– Context, pros and cons, the ALFRED reactor

– Modelica language, causal and acausal approach, simulation environment

Core, steam generator, primary and secondary loop models

– Pairing selection, controllers tuning, operational transient simulation, startup and logic control, grid connection

– Spatial neutronics model

Lead favorable features

High boiling point

High density

No exothermic reaction with water/air

Volatile FP retention

Good shielding properties

Low moderation – High P/D

GIF Objectives

Safety: . Reduced risk of core voiding . Dispertion of fuel against compaction . Large thermal inertia . Mitigation of core blockage . Natural circulation capabilities

Plant semplification & Economics: . No intermediate HX

Sustanability: . Closed fuel cycle . Efficient conversion of fertile U . MA management

Lead drawbacks

Corrosion of structural material

Erosion of structural material

High melting point

Opacity High density

Problems for:

Material technology: . Need for coolant chemical control and/or cladding coating

Mechanical design

Refuelling

. 300 MWth – 125 MWe

. Pool reactor – 8 loops

. Scaled-down demonstrator for ELFR (European Lead Fast Reactor)

LFR embodies Gen-IV key concepts of economics, sustainability, safety & reliability, proliferation resistance and physical protection

Power 300 MWth

Primary circuit 400-480 °C

Secondary circuit 335-450 °C

Need to develop specific simulation tools allow improving the control system design

Lead-cooled Fast Reactor (Gen–IV) Innovative reactor concept

Trade-off between the control design requirements and the characterization of the system dynamics

Control strategies retrieved from LWRs and SFRs not suitable due to the different features of LFRs

Multi-steps and multi-disciplinary process

. Fast-running

. Low order

. Easy to linearize – possibility to employ the analysis tool of LTI systems

. ODE based

. Comprehensive behaviour of the system

. Possibility to couple with the control system simulator

Different purposes: «to provide insight and understanding in the operational characteristics, reactivity control systems, safety systems, and response to transients and accident situations for a variety of common nuclear power plants.”

Common requirements: PC-based = fast running & ODE based provide insight … = comprehensive behavior of the system

Possible adoption of the same modelling approach

Great potential for plant simplifications and higher efficiencies, introducing additional safety concerns and design challenges – Innovative reactor concept

Need to develop specific simulation tools allow improving control system design – Different reactor, different control strategies

Control-oriented simulators demand specific desiderata to fulfil

Modelling options Data-driven | needs input-output info

Model-based (first principles) | requires physics knowledge

• Modularity • Openness • Efficiency • Multi-physics domain

• Hierarchical structure + Inheritance • Abstraction • Encapsulation

1. It is a modelling language designed for the study of engineering system dynamics

2. It is declarative, focussing on what the model should describe, rather than how the model is solved

3. It is equation-based, modelling (DAE systems) in terms of physical/engineering principles

4. Physical modeling, model components can correspond to physical objects in the real world

The Modelica features are also suitable for educational purposes

model Pendulum "Planar Pendulum" constant Real PI = 3.141592653589793; parameter Real m = 1; parameter Real g = 9.81; parameter Real l = 0.5; Real lambda; Real q1(start = 0.5); Real q2(start = 0); Real v1; Real v2;

equation m*der(v1) = -q1*lambda; m*der(v2) = m*g - q2*lambda; der(q1) = v1; der(q2) = v2; 0 = q1^2 + q2^2 - l^2; end Pendulum

equation m*der(v1) = -q1*lambda; m*der(v2) = m*g - q2*lambda; der(q1) = v1; der(q2) = v2; 0 = q1^2 + q2^2 - l^2;

F 𝑥 , 𝑥, 𝑡, 𝑢 = 0 General implicit form of DAE

Explicit ODE 𝑥 = 𝐹(𝑥, 𝑡, 𝑢)

. The input/output declaration have to be established a priori . Equations have to be rewritten in state space representation . Low flexibility and reusability . Block diagram representation (topology altered) . Easy to linearize . Lower computational cost (efficient integration algorithm for ODE)

𝑥 = 𝐹(𝑥, 𝑡, 𝑢) 𝑦 = 𝐺(𝑥, 𝑡, 𝑢)

𝑥 = 𝐴𝑥 + 𝐵𝑢𝑦 = 𝐶𝑥 + 𝐷𝑢

. the equations are not specified with the classic input/output declaration, granting a more flexible and efficient data flow

m*der(v1) = -q1*lambda;

der(v1) = -q1*lambda/m;

m*der(v1) + q1*lambda = 0;

Modelica code

Translator

Analyzer

Optimizer

Code generator

C Compiler

Simulation

. causality remains unspecified as long as equations have to be solved (higher computational cost since the equations have to be symbolically manipulated)

. More realistic description (component-oriented), modularity and possibility of easily reusing previously developed model

Component/device

. Each Icon represents a physical component. (electrical resistance, mechanical device, pump, ...)

Connection

. A connection line represents the actual physical coupling (wire, fluid flow, heat flow, ...)

Connector

. A component consists of connected sub-components (= hierarchical structure) and/or is described by equations.

Schematics

model Resistor "Ideal electrical resistor" extends OnePort; parameter Real R(unit="Ohm") "Resistance"; equation R*i = v; end Resistor;

model Capacitor "Ideal electrical capacitor" extends OnePort; parameter Real C(unit="F") "Capacitance"; equation C*der(v) = i; end Capacitor;

121

11 mmdt

dLL

dt

dA inls

1111211

111 swsCsinlininlls TThDhmhmdt

dPL

dt

dLhhL

dt

dA

1111,21

121111 wppCpoutoutpppppppp TTLhDTTC

dt

dLTTL

dt

dCA

111111111

11211

11 swsCsinwppCpoutw

wwwww TTLhDTTLhDdt

dTLTT

dt

dLCA

Causal approach: Moving-Boundary model

Mass + energy balance for the different region (water, lead, wall)

Acausal approach: Object-oriented model (1-D)

T_ste

am

_sens

T

Water_Pump

w 0 T

Lead_S

ide

Wate

r_S

ide

Int

ExtM

eta

lWall S

wap

Conv_W

ate

r

fluid

sid

e

Conv_Lead

fluid

sid

e

P

Sink

p0h

Lead_Pum

pT

P_sens

p

P

Sink1

p0h

T_le

ad_sens

T

T_Steam

G_Pb

T_Pb_in

T_w ater_in

G_w ater

Pressure

T_Pb_out

Graphical editor for Modelica models

Modelica simulation environment (free or commercial)

Translation of Modelica models in C-Code, Simulation, and interactive scripting (plot, freq. resp., ...)

Modelica Simulation-environment (free or commercial)

Textual description on file (equations, "schematic", animation)

Free Modelica language

Flexible simulation environment • Modelling – from core to grid • Simulation – scenario analysis • Control – feedback control, discrete event

system

natural choice for the system modelling in control field

. Modelling language

. Equation based

. Object-oriented

. Declarative

. Acausal approach

. Physical modelling/component-oriented

. Free (OpenModelica) or Commercial (Dymola)

. Graphical editor

. Translation of Modelica code in C-code, simulation and post-processing

Improve of the educational simulator in reference to user interaction, model openness,

modularity, efficiency

Main models: . Turbine . SG . Core (Neutronics and T/H) . Pool

Nuclear components created ad hoc, conventional ones taken from ThermoPower library [3]

Ref. R. Ponciroli et al., 2014. Object-Oriented modelling and simulation for the ALFRED dynamics. Prog. Nucl. Energ., 71, 15-29.

CORE

SG

Three main subsystem: . Kinetics (point reactor Kinetics) . FuelRods (1D Heat Transfer) . LeadTube (1D Heat Transfer)

Connectors represent lead flow and temperature/power information

Point reactor kinetics (one neutron energy group and 8 delayed precursor groups) + possibility to represent the subcritical mode

Essential for describing reactor behavior during start-up: . Insertion of neutron source in a subcritical system; . Effect of a reactivity ramp in a subcritical system; . Neutron kinetics in the first phase of criticality when no temperature feedbacks are present

equation RhoTot = Rho + RhoExt.Rho der(N) = sum(KP.beta)/KP.LAMBDA*((RhoTot - 1)*N + KP.beta/sum(KP.beta) *D) + S for i in 1:KP.NPG loop der(D[i]) = KP.lambda[i]*(N - D[i]) end for;

𝑑𝑛

𝑑𝑡=

𝜌 − 𝛽

Λ𝑛 + 𝜆𝑖𝑐𝑖

8

𝑖=1

+ 𝑞

𝑑𝑐𝑖𝑑𝑡

=𝛽𝑖

Λ𝑛 − 𝜆𝑖𝑐𝑖 𝑖 = 1,… , 8

0,0,

1

2

0,0 ln1.1)( llWZCCCZD

f

D

f

DllL TTTTT

TKTTt

0,,,0,0,0, inlinlDiaCCFZllWRCCCR TTTTTTTT

SR

SRSR

SRCRCRCRCRCRoutloutlPadL

xhADChBATT

sin0,,,

. Doppler effect (effective Doppler temp)

. Lead expansion

. Axial and radial clad expansion

. Axial and radial wrapper expansion

. Axial fuel expansion

. Diagrid and pad (flowering) expansion

. Ad hoc formulation for the user reactivity (CR and SR)

𝑑𝑓𝑐𝑓𝜕𝑇𝑓

𝜕𝑡=

1

𝑟

𝜕

𝜕𝑟 𝑟𝑘𝑓

𝜕𝑇𝑓

𝜕𝑟 + 𝑞′′′

𝜕

𝜕𝑟 𝑟𝑘𝑔

𝜕𝑇𝑔

𝜕𝑟 = 0

𝑑𝑐𝑐𝑐𝜕𝑇𝑐𝜕𝑡

=1

𝑟

𝜕

𝜕𝑟 𝑟𝑘𝑐

𝜕𝑇𝑐𝜕𝑟

The model describes the thermal behaviour of the fuel pins

Only the radial heat transfer has been considered, thus disregarding both the axial and the circumferential thermal diffusion

,

,

Time-dependent Fourier equation discretized in five radial regions (cladding, gaseous gap and three fuel zones) and N axial nodes

𝐴𝜕𝑑

𝜕𝑡+

𝜕𝑤

𝜕𝑥= 0

𝜕𝑤

𝜕𝑡+ 𝐴

𝜕𝑝

𝜕𝑥+ 𝑑𝑔𝐴

𝜕𝑧

𝜕𝑥+

𝐶𝑓𝜔

2𝑑𝐴2 𝑤 𝑤 = 0

𝑑𝐴𝜕ℎ

𝜕𝑡+ 𝑑𝐴𝑢

𝜕ℎ

𝜕𝑥− 𝐴

𝜕𝑝

𝜕𝑡= 𝜔𝜑

The model describes the coolant through the core channels

1-dimensional single-phase fluid flow with heat transfer from the fuel boundary

Distributed-parameter mass, momentum and energy conservation equations discretized by employing a finite volume method.

Distributed pressure drop in each section is accounted by Blausius correlation

Dummy assemblies considered to reproduce a correct pressure field during start up

Bayonet-tube design

1-D description of the actual geometry reproduced by means of different tube models connected together (reusability + inheritance)

The first part of water flow in the SG has been neglected

Concentric tube bundles in a counter-current flow configuration, pressure drops concentrated at the bayonet bottom

A two-phase homogeneous model for water, single phase fluid for lead

. Hot and cold pool – free-surface cylindrical lead tank (thermal inertia & pressure)

. Hot and cold legs – LeadTube components (time delay due to transport)

. Lead pump – ideal flow rate regulator

. Attemperator – reduced water mass flow rate at saturation conditions

. Turbine – two stages (HP & LP), choke flow condition, turbine admission valve . Bypass – water flow directly disposed to the condenser

ppdAw vcvv )(

G_att

plant_Entire_StartUp_Subcritical_Heat_370_h

kv

ByPass

G_Pb

h_CR

h_SR

S

h_S_in

G_w ater

T_cold_leg

T_Steam

Pressure

Th_Pow er

G_ByPass

T_hot_leg

G_Turbine

Reactivity

Simulation of 2500 s requiring a computational time of less than 30 s (2.20 GHz with 8 GB memory)

SG pressure

SG outlet temperature

Reactivity

Thermal

Power

Average fuel temperature

Steam temperature

Simulation of 2500 s requiring a computational time of less than 30 s (2.20 GHz with 8 GB memory)

SG pressure

SG outlet temperature

Reactivity

Thermal

Power

Core outlet temperature

Steam temperature

Nuclear components created ad hoc, conventional ones taken from ThermoPower library

. Core

Kinetics (Point-reactor Kinetics), FuelRods, LeadTube (1D description)

. Steam Generator

1D description, two-phase homogeneous model for water, single phase for lead

. Primary loop model

Hot and cold pool, hot and cold legs, lead pump

. Secondary loop model

Attemperator, turbine, bypass

Simulation of 2500 s requiring a computational time of less than 30 s (2.20 GHz with 8 GB memory) – Fast-running & PC-based

The model is used in the decentralized control scheme finalization for: . Pairing selection (on linearized model) . PID tuning (on linearized model) . Test the performance control schemes (on non-linear model)

Ref: R. Ponciroli et al., 2014. “A preliminary approach to the ALFRED reactor control strategy”. Prog. Nucl. Energ., 73, 113-128.

)()()(

)()()(

tuBtxAtx

tuDtxCty

Relative Gain Array Algorithm

Control/Controlled variable pairing T_feed G_water h_CR G_Pb kv

T_steam 0.4169 0.0082 0.1729 0.0274 -0.0006

T_fuel 0.0478 0.0003 0.2683 -0.0008 -0.0002

Pressure 0.0000 -0.0021 -0.0000 -0.0000 0.9989

G_turbine -0.0000 0.9986 -0.0000 -0.0000 -0.0000

T_cold_leg 0.1597 -0.0019 0.0741 0.5911 0.0007

Th_power 0.2757 -0.0007 0.4267 -0.0018 0.0004

T_hot_leg 0.1000 -0.0024 0.0581 0.3841 0.0009

The analysis tools of Linear Time-Invariant systems can be applied

Control loop Controller parameters Controller performance

Controlled variable Control variable Kp Ki Phase Margin [°] Cut-off frequency [rad s-1]

T_cold_leg [°C] G_water [kg s-1] -2 -1·10-2 99 3.370·10-3

Th_power [W] h_CR [cm] -2·10-11 -4·10-11 110 3.323·10-3

Pressure [Pa] kv [-] -3·10-7 -1·10-8 104 0.5418

T_steam [°C] G_att [kg s-1] -0.1 -5·10-2 93 0.0833

It is possible to export the Dymola model in Simulink

Ref. R. Ponciroli et al., 2014. Development of the ALFRED reactor full power mode control system. Prog. Nucl. Energ. (submitted).

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

Cold leg

temperature

Thermal power

Steam temperature

SG pressure

(a)

(b)

Thermal power

(c)

(d)

Steam temperature

(a)

(b)

Cold leg

temperature

SG pressure

Synchronized Petri Net for ALFRED reactor

start-up sequence

Place description

Event description

Ref: R. Ponciroli et al., 2013. Petri net approach for a Lead-cooled Fast Reactor startup design, International Conference on Fast

Reactors and Related Fuel Cycles: Safe Technologies and Sustainable Scenarios (FR13), Paris, 4-7 March 2013.

Thermal power

Reactivity

Cold leg

temperature

SG pressure

Hot leg

temperature

Steam temperature

CRs position

SRs position

Feedwater mass flow

rate

Feedwater temperature

.Logic control – Master system Ensure that sequences of activities are carried out according to the occurrence of certain events

. Modulating control – Slave system Force the controlled variables to follow the corresponding set-points.

Adoption of the ALFRED reactor within the electrical grid

Present electrical grid: not only predictable time demands, but also demand arising from discontinuous power supply due to RES.

Restoring the power balance so as to stabilize the frequency on the grid. The aim is to limit the frequency deviation from the nominal value.

UCTE requirements: Max deviation: ± 800mHz Time constants: 30 s

54.81 𝑀𝑊𝑒𝑙 ; 132.27 𝑀𝑊𝑒𝑙

The gap between the generated electric power and the power absorbed by the loads determines the variation of the rotation speed of the turbo-generator and

therefore the frequency of the grid,

Frequency profile in the synchronous grid of Continental Europe.

. Pairing selection – RGA algorithm

. Controller tuning – grating the asymptotic robust stability

. Test the performance control schemes – on non-linear model

. Petri net approach

. Logic control

. Simulation of ALFRED in the electric grid with renewable plants and the different control strategy

This modelling approach is suitable “to provide insight and understanding” of the controlled behavior of the reactor which can be of interest for educational purposes

Neutronics usually modeled by point-wise kinetics

. No info on spatial distribution of the flux . Simplified evaluation of the temperature feedbacks (constant coefficient) . No possibility of exploiting all the capacities of advanced control schemes

taking into account the temperature and flux spatial distribution

0D modelling

NEW!

NEW!

NEW!

NEW!

NEW!

NEW!

NEW!

NEW!

NEW!

NEW!

NEW!

NEW!

NEW!

NEW!

NEW!

NEW!

NEW!

NEW!

𝑑𝑛

𝑑𝑡=

𝜌 − 𝛽

Λ𝑛 + 𝜆𝑖𝑐𝑖

8

𝑖=1

+ 𝑞

𝑑𝑐𝑖𝑑𝑡

=𝛽𝑖

Λ𝑛 − 𝜆𝑖𝑐𝑖 𝑖 = 1,… , 8

0-D description of the main

physics of a nuclear reactor

Classic approach in

control-oriented simulator

Neutron flux spatial dependence as sum of the

eigenfunctions of the neutron diffusion equation (PDE)

𝑉−1𝜕𝜙

𝜕𝑡= 𝛻 ∙ 𝐷𝛻𝜙 − Σ𝑎𝜙 − Σ𝑠𝜙 + 1 − 𝛽 𝜒𝑝𝐹

𝑇𝜙 + 𝜆𝑚𝜒𝑑𝐶𝑚

𝑗

ℒ𝜓 =1

𝑘 ℳ𝜓 ℒ = 𝛻 ∙ 𝐷𝛻 − Σ𝑎 − Σ𝑠 ℳ=𝜒𝑝𝐹

𝑇

𝜙 = 𝑑𝑖𝑎𝑔 𝜓 𝑟

𝑁

𝑛 𝑡

The dynamics behaviour of the neutron flux is reduced to the study of the

time-dependent coefficient (related to each eigenfunction), and it can be

represented by a set of Ordinary Differential Equations (ODEs)

• Temperature feedbacks (or perturbations ) are locally calculated improving

the model accuracy

• Monitoring of the spatial distribution of the flux improving the model detail

𝑉 𝑚 ∙ 𝑛 𝑚

𝑁

𝑚=1

= −𝐿 𝑚 − 𝛿𝐿 𝑚 + 1 − 𝛽 ∙ 𝑀 𝑚 + 𝛿𝑀 𝑚

𝑁

𝑚=1

∙ 𝑛𝑚 + 𝜆𝑗𝑐 𝑗

8

𝑗=1

𝑐 𝑗 = 𝛽𝑗𝑋 𝑀 𝑚 + 𝛿𝑀 𝑚 ∙ 𝑛𝑚

𝑁

𝑚=1

− 𝜆𝑗𝑐 𝑗 𝑗 = 1 ÷ 8

𝐿 𝑚 = 𝜓 † ∙ 𝐿𝜓𝑚𝑑𝛺 𝑀 𝑚 = 𝜓

† ∙ 𝑀𝜓𝑚𝑑𝛺 𝑉 𝑚 = 𝜓 † ∙ 𝑉−1 ∙ 𝜓𝑚𝑑𝛺

e.g., possibility to develop an optimal control of the control rod movement

𝜙 = 𝑑𝑖𝑎𝑔 𝜓 𝑟

𝑁

𝑛 𝑡

• Eigenfunctions and adjoint problem eigenfunctions (# 10)

• 6 energy groups

• 7 radial coarse zones: lead, clad, gap, 3 x fuel (inner, central, outer), central void

• 10 axial slices

• Temperature dependence of the cross sections (from SERPENT) for each coarse

zone and slice

• Calculations of the integrals

Σ 𝑇, 𝜌 =𝜌

𝜌0Σ0 + 𝛼 ∙ 𝑙𝑜𝑔

𝑇

𝑇0

𝛿𝐿 𝑚 𝑑, 𝑇 = 𝜓 † ∙ 𝛿𝐿 𝑑, 𝑇 𝜓𝑚𝑑𝛺 =

= 𝐷 𝑑𝑧, 𝑇𝑧 𝛻𝜓 † ∙ 𝛻𝜓𝑚 𝑑𝛺𝑧 + 𝛾𝑟 𝜓

† ∙ 𝜓𝑚𝑑𝑆

𝜕𝛺𝑟

+ 𝛾𝑎 𝜓 † ∙ 𝜓𝑚𝑑𝑆 +

𝜕𝛺𝑎𝑧

+𝛴𝑎 𝑑𝑧, 𝑇𝑧 𝜓 † ∙ 𝜓𝑚 𝑑𝛺𝑧 + 𝛴𝑠 𝑑𝑧, 𝑇𝑧 𝜓

† ∙ 𝜓𝑚 𝑑𝛺𝑧

Ref: S. Lorenzi et al., 2014. Development of a spatial neutronics model for control-oriented dynamics simulation. Proceedings of the

2014 22nd International Conference on Nuclear Engineering (ICONE22), July 7-11, 2014, Prague, Czech Republic.

(A) (B)

Reactivity comparison

1. Uniform temperature decrease, ∆Tf1= ∆Tf2 = ∆Tf3= ∆Tl=-50 K; 2. Temperature enhancement in a single pin and in the 5th axial slice, i.e.,

∆Tf1=+400 K, ∆Tf2=+300 K, ∆Tf3=+200 K, ∆Tl=+100 K; 3. Shutdown scenario: all the temperature are set equal to the inlet lead

temperature, i.e., T=673.15 K.

From stationary case:

Neutronics modeling approach Reactivity inserted (pcm)

Case 1 Case 2 Case 3 Neutron diffusion PDE (reference) 78.4 -25.5 1267.6 Modal Neutronics Model (MNM) 78.8 -25.6 1268.9 Point Kinetics (PK) 81 -18.9 1363.8

Transient comparison – Increase of the inlet lead temperature (20°C)

0 10 20 30 40 5040

40.5

41

41.5

Time (s)

Pin

pow

er

(kW

)

Neutron diffusion PDE modelModal neutronics model

0 10 20 30 40 50-5

-4

-3

-2

-1

0

1

2

Time (s)

Reactivity (

pcm

)

0 10 20 30 40 50-5

0

5

10

15

20

Time(s)

Tem

pera

ture

vari

ation (

°C)

Inner fuel

Central fuel

Outer fuel

Lead

Neutron diffusion PDE model Modal neutronics model

Neutronics modeling approach Computational time

Neutron diffusion PDE (reference) 40 h

Modal Neutronics Model (MNM) N=1 N=3 N=5 N=10

15 s 25 s 45 s 200 s

The MNM run with a laptop (2.20 GHz, 8 GB RAM). The neutron diffusion PDE solved with a workstation (8 x 2.8 GHz, 64 GB RAM).

• ODEs system

der(N) = (A1() + Pert(T))*N + A2()*C;

der(C) = A3()*N + A4()*C;

• Thermal feedback

Pert=Coeff_T_F(T_fuel,T_fuel_ref)*A1FT()+Coeff_T_L(T_lead,T_lead_ref)*A1PLT()

• C library for the matrices

(calculated through Matlab script)

A1 [60,60]; A2 [60,480]; A3 [480,60]; A4 [480,480]; Pert [420,60,60]

• 6 energy groups, 8 precursors group, 3

pin, 4 radial coarse zones, 10 axial slices

• Pin model improved

. FA homogenization

. Modal approach not sufficient for Control Rod movement

Reduction order technique aimed at using low dimensional approximations of high dimensional system according to max energy/info criterion.

taking into account the temperature and flux spatial distribution

Modal approach: Neutron flux spatial dependence as sum of the eigenfunctions of the neutron diffusion equation (PDE)

Proper Orthogonal Decomposition: Reduction order technique aimed at using low dimensional approximations of high dimensional system according to max energy/info criterion.

Approach suitable for advanced 3D models that potentially could be implemented in PC-based educational simulators

An object-oriented simulation tool for the analysis of the ALFRED plant

behaviour has been developed

The Modelica language has been adopted due to its modelling feature and the control simulator desiderata

The model has been adopted as analysis tool in all the several step of

the control system design

Further improvement are foreseen in order to enhance the accuracy of the

model, i.e. Spatial Neutronics

An equation based object-oriented approach could help to improve user interaction, model openness, modularity, efficiency

Suitable for advanced 3D models in PC-based simulators

“to provide insight and understanding” of the controlled reactor behavior

. Cammi, A. et al., 2005. Object-oriented modelling, simulation and control of IRIS nuclear power plant with Modelica. Proceedings of the 4th International Modelica Conference, Hamburg, Germany, March 7-8, 2005. . Fritzson, P. Principles of Object-Oriented Modeling and Simulation with Modelica 2.1. Wiley-IEEE Press, 2004. . http://www.modelica.org.

[1] P. Fritzson, 2011. Modelica – A Cyber-Physical Modeling Language and the OpenModelica Environment. In proceeding of: Proceedings of the 7th International Wireless Communications and Mobile Computing Conference, IWCMC 2011, Istanbul, Turkey, 4-8 July, 2011 [2] L. Petzold, 1982. Differential/Algebraic Equations are not ODE’s. J. ScI. STAT. COMPUT. Vol. 3, No. 3, 1982 [3] Casella, F., Leva, A., 2006. Modeling of thermo-hydraulic power generation processes using Modelica. Math. Comp. Model. Dyn. Syst. 12 (1), 19-33.

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Object-oriented and Modelica

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Modal Method and POD

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The example of Slide 15 is taken from the slides of the course “Tecniche e strumenti di simulazione held“ by Prof. G. Ferretti, Politecnico di Milano. Slide 19 and 24 are taken from the slides of the presentation “Modelica Overview“ , available at www.modelica.org. Copyright © 2005-2009, Martin Otter. The material is provided "as is" without any warranty. It is licensed under the CC-BY-SA (Creative Commons Attribution-Sharealike 3.0 Unported) License, see http://creativecommons.org/licenses/by-sa/3.0/legalcode. The example of Slide 21 is taken from “Modelica tutorial“, ver 1.4, available at www.modelica.org.