Reduced Order Model based Optimally Tuned Fractional Order PID controller for Pressurized Water Nuclear Reactor

Transcription

1 Preprint of the rd IFAC Conference on Advance in Proportional- Integral-Derivative Control, Ghent, Belgium, May 9-11, 18 FrIS.1 Reduced Order Model baed Optimally Tuned Fractional Order PID controller for Preurized Water Nuclear Reactor M. Santhiya*, Anuj Abraham $, N. Pappa #, M. Chitra # *Aitant Profeor, Kongu Engineering College, Erode, India. anthiyain@gmail.com $ Aitant Profeor, Department of Applied Electronic & Intrumentation, Rajagiri School of Engineering & Technology, APJ Abdul Kalam Technological Univerity, Kerala. India. anuj1986aei@gmail.com # Department of Intrumentation Engineering, Madra Intitute of Technology, Anna Univerity, Chennai, India. npappa@rediffmail.com, chitramurugan.g@gmail.com Abtract: In thi paper, the reduced model of the Preurized Water Nuclear Reactor (PWR) i derived baed on the point kinetic equation and thermal equilibrium relation. The power level of the nuclear reactor i controlled by adjuting the inertion reactivity of the rod. Several controller uch a Genetic Algorithm baed PID controller (GAPID), Fractional Order PID controller (FOPID) and Genetic Algorithm baed Fractional Order PID Controller (GAFOPID) are ued to control the power level of the PWR type of nuclear reactor. The imulation reult depict that the Genetic Algorithm baed Fractional Order PID Controller (GAFOPID) how the atifactory repone than other control technique. Keyword: reactor power, reactivity, controller, genetic algorithm, performance indice. 1. INTRODUCTION Nowaday, nuclear reactor i a major component of a nuclear power plant and adopt integrated deign. Over the year, one of the everal nuclear engineering problem i to control the reactor power level. The reactor power control i important from the tandpoint of afety concern and for regular and appropriate operation of nuclear power plant, Guimarãe, et al., (11). Thi alo include the reliability and cleanline of nuclear plant over the globe in indutrial technology. Gupta, A et al., (17). The dynamic behavior of the Preurized Water Reactor nuclear reactor i very complex and non-linear. Becaue of thi non-linear tructure, the reactor dynamic change with time according to power level, Alekei et al., (11). Hence it i neceary to ue non-linear controller for controlling the power level of nuclear reactor. Conventional reactor regulating ytem control the average temperature of the reactor core according to it referenced temperature determined by the turbine load, Da Cota et al., (11). Thi control ha diadvantage of controlling under low power demand variation. It i hard to get the atifying performance with the claic control trategy to control nuclear reactor power. Advanced intelligent control give a bright future to nonlinear time dependent control ytem. In recent year, there ha been a growing interet and reearch in the deign of intelligent ytem uing oft computing methodologie uch a Fuzzy Logic (FL), Artificial Neural Network (ANN), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Evolutionary Strategy (ES), Simulated Annealing (SA), and o on for nuclear engineering problem. The paper i organized a follow. In Section, the mathematical model of the nuclear reactor i preented. In Section, the controller deign i preented. Concluion i given in Section 4.. MATHEMATICAL MODEL OF PWR TYPE OF NUCLEAR REACTOR The nuclear reactor i an uncertain, nonlinear ytem and it parameter varie with time a a function of operating power level, fuel burn up and control rod worth..1 Model A reduced model of a PWR type of nuclear reactor i derived baed on the point kinetic and on thermal hydraulic equilibrium relation. The kinetic equation are ued to decribe the neutron balance in the core and the thermal equilibrium relation are ued to decribe the energy exchange between the different loop, German, G et al., (1). The plant geometry and the operating condition play an important role in deciding the number of effective parameter. Thu, thee parameter need to be calibrated for each plant and ituation conidered, Hetrick, L et al., (1971). The block diagram of the nuclear reactor model i hown in the Fig. 1 Fig. 1. Block diagram of Nuclear reactor model 18 International Federation of Automatic Control 669

2 Preprint of the rd IFAC Conference on Advance in Proportional- Integral-Derivative Control, Ghent, Belgium, May 9-11, 18 The reduced model can be written a follow: dp = β P [α dt Ʌ f(t f T f ) + α av (T av T av ) + 6 ρ in 1] + λ i c i i=1 (1) ß 4 =.64 X 1-4 λ 4 =.11 X 1-1 ß 5 = 8. X 1-4 λ 5 = 1.4 ß 6 =1.69 X 1-4 λ 6 =.87 ß=6.5 X 1 - λ =5 X 1-4 dc i dt P = β i λ Ʌ ic i Where i = 1 to 6. dt f dt dt av dt = A T 1(T f T av ) + A f Tav 1 P P = A (T f T av ) T f T av A T (T av T in ) av T in The contant A 1 and A are given by the following relation: UA A 1 = A M f C = UA pf M C C pc (5) where P denote the power level of the reactor (MW), β, β i, λ i, Ʌ are effective nuclear contant, C i i amplitude related with the concentration of the i th group of neutron precuror (atom/cm ) T f i the average temperature of the fuel in the veel (K) T av i the average temperature of the coolant in the veel (K) A i the area of the core (cm ), U i an effective heat tranfer parameter for the core (MW/ K), M f i the effective fuel ma (ton), C pf i the pecific heat coefficient for the fuel (MJ/ K), M C i the effective coolant ma (ton), C pc i the pecific heat coefficient for the coolant (MJ/ K), T out and T in are the temperature of the coolant at the outlet and at the inlet of the veel (K), T f and T av are the temperature of the fuel and of the coolant at the initial teady tate (K), ρ in i the inertion of reactivity that generate the tranient (dollar), α f and α av are the reactivity feedback coefficient aociated to the temperature of the fuel and of the coolant, repectively (( k/k/ K). The numerical value conidered for the neutronic contant (Concentration of the Six Neutron Precuror Group and Relative Decay Rate) and the teady tate data are given in Table 1 and Table repectively. Table 1. Neutronic Contant ß value λ value ß 1 =.47 X 1-4 λ 1 = 1.7 X 1 - ß = 1.8 X 1 - λ =.17 X 1 - ß =1. X 1 - λ = 1.15 X 1-1 () () (4) Table.Value of the dependent variable at Steady State Variable at teady tate Value P (MW) T f (K) T av (K) T in (K) 5 α f x 1 - α av -.7 x 1 - A 1.77 x 1-1 A 4.1 x 1-1 In Table, the main parameter of the reactor are reported. Table. Reactor parameter Parameter Value Fuel rod/aembly 14 x 14 Fuel rod diameter 1.7 x1 - m Fuel rod aembly ize.199 m Fuel thermal conductivity.461 W/m/K Fuel pecific heat.48 x 1 J/kg/K Fuel pellet diameter 1.16 x 1 - m Fuel denity 1.8 x 1 kg/m Clad thermal conductivity 15. W/m/K Clad pecific heat.48 x 1 J/kg/K Clad denity x 1 kg/m Clad thickne.54 x 1 - m Fuel-to-clad (gap) x 1 heat tranfer W/m /K coefficient Preure 15.8 MPa Average inlet ma flux.6 x 1 kg/m / Coolant inlet temperature 5.15 K Initial thermal power MW 67

3 POWER (MW) Preprint of the rd IFAC Conference on Advance in Proportional- Integral-Derivative Control, Ghent, Belgium, May 9-11, 18. CONTROLLER DESIGN FOR NUCLEAR REACTOR Well-developed analytical technique that require accurate modeling of the proce under control have been ued uccefully a a mean of control for year. However, not all of the procee can be modeled with ufficient accuracy, and in ome cae noiy operating condition or drift in proce variable may render the controller developed in thi fahion uele, Torabi, Keivan et al. (11). In thi work, variou controller are developed for a validated model of the PWR type of nuclear reactor. The deired performance criteria choen for power level are, Minimum Overhoot (Relative) Quicker Settling Time (% band)..1 PID Controller optimized by Genetic Algorithm (GAPID) The tructure of the PID controller i a follow: G ( ) K The bet repone can be achieved by proper tuning of K P, K I and K D value of PID controller. Hence to obtain the bet value of K P, K I and K D, optimization algorithm are ued. Genetic algorithm i one of the nontraditional optimization method. GA trie to imitate natural genetic and natural election. Survival of the fittet i the main philoophy behind the Genetic Algorithm. The genetic algorithm olve optimization problem by mimicking the principle of biological evolution, repeatedly modifying a population of individual point uing rule modeled on gene combination in biological reproduction. The parameter of the Genetic Algorithm ued for tuning PID controller are hown in the table 4. Table 4. Parameter of the Genetic Algorithm Parameter Value Number of population 1 Number of generation 5 Number of parameter to be optimized c Croover probability.99 p K I K D (6) TIME () Fig.. Servo repone of the nuclear reactor uing GAPID controller The ettling time of thi controller i around econd and it contain overhoot of about 15 MW.. Fractional Order PID Controller GAPID PID controller belong to the dominating form of feedback indutrial controller and there i a continuou effort to improve their quality and robutne. In recent year, there i an increaing number of tudie related to the application of fractional controller in many area of cience and engineering. Fractional Order PID (FO-PID) controller could benefit the indutry ignificantly with a wide pread impact when FO-PID parameter tuning technique have been well developed. Thi fact i due to a better undertanding of the Fractional Calculu (FC), Marzio, Mareguerra et al. (4). The FC concept are adapted to frequency-baed method. The introduction of fractional order calculu idea to conventional controller deign extend the opportunity of added performance improvement. Reearch activitie are now focued to develop new tuning rule for fractional controller for real ytem. Some of thee technique are baed on an extenion of the claical PID control theory. Clearly, depending on the value of the order λ and μ, we get an infinite number of choice for controller type (defined through the (λ, μ) plane). Conventional ytem are derived from differential equation of integer order wherea fractional order ytem are derived from fractional order differential equation (Zarabadipour, H et al.). Since PID control i popular in many indutry ection, PI λ D μ controller hould provide additional potential to achieve better performance. The tructure of the FOPID controller i hown in Fig.. Mutation probability.1 The optimal value of K P, K I and K D obtained by Genetic Algorithm are 1.5,.91 and.61 repectively. The repone of the PWR type of nuclear reactor controlled by GAPID i hown in the following Fig.. Fig.. Fractional Order PID control tructure G ( ) K c p K I K D (7) 671

4 Power (MW) Phae (deg) Magnitude (db) Preprint of the rd IFAC Conference on Advance in Proportional- Integral-Derivative Control, Ghent, Belgium, May 9-11, 18 The overall cloed loop tranfer function can be written a, y G ( ) G ( ) C P r 1 G ( ) G ( ) C Fractional Order Cloed Loop Characteritic Polynomial (FOCP) i given by, K P 1 G ( ) G ( ) C On ubtituting the value of G C () and G P (), the final value of K P,K I are derived a below, (in( ) * ( )) * ( ( d n d n ) d n ) 1 1 (co( ) * ( )) * ( ( d n d n ) ( d n )) ( ) * [ ( n d ) ( n d n d ) K I (11) ( n ) ( n ) * (in( ) * ( )) 1 ω change from to ω maximum, whoe value i determined by ubtituting K I =.Uing equation, K P and K I value are calculated for each value of λ (varying from.1 to.9, in tep of.1), by ubtituting ω from to ω maximum. A tability curve in the K P -K I plane i contructed for each λ. All region bounded in between tability curve and the tability line i repreented a Global Stability Region. From the Global Stability Region, the average value of K P and K I correponding to the each value of λ i obtained Among thee value, the bet fit of K P average and K I average and correponding λ are identified by mean of optimization technique. The 9 th order tranfer function model i obtained from the mathematical model of the nuclear reactor a hown below, G P 96 8 P ( n ) ( n ) * (in( ) * ( )) e e5 ( ) e P e e (8) (9) (1) (1) The reduced econd order tranfer function i hown below, From Fig. 4, it i evident that the reduced econd order tranfer function i validated againt the original model and 9 th order model. The frequency repone of the tranfer function i alo validated againt the 9 th order tranfer function G P (1) ( ) Bode Diagram Frequency 1 1 (rad/ec) 1 Frequency (rad/) Fig. 5. Frequency of the original and reduced model 9th order nd order When K I =, ω maximum i found to be.48. We will fix the value of K D and find the tability region in the (K P, K I ) plane. The value of K D i fixed a.91 and μ a.968 from the tability plane, Subhranu Padhee et al. (11). The average value of K P and K I for each value of λ i calculated and ISE, IAE of each tuning value are found out. From the value of ISE and IAE, the value of λ which give minimum IAE and ISE i choen a final K P and K I value. The performance indice comparion of variou value of λ i hown in Table 5. Table 5. Performance indice comparion of variou value of λ th ORDER TF ORIGINAL nd ORDER TF λ K P average K I average ISE IAE Time () Fig. 4. Validation of the Tranfer function The 9 th order tranfer function model i reduced into econd order tranfer function by uing Hankel ingular value-baed model reduction (Anuj et.al 17)

5 Power (MW) Power (MW) POWER (MW) POWER (MW) Preprint of the rd IFAC Conference on Advance in Proportional- Integral-Derivative Control, Ghent, Belgium, May 9-11, 18 The repone of the Fractional Order PID controller for controlling the nuclear reactor power i hown in Fig. 6. FOPID GAFOPID TIME () TIME () Fig. 7. Repone of the GAFOPID controller Fig 6. Servo repone of the nuclear reactor uing FOPID controller The ettling time of FOPID controller i about 1 ec.. Genetic Algorithm Baed Fractional Order PID Controller The workability of genetic algorithm (GA) i baed on Darwinian theory of urvival of the fittet. Genetic algorithm (GA) may contain a chromoome, a gene, et of population, fitne, fitne function, breeding, mutation and election. Genetic algorithm (GA) begin with a et of olution repreented by chromoome, called population. Solution from one population are taken and ued to form a new population, which i motivated by the poibility that the new population will be better than the old one. Further, olution are elected according to their fitne to form new olution, that i, offpring. In thi work, Genetic Algorithm i ued to optimize the FOPID parameter uch a K P, K I, K D, λ, and μ. The parameter of the Genetic Algorithm ued for optimizing the FOPID controller are hown in Table 6. Table 6. Parameter of the Genetic Algorithm Parameter Value The ettling time of GAFOPID controller i le than which i very le when compared to other controller. The ervo operation comparion of variou controller are hown Fig Time () Fig. 8. Comparion of Servo Repone GAFOPID FOPID GAPID SET The fuel temperature of the reactor act a a diturbance variable. At 17, the fuel temperature i increaed by 1 o K. The regulatory repone of the variou controller i hown in the Fig. 9. The GAFOPID controller reject the diturbance effectively than other controller. Number of population 5 Number of generation 5 Number of parameter to be optimized 5 Croover probability.99 Mutation probability FOPID GAFOPID GAPID The optimal value obtained from GA are K P =.499, K I =.8878, λ=.7691, K D =.91 and μ=.968. The repone of the Genetic Algorithm baed Fractional Order PID controller (GAFOPID) i hown in Fig Time () Fig. 9. Comparion of Regulatory Repone The performance indice of variou controller are tabulated in Table 7. 67

6 Preprint of the rd IFAC Conference on Advance in Proportional- Integral-Derivative Control, Ghent, Belgium, May 9-11, 18 Table 7. Comparion of Performance Indice Type of Controller ISE IAE Settling time () GAFOPID FOPID GAPID Marzio Mareguerra, Enrico Zio, Raffaele Canetta (4), Uing genetic algorithm for calibrating implified model of nuclear reactor dynamic, Annal of Nuclear Energy, 1, Torabi Keivan, Safarzadeh Omid, Rahimi-Moghaddam Abdolfazl (11). Robut control of the PWR core power uing quantitative feedback theory, IEEE Tran Nucl. Sci., 58(1): Subhranu Padhee, Abhinav Gautam, Yaduvir Singh, and Gagandeep Kaur (11). A Novel Evolutionary Tuning Method for Fractional Order PID Controller, Int. J. of Soft Computing and Eng. (IJSCE), 1(), CONCLUSIONS The reduced model of the PWR type of nuclear reactor i developed by uing the neutron balance equation and the heat exchange equation. The nuclear reactor i controlled by uing Genetic Algorithm baed PID controller, Fractional Order PID controller and Fractional Order PID controller optimized by Genetic Algorithm. From the reult, it i clear that the Fractional Order PID controller optimized by Genetic Algorithm how the atifactory performance when it i compared to other controller trategie in both ervo and regulatory level control. REFERENCES Alekei Tepljakov, Eduard Petlenkov and Juri Belikov (11). FOMCON: Fractional-Order Modeling and Control Toolbox for MATLAB, 18 th Int.Conf.in Mixed Deign of Integrated Circuit and Sytem, Gliwice, Poland. Anuj Abraham, Pappa, N, Shanmugha Priya, M, Mary Hexy, (17). Predictive control deign for a MIMO multivariable proce uing order reduction technique. Int. J. of Mod. and Sim., T & F, 7(4), Da Cota Rafael Gome, de Abreu Mol Antônio Carlo, de Carvalho Paulo Victor R, Lapa Celo Marcelo Franklin (11). An efficient neuro-fuzzy approach to nuclear power plant tranient identification, Ann Nucl Energy,8(6): Guimarãe Antonio Céar Ferreira, Lapa Celo Marcelo Franklin, Moreira Maria de Lourde (11). Fuzzy methodology applied to probabilitic afety aement for digital ytem in nuclear power plant, Nucl. Eng. De., 41(9): Gupta A Boe D, Banerjee S, Kumar M, Marathe PP, Mukhopadhyay S (17). An Interval Approach to Nonlinear Controller Deign for Load-Following Operation of a Small Modular Preurized Water Reactor, IEEE Tranaction on Nuclear Science, 64(9), German G. Theler, Fabian J. Bonetto (1). On the tability of the point reactor kinetic equation, Nuclear Engineering and Deign, 4, Hetrick David L. (1971). Dynamic of nuclear reactor. The Univerity of Chicago Pre. H.Zarabadipour, H.Emadi, Reduced optimal controller deign for the nuclear reactor power control ytem, Nucl.Eng. De, 19,