Modeling and model identification of a pressurizer at the Paks Nuclear Power Plant

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1 Modeling and model identification of a pressurizer at the aks Nuclear ower lant István Varga, ábor Szederkényi, Katalin M. Hangos, József Bokor Systems and Control Laboratory Computer and Automation Research Institute Budapest, Hungary SYSID 2006, Newcastle p. /7

2 ressurizer Engineering model Water energy balance du dt = c pmt I c p mt + K W (T W T) + Wall energy balance 4 i= W HE χ i du W dt Constitutive equations = K W (T T W ) W loss U = c p MT, U W = C pw T W, SYSID 2006, Newcastle p. 8/7

3 ressurizer State-space model ressure equation p = h(t) = eϕ(t) 00 State-space model:, ϕ(t) = c 0 + c T + c 2 T 2 + c 3 T 3 x = Ā x + Bū + Ē d x = [T T W ] T, ū {0,, 2, 3, 4}, d = [TI W loss ] T ȳ = [ 0 ] x, x = h (p) SYSID 2006, Newcastle p. 9/7

4 ressurizer arameters State space equation matrices: B = Ā = W HE c p M 0 m M K W c p M K W C pw, Ē = K W c p M K W C pw m M 0 0 C pw. arameters with known values: W HE, c p, M, m 2. arameter with an acceptable initial guess: W loss 3. arameters with completely unknown values: C pw, K W. SYSID 2006, Newcastle p. 0/7

5 Identification reliminary steps Scaling: to obtain a dimensionless form Measured input and output: scaled u Time [h] y Time [h] Sufficient excitation: by the old "on-off" controller SYSID 2006, Newcastle p. /7

6 Identification Structure validation ESS method: set of candidate model structures y(k) = n r i= b i z nc + a i z u(k d) + Investigated structures (with d = 0): i= q i z + q 2i z 2 + p i z + p 2i z 2u(k d) n r =,n c = 0; n r = 2,n c = 0; n r = 3,n c = 0 n r = 4,n c = 0; n r =,n c = ; n r =,n c = 2 n r = 3,n c = ; n r = 2,n c = Result of model structure validation: physical model structure (n r = 2,n c = 0) confirmed (M is known) SYSID 2006, Newcastle p. 2/7

7 Identification Initial step Criterion functions: V (N,θ) = N (y(i) ŷ(i)) N 2, V 2 (N,θ) = i= N N (p(i) ˆp(i)) 2 i= arameters to be estimated: [ θ = C pw K W ] T W loss, (x2 (0)) Method: Nelder-Mead simplex search parameter lower end (%) upper end (%) Results: C pw K W W loss SYSID 2006, Newcastle p. 3/7

8 Identification Iterative procedure for refining θ = [C pw K W W loss ] T and θ 2 = [c 0 c c 2 c 3 ] T. i:=0; set the values of ˆθ (0) and ˆθ 2 (0) 2. Using the initial values ˆθ 2 (i), estimate θ 2 with fixed θ := ˆθ (i) applying the simplex search method and criterion function V = V + V 2 ; results in ˆθ 2 (i + ). 3. Invert the temperature from the pressure measurements using ˆθ 2 (i + ). 4. Using the initial values ˆθ (i) estimate θ with fixed θ 2 := ˆθ 2 (i + ) applying the simplex search method and criterion function V = V + V 2 ; results in ˆθ (i + ). 5. If sufficient convergence in V, then stop, else i:=i+; go to step (2). SYSID 2006, Newcastle p. 4/7

9 Identification Results Scaled model matrices: A = 0 3¾ ,B =¾ ,E =¾ Updated pressure-temperature curve coefficients: c 0 = , c = c 2 = , c 3 = Model validation output measured output simulated output SYSID 2006, Newcastle p. 5/7

10 Conclusion Dynamic model: low order, based on first engineering principles Advantage: good initial guesses were available Model structure validation by ESS method arameter estimation: nonlinear in parameters, simplex method physical parameters: small number, in the state equations pressure-temperature curve parameters: in the output equation simple iterative scheme followed by an initial step SYSID 2006, Newcastle p. 6/7

11 Aim and the physical system Aim: developing a dynamic model of the primary circuit of VVER plants for controller design purposes. Model has to be minimal and its parameters have to be physical meaning. Valve positiont ressurizer: ressure controller Heating power ressurizer, R 23 bar 325 C l R S Steam generator S Steam 46 bar, 260 C 450 t/h, 0,25% l S m S,in m S,out Steam generator: Level controller Base signal R N 297 C Valve position T C,HL Reactor,R Reactor power controller v Main hidraulic pump C T C,CL Inlet secundary water 222 C ressurizer: Level controller Base signal Correction signal reheater Fazekas Cs., Szederkényi., Hangos M.K. SZTAKI , Nov. 5 - p. 3/38

12 Model: State space model dn dt dm C dt dt C dt p v 2 + p 2 v + p 3 = N + S (25) Λ = m in m out (26) = p, C m in T C,I T C +c p, C m out 5 + W R c p, C M C c 6 K T,S, T C T W a Kloss, C T C T out,c (27) dm S dt dt S dt = m S,in m S,out (28) = c L p,s S c L M p,s m S,in T SSW T L S +cp,s m S,out T S dt W dt dt R dt = = m S,out E evap,s + K T,S,2 T W T S b Kloss,S T S T out,s (29) c p,w M W K T,S, T C T W a KT,S,2 T W T S b (30) c p, R M R χ m R >0 c p, C m R T C,HL + χ m R <0 c p, R m R T R W loss, R + W heat, R c p, R m R T R (3) W R = c Ψ N (32) p S = p T (T S ) (33) l R = A R M C ϕ(t C ) V C 0 (34) p R = p T (T R ) (35) Fazekas Cs., Szederkényi., Hangos M.K. SZTAKI , Nov. 5 - p. 8/38

13 Model: Variables and parameters Variables State variables: N, M C, T C, T R, M S, T S, T W Input variables: v, m in, m S,in, W heat, R Disturbances: m out, m S,out, T S,SW, T C,I Output variables: N (W R ), l R (M C ), p R, p S Constants Reactor: parameters of nuclear physics: other: c Ψ. Λ, S; parameters of the control rod: p, p 2, p 3 ; Liquid in the primary circuit: c p, C, K T,S,, K loss, C, a. Steam generator: c L p,s, K T,S,2, K loss,s, b. Wall of the steam generator: c p,w M W. ressurizer: c p, R, W loss, R, V 0 C. Known constants are T out, C = 00 o C, T out,s = 260 o C (in case of the unit and 3) and T out,s = 259 o C (in case of the unit 4) Fazekas Cs., Szederkényi., Hangos M.K. SZTAKI , Nov. 5 - p. 9/38

14 E: Decomposition It is seen from the state equations (25)-(3) that the parameters in the neutron flux balance equation (25) can be estimated independently of the parameters of the other operating units. Then the coupled equations (26)-(30) describing the dynamics of the liquid in the primary circuit, in the steam generator and the dynamics of the tube-wall of the steam generator form another component. The third component is the pressurizer that depends on the dynamics of the liquid in the primary circuit. T W STEAM ENERATOR T C TUBE-WALL T S REACTOR W R RIMARY CIRCUIT T W M C RESSURIZER T C Fazekas Cs., Szederkényi., Hangos M.K. SZTAKI , Nov. 5 - p. 20/38

15 Results: ressurizer A good quality and long measured data sequence from the Unit computer exists describing the dynamics of pressurizer using the old "on-off type" pressure controller, this data sequence is used for parameter estimation. Old data arameter Unit Time span: 8.88 h c p,r J/kg/K W loss,r W Error Fazekas Cs., Szederkényi., Hangos M.K. SZTAKI , Nov. 5 - p. 30/38

16 Results: ressurizer Measured Simulated by model Temperature Temperature [C] Time [h] Fazekas Cs., Szederkényi., Hangos M.K. SZTAKI , Nov. 5 - p. 3/38

17 Results: Quality of estimates 3.72 x x W loss,r Error value c p,r [J/kg/K] Fazekas Cs., Szederkényi., Hangos M.K. SZTAKI , Nov. 5 - p. 34/38 x 0 5 W loss,r c [J/kg/K] p,r

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