grand Challenge Application ployed the CM-5 at NCSA under Grant MCA94P02.
method.
as flexibility
and
[3].
by a set of nonlinear
of [7] to develop
to compute
such as the Newton-Raphson
many
by the Green-Ampt
for several
watershed
mathematical
landscape
described
Event
[1,2]. have
methods scale
of studies
are several
to afford
for large model
to a Discrete
we need
There
shown
simulation
topic
is to divide
of waterways
by iteration
been
system
process
source. and
solved
continuous converted
complex
by a network
are generally simulation
has been the main
this
as a point
a point
over continuous
can be directly
stracting
are connected
it can be considered
infiltration
considering
in a watershed
in modeling
differential
equations
[4-61 model event
for infiltration
[8]. A fuzzy
without
simulation system
[7]. by ab-
is
used
Group Grant ASC-9318169 an.d em-
Typeset by A&-TEX 215
B. P. ZEIGLER et al.
216
to solve the Green-Ampt equation for significant events of the infiltration process without using iteration methods. Asynchronous Genetic Algorithms (AGAS) [9] are used to design the fuzzy system. Mapping the infiltration process to a high fidelity DEVS representation not only provides an efficient simulation model but also affords several insights into the infiltration process itself. Therefore the methodology is of interest to hydrologists and to scientists in general seeking effective models of complex systems.
2. RAINFALL
RUNOFF
PROCESS
Rainfall runoff in a watershed is a complex process. Many factors influence this process, including the conditions of the soil surface and its vegetative cover, the properties of the soil such as its porosity and hydraulic conductivity, and the current moisture content of the soil. Partly because of limitations During a rainstorm, the rate of rainfall changes constantly. in measuring equipment,
we commonly approximate
this rate change with a finite number of
relatively short pulses. Each pulse is assumed to have a constant rate, but the rate changes from pulse to pulse. This sequence of rainfall pulses is both temporally and spatially distributed. Infiltration is the process by which portions of the rain that are not intercepted by plants or surface litter enter the soil. The infiltration rate is not constant. Its pattern responds to the variation in rainfall rates and to the accumulated infiltration amount. Most mathematical models of infiltration are represented by nonlinear differential equations and they are usually solved by iteration methods. Evapotmsportation
Rainfall
t
A!
surface
bedrock bedrock
Figure 1. Cellular space representation
of a watershed
Figure 1 shows a model of the rainfall runoff process of a watershed. We divide a large watershed into many small cells so that each cell can be considered as a point source. A large watershed may have millions of cells and it is impossible or impractical to calculate the infiltration rate of each cell by solving the nonlinear differential equations using iteration methods. In this article we show how a DEVS approximation of infiltration can be a solution of this problem.
3. GREEN-AMPT
INFILTRATION
MODEL
The Green-Ampt equation has became widely used to compute infiltration in catchment-scale hydrologic models [lO,ll]. In addition to the fact that the parameters in the equation have
DEW
Approximation
217
physical significance, experimental works have been completed or are underway to obtain for the parameters based on soil texture and on the effects of management [12]. The rate form of the Green-Ampt
equation
for the one stage case of initially
ponded
values
condition
is
where fc(t) = infiltration
capacity
(L/T),
K, = effective saturated
hydraulic
conductivity
(L/T),
?,b = average capillary potential at the wetting F(t) = cumulative infiltrated depth(l) (note: ables).
The soil moisture
front (L), 6d = soil moisture deficit (L/L), and L and T represent length and time for all varideficit can be computed as 6d = 6, - & = “I (S,,,
where
8, = volumetric
(L/L),
7 = soil porosity,
Recognizing
that
water content and S,,,
fc(t) = q,
at saturation
(L/L),
and Si are maximum we integrate
F(t)
- &) ,
(2)
8, = initial and initial
this relation
volumetric
water content
values of relative
saturation.
to obtain
[ F(t) 1.
= K,t -t $ti,l In 1 + -
+sd
Equation Raphson
(3) is normally solved numerically iteration method.
for successive
increments
of time using the Newton-
Stage S 1: No Runoff Stage S2: Transition Stage S3: Constant Runoff
10.0
K,+& Ke .-
tc
Figure 2. Green-Ampt
Time( hours )
infiltration model.
Figure 2 shows the infiltration capacity and the rainfall excess during a constant rainfall. There is no rainfall excess until the rainfall intensity becomes larger than the infiltration capacity. The rainfall intensity is larger than the infiltration capacity after the time to ponding (tp) where the rainfall intensity equals to the infiltration capacity. Equation (1) shows that the infiltration
B. P. ZEIGLER et al.
218
capacity fC asymptotically approaches to K,, constant after the time to constant runoff (tc) divide the infiltration process into three stages: runoff and a stage with constant runoff using t,
SYSTEM
and the rainfall excess can be considered as a wh ere fC becomes K, + E for small E. We can a stage without runoff, a stage with transitional and t, as in Figure 2.
4. DISCRETE EVENT SPECIFICATION FORMALISM
The DEVS formalism introduced by Zeigler [4] provides a means of specifying a mathematical object called a system. Basically, a system has a time base, inputs, states, outputs, and functions for determining next states [4-61. The state of a system summarizes the information concerning past inputs that is needed to determine the response of the system to subsequent inputs [5]. The DEVS formalism focuses on the changes of state variables and generates time segments that are piecewise constant. In the DEVS formalism, one must specify basic models from which larger ones are built, and describe how these models are connected together in hierarchical fashion. In this formalism, basic models are defined as follows. A DEVS model is a structure:
where l l l l l l l
l
X is the set of external (input) event types. S is the sequential state set. Y is the output set. hint : S + S, the internal transition function. Lt 1Q x X + S, the external transition jknction. Q is the total state set = {(s,e) 1s E S, 0 5 e < ta(s)}. ta:S--, Rof,, the time advance function. This function determines the maximum time during which the system can remain in the current state. If time ta(s) elapses without the occurrence of external events, an internal state transition transpires based on 6i,t, immediately preceded by an output generation using the function, X. The external event occurring at elapsed time e after the last transition evokes a transition determined by Gext. X : S + Y, the output function.
A basic model, called atomic model, contains the following information: l l l
l
l
l
l
the set of input ports from which external events are received; the set of output ports to which external events are sent; the set of state variables and parameters: two state variables are usually present-phase and sigma (in the absence of external events the system stays in the current phase for the time given by sigma); the internal transition function, which specifies the next state to which the system will transit after the time given by sigma has elapsed; the external transition function, which specifies how the system changes states when an input is received and places the system in a new phase and sigma, thus scheduling it for a next internal transition; the output function, which generates an external output just before an internal transition takes place; the time advance function, which controls the timing of internal transitions-when the sigma state variable is present, this function just returns the value of sigma.
DEVS
Approximation
219
5. DEVS REPRESENTATION FOR INFILTRATION The infiltration process in Section 3 can be described by the DEVS formalism in an efikient way if the following can be efficiently calculated: (1) The time to ponding (tto_ponding)from any time t where the rainfall intensity changes in stage Sl. This time is a function of the rainfall intensity and the cumulative infiltrated depth at t. (2) The time to constant runoff from t, (tto_const= tc - tP) where the rainfall excess can be considered as a constant. This time is only a function of the cumulative infiltrated depth at t,. (3) The cumulative infiltrated depth F(t,.). A DEVS model M for infiltration can be defined as M = (X, S, Y76int3 fiext, X7 ta) ) where X = {Tin 1Tin = input rainfall intensity (L/T)}, Y = {g 1y = runoff (L/T)}, 5’ = {s / s = (pha=, Ww rcur, T,, F, Tteconst)), rcur = current, rainfall intensity (L/T), r, = rainfall excess(L), F = cumulative infiltrated depth (L), and Tto-const = remaining time to constant runoff. In Section 3, we divided the infiltration process into three stages-NoRunoff (stage Sl), Frunsition (stage S2), and ConstantRunoff (stage S3). We define phase as one of NoRunoff, Transition, and ConstantRunoff for each stage Sl, S2, and S3, respectively. In addition to these three phases we need two more phases, Fran&ion” and ConstantRunoff’, to generate the output when the rainfall intensity changes during stages 52 and S3. The internal transition function 6i,t, the external transition function dext, and the time advance function ta for the DEVS model M are shown below. l
The internal transition function 6i,t,(s) =
s’, where s’ =
(phase’, sigma’, &,
r:,, F’,
T’to-const) When phase = NoRunoff F’ = F + r,,, * sigma r’e = rcur * ttwconst(F’) - (F(k) - F’) kmmst (W phase’ = Transition sigma’ = ta(s) (Variables not shown are unchanged.) When phase = Fran&ion F’ = F(t,) Ir, - rcUr- K, phase’ = ConstantRunoff sigma’ = k(s) When phase = l’bansitiori phase’ = Transition
l
sigma’ = Tto-const When phase = ConstantRunoff’ phase’ = ConstantRunoff sigma’ = b(s) The external transition function &,t(s, e, x) T’to-const) When phase = NoRunoff F’ = F f rcur * e r’CUT= Tin sigma’ = k(s)
= s’, where s’ = (phase’, sigma’, riur, ri, F’.
B. P. ZEIGLER
220
et al.
When phase = Fran&ion T’to-const = sigma - e r’e = r e - (rr~u,- ?n) r’CUT= Tin phase’ = Transition” sigma’ = 0
When phase = ConstantRunoff F’=F+K,*e r’CUT= Tin r’ = r’ - K
l
pehase”: ConltantRunofl” sigma’ = 0 The time advance function ta(s): When phase = NoRunofl ta(s) = tto-ponding(rcur,F) When phase = Transition
l
ta(s) = tto-const(F) When phase = ConstantRunofl ta(s) = cx2 The output function X(s): At the end of phase = NoRunoff rcUr* tto-const(J’) - (J’(td - F) X(s) = tto-const(F) At the end of phase = Tkansitiod’ X(s) = r, At the end of phase = Bans&on X(s) = r,,, - K, At the end of phase = ConstantRunoff X(s) = r,
The operation of the model M is as follows: (1) At time t = 0, the phase is NoRuno# and the next event is scheduled as sigma = w h ere r is the rainfall intensity and F is the cumulative infiltrated depth tt~pondidr,F)r at t = 0. (2) If the rainfall intensity changes at time ti during stage Sl, the next event is rescheduled a.s sigma = ho-ponding(r, F). (3) Note that between t, and t, the rainfall excess, and therefore the output runoff, varies. A DEVS model, however, can only approximate this curve by finite number of outputs. If the rainfall intensity does not change during stage S2, the output at t, is to represent the rainfall excess between t, and t,. Conservation of mass requires that r, x (tc tP) = total runoff. Therefore, at time t = t,, the rainfall excess is approximated as r, = T*tto-=onat(Fp)-(F(tc)-~~) tt,.c0net(F,) ’ where r is the rainfall intensity, tto-const(Fp) = t, - t,, FP is the cumulative infiltrated depth at t,, and F(t,) is the cumulative infiltrated depth at time t,. The next internal event is scheduled as sigma = tto_const(Fp). (4) Consider a rainfall intensity change at tl(t, 5 tl 5 tc). We update sigma = sigma - e, where e is time elapsed since the last internal or external model event. The rainfall excess r: is recalculated as r: = r, - (rcur - Till) (recall that r cUT= current rainfall intensity and Tin = new input rainfall intensity). Note that the rainfall intensity change only affects the rainfall excess but not the infiltration process [8]. (5) At time t = t,, the rainfall excess is calculated as r,,, - K,. The next event is scheduled as sigma = 03, i.e., the model will remain passive unless activated by an external event.
DEVS
221
Approximation
(6) The rainfall intensity change at time t during stage S3 recalculates the rainfall excess r: as r: = r - K,, where r is the rainfall intensity at time t. To realize the above DEVS model we need to represent the Green-Amp solution for tto_ponding, Although the DEVS model for infiltration approximates the rainfall excess tto-const, and F(t,). for the transitional stage as finite number of outputs, this is not a major source of error for a, long term simulation of a large watershed represented by a grid system of small cells-the intended application. Stage S 1: No Runoff Stage S2: Transition Stage S3: Constant Runoff
s3
i
r-K,
r : rainfall intensity
i
--- --------I,-~.~~~:~~~!~.-.....
.....*........................
Time(hours) Figure 3. DEVS approximation model behavior. (Solid and dashed curves represent the outputs of the DEVS and continuous model, respectively.) Figure 3 shows the behavior of the DEVS model for infiltration during a constant rainfall compared to that of the continuous system model. For constant rainfall input, the DEVS model approximates the continuous curve of rainfall excess in two steps. Since only two computations are needed, simulation of the DEVS model is more efficient than that of the continuous model which may need thousands of small steps.
6. IMPLEMENTING A DEVS MODEL USING A FUZZY SYSTEM In Section 5, we presented a DEVS model for infiltration, but we need solve the Green-Ampt equation for three unknowns, tto_ponding,tto_constrand F(t,), to implement it. We can analytically solve the Green-Ampt equation for tto_pondingin the case that the duration of a rainfall event is divided into many short periods in such a way that within each period the rainfall intensity is essentially constant [13]. Assuming that the rainfall intensity changes from rprev to rC,, at time t for t < t,, we can calculate the time to ponding tto_pondingsince we know that the infiltration capacity (fc(tp)) and the cumulative infiltrated depth (F,) at t, should be r,,,. and respectively. If the rainfall intensity changes more than once before time t, rprevt+ r,,,k-p0nding, then Fp should be F(t) + r,,, x tto_ponding, where F(t) is the cumulative infiltrated depth at time t. Using equation (l), the time to ponding from any time t can be calculated as
ho-pending
and
ho-pending
at time t.
(rcur,
F(t))
=
KX(t)
+ K,lCl& rcur (rcur
-
r,&‘(t)
Ke)
’
is a function of the rainfall intensity rcur and the cumulative infiltrated depth F
222
B. P. ZEIGLER et al.
We defined the time t, as the instant when the infiltration capacity from equation (l), the cumulative infiltrated depth F(t,) is
f (tc) becomes K, + E, and
K,@, -. E
(5)
The Green-Ampt
equation allows us to solve it for the time to ponding and the cumulative infiltrated depth F(t,) analytically, but it does not allow us to solve it for tto-const which is a function of Fp. There are several ways to solve this problem. One possible solution is to use approximation forms of the Green-Ampt equation which allow analytical solution for tto-const( Fp) (see [14]). The second one is to maintain a look-up table which contains all solutions for every possible input event. However this scheme requires a lot of memory. Another way is to maintain the approximate solutions using different types of computational structure such as neural networks or fuzzy systems. In this paper we use a fuzzy system described in Appendix A to hold the solutions of tto_const(Fp) for the whole range of Fp. The range of Fp can be calculated from equation (1) for a given soil, a range of input rainfall intensity, and an infiltration capacity at t,. At t,, the rainfall intensity T is infiltrated depth F is the ponding depth Fp. From equation (l),
fc and
the cumulative
Thus, Fpmin =
KeyK Tmax
, e
(f-3)
where Fp min is the minimum ponding depth and Tmax is the maximum rainfall intensity. Let the infiltration capacity and the cumulative infiltrated depth at t, be fc(tc) and F(t,), respectively; then the maximum ponding depth F,,,, is Ke$dd/(fc(tc)- Ke) since Fp 5 F(t,). If we define
fc(tc) as K, + E,then %.?(led
Fpmax = -.
(7)
E
Given the range of Fp as in equations (6) and (7), we can solve the Green-Ampt equation for t to-const for a given number of training input points and design a fuzzy system that approximates the solutions using the AGA optimizer.
7. EXPERIMENTAL
RESULTS
Green-Ampt parameter values for a silt soil are K, = 5.0 (mm/h), 111= 190.0 (mm), and q = 0.42 [12]. Let th e maximum rainfall intensity rmax be 200.0 (mm/h), S,,, be 1.0 and S’i be 0.5; then Fpmin is 1.02 (mm) from equation (6). We chose E to be K, x 0.2 to keep t, close to is calculated as 199.50 (mm) using equation (7). The possible range for the one day, and F,,,, output(tto_const) can be obtained by solving the Green-Ampt equation using an iteration method. As shown in Figure 4, tto_consthas a value between 0.0 (hours) and 32.0 (hours). It may happen that a better fuzzy approximation is obtained by locating the center of some membership functions outside the range calculated above. Therefore, we extended the search space by 60% and defined three fuzzy regions for -60.0 < Fp 5 260 and -9.6 5 tto_const 5 41.6. We then optimized the membership functions and rules at the same time using the AGA optimizer described in Appendix B. Figure 4 shows the solution for tto_const by the fuzzy system compared to the solution by the Newton-Raphson method, which is the target of the fuzzy system, and the solution by the two-term Taylor series approximation. The experimental results show that the fuzzy system can solve the Green-Ampt equation for tto_const within 0.3 hours maximum error. The fuzzy system approximates tto-const better than
223
DEVS Approximation
tI
30.0-
11
25.0&
L
2o.ow
15.ow
IOK
5.0-
o.o100.0
50.0
0.0
I- Fp(mm)
200.0
150.0
Figure 4. The time to constant runoff t to_const for the silt soil. (Dashed, solid and dotted curves are obtained for the fuzzy system, Newton-Raphson method and twoterm Taylor series approximation, respectively.)
c--7
-200.0
-100.0
0.0
100.0
200.0
LA
300.0
Gil.0
Figure 5. The membership functions of the fuzzy system. (SM, ME, LA stand for small, medium and large, respectively.)
B. P. ZEIGLER
224
et al.
execution time(hours)
:I_
rm
5.00
/ r'
3.00
\ I I
2.00
1.00
1,
\, \ L
0.00 O.&l
x
20.00
40.00
60.00
number of processors Figure 6. The execution time to optimize the fuzzy system on the CM-5 massively parallel computer (measured for l,OOO,OOOevaluations).
the two-term Taylor series approximation of the Green-Ampt equation which recently appeared in the literature [14]. The latter suffers 3.5 hours maximum error as shown in Figure 4. The fuzzy system was trained using only 100 data points while the curve shown in Figure 4 represents the output for 1,000 input points. The membership functions of the fuzzy approximation are shown in Figure 5 and the rules are as follows: l l l
Rule 1: If Fp is Small, tto_const is 35.2 hours. Rule 2: If Fp is Medium, tto_const is 31.7 hours. Rule 3: If Fp is Large, tto_const is -7.2 hours.
We need about l,OOO,OOOevaluations to optimize the fuzzy system. Execution times for this task for various processor sets on the CM-5 massively parallel computer are shown in Figure 6. While it takes about 7 hours to complete an optimization run using one processor, this time is reduced to approximately 11 minutes using 64 processors. Larger numbers of processors did not improve the search time in this application. But the approximate speedup of 35 times is certainly significant in establishing the practicality of the approach.
8. CONCLUSIONS
AND
FUTURE
WORK
We have devised a DEVS model for infiltration described by the Green-Ampt equation and shown that this model can be realized using a fuzzy system approximation designed by GA optimization on a CM-5 supercomputer. The fuzzy system outperforms the two-term Taylor series approximation proposed in [14] on the same data set. The approach using fuzzy approximation requires off-line training using GAS. However it has an important benefit. Real world observed data, alone or combined with that generated by a mathematical model, can be used to train the fuzzy membership functions. In contrast, the Taylor series approximation requires a mathematically tractable model. In this paper we have shown that fuzzy systems can represent the time to constant runoff for any one type of soil. However, if this technique were to be used in a watershed consisting of many different types of soil, one fuzzy approximation for each would be required. So it remains
225
DEVS Approximation of interest types
to design
a fuzzy system
of soil with just
dimensions
to represent
that
can represent
one set of membership
functions
the time to constant and employing
runoff
for different
one or more additional
soil characteristics.
Writing the DEVS model forced us to consider several state-input conditions that are not typically considered in the hydrology literature. By forcing us to provide behaviors under these conditions, the DEVS abstraction also stimulated
methodology
us to plan new experiments
afforded new insights
into underlying
to fill in gaps in our understanding
real processes.
It
of basic hydrologic
processes.
APPENDIX FUZZY
A
SYSTEMS
The basic idea of fuzzy systems or fuzzy rule based systems centers around the labeling process, in which the input is translated into a label [15]. With expert supplied membership functions for labels and a set of rules, inputs can be fuzzified and through fuzzy logic eventually defuzzijied to generate outputs. It is important to note that the transitions between labels are not abrupt and an input value might belong to several label regions. The fuzzification, inferencing and defuzzification processes can be parallelized. For example, an input signal can be fuzzified by matching all membership functions simultaneously against the incoming value. In this way, fuzzy processing can be viewed as a parallel neural network where each neuron represents a fuzzy membership function and each link represents the weight of a fuzzy rule as in Figure 7 [16,17]. __
-1 ;
--~ I-- \ ~- -------.
4F
1
NL
: NS ‘)I’
ZE”;’
PS
‘)
PL
)i
hput SignalB layer 1
layer 2
layer3
layer4
(a) Fuzzy logic processor
layer5
1:PL
2:PS
3:ZE
4:NS
5:NL
(b) Fuzzy rule table
Figure 7. Fuzzy inference network and fuzzy subspaces. While an earlier fuzzy system [18] was implemented in rule-based (if-then) form, the current fuzzy system employs a parallel inferencing network structure. Due to such parallelization, it can provide better real-time performance. Recently there has been research in developing well-performing fuzzy membership functions without help of human expertise [19]. To do this, it is necessary to employ computeraided optimization. Tuning the membership functions and finding a set of rules are tasks that are difficult to do manually. A probabilistic optimization method utilizing evolution strategies,, such as GAS, can be employed to reliably find optirnal membership functions. However, optimizing multiparameter systems where each experiment requires a simulation can be very time consuming. Therefore, we developed new forms of GAS which are especially oriented to simulation--based optimization on parallel computers [9,20].
I
B. P. ZEIGLER et al.
226
APPENDIX GENETIC
B
ALGORITHMS
The
GA is a probabilistic algorithm which maintains a population of individuals, P(t) = represents a potential solution to the problem at . * * , zCn(t) for iteration t. Each individual hand, and is implemented as some (possibly complex) data structure S. Each solution xi(t) is evaluated to give some measure of its fitness. Then new population (iteration t + 1) is formed by selecting more fit individuals (select step). Some members of a new population undergo transformation (recombine step) by means of “genetic” operators to form new solutions. There are higher order transformations cj (crossover type as shown in Figures 8a and b), which create new individuals by combining parts from several (two or more) individuals and unary transformations rni (mutation type as shown in Figures 8c and d), which create new individuals by a small the search change in a single individual (m : S -+ S) [21]. After some number of generations converges and is successful if the best individual represents the global optimum solution. x1(t),
crossover site I
A~lOOlOOOOlllOOOO B
(b) after crossover
(a) before crossover
mutation site 6 A
100100001110000 (c)
110100001110000~
A
(d) after mutation
before mutation Figure 8. Genetic operator:
crossover and mutation.
Parallel GAS (PGAs) were investigated to reduce search time in real applications [20]. We improved upon PGAs by employing the AGA which is especially oriented to massively parallel processing and does not need to be synchronized by generations to create successive populations 191. In such a multiprocessor environment, individuals in the AGA are evaluated concurrently, and a controller updates the genetic population continuously as evaluation results become known. This is important in our methodology, since evaluations performed by simulations can be highly variable in their execution times. The optimization scheme was first simulated in the Chez-Scheme environment on a workstation and its performance was evaluated with various test functions [9]. Recently, the scheme was ported to the CM-5 massively parallel computer. The performance benefits predicted by simulation were remarkably well corroborated [22].
(new individual, fitness)
(control processor) Figure 9. The block diagram of asynchronous nection machine CM-5.
(processing nodes) genetic algorithms operation
in con-
227
DEVS Approximation
Figure 9 shows the operation of AGAs in the Connection Machine CM-5. Complex evaluation functions, such as the fuzzy system evaluation module, can be implemented inside the CM-5 processing nodes (GA-agent). When execution starts, genetic parameters, such as crossover and mut,ation probability, string size and population size, are initialized in the control processor (GAcontroller). Whenever there are idle processing nodes available, the control processor requests new individuals from the gene-pool and sends them to the processing node. When an individual is returned to the control processor after evaluat,ion, it updates the gene-pool. If other individuals arrive at the processor during the update operation, they are stored in the message queue until the current update is completed. The newly evaluated individuals replace their parents if their fitness is higher than that of their parents. The selection for new individual is performed based on t,he roulette wheel method-slot sized according to fitness [23].
I
~_Estimation Perforjance
AGA Optimizer
.
Membership Functions
.... .
~_._._ Training
Inputs
.
.
Fuzzy System
I
Simulation Model (Target System)
Figure 10. GA optimization
of a fuzzy system module.
Figure 10 shows the interconnection of the fuzzy system, simulation model and AGA optimizer. The fuzzy system estimates the simulation model of the target system and the membership functions and rules of the fuzzy system are optimized by the AGA optimizer.
REFERENCES I.
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