Research Article Nonlinear Fuzzy Model Predictive...

Research ArticleNonlinear Fuzzy Model Predictive Control for a PWRNuclear Power Plant

Xiangjie Liu and Mengyue Wang

State Key Laboratory of Alternate Electrical Power System with Renewable Energy SourcesNorth China Electric Power University Beijing 102206 China

Correspondence should be addressed to Xiangjie Liu liuxjncepueducn

Received 22 March 2014 Accepted 7 May 2014 Published 2 June 2014

Academic Editor Chengjin Zhang

Copyright copy 2014 X Liu and M WangThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Reliable power and temperature control in pressurized water reactor (PWR) nuclear power plant is necessary to guarantee highefficiency and plant safety Since the nuclear plants are quite nonlinear the paper presents nonlinear fuzzy model predictive control(MPC) by incorporating the realistic constraints to realize the plant optimization T-S fuzzy modeling on nuclear power plantis utilized to approximate the nonlinear plant based on which the nonlinear MPC controller is devised via parallel distributedcompensation (PDC) scheme in order to solve the nonlinear constraint optimization problem Improved performance comparedto the traditional PID controller for a TMI-type PWR is obtained in the simulation

1 Introduction

In nuclear power plants steam energy is produced in thenuclear reactor from continuous fission of the atoms of thefuelThe steam is then used to drive the turbine and generatorto produce electricity

Since burning fossil fuels in thermal power plant can havesevere environment problem the development of nuclearpower plants should be encouraged For example in Chinaalthough the supercritical and ultrasupercritical technique aswell as integrated gasification combined cycle (IGCC) plantshave been well developed the coal consumption still reachesabout 391 million tons in 2012 which comprised over 70 ofnational primary energy Fine particles in the air measuringless than 25 micrometres reached 993 micrograms per cubicmeter in Beijing on 12 January 2013 compared with the limitof 25 published by the WHO guidelines

Since nuclear power plants are the complex and nonlinearsystems it is a great challenge to control the power andtemperature of the nuclear reactor especially when wide-range power variations occur in the load following condi-tion Thus various advanced control schemes have appearedduring the past two decades for example the observer-based optimal state feedback assisted control [1 2] thelinear quadratic Gaussian with loop transfer recovery control

[3 4] the nonlinear control approach [5 6] and the neuralnetworkfuzzy approaches [7 8]

Modern nuclear power plants should respond to theload demand on the power grid which demand high plantoperation performance subject to various kinds of con-straints Meanwhile nuclear safety and radioactive pollutionprevention have long been much concerned problem It istherefore extremely important to reach the economical andsafe operation to maximize the thermal efficiency of thenuclear power plant Under these aims model predictivecontrol (MPC) shows its obvious advantage since it isan advanced model-based control scheme It performs anoptimization procedure to calculate optimal control actionsat every sampling period based on an explicit process modelsubject to process input output and state constraints Sofar MPC has been well constituted for thermal power plantcontrol [9ndash11] and also in nuclear power plant water-levelcontrol [12ndash14] Paper [15] reports a MPC application in athree-dimensional nuclear reactor analysis code for nuclearreactor power

Nuclear power plants are generally nonlinear due tothe frequent changes of the operating point right acrossthe whole operation range In general the nonlinear modelpredictive control (NMPC) also online solves an optimizationproblem by using the sequential quadratic program (SQP)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 908526 10 pageshttpdxdoiorg1011552014908526

2 Mathematical Problems in Engineering

The resulting nonlinear programming problems are usuallynonconvex and the online computational burden is generallylarge Since the nonlinearities vary with operating powerlevels as well as ageing effects [5 16] fuzzy models areused to express the plant dynamic MPCs are realized in aparallel distributed compensation (PDC) scheme that is foreach local model a MPC is designed The overall controllerwhich is nonlinear in nature is similarly constructed bycombining local controllers via fuzzy inference Simula-tions on the three-mile island- (TMI-) type pressurizedwater reactor (PWR) show the effectiveness of the proposedmethod

2 The Plant Description

As shown in Figure 1 nuclear fission reaction in the reactorreleases tremendous energy which is transferred from thereactor to the steam generator by the coolant heating thewater in the steam generator to generate abundant steamThen the turbine is driven by steam to generate electricityThe main objective is to control the power and temperatureof the reactor by inserting or elevating the control rodsConventionally reactor control system controls the averagetemperature of the reactor core coolant to track the referencetemperature which is proportional to the turbine load inorder to guarantee well matching between the reactor powerlevel and the load demand at the same timeThis temperaturecontrol strategy considering not only high thermal efficiencybut also technical limits and economical and safe operationis widely used in modern PWR power plants

However the plant is complex and highly nonlinearwhose parameters vary with the operation conditions Mean-while realistic constraints in the system may lead to actuatorsaturation The performance of the conventional controllersis often unsatisfactory Besides unpredictable disturbancesin the temperature measurement system will induce fluctua-tions on the measuring values of coolant temperature whichmay lead to the abnormal control rod action and thus mustbe considered in the controller design The proposed Takagi-Sugeno fuzzy modeling method is used to approximatethe nonlinear plant based on which the nonlinear MPCcontroller is devised via PDC scheme in order to solve thenonlinear and constraint problems in the reactor power andtemperature control system

3 T-S Fuzzy Modeling

The reactor model considered is the point kinetics with onedelayed neutron group and also coolant and fuel temperaturefeedback It is typical for a TMI-type PWR at the middle ofthe fuel cycle rated at 2500MW [2 7]

119889

119889119905119899119903=120575120588 minus 120573

Λ119899119903+ 120582119888119903

119889

119889119905119888119903= 120582119899119903minus 120582119888119903

To turbine

Feedwater

Steam

(1) Control rod(2) Containment shell(3) Reactor core(4) Pressurizer

(5) Hot leg(6) Cold leg(7) Steam generator(8) Reactor coolant pump

1

2

3

4

7

8

5

6

Figure 1 PWR and steam generation process

119889

119889119905119879119891=1198911198911198751198860

120583119891

119899119903minusΩ

120583119891

119879119891

+Ω

2120583119891

119879119897+Ω

2120583119891

119879119890

119889

119889119905119879119897=(1 minus 119891

119891) 1198751198860

120583119888

119899119903+Ω

120583119888

119879119891

minus(2119872 + Ω)

2120583119888

119879119897+(2119872 minus Ω)

2120583119888

119879119890

119889

119889119905120575120588119903= 119866119903119911119903

120575120588 = 120575120588119903+ 120572119891(119879119891minus 1198791198910) +

120572119888

2(119879119897minus 1198791198970)

+120572119888

2(119879119890minus 1198791198900)

(1)

The symbols in the above equations are demonstrated inthe Nomenclature section

Define

120575119899119903= 119899119903minus 1198991199030

120575119888119903= 119888119903minus 1198881199030

120575119879119891= 119879119891minus 1198791198910

120575119879119897= 119879119897minus 1198791198970

120575120588119903= 120588119903minus 1205881199030

(2)

Mathematical Problems in Engineering 3

where the symbol 120575 indicates a deviation about an equilib-rium point 119899

1199030 1198881199030 1198791198910 1198791198970 and 120588

1199030correspond to the values

of 119899119903 119888119903 119879119891 119879119897 and 120588

119903at an equilibrium point respectively

Choose the state input and output vector respectively asfollows

x = [120575119899119903120575119888119903120575119879119891120575119879119897120575120588119903]119879

119906 = 119911119903

y = [120575119899119903120575119879119897]119879

(3)

The model can be linearized at the equilibrium pointusing the perturbation theory which is valid only for 120575119899

119903≪

1198991199030 The linear model can be written as the following state

space formx = Ax + B119906

y = Cx(4)

The corresponding matrices A B and C are

A =

[[[[[[[[[[[[[[

[

minus120573

Λ

120573

Λ

119899119894

1199030120572119891

Λ

119899119894

1199030120572119888

2Λ

119899119894

1199030

Λ

120582 minus120582 0 0 0

1198911198911198750119886

120583119891

0 minusΩ

120583119891

Ω

2120583119891

0

(1 minus 119891119891) 1198750119886

120583119888

0Ω

120583119888

minus2119872 + Ω

2120583119888

0

0 0 0 0 0

]]]]]]]]]]]]]]

]

B = [0 0 0 0 119866119894

119903]119879

C = [1 0 0 0 0

0 0 0 1 0]

(5)

where 1198991198941199030

is the power level and 119866119894

119903is the control rod

worth for the 119894th (119894 = 1 2 9) operating point whichis defined in Table 1 The linear model (4) is defined ateach operating point (119899119894

1199030 119866119894

119903) and the parameters in (6)

are functions of the operating point [7 17] The remainingconstant parameters are shown in Table 2 Consider thefollowing

120572119891(1198991199030) = (119899

1199030minus 424) times 10

minus5120575119896

119896∘C

120572119888(1198991199030) = (minus40119899

1199030minus 173) times 10

minus5120575119896

119896∘C

120583119888(1198991199030) = (

160

91198991199030+ 54022) MW ∘C

Ω (1198991199030) = (

5

31198991199030+ 49333) MW ∘C

119872 (1198991199030) = (280119899

1199030+ 740) MW ∘C

(6)

The sampling period is selected to be 1 s and then ninediscrete-time local models of (4) can be described as follows

x (119896 + 1) = G119894x (119896) +H

119894119906 (119896)

y (119896) = C119894x (119896)

(119894 = 1 2 9)

(7)

where

G1=

[[[[[

[

542 times 10minus1

451 times 10minus1

minus140 times 10minus2


375 times 101

112 times 10minus3

999 times 10minus1



309 times 10minus2

654 times 10minus1

217 times 10minus1

991 times 10minus1


180 times 101

240 times 10minus2

792 times 10minus3

663 times 10minus4

985 times 10minus1

658 times 10minus1

0 0 0 0 1

]]]]]

]

G2=

[[[[[

[

547 times 10minus1

452 times 10minus1



751

113 times 10minus3

999 times 10minus1



619 times 10minus3

656 times 10minus1

217 times 10minus1

997 times 10minus1


361

270 times 10minus2

893 times 10minus3

845 times 10minus4

986 times 10minus1

148 times 10minus1

0 0 0 0 1

]]]]]

]

G6=

[[[[[

[

547 times 10minus1

452 times 10minus1



751 times 101

113 times 10minus3

999 times 10minus1



619 times 10minus2

656 times 10minus1

217 times 10minus1

996 times 10minus1


361 times 101

211 times 10minus2

696 times 10minus3

874 times 10minus4

985 times 10minus1

116

0 0 0 0 1

]]]]]

]

G1= G4= G8 G

2= G3= G9 G

5= G6= G7


H1=

[[[[[

[

299 times 10minus3

160 times 10minus6

914 times 10minus4

333 times 10minus5

145 times 10minus4

]]]]]

]

H4=

[[[[[

[

289 times 10minus4

200 times 10minus7

883 times 10minus5

360 times 10minus6

700 times 10minus5

]]]]]

]

H8=

[[[[[

[

120 times 10minus2

630 times 10minus6

366 times 10minus3

117 times 10minus4

290 times 10minus4

]]]]]

]

H1= H2= H6 H

3= H4= H5 H

7= H8= H9

C1= [

1 0 0 0 0

0 0 0 1 0]

C1= C2= C3= C4= C5= C6= C7= C8= C9

(8)

Table 1 The definition of the operating points

119866119903

1198991199030

01 05 100029 Point 3 Point 4 Point 500145 Point 2 Point 1 Point 600070 Point 9 Point 8 Point 7

Table 2 The constant parameters in the system matrices

120582 015 sminus1

120573 0006019119891119891

092Λ 00001 s120583119891

236MWs∘C1198751198860

25000MW

According to Table 1 power level 1198991199030and control rod worth

119866119903are selected as the premise variables 119899

1199030and119866

119903are defined

in region [0 1] and region [0 0029] respectively 1198991199030

and119866119903are divided into three fuzzy sets denoted respectively as

ldquolow (L)rdquo ldquomedium (M)rdquo and ldquohigh (H)rdquo The correspondingunivariate membership functions of 119899

1199030or 119866119903are defined

by triangle membership functions as shown in Figure 2Thus the space is divided into nine fuzzy subspaces givenas follows

119877(119894)

(119894 = 1 2 9)

IF 1199111(119896) is 119865119894

1and 119911

2(119896) is 119865119894

2

THEN x119894(119896 + 1) = G

119894x119894(119896) +H

119894119906119894(119896)

y119894(119896) = C

119894x119894(119896)

(9)

where 119877(119894) denotes the 119894th fuzzy inference rule 119911119895(119896) (119895 =

1 2) are premise variables for 1198991199030and119866

119903 119865119894119895denotes the fuzzy

set partitioned for the 119895th premise variable which is valid inthe 119894th rule (119866

119894 119867119894 119862119894) represents the consequent 119894th linear

model deduced at the 119894th equilibrium point in (7)

000

70

0145

002

90

1

1

01 05 10

0

0

Subspace 6Subspace 7Subspace 1Subspace 8

A120572

120572

LM

H

L M H

Gr

nr0

Figure 2The division of fuzzy space and the definition of univariatemembership functions

By using the center of gravity defuzzification method[10 18] the final output of the nonlinear T-S model with ninefuzzy rules is

x (119896 + 1) =9

sum

119894=1

120583119894x119894(119896 + 1) =

9

sum

119894=1

120583119894(G119894x119894(119896) +H

119894119906119894(119896))

y (119896) =9

sum

119894=1

120583119894y119894(119896) =

9

sum

119894=1

120583119894C119894x119894(119896)

(10)

where

120583119894=

120583119894

sum9

119894=1120583119894

(11)

120583119894=

2

prod

119895=1

120572119865119894

119895

(119911119895(119896)) (12)


0

1

00005001001500200250030

080907060504030201

08 0907060504030201

1

Gr nr0

Figure 3 Multivariate membership functions in the fuzzy subspace1

The 120572119865119894

119895

(119911119895(119896)) denotes the univariate membership func-

tion of premise variable 119911119895(119896) with respect to the fuzzy set

119865119894

119895 120583119894is the multivariate membership function obtained by

the product of the outputs of the two univariate membershipfunctions as shown in Figure 3

Four rules are activated each time instant At point A inFigure 2 these four rules are as follows

1198771 if 119866

119903(119896) is119872 and 119899

1199030(119896) is 119872

thenx1(119896 + 1) = G

1x1(119896) +H

11199061(119896)

y1(119896) = C

1x1(119896)

1198776 if 119866

119903(119896) is119872 and 119899

1199030(119896) is119867

thenx6(119896 + 1) = G

6x6(119896) +H

61199066(119896)

y6(119896) = C

6x6(119896)

1198777 if 119866

119903(119896) is119867 and 119899

1199030(119896) is119867

thenx7(119896 + 1) = G

7x7(119896) +H

71199067(119896)

y7(119896) = C

7x7(119896)

1198778 if 119866

119903(119896) is119867 and 119899

1199030(119896) is119872

thenx8(119896 + 1) = G

8x8(119896) +H

81199068(119896)

y8(119896) = C

8x8(119896)

The membership functions are shown in Figure 4

4 The Nonlinear Fuzzy MPC

41 The Local Linear MPC The local linear model (7) can beexpressed as follows

x (119896) = Gx (119896 minus 1) +H119906 (119896 minus 1)

y (119896 minus 1) = Cx (119896 minus 1) (13)

0

1

00005001001500200250030

080907060504030201

08 0907060504030201

1

Gr nr0

Figure 4 Multivariate membership functions activated at point A

Subtracting (13) from (7) can result in

Δx (119896 + 1) = GΔx (119896) +HΔ119906 (119896)

Δy (119896) = CΔx (119896) (14)

where Δx(119896) = x(119896) minus x(119896 minus 1) Δ119906(119896) = 119906(119896) minus 119906(119896 minus 1) andΔy(119896) = y(119896) minus y(119896 minus 1)

Hence Δy(119896 + 1) = CΔx(119896 + 1) = CGΔx(119896) + CHΔ119906(119896)Choosing a new state variable vector x1015840(119896) =

[Δx(119896)119879 y(119896)119879]119879 can result in

[Δx (119896 + 1)y (119896 + 1) ] = [

G 0CG I] [

Δx (119896)y (119896) ] + [

HCH]Δ119906 (119896)

y (119896) = [0 I] [Δx (119896)y (119896) ]

(15)

Denote (15) as the following normal form

x1015840 (119896 + 1) = G1015840x1015840 (119896) +H1015840119906 (119896)

y (119896) = C1015840x1015840 (119896) (16)

where x1015840(119896) isin 1198771198991015840

(1198991015840

= 119899 + 119901) G1015840 = [ G 0CG I ] H

1015840

= [ HCH ] and

C1015840 = [0 I] 119899 and 119901 are respectively the number of the statevariables and the output variables

The predictive control is obtained by minimizing thefollowing cost function

119869 =1003817100381710038171003817Y minus r

119904

10038171003817100381710038172

Q + ΔU2

R (17)

The control is obtained by minimizing the cost functionat each time instant 119896

Define

Y = [y(119896 + 1 | 119896)119879y(119896 + 2 | 119896)119879 sdot sdot sdot y(119896 + 119873119875| 119896)119879

]119879

r119904= [r(119896 + 1)119879r(119896 + 2)119879 sdot sdot sdot r(119896 + 119873

119875)119879

]119879

ΔU = [Δ119906 (119896) Δ119906 (119896 + 1) sdot sdot sdot Δ119906 (119896 + 119873119888minus 1)]119879


Q =

[[[[

[

Q (1) 0 sdot sdot sdot 0

0 Q (2) sdot sdot sdot 0

d

0 0 sdot sdot sdot Q (119873

119901)

]]]]

]

R =[[[[

[

R (0) 0 sdot sdot sdot 0

0 R (1) sdot sdot sdot 0

d

0 0 sdot sdot sdot R (119873

119888minus 1)

]]]]

]

(18)

where Q and R are the weighting matrices for the pre-diction error and the control respectively 119873

119901and 119873

119888are

respectively the costing horizon for the prediction errorand the control and y(119896 + 119895 | 119896)

119879 and r(119896 + 119895)119879 (119895 =

1 2 119873119901) are the optimum 119895th step ahead prediction of

the system output on data up to time instant 119896 and the119895th step ahead reference trajectory respectively ΔU is thevector composed of the future control increments whichis obtained by computing the derivative of 119869 expressed asfollows

ΔU = (R +Φ119879QΦ)minus1

Φ119879Q (r119904minus Fx (119896)) (19)

where

F =

[[[[[[[

[

C1015840G1015840

C1015840G10158402

C1015840G10158403

C1015840G1015840119873119901

]]]]]]]

]

Φ =

[[[[[[[

[

C1015840H1015840 0 0 sdot sdot sdot 0C1015840G1015840H1015840 C1015840H1015840 0 sdot sdot sdot 0C1015840G10158402H1015840 C1015840G1015840H1015840 C1015840H1015840 sdot sdot sdot 0

d

C1015840G1015840119873119901minus1H1015840 C1015840G1015840119873119901minus2H1015840 C1015840G1015840119873119901minus3H1015840 sdot sdot sdot C1015840G1015840119873119901minus119873119888H1015840

]]]]]]]

]

(20)

The optimized 119873119888steps ahead control ΔU is computed

and only the first step ahead control Δ119906(119896) is implementedusing a receding horizon principle [10] giving

Δ119906 (119896) = [1 0 sdot sdot sdot 0]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873119888

ΔU (21)

further 119906(119896) can be obtained by

119906 (119896) = 119906 (119896 minus 1) + Δ119906 (119896) (22)

The input constraint is expressed as follows

Uminle U le Umax

(23)

U can be expressed by ΔU as follows

[[[[[[

[

119906 (119896)

119906 (119896 + 1)

119906 (119896 + 2)

119906 (119896 + 119873

119888minus 1)

]]]]]]

]

=

[[[[[[

[

1

1

1

1

]]]]]]

]

119906 (119896 minus 1) +

[[[[[[

[

1 0 0 sdot sdot sdot 0



d


]]]]]]

]

[[[[[[

[

Δ119906 (119896)

Δ119906 (119896 + 1)

Δ119906 (119896 + 2)

Δ119906 (119896 + 119873

119888minus 1)

]]]]]]

]

(24)

Define

C1=

[[[[[[

[

1

1

1

1

]]]]]]

]

C2=

[[[[[[

[




d


]]]]]]

]

120595 = [minusC2

C2

] 120574 = [minusUmin

+ C1119906 (119896 minus 1)

Umaxminus C1119906 (119896 minus 1)

]

(25)

and constraint (23) can be rewritten as follows

120595ΔU le 120574 (26)

Hence the constrained optimization problem can bedescribed as

min 119869 =1003817100381710038171003817Y minus r

119904

10038171003817100381710038172

Q + ΔU2

R

st 120595ΔU le 120574

(27)

which can be solved by quadratic programming (QP)method

42 The Nonlinear Fuzzy MPC Based on the T-S fuzzymodel the controller is devised via the parallel distributedcontrol (PDC) scheme [19 20] In this scheme the overallcontroller which is naturally nonlinear is a fuzzy combi-nation of each individual linear controller designed basedon the corresponding local model Here a nonlinear MPCcontroller consisting of nine local MPC controllers is con-stituted as shown in Figure 5 For each local linear modela conventional MPC controller is designed independently


Table 3 The parameters of the nine local controllers

Parameters 119894th controller1 2 3 4 5 6 7 8 9

119902 1 1 1 1 1 1 1 1 1119903 007 001 002 007 01 01 01 005 01119873119901

380 430 400 340 290 300 325 400 460119873119888

50 45 45 50 60 60 60 40 30

MPC1

MPC9

Nonlinear plant

Temperature of coolant leaving

the reactorSet point

Controller

Tl0

u(k) Tl

1205831

1205839

u1

u9

+

Figure 5 The schematic diagram of fuzzy MPC system

Consequently the final control is the weighted sum of thecontrol obtained from nine local MPC controllers Considerthe following

119906 (119896) =

9

sum

119894=1

120583119894119906119894(119896) (28)

where 119906119894(119896) is the control of the 119894th local predictive controller

and 120583119894is defined previously in (11) Since fuzzy logics dom-

inate the switching between local controllers the resultingcontrol can be smooth by properly selecting the weights 120583

119894

[10]

5 The Application to the PWR NuclearPower Plant

In using the proposed fuzzy MPC the reference temperaturefunction of power level is defined as follows

1198791198970= 290 + 2451119899

1199030 (29)

Figure 6 shows this relationshipAssume that the maximum speed of control rod is 02

fraction of core length per second Hence the constraint onthe control is considered as minus02 le 119906 le 02 which is involvedin the optimization algorithm of MPC

Define

Q = 119902 times

[[[[

[

I2times2

0 sdot sdot sdot 00 I2times2

sdot sdot sdot 0

d

0 0 sdot sdot sdot I2times2

]]]]

]119873119901times119873119901

(30)

and R = 119903 times I119873119888times119873119888

where 119902 and 119903 are the two coefficientsAssume the control rod worth119866

119903to be 00145 the parameters

of the nine local controllers are listed in Table 3

20 40 60 80 100

290

31451

Power level ()0

300

310

Tem

pera

ture

(∘C)

Tl

Figure 6 Reference temperature function of power level

The performance of the nonlinear fuzzy MPC can beevaluated in three cases

Case A Full power operation that is 100 rarr 90 rarr

100 step changes in power level Comparison of theperformance between the proposed fuzzy MPC and theconventional PID controller is presented in Figures 7(a) and7(b)The temperature and the power stabilize to their respectset points in an acceptable time whereas the nonlinear fuzzyMPC reduces the fluctuation obviously In real-time nuclearpower control inserting the control rod can slow down therelease of energy in the reactor so that the temperature andpower level are decreased while lifting the control rod speedwill raise the temperature and power level

Case B Low power operation that is 20 rarr 10 rarr 20step changes in power level Comparison of the performancebetween the proposed controller and the conventional PIDcontroller is presented in Figures 8(a) and 8(b) showingthe obvious advantage of the proposed fuzzy MPC It isnoticed that the response time is longer than that in thefull power operation and the resulting fluctuation is moresevere

In both cases the constraint on the control rod speed canbe handled by the proposed fuzzy MPC effectively while thePID controller cannot achieve it as shown in Figures 7(c) and8(c)


24 30 36 42 48 54 60 66 723115

312

3125

313

3135

314

3145

315

Time (s)

Coo

lant

tem

pera

ture

Nonlinear fuzzy MPCPID

Reference trajectory

(a)

24 30 36 42 48 54 60 66 72088

09

092

094

096

098

1

102

Pow

er le

vel

Reference trajectoryNonlinear fuzzy MPC

PID

Time (s)

(b)


24 30 36 42 48 54 60 66 72

002040608

Con

trol r

od sp

eed

Time (s)

minus02

minus04

minus06

minus08

(c)

Figure 7 Comparison of the performance between the nonlinearfuzzy MPC and the conventional PID controller (100 rarr 90 rarr

100 power level change) (a) coolant temperature (unit ∘C) (b)power level and (c) control rod speed

Case C Wide-range load following that is 100 rarr

10 rarr 100 ramp changes in power level with 5minrate as shown in Figure 9 With the constraint handlingon the control rod speed by the fuzzy MPC satisfactorytracking performance is obtained for both temperature andpower

24 30 36 42 48 54 60 66 72292

2925

293

2935

294

2945

295

2955

Coo

lant

tem

pera

ture

Time (s)

Reference trajectory Nonlinear fuzzy MPCPID

(a)

30 36 42 48 54 60 66 72008

01

012

014

016

018

02

022

Time (s)

Pow

er le

vel



(b)

Time (s)24 30 36 42 48 54 60 66 72

004081216

2

Con

trol r

od sp

eed


minus08

minus04

minus12

minus16

minus2

(c)

Figure 8 Comparison of the performance between the nonlinearfuzzy controller and the conventional PID controller (20 rarr

10 rarr 20 power level change) (a) coolant temperature (unit∘C) (b) power level and (c) control rod speed

6 Conclusions

The paper constituted a nonlinear MPC controller by incor-porating fuzzy modeling technique Feasible optimal solu-tions have been acquired under plant nonlinearity andconstraints The proposed nonlinear fuzzy MPC is simulatedin power and temperature control for a three-mile island-


0 12 24 36 48 60 72290

295

300

305

310

315

Time (s)

Coo

lant

tem

pera

ture

Reference trajectoryCoolant temperature

(a)

Time (s)0 12 24 36 48 60 72

010203040506070809

111

Pow

er le

vel

Reference trajectoryPower level

(b)

Time (s)0 12 24 36 48 60 72

0005

01015

02

Con

trol r

od sp

eed

minus02

minus015

minus01

minus005

(c)

Figure 9 The performance for the nonlinear fuzzy controller(100 rarr 10 rarr 100 ramp changes in power level with 5minrate) (a) coolant temperature (unit∘C) (b) power level and (c)control rod speed

(TMI-) type PWR Better performance is obtained whileit is compared with the PID controller Load following isaccomplished under constraints on the control rod

Nomenclature

119899119903 Neutron density relative to density at ratedcondition

119888119903 Precursor density relative to density at ratedcondition

120575120588 Reactivity120575120588119903 Reactivity due to the control rod

120573 Fraction of delayed fission neutronsΛ Effective prompt neutron lifetime120582 Effective precursor density119879119891 Average reactor fuel temperature

119891119891 Fraction of reactor power deposited in fuel

120583119891 Heat capacity of the fuel

1198751198860 Rated power level

Ω Heat transfer coefficient between fuel andcoolant

119879119897 The temperature of coolant leaving the

reactor119879119890 The temperature of coolant entering the

reactor120583119888 Heat capacity of the coolant

119872 Mass flow rate times heat capacity of thewater

119866119903 Total reactivity worth of the rod

119911119903 The control rod speed (fraction of core

length per second)120572119891 Fuel temperature reactivity coefficient

120572119888 Coolant temperature reactivity coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by National Natural Science Foun-dation of China under Grants nos 60974051 and 61273144and by Natural Science Foundation of Beijing under Grantno 4122071

References

[1] R M Edwards K Y Lee and A Ray ldquoRobust optimal controlof nuclear reactors and power plantsrdquo Nuclear Technology vol98 no 2 pp 137ndash148 1992

[2] R M Edwards K Y Lee and M A Schultz ldquoState feedbackassisted classical control an incremental approach to controlmodernization of existing and future nuclear reactors andpower plantsrdquo Nuclear Technology vol 92 no 2 pp 167ndash1851990

[3] H Arab-Alibeik and S Setayeshi ldquoImproved temperature con-trol of a PWRnuclear reactor using an LQGLTR based control-lerrdquo IEEE Transactions on Nuclear Science vol 50 no 1 pp 211ndash218 2003

[4] A Ben-Abdennour R M Edwards and K Y Lee ldquoLQGLTRrobust control of nuclear reactors with improved temperatureperformancerdquo IEEE Transactions on Nuclear Science vol 39 no6 pp 2286ndash2294 1992

[5] Z Dong J Feng and X Huang ldquoNonlinear observer-basedfeedback dissipation load-following control for nuclear reac-torsrdquo IEEE Transactions on Nuclear Science vol 56 no 1 pp272ndash285 2009


[6] H Eliasi M B Menhaj and H Davilu ldquoRobust nonlinearmodel predictive control for a PWR nuclear power plantrdquoProgress in Nuclear Energy vol 54 no 1 pp 177ndash185 2012

[7] C-C Ku K Y Lee and R M Edwards ldquoImproved nuclearreactor temperature control using diagonal recurrent neuralnetworksrdquo IEEE Transactions on Nuclear Science vol 39 no 6pp 2298ndash2308 1992

[8] P Ramaswamy R M Edwards and K Y Lee ldquoFuzzy logiccontroller for nuclear power plantrdquo in Proceedings of the 2ndInternational Forum on Applications of Neural Networks toPower Systems (ANNP rsquo93) pp 29ndash34 Yokohama Japan 1993

[9] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[10] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[11] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boiler-turbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[12] M G Na ldquoAuto-tuned PID controller using a model predictivecontrol method for the steam generator water levelrdquo IEEETransactions on Nuclear Science vol 48 no 5 pp 1664ndash16712001

[13] M G Na Y R Sim and Y J Lee ldquoDesign of an adaptivepredictive controller for steam generatorsrdquo IEEE Transactionson Nuclear Science vol 50 no 2 pp 186ndash193 2003

[14] M V Kothare B Mettler M Morari P Bendotti and C-MFalinower ldquoLevel control in the steam generator of a nuclearpower plantrdquo IEEE Transactions on Control Systems Technologyvol 8 no 1 pp 55ndash69 2000

[15] M G Na I J Hwang and Y J Lee ldquoDesign of a fuzzy modelpredictive power controller for pressurized water reactorsrdquoIEEE Transactions on Nuclear Science vol 53 no 3 pp 1504ndash1514 2006

[16] M G Na S H Shin and W C Kim ldquoA model predictivecontroller for nuclear reactor powerrdquo Journal of the KoreanNuclear Society vol 35 no 5 pp 399ndash411 2003


[18] J Shi J Wang and YWang ldquoT-S fuzzymodeling for predictingchaotic time series based on unscented Kalman filter approachrdquoin Proceedings of the 8th International Conference on FuzzySystems and Knowledge Discovery (FSKD rsquo11) pp 722ndash726Shanghai China July 2011

[19] Z Zhang and Z Sun ldquoT-S fuzzy model in discrete form andits application to identification of a plane manipulator with twodegree of freedomrdquo in Proceedings of the IEEE InternationalConference on Intelligent Processing Systems (ICIPS rsquo97) vol 1pp 368ndash372 Beijing China October 1997

[20] X Luan A Young W Han and Y Zhai ldquoLoad-followingcontrol of nuclear reactors based on Takagi-Sugeno fuzzymodelrdquo in Proceedings of the 18th IFAC World Congress pp8253ndash8258 Milano Italy September 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The resulting nonlinear programming problems are usuallynonconvex and the online computational burden is generallylarge Since the nonlinearities vary with operating powerlevels as well as ageing effects [5 16] fuzzy models areused to express the plant dynamic MPCs are realized in aparallel distributed compensation (PDC) scheme that is foreach local model a MPC is designed The overall controllerwhich is nonlinear in nature is similarly constructed bycombining local controllers via fuzzy inference Simula-tions on the three-mile island- (TMI-) type pressurizedwater reactor (PWR) show the effectiveness of the proposedmethod

2 The Plant Description

As shown in Figure 1 nuclear fission reaction in the reactorreleases tremendous energy which is transferred from thereactor to the steam generator by the coolant heating thewater in the steam generator to generate abundant steamThen the turbine is driven by steam to generate electricityThe main objective is to control the power and temperatureof the reactor by inserting or elevating the control rodsConventionally reactor control system controls the averagetemperature of the reactor core coolant to track the referencetemperature which is proportional to the turbine load inorder to guarantee well matching between the reactor powerlevel and the load demand at the same timeThis temperaturecontrol strategy considering not only high thermal efficiencybut also technical limits and economical and safe operationis widely used in modern PWR power plants

However the plant is complex and highly nonlinearwhose parameters vary with the operation conditions Mean-while realistic constraints in the system may lead to actuatorsaturation The performance of the conventional controllersis often unsatisfactory Besides unpredictable disturbancesin the temperature measurement system will induce fluctua-tions on the measuring values of coolant temperature whichmay lead to the abnormal control rod action and thus mustbe considered in the controller design The proposed Takagi-Sugeno fuzzy modeling method is used to approximatethe nonlinear plant based on which the nonlinear MPCcontroller is devised via PDC scheme in order to solve thenonlinear and constraint problems in the reactor power andtemperature control system

3 T-S Fuzzy Modeling

The reactor model considered is the point kinetics with onedelayed neutron group and also coolant and fuel temperaturefeedback It is typical for a TMI-type PWR at the middle ofthe fuel cycle rated at 2500MW [2 7]

119889

119889119905119899119903=120575120588 minus 120573

Λ119899119903+ 120582119888119903

119889

119889119905119888119903= 120582119899119903minus 120582119888119903

To turbine

Feedwater

Steam

(1) Control rod(2) Containment shell(3) Reactor core(4) Pressurizer

(5) Hot leg(6) Cold leg(7) Steam generator(8) Reactor coolant pump

1

2

3

4

7

8

5

6

Figure 1 PWR and steam generation process

119889

119889119905119879119891=1198911198911198751198860

120583119891

119899119903minusΩ

120583119891

119879119891

+Ω

2120583119891

119879119897+Ω

2120583119891

119879119890

119889

119889119905119879119897=(1 minus 119891

119891) 1198751198860

120583119888

119899119903+Ω

120583119888

119879119891

minus(2119872 + Ω)

2120583119888

119879119897+(2119872 minus Ω)

2120583119888

119879119890

119889

119889119905120575120588119903= 119866119903119911119903

120575120588 = 120575120588119903+ 120572119891(119879119891minus 1198791198910) +

120572119888

2(119879119897minus 1198791198970)

+120572119888

2(119879119890minus 1198791198900)

(1)

The symbols in the above equations are demonstrated inthe Nomenclature section

Define

120575119899119903= 119899119903minus 1198991199030

120575119888119903= 119888119903minus 1198881199030

120575119879119891= 119879119891minus 1198791198910

120575119879119897= 119879119897minus 1198791198970

120575120588119903= 120588119903minus 1205881199030

(2)



1199030 1198881199030 1198791198910 1198791198970 and 120588


of 119899119903 119888119903 119879119891 119879119897 and 120588



x = [120575119899119903120575119888119903120575119879119891120575119879119897120575120588119903]119879

119906 = 119911119903

y = [120575119899119903120575119879119897]119879

(3)


119903≪



y = Cx(4)


A =

[[[[[[[[[[[[[[

[

minus120573

Λ

120573

Λ

119899119894

1199030120572119891

Λ

119899119894

1199030120572119888

2Λ

119899119894

1199030

Λ

120582 minus120582 0 0 0

1198911198911198750119886

120583119891

0 minusΩ

120583119891

Ω

2120583119891

0

(1 minus 119891119891) 1198750119886

120583119888

0Ω

120583119888

minus2119872 + Ω

2120583119888

0

0 0 0 0 0

]]]]]]]]]]]]]]

]

B = [0 0 0 0 119866119894

119903]119879

C = [1 0 0 0 0

0 0 0 1 0]

(5)

where 1198991198941199030




1199030 119866119894



120572119891(1198991199030) = (119899

1199030minus 424) times 10

minus5120575119896

119896∘C

120572119888(1198991199030) = (minus40119899

1199030minus 173) times 10

minus5120575119896

119896∘C

120583119888(1198991199030) = (

160

91198991199030+ 54022) MW ∘C

Ω (1198991199030) = (

5

31198991199030+ 49333) MW ∘C

119872 (1198991199030) = (280119899

1199030+ 740) MW ∘C

(6)


x (119896 + 1) = G119894x (119896) +H

119894119906 (119896)

y (119896) = C119894x (119896)

(119894 = 1 2 9)

(7)

where

G1=

[[[[[

[

542 times 10minus1

451 times 10minus1



375 times 101

112 times 10minus3

999 times 10minus1



309 times 10minus2

654 times 10minus1

217 times 10minus1

991 times 10minus1


180 times 101

240 times 10minus2

792 times 10minus3

663 times 10minus4

985 times 10minus1

658 times 10minus1

0 0 0 0 1

]]]]]

]

G2=

[[[[[

[

547 times 10minus1

452 times 10minus1



751

113 times 10minus3

999 times 10minus1



619 times 10minus3

656 times 10minus1

217 times 10minus1

997 times 10minus1


361

270 times 10minus2

893 times 10minus3

845 times 10minus4

986 times 10minus1

148 times 10minus1

0 0 0 0 1

]]]]]

]

G6=

[[[[[

[

547 times 10minus1

452 times 10minus1



751 times 101

113 times 10minus3

999 times 10minus1



619 times 10minus2

656 times 10minus1

217 times 10minus1

996 times 10minus1


361 times 101

211 times 10minus2

696 times 10minus3

874 times 10minus4

985 times 10minus1

116

0 0 0 0 1

]]]]]

]

G1= G4= G8 G

2= G3= G9 G

5= G6= G7


H1=

[[[[[

[

299 times 10minus3

160 times 10minus6

914 times 10minus4

333 times 10minus5

145 times 10minus4

]]]]]

]

H4=

[[[[[

[

289 times 10minus4

200 times 10minus7

883 times 10minus5

360 times 10minus6

700 times 10minus5

]]]]]

]

H8=

[[[[[

[

120 times 10minus2

630 times 10minus6

366 times 10minus3

117 times 10minus4

290 times 10minus4

]]]]]

]

H1= H2= H6 H

3= H4= H5 H

7= H8= H9

C1= [

1 0 0 0 0

0 0 0 1 0]

C1= C2= C3= C4= C5= C6= C7= C8= C9

(8)


119866119903

1198991199030



120582 015 sminus1

120573 0006019119891119891

092Λ 00001 s120583119891

236MWs∘C1198751198860

25000MW



1199030and119866

119903are defined




1199030or 119866119903are defined


119877(119894)

(119894 = 1 2 9)

IF 1199111(119896) is 119865119894

1and 119911

2(119896) is 119865119894

2

THEN x119894(119896 + 1) = G

119894x119894(119896) +H

119894119906119894(119896)

y119894(119896) = C

119894x119894(119896)

(9)







000

70

0145

002

90

1

1

01 05 10

0

0


A120572

120572

LM

H

L M H

Gr

nr0



x (119896 + 1) =9

sum

119894=1

120583119894x119894(119896 + 1) =

9

sum

119894=1

120583119894(G119894x119894(119896) +H

119894119906119894(119896))

y (119896) =9

sum

119894=1

120583119894y119894(119896) =

9

sum

119894=1

120583119894C119894x119894(119896)

(10)

where

120583119894=

120583119894

sum9

119894=1120583119894

(11)

120583119894=

2

prod

119895=1

120572119865119894

119895

(119911119895(119896)) (12)


0

1

00005001001500200250030

080907060504030201

08 0907060504030201

1

Gr nr0


The 120572119865119894

119895



119865119894




1198771 if 119866

119903(119896) is119872 and 119899

1199030(119896) is 119872

thenx1(119896 + 1) = G

1x1(119896) +H

11199061(119896)

y1(119896) = C

1x1(119896)

1198776 if 119866

119903(119896) is119872 and 119899

1199030(119896) is119867

thenx6(119896 + 1) = G

6x6(119896) +H

61199066(119896)

y6(119896) = C

6x6(119896)

1198777 if 119866

119903(119896) is119867 and 119899

1199030(119896) is119867

thenx7(119896 + 1) = G

7x7(119896) +H

71199067(119896)

y7(119896) = C

7x7(119896)

1198778 if 119866

119903(119896) is119867 and 119899

1199030(119896) is119872

thenx8(119896 + 1) = G

8x8(119896) +H

81199068(119896)

y8(119896) = C

8x8(119896)




x (119896) = Gx (119896 minus 1) +H119906 (119896 minus 1)

y (119896 minus 1) = Cx (119896 minus 1) (13)

0

1

00005001001500200250030

080907060504030201

08 0907060504030201

1

Gr nr0



Δx (119896 + 1) = GΔx (119896) +HΔ119906 (119896)

Δy (119896) = CΔx (119896) (14)



[Δx(119896)119879 y(119896)119879]119879 can result in

[Δx (119896 + 1)y (119896 + 1) ] = [

G 0CG I] [

Δx (119896)y (119896) ] + [

HCH]Δ119906 (119896)

y (119896) = [0 I] [Δx (119896)y (119896) ]

(15)


x1015840 (119896 + 1) = G1015840x1015840 (119896) +H1015840119906 (119896)

y (119896) = C1015840x1015840 (119896) (16)

where x1015840(119896) isin 1198771198991015840

(1198991015840

= 119899 + 119901) G1015840 = [ G 0CG I ] H

1015840

= [ HCH ] and



119869 =1003817100381710038171003817Y minus r

119904

10038171003817100381710038172

Q + ΔU2

R (17)


Define


]119879


119875)119879

]119879



Q =

[[[[

[



d


119901)

]]]]

]

R =[[[[

[



d


119888minus 1)

]]]]

]

(18)


119901and 119873

119888are


119879 and r(119896 + 119895)119879 (119895 =




Φ119879Q (r119904minus Fx (119896)) (19)

where

F =

[[[[[[[

[

C1015840G1015840

C1015840G10158402

C1015840G10158403

C1015840G1015840119873119901

]]]]]]]

]

Φ =

[[[[[[[

[


d


]]]]]]]

]

(20)



Δ119906 (119896) = [1 0 sdot sdot sdot 0]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873119888

ΔU (21)


119906 (119896) = 119906 (119896 minus 1) + Δ119906 (119896) (22)


Uminle U le Umax

(23)


[[[[[[

[

119906 (119896)

119906 (119896 + 1)

119906 (119896 + 2)

119906 (119896 + 119873

119888minus 1)

]]]]]]

]

=

[[[[[[

[

1

1

1

1

]]]]]]

]

119906 (119896 minus 1) +

[[[[[[

[




d


]]]]]]

]

[[[[[[

[

Δ119906 (119896)

Δ119906 (119896 + 1)

Δ119906 (119896 + 2)

Δ119906 (119896 + 119873

119888minus 1)

]]]]]]

]

(24)

Define

C1=

[[[[[[

[

1

1

1

1

]]]]]]

]

C2=

[[[[[[

[




d


]]]]]]

]

120595 = [minusC2

C2


+ C1119906 (119896 minus 1)


]

(25)


120595ΔU le 120574 (26)


min 119869 =1003817100381710038171003817Y minus r

119904

10038171003817100381710038172

Q + ΔU2

R

st 120595ΔU le 120574

(27)






119902 1 1 1 1 1 1 1 1 1119903 007 001 002 007 01 01 01 005 01119873119901

380 430 400 340 290 300 325 400 460119873119888

50 45 45 50 60 60 60 40 30

MPC1

MPC9

Nonlinear plant



Controller

Tl0

u(k) Tl

1205831

1205839

u1

u9

+



119906 (119896) =

9

sum

119894=1

120583119894119906119894(119896) (28)




119894

[10]



1198791198970= 290 + 2451119899

1199030 (29)



Define

Q = 119902 times

[[[[

[

I2times2


sdot sdot sdot 0

d


]]]]

]119873119901times119873119901

(30)





20 40 60 80 100

290

31451

Power level ()0

300

310

Tem

pera

ture

(∘C)

Tl








24 30 36 42 48 54 60 66 723115

312

3125

313

3135

314

3145

315

Time (s)

Coo

lant

tem

pera

ture



(a)

24 30 36 42 48 54 60 66 72088

09

092

094

096

098

1

102

Pow

er le

vel


PID

Time (s)

(b)


24 30 36 42 48 54 60 66 72

002040608

Con

trol r

od sp

eed

Time (s)

minus02

minus04

minus06

minus08

(c)





24 30 36 42 48 54 60 66 72292

2925

293

2935

294

2945

295

2955

Coo

lant

tem

pera

ture

Time (s)


(a)

30 36 42 48 54 60 66 72008

01

012

014

016

018

02

022

Time (s)

Pow

er le

vel



(b)

Time (s)24 30 36 42 48 54 60 66 72

004081216

2

Con

trol r

od sp

eed


minus08

minus04

minus12

minus16

minus2

(c)



6 Conclusions



0 12 24 36 48 60 72290

295

300

305

310

315

Time (s)

Coo

lant

tem

pera

ture


(a)

Time (s)0 12 24 36 48 60 72

010203040506070809

111

Pow

er le

vel


(b)

Time (s)0 12 24 36 48 60 72

0005

01015

02

Con

trol r

od sp

eed

minus02

minus015

minus01

minus005

(c)



Nomenclature



















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199030 1198881199030 1198791198910 1198791198970 and 120588


of 119899119903 119888119903 119879119891 119879119897 and 120588



x = [120575119899119903120575119888119903120575119879119891120575119879119897120575120588119903]119879

119906 = 119911119903

y = [120575119899119903120575119879119897]119879

(3)


119903≪



y = Cx(4)


A =

[[[[[[[[[[[[[[

[

minus120573

Λ

120573

Λ

119899119894

1199030120572119891

Λ

119899119894

1199030120572119888

2Λ

119899119894

1199030

Λ

120582 minus120582 0 0 0

1198911198911198750119886

120583119891

0 minusΩ

120583119891

Ω

2120583119891

0

(1 minus 119891119891) 1198750119886

120583119888

0Ω

120583119888

minus2119872 + Ω

2120583119888

0

0 0 0 0 0

]]]]]]]]]]]]]]

]

B = [0 0 0 0 119866119894

119903]119879

C = [1 0 0 0 0

0 0 0 1 0]

(5)

where 1198991198941199030




1199030 119866119894



120572119891(1198991199030) = (119899

1199030minus 424) times 10

minus5120575119896

119896∘C

120572119888(1198991199030) = (minus40119899

1199030minus 173) times 10

minus5120575119896

119896∘C

120583119888(1198991199030) = (

160

91198991199030+ 54022) MW ∘C

Ω (1198991199030) = (

5

31198991199030+ 49333) MW ∘C

119872 (1198991199030) = (280119899

1199030+ 740) MW ∘C

(6)


x (119896 + 1) = G119894x (119896) +H

119894119906 (119896)

y (119896) = C119894x (119896)

(119894 = 1 2 9)

(7)

where

G1=

[[[[[

[

542 times 10minus1

451 times 10minus1



375 times 101

112 times 10minus3

999 times 10minus1



309 times 10minus2

654 times 10minus1

217 times 10minus1

991 times 10minus1


180 times 101

240 times 10minus2

792 times 10minus3

663 times 10minus4

985 times 10minus1

658 times 10minus1

0 0 0 0 1

]]]]]

]

G2=

[[[[[

[

547 times 10minus1

452 times 10minus1



751

113 times 10minus3

999 times 10minus1



619 times 10minus3

656 times 10minus1

217 times 10minus1

997 times 10minus1


361

270 times 10minus2

893 times 10minus3

845 times 10minus4

986 times 10minus1

148 times 10minus1

0 0 0 0 1

]]]]]

]

G6=

[[[[[

[

547 times 10minus1

452 times 10minus1



751 times 101

113 times 10minus3

999 times 10minus1



619 times 10minus2

656 times 10minus1

217 times 10minus1

996 times 10minus1


361 times 101

211 times 10minus2

696 times 10minus3

874 times 10minus4

985 times 10minus1

116

0 0 0 0 1

]]]]]

]

G1= G4= G8 G

2= G3= G9 G

5= G6= G7


H1=

[[[[[

[

299 times 10minus3

160 times 10minus6

914 times 10minus4

333 times 10minus5

145 times 10minus4

]]]]]

]

H4=

[[[[[

[

289 times 10minus4

200 times 10minus7

883 times 10minus5

360 times 10minus6

700 times 10minus5

]]]]]

]

H8=

[[[[[

[

120 times 10minus2

630 times 10minus6

366 times 10minus3

117 times 10minus4

290 times 10minus4

]]]]]

]

H1= H2= H6 H

3= H4= H5 H

7= H8= H9

C1= [

1 0 0 0 0

0 0 0 1 0]

C1= C2= C3= C4= C5= C6= C7= C8= C9

(8)


119866119903

1198991199030



120582 015 sminus1

120573 0006019119891119891

092Λ 00001 s120583119891

236MWs∘C1198751198860

25000MW



1199030and119866

119903are defined




1199030or 119866119903are defined


119877(119894)

(119894 = 1 2 9)

IF 1199111(119896) is 119865119894

1and 119911

2(119896) is 119865119894

2

THEN x119894(119896 + 1) = G

119894x119894(119896) +H

119894119906119894(119896)

y119894(119896) = C

119894x119894(119896)

(9)







000

70

0145

002

90

1

1

01 05 10

0

0


A120572

120572

LM

H

L M H

Gr

nr0



x (119896 + 1) =9

sum

119894=1

120583119894x119894(119896 + 1) =

9

sum

119894=1

120583119894(G119894x119894(119896) +H

119894119906119894(119896))

y (119896) =9

sum

119894=1

120583119894y119894(119896) =

9

sum

119894=1

120583119894C119894x119894(119896)

(10)

where

120583119894=

120583119894

sum9

119894=1120583119894

(11)

120583119894=

2

prod

119895=1

120572119865119894

119895

(119911119895(119896)) (12)


0

1

00005001001500200250030

080907060504030201

08 0907060504030201

1

Gr nr0


The 120572119865119894

119895



119865119894




1198771 if 119866

119903(119896) is119872 and 119899

1199030(119896) is 119872

thenx1(119896 + 1) = G

1x1(119896) +H

11199061(119896)

y1(119896) = C

1x1(119896)

1198776 if 119866

119903(119896) is119872 and 119899

1199030(119896) is119867

thenx6(119896 + 1) = G

6x6(119896) +H

61199066(119896)

y6(119896) = C

6x6(119896)

1198777 if 119866

119903(119896) is119867 and 119899

1199030(119896) is119867

thenx7(119896 + 1) = G

7x7(119896) +H

71199067(119896)

y7(119896) = C

7x7(119896)

1198778 if 119866

119903(119896) is119867 and 119899

1199030(119896) is119872

thenx8(119896 + 1) = G

8x8(119896) +H

81199068(119896)

y8(119896) = C

8x8(119896)




x (119896) = Gx (119896 minus 1) +H119906 (119896 minus 1)

y (119896 minus 1) = Cx (119896 minus 1) (13)

0

1

00005001001500200250030

080907060504030201

08 0907060504030201

1

Gr nr0



Δx (119896 + 1) = GΔx (119896) +HΔ119906 (119896)

Δy (119896) = CΔx (119896) (14)



[Δx(119896)119879 y(119896)119879]119879 can result in

[Δx (119896 + 1)y (119896 + 1) ] = [

G 0CG I] [

Δx (119896)y (119896) ] + [

HCH]Δ119906 (119896)

y (119896) = [0 I] [Δx (119896)y (119896) ]

(15)


x1015840 (119896 + 1) = G1015840x1015840 (119896) +H1015840119906 (119896)

y (119896) = C1015840x1015840 (119896) (16)

where x1015840(119896) isin 1198771198991015840

(1198991015840

= 119899 + 119901) G1015840 = [ G 0CG I ] H

1015840

= [ HCH ] and



119869 =1003817100381710038171003817Y minus r

119904

10038171003817100381710038172

Q + ΔU2

R (17)


Define


]119879


119875)119879

]119879



Q =

[[[[

[



d


119901)

]]]]

]

R =[[[[

[



d


119888minus 1)

]]]]

]

(18)


119901and 119873

119888are


119879 and r(119896 + 119895)119879 (119895 =




Φ119879Q (r119904minus Fx (119896)) (19)

where

F =

[[[[[[[

[

C1015840G1015840

C1015840G10158402

C1015840G10158403

C1015840G1015840119873119901

]]]]]]]

]

Φ =

[[[[[[[

[


d


]]]]]]]

]

(20)



Δ119906 (119896) = [1 0 sdot sdot sdot 0]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873119888

ΔU (21)


119906 (119896) = 119906 (119896 minus 1) + Δ119906 (119896) (22)


Uminle U le Umax

(23)


[[[[[[

[

119906 (119896)

119906 (119896 + 1)

119906 (119896 + 2)

119906 (119896 + 119873

119888minus 1)

]]]]]]

]

=

[[[[[[

[

1

1

1

1

]]]]]]

]

119906 (119896 minus 1) +

[[[[[[

[




d


]]]]]]

]

[[[[[[

[

Δ119906 (119896)

Δ119906 (119896 + 1)

Δ119906 (119896 + 2)

Δ119906 (119896 + 119873

119888minus 1)

]]]]]]

]

(24)

Define

C1=

[[[[[[

[

1

1

1

1

]]]]]]

]

C2=

[[[[[[

[




d


]]]]]]

]

120595 = [minusC2

C2


+ C1119906 (119896 minus 1)


]

(25)


120595ΔU le 120574 (26)


min 119869 =1003817100381710038171003817Y minus r

119904

10038171003817100381710038172

Q + ΔU2

R

st 120595ΔU le 120574

(27)






119902 1 1 1 1 1 1 1 1 1119903 007 001 002 007 01 01 01 005 01119873119901

380 430 400 340 290 300 325 400 460119873119888

50 45 45 50 60 60 60 40 30

MPC1

MPC9

Nonlinear plant



Controller

Tl0

u(k) Tl

1205831

1205839

u1

u9

+



119906 (119896) =

9

sum

119894=1

120583119894119906119894(119896) (28)




119894

[10]



1198791198970= 290 + 2451119899

1199030 (29)



Define

Q = 119902 times

[[[[

[

I2times2


sdot sdot sdot 0

d


]]]]

]119873119901times119873119901

(30)





20 40 60 80 100

290

31451

Power level ()0

300

310

Tem

pera

ture

(∘C)

Tl








24 30 36 42 48 54 60 66 723115

312

3125

313

3135

314

3145

315

Time (s)

Coo

lant

tem

pera

ture



(a)

24 30 36 42 48 54 60 66 72088

09

092

094

096

098

1

102

Pow

er le

vel


PID

Time (s)

(b)


24 30 36 42 48 54 60 66 72

002040608

Con

trol r

od sp

eed

Time (s)

minus02

minus04

minus06

minus08

(c)





24 30 36 42 48 54 60 66 72292

2925

293

2935

294

2945

295

2955

Coo

lant

tem

pera

ture

Time (s)


(a)

30 36 42 48 54 60 66 72008

01

012

014

016

018

02

022

Time (s)

Pow

er le

vel



(b)

Time (s)24 30 36 42 48 54 60 66 72

004081216

2

Con

trol r

od sp

eed


minus08

minus04

minus12

minus16

minus2

(c)



6 Conclusions



0 12 24 36 48 60 72290

295

300

305

310

315

Time (s)

Coo

lant

tem

pera

ture


(a)

Time (s)0 12 24 36 48 60 72

010203040506070809

111

Pow

er le

vel


(b)

Time (s)0 12 24 36 48 60 72

0005

01015

02

Con

trol r

od sp

eed

minus02

minus015

minus01

minus005

(c)



Nomenclature



















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










H1=

[[[[[

[

299 times 10minus3

160 times 10minus6

914 times 10minus4

333 times 10minus5

145 times 10minus4

]]]]]

]

H4=

[[[[[

[

289 times 10minus4

200 times 10minus7

883 times 10minus5

360 times 10minus6

700 times 10minus5

]]]]]

]

H8=

[[[[[

[

120 times 10minus2

630 times 10minus6

366 times 10minus3

117 times 10minus4

290 times 10minus4

]]]]]

]

H1= H2= H6 H

3= H4= H5 H

7= H8= H9

C1= [

1 0 0 0 0

0 0 0 1 0]

C1= C2= C3= C4= C5= C6= C7= C8= C9

(8)


119866119903

1198991199030



120582 015 sminus1

120573 0006019119891119891

092Λ 00001 s120583119891

236MWs∘C1198751198860

25000MW



1199030and119866

119903are defined




1199030or 119866119903are defined


119877(119894)

(119894 = 1 2 9)

IF 1199111(119896) is 119865119894

1and 119911

2(119896) is 119865119894

2

THEN x119894(119896 + 1) = G

119894x119894(119896) +H

119894119906119894(119896)

y119894(119896) = C

119894x119894(119896)

(9)







000

70

0145

002

90

1

1

01 05 10

0

0


A120572

120572

LM

H

L M H

Gr

nr0



x (119896 + 1) =9

sum

119894=1

120583119894x119894(119896 + 1) =

9

sum

119894=1

120583119894(G119894x119894(119896) +H

119894119906119894(119896))

y (119896) =9

sum

119894=1

120583119894y119894(119896) =

9

sum

119894=1

120583119894C119894x119894(119896)

(10)

where

120583119894=

120583119894

sum9

119894=1120583119894

(11)

120583119894=

2

prod

119895=1

120572119865119894

119895

(119911119895(119896)) (12)


0

1

00005001001500200250030

080907060504030201

08 0907060504030201

1

Gr nr0


The 120572119865119894

119895



119865119894




1198771 if 119866

119903(119896) is119872 and 119899

1199030(119896) is 119872

thenx1(119896 + 1) = G

1x1(119896) +H

11199061(119896)

y1(119896) = C

1x1(119896)

1198776 if 119866

119903(119896) is119872 and 119899

1199030(119896) is119867

thenx6(119896 + 1) = G

6x6(119896) +H

61199066(119896)

y6(119896) = C

6x6(119896)

1198777 if 119866

119903(119896) is119867 and 119899

1199030(119896) is119867

thenx7(119896 + 1) = G

7x7(119896) +H

71199067(119896)

y7(119896) = C

7x7(119896)

1198778 if 119866

119903(119896) is119867 and 119899

1199030(119896) is119872

thenx8(119896 + 1) = G

8x8(119896) +H

81199068(119896)

y8(119896) = C

8x8(119896)




x (119896) = Gx (119896 minus 1) +H119906 (119896 minus 1)

y (119896 minus 1) = Cx (119896 minus 1) (13)

0

1

00005001001500200250030

080907060504030201

08 0907060504030201

1

Gr nr0



Δx (119896 + 1) = GΔx (119896) +HΔ119906 (119896)

Δy (119896) = CΔx (119896) (14)



[Δx(119896)119879 y(119896)119879]119879 can result in

[Δx (119896 + 1)y (119896 + 1) ] = [

G 0CG I] [

Δx (119896)y (119896) ] + [

HCH]Δ119906 (119896)

y (119896) = [0 I] [Δx (119896)y (119896) ]

(15)


x1015840 (119896 + 1) = G1015840x1015840 (119896) +H1015840119906 (119896)

y (119896) = C1015840x1015840 (119896) (16)

where x1015840(119896) isin 1198771198991015840

(1198991015840

= 119899 + 119901) G1015840 = [ G 0CG I ] H

1015840

= [ HCH ] and



119869 =1003817100381710038171003817Y minus r

119904

10038171003817100381710038172

Q + ΔU2

R (17)


Define


]119879


119875)119879

]119879



Q =

[[[[

[



d


119901)

]]]]

]

R =[[[[

[



d


119888minus 1)

]]]]

]

(18)


119901and 119873

119888are


119879 and r(119896 + 119895)119879 (119895 =




Φ119879Q (r119904minus Fx (119896)) (19)

where

F =

[[[[[[[

[

C1015840G1015840

C1015840G10158402

C1015840G10158403

C1015840G1015840119873119901

]]]]]]]

]

Φ =

[[[[[[[

[


d


]]]]]]]

]

(20)



Δ119906 (119896) = [1 0 sdot sdot sdot 0]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873119888

ΔU (21)


119906 (119896) = 119906 (119896 minus 1) + Δ119906 (119896) (22)


Uminle U le Umax

(23)


[[[[[[

[

119906 (119896)

119906 (119896 + 1)

119906 (119896 + 2)

119906 (119896 + 119873

119888minus 1)

]]]]]]

]

=

[[[[[[

[

1

1

1

1

]]]]]]

]

119906 (119896 minus 1) +

[[[[[[

[




d


]]]]]]

]

[[[[[[

[

Δ119906 (119896)

Δ119906 (119896 + 1)

Δ119906 (119896 + 2)

Δ119906 (119896 + 119873

119888minus 1)

]]]]]]

]

(24)

Define

C1=

[[[[[[

[

1

1

1

1

]]]]]]

]

C2=

[[[[[[

[




d


]]]]]]

]

120595 = [minusC2

C2


+ C1119906 (119896 minus 1)


]

(25)


120595ΔU le 120574 (26)


min 119869 =1003817100381710038171003817Y minus r

119904

10038171003817100381710038172

Q + ΔU2

R

st 120595ΔU le 120574

(27)






119902 1 1 1 1 1 1 1 1 1119903 007 001 002 007 01 01 01 005 01119873119901

380 430 400 340 290 300 325 400 460119873119888

50 45 45 50 60 60 60 40 30

MPC1

MPC9

Nonlinear plant



Controller

Tl0

u(k) Tl

1205831

1205839

u1

u9

+



119906 (119896) =

9

sum

119894=1

120583119894119906119894(119896) (28)




119894

[10]



1198791198970= 290 + 2451119899

1199030 (29)



Define

Q = 119902 times

[[[[

[

I2times2


sdot sdot sdot 0

d


]]]]

]119873119901times119873119901

(30)





20 40 60 80 100

290

31451

Power level ()0

300

310

Tem

pera

ture

(∘C)

Tl








24 30 36 42 48 54 60 66 723115

312

3125

313

3135

314

3145

315

Time (s)

Coo

lant

tem

pera

ture



(a)

24 30 36 42 48 54 60 66 72088

09

092

094

096

098

1

102

Pow

er le

vel


PID

Time (s)

(b)


24 30 36 42 48 54 60 66 72

002040608

Con

trol r

od sp

eed

Time (s)

minus02

minus04

minus06

minus08

(c)





24 30 36 42 48 54 60 66 72292

2925

293

2935

294

2945

295

2955

Coo

lant

tem

pera

ture

Time (s)


(a)

30 36 42 48 54 60 66 72008

01

012

014

016

018

02

022

Time (s)

Pow

er le

vel



(b)

Time (s)24 30 36 42 48 54 60 66 72

004081216

2

Con

trol r

od sp

eed


minus08

minus04

minus12

minus16

minus2

(c)



6 Conclusions



0 12 24 36 48 60 72290

295

300

305

310

315

Time (s)

Coo

lant

tem

pera

ture


(a)

Time (s)0 12 24 36 48 60 72

010203040506070809

111

Pow

er le

vel


(b)

Time (s)0 12 24 36 48 60 72

0005

01015

02

Con

trol r

od sp

eed

minus02

minus015

minus01

minus005

(c)



Nomenclature



















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

1

00005001001500200250030

080907060504030201

08 0907060504030201

1

Gr nr0


The 120572119865119894

119895



119865119894




1198771 if 119866

119903(119896) is119872 and 119899

1199030(119896) is 119872

thenx1(119896 + 1) = G

1x1(119896) +H

11199061(119896)

y1(119896) = C

1x1(119896)

1198776 if 119866

119903(119896) is119872 and 119899

1199030(119896) is119867

thenx6(119896 + 1) = G

6x6(119896) +H

61199066(119896)

y6(119896) = C

6x6(119896)

1198777 if 119866

119903(119896) is119867 and 119899

1199030(119896) is119867

thenx7(119896 + 1) = G

7x7(119896) +H

71199067(119896)

y7(119896) = C

7x7(119896)

1198778 if 119866

119903(119896) is119867 and 119899

1199030(119896) is119872

thenx8(119896 + 1) = G

8x8(119896) +H

81199068(119896)

y8(119896) = C

8x8(119896)




x (119896) = Gx (119896 minus 1) +H119906 (119896 minus 1)

y (119896 minus 1) = Cx (119896 minus 1) (13)

0

1

00005001001500200250030

080907060504030201

08 0907060504030201

1

Gr nr0



Δx (119896 + 1) = GΔx (119896) +HΔ119906 (119896)

Δy (119896) = CΔx (119896) (14)



[Δx(119896)119879 y(119896)119879]119879 can result in

[Δx (119896 + 1)y (119896 + 1) ] = [

G 0CG I] [

Δx (119896)y (119896) ] + [

HCH]Δ119906 (119896)

y (119896) = [0 I] [Δx (119896)y (119896) ]

(15)


x1015840 (119896 + 1) = G1015840x1015840 (119896) +H1015840119906 (119896)

y (119896) = C1015840x1015840 (119896) (16)

where x1015840(119896) isin 1198771198991015840

(1198991015840

= 119899 + 119901) G1015840 = [ G 0CG I ] H

1015840

= [ HCH ] and



119869 =1003817100381710038171003817Y minus r

119904

10038171003817100381710038172

Q + ΔU2

R (17)


Define


]119879


119875)119879

]119879



Q =

[[[[

[



d


119901)

]]]]

]

R =[[[[

[



d


119888minus 1)

]]]]

]

(18)


119901and 119873

119888are


119879 and r(119896 + 119895)119879 (119895 =




Φ119879Q (r119904minus Fx (119896)) (19)

where

F =

[[[[[[[

[

C1015840G1015840

C1015840G10158402

C1015840G10158403

C1015840G1015840119873119901

]]]]]]]

]

Φ =

[[[[[[[

[


d


]]]]]]]

]

(20)



Δ119906 (119896) = [1 0 sdot sdot sdot 0]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873119888

ΔU (21)


119906 (119896) = 119906 (119896 minus 1) + Δ119906 (119896) (22)


Uminle U le Umax

(23)


[[[[[[

[

119906 (119896)

119906 (119896 + 1)

119906 (119896 + 2)

119906 (119896 + 119873

119888minus 1)

]]]]]]

]

=

[[[[[[

[

1

1

1

1

]]]]]]

]

119906 (119896 minus 1) +

[[[[[[

[




d


]]]]]]

]

[[[[[[

[

Δ119906 (119896)

Δ119906 (119896 + 1)

Δ119906 (119896 + 2)

Δ119906 (119896 + 119873

119888minus 1)

]]]]]]

]

(24)

Define

C1=

[[[[[[

[

1

1

1

1

]]]]]]

]

C2=

[[[[[[

[




d


]]]]]]

]

120595 = [minusC2

C2


+ C1119906 (119896 minus 1)


]

(25)


120595ΔU le 120574 (26)


min 119869 =1003817100381710038171003817Y minus r

119904

10038171003817100381710038172

Q + ΔU2

R

st 120595ΔU le 120574

(27)






119902 1 1 1 1 1 1 1 1 1119903 007 001 002 007 01 01 01 005 01119873119901

380 430 400 340 290 300 325 400 460119873119888

50 45 45 50 60 60 60 40 30

MPC1

MPC9

Nonlinear plant



Controller

Tl0

u(k) Tl

1205831

1205839

u1

u9

+



119906 (119896) =

9

sum

119894=1

120583119894119906119894(119896) (28)




119894

[10]



1198791198970= 290 + 2451119899

1199030 (29)



Define

Q = 119902 times

[[[[

[

I2times2


sdot sdot sdot 0

d


]]]]

]119873119901times119873119901

(30)





20 40 60 80 100

290

31451

Power level ()0

300

310

Tem

pera

ture

(∘C)

Tl








24 30 36 42 48 54 60 66 723115

312

3125

313

3135

314

3145

315

Time (s)

Coo

lant

tem

pera

ture



(a)

24 30 36 42 48 54 60 66 72088

09

092

094

096

098

1

102

Pow

er le

vel


PID

Time (s)

(b)


24 30 36 42 48 54 60 66 72

002040608

Con

trol r

od sp

eed

Time (s)

minus02

minus04

minus06

minus08

(c)





24 30 36 42 48 54 60 66 72292

2925

293

2935

294

2945

295

2955

Coo

lant

tem

pera

ture

Time (s)


(a)

30 36 42 48 54 60 66 72008

01

012

014

016

018

02

022

Time (s)

Pow

er le

vel



(b)

Time (s)24 30 36 42 48 54 60 66 72

004081216

2

Con

trol r

od sp

eed


minus08

minus04

minus12

minus16

minus2

(c)



6 Conclusions



0 12 24 36 48 60 72290

295

300

305

310

315

Time (s)

Coo

lant

tem

pera

ture


(a)

Time (s)0 12 24 36 48 60 72

010203040506070809

111

Pow

er le

vel


(b)

Time (s)0 12 24 36 48 60 72

0005

01015

02

Con

trol r

od sp

eed

minus02

minus015

minus01

minus005

(c)



Nomenclature



















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Q =

[[[[

[



d


119901)

]]]]

]

R =[[[[

[



d


119888minus 1)

]]]]

]

(18)


119901and 119873

119888are


119879 and r(119896 + 119895)119879 (119895 =




Φ119879Q (r119904minus Fx (119896)) (19)

where

F =

[[[[[[[

[

C1015840G1015840

C1015840G10158402

C1015840G10158403

C1015840G1015840119873119901

]]]]]]]

]

Φ =

[[[[[[[

[


d


]]]]]]]

]

(20)



Δ119906 (119896) = [1 0 sdot sdot sdot 0]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873119888

ΔU (21)


119906 (119896) = 119906 (119896 minus 1) + Δ119906 (119896) (22)


Uminle U le Umax

(23)


[[[[[[

[

119906 (119896)

119906 (119896 + 1)

119906 (119896 + 2)

119906 (119896 + 119873

119888minus 1)

]]]]]]

]

=

[[[[[[

[

1

1

1

1

]]]]]]

]

119906 (119896 minus 1) +

[[[[[[

[




d


]]]]]]

]

[[[[[[

[

Δ119906 (119896)

Δ119906 (119896 + 1)

Δ119906 (119896 + 2)

Δ119906 (119896 + 119873

119888minus 1)

]]]]]]

]

(24)

Define

C1=

[[[[[[

[

1

1

1

1

]]]]]]

]

C2=

[[[[[[

[




d


]]]]]]

]

120595 = [minusC2

C2


+ C1119906 (119896 minus 1)


]

(25)


120595ΔU le 120574 (26)


min 119869 =1003817100381710038171003817Y minus r

119904

10038171003817100381710038172

Q + ΔU2

R

st 120595ΔU le 120574

(27)






119902 1 1 1 1 1 1 1 1 1119903 007 001 002 007 01 01 01 005 01119873119901

380 430 400 340 290 300 325 400 460119873119888

50 45 45 50 60 60 60 40 30

MPC1

MPC9

Nonlinear plant



Controller

Tl0

u(k) Tl

1205831

1205839

u1

u9

+



119906 (119896) =

9

sum

119894=1

120583119894119906119894(119896) (28)




119894

[10]



1198791198970= 290 + 2451119899

1199030 (29)



Define

Q = 119902 times

[[[[

[

I2times2


sdot sdot sdot 0

d


]]]]

]119873119901times119873119901

(30)





20 40 60 80 100

290

31451

Power level ()0

300

310

Tem

pera

ture

(∘C)

Tl








24 30 36 42 48 54 60 66 723115

312

3125

313

3135

314

3145

315

Time (s)

Coo

lant

tem

pera

ture



(a)

24 30 36 42 48 54 60 66 72088

09

092

094

096

098

1

102

Pow

er le

vel


PID

Time (s)

(b)


24 30 36 42 48 54 60 66 72

002040608

Con

trol r

od sp

eed

Time (s)

minus02

minus04

minus06

minus08

(c)





24 30 36 42 48 54 60 66 72292

2925

293

2935

294

2945

295

2955

Coo

lant

tem

pera

ture

Time (s)


(a)

30 36 42 48 54 60 66 72008

01

012

014

016

018

02

022

Time (s)

Pow

er le

vel



(b)

Time (s)24 30 36 42 48 54 60 66 72

004081216

2

Con

trol r

od sp

eed


minus08

minus04

minus12

minus16

minus2

(c)



6 Conclusions



0 12 24 36 48 60 72290

295

300

305

310

315

Time (s)

Coo

lant

tem

pera

ture


(a)

Time (s)0 12 24 36 48 60 72

010203040506070809

111

Pow

er le

vel


(b)

Time (s)0 12 24 36 48 60 72

0005

01015

02

Con

trol r

od sp

eed

minus02

minus015

minus01

minus005

(c)



Nomenclature



















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119902 1 1 1 1 1 1 1 1 1119903 007 001 002 007 01 01 01 005 01119873119901

380 430 400 340 290 300 325 400 460119873119888

50 45 45 50 60 60 60 40 30

MPC1

MPC9

Nonlinear plant



Controller

Tl0

u(k) Tl

1205831

1205839

u1

u9

+



119906 (119896) =

9

sum

119894=1

120583119894119906119894(119896) (28)




119894

[10]



1198791198970= 290 + 2451119899

1199030 (29)



Define

Q = 119902 times

[[[[

[

I2times2


sdot sdot sdot 0

d


]]]]

]119873119901times119873119901

(30)





20 40 60 80 100

290

31451

Power level ()0

300

310

Tem

pera

ture

(∘C)

Tl








24 30 36 42 48 54 60 66 723115

312

3125

313

3135

314

3145

315

Time (s)

Coo

lant

tem

pera

ture



(a)

24 30 36 42 48 54 60 66 72088

09

092

094

096

098

1

102

Pow

er le

vel


PID

Time (s)

(b)


24 30 36 42 48 54 60 66 72

002040608

Con

trol r

od sp

eed

Time (s)

minus02

minus04

minus06

minus08

(c)





24 30 36 42 48 54 60 66 72292

2925

293

2935

294

2945

295

2955

Coo

lant

tem

pera

ture

Time (s)


(a)

30 36 42 48 54 60 66 72008

01

012

014

016

018

02

022

Time (s)

Pow

er le

vel



(b)

Time (s)24 30 36 42 48 54 60 66 72

004081216

2

Con

trol r

od sp

eed


minus08

minus04

minus12

minus16

minus2

(c)



6 Conclusions



0 12 24 36 48 60 72290

295

300

305

310

315

Time (s)

Coo

lant

tem

pera

ture


(a)

Time (s)0 12 24 36 48 60 72

010203040506070809

111

Pow

er le

vel


(b)

Time (s)0 12 24 36 48 60 72

0005

01015

02

Con

trol r

od sp

eed

minus02

minus015

minus01

minus005

(c)



Nomenclature



















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










24 30 36 42 48 54 60 66 723115

312

3125

313

3135

314

3145

315

Time (s)

Coo

lant

tem

pera

ture



(a)

24 30 36 42 48 54 60 66 72088

09

092

094

096

098

1

102

Pow

er le

vel


PID

Time (s)

(b)


24 30 36 42 48 54 60 66 72

002040608

Con

trol r

od sp

eed

Time (s)

minus02

minus04

minus06

minus08

(c)





24 30 36 42 48 54 60 66 72292

2925

293

2935

294

2945

295

2955

Coo

lant

tem

pera

ture

Time (s)


(a)

30 36 42 48 54 60 66 72008

01

012

014

016

018

02

022

Time (s)

Pow

er le

vel



(b)

Time (s)24 30 36 42 48 54 60 66 72

004081216

2

Con

trol r

od sp

eed


minus08

minus04

minus12

minus16

minus2

(c)



6 Conclusions



0 12 24 36 48 60 72290

295

300

305

310

315

Time (s)

Coo

lant

tem

pera

ture


(a)

Time (s)0 12 24 36 48 60 72

010203040506070809

111

Pow

er le

vel


(b)

Time (s)0 12 24 36 48 60 72

0005

01015

02

Con

trol r

od sp

eed

minus02

minus015

minus01

minus005

(c)



Nomenclature



















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 12 24 36 48 60 72290

295

300

305

310

315

Time (s)

Coo

lant

tem

pera

ture


(a)

Time (s)0 12 24 36 48 60 72

010203040506070809

111

Pow

er le

vel


(b)

Time (s)0 12 24 36 48 60 72

0005

01015

02

Con

trol r

od sp

eed

minus02

minus015

minus01

minus005

(c)



Nomenclature



















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Nonlinear Fuzzy Model Predictive...

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