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β Context, pros and cons, the ALFRED reactor
β Modelica language, causal and acausal approach, simulation environment
Core, steam generator, primary and secondary loop models
β Pairing selection, controllers tuning, operational transient simulation, startup and logic control, grid connection
β 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 β 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 β 125 MWe
. Pool reactor β 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 Β°C
Secondary circuit 335-450 Β°C
Need to develop specific simulation tools allow improving the control system design
Lead-cooled Fast Reactor (GenβIV) 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 β 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: Β«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.β
Common requirements: PC-based = fast running & ODE based provide insight β¦ = 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 β Innovative reactor concept
Need to develop specific simulation tools allow improving control system design β 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
β’ Modularity β’ Openness β’ Efficiency β’ Multi-physics domain
β’ Hierarchical structure + Inheritance β’ Abstraction β’ 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 π₯ , π₯, π‘, π’ = 0 General implicit form of DAE
Explicit ODE π₯ = πΉ(π₯, π‘, π’)
. 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)
π₯ = πΉ(π₯, π‘, π’) π¦ = πΊ(π₯, π‘, π’)
π₯ = π΄π₯ + π΅π’π¦ = πΆπ₯ + π·π’
. 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 β’ Modelling β from core to grid β’ Simulation β scenario analysis β’ Control β 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;
ππ
ππ‘=
π β π½
Ξπ + ππππ
8
π=1
+ π
πππππ‘
=π½π
Ξπ β ππππ π = 1,β¦ , 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)
πππππππ
ππ‘=
1
π
π
ππ πππ
πππ
ππ + πβ²β²β²
π
ππ πππ
πππ
ππ = 0
πππππππππ‘
=1
π
π
ππ πππ
πππππ
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
π΄ππ
ππ‘+
ππ€
ππ₯= 0
ππ€
ππ‘+ π΄
ππ
ππ₯+ πππ΄
ππ§
ππ₯+
πΆππ
2ππ΄2 π€ π€ = 0
ππ΄πβ
ππ‘+ ππ΄π’
πβ
ππ₯β π΄
ππ
ππ‘= ππ
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 β free-surface cylindrical lead tank (thermal inertia & pressure)
. Hot and cold legs β LeadTube components (time delay due to transport)
. Lead pump β ideal flow rate regulator
. Attemperator β reduced water mass flow rate at saturation conditions
. Turbine β two stages (HP & LP), choke flow condition, turbine admission valve . Bypass β 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) β 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. βA preliminary approach to the ALFRED reactor control strategyβ. 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 [Β°] Cut-off frequency [rad s-1]
T_cold_leg [Β°C] G_water [kg s-1] -2 -1Β·10-2 99 3.370Β·10-3
Th_power [W] h_CR [cm] -2Β·10-11 -4Β·10-11 110 3.323Β·10-3
Pressure [Pa] kv [-] -3Β·10-7 -1Β·10-8 104 0.5418
T_steam [Β°C] G_att [kg s-1] -0.1 -5Β·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 β Master system Ensure that sequences of activities are carried out according to the occurrence of certain events
. Modulating control β 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: Β± 800mHz Time constants: 30 s
54.81 ππππ ; 132.27 ππππ
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 β RGA algorithm
. Controller tuning β grating the asymptotic robust stability
. Test the performance control schemes β 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 βto provide insight and understandingβ 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!
ππ
ππ‘=
π β π½
Ξπ + ππππ
8
π=1
+ π
πππππ‘
=π½π
Ξπ β ππππ π = 1,β¦ , 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)
πβ1ππ
ππ‘= π» β π·π»π β Ξ£ππ β Ξ£π π + 1 β π½ πππΉ
ππ + πππππΆπ
π
βπ =1
π β³π β = π» β π·π» β Ξ£π β Ξ£π β³=πππΉ
π
π = ππππ π π
π
π π‘
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)
β’ Temperature feedbacks (or perturbations ) are locally calculated improving
the model accuracy
β’ Monitoring of the spatial distribution of the flux improving the model detail
π π β π π
π
π=1
= βπΏ π β πΏπΏ π + 1 β π½ β π π + πΏπ π
π
π=1
β ππ + πππ π
8
π=1
π π = π½ππ π π + πΏπ π β ππ
π
π=1
β πππ π π = 1 Γ· 8
πΏ π = π β β πΏππππΊ π π = π
β β πππππΊ π π = π β β πβ1 β ππππΊ
e.g., possibility to develop an optimal control of the control rod movement
π = ππππ π π
π
π π‘
β’ Eigenfunctions and adjoint problem eigenfunctions (# 10)
β’ 6 energy groups
β’ 7 radial coarse zones: lead, clad, gap, 3 x fuel (inner, central, outer), central void
β’ 10 axial slices
β’ Temperature dependence of the cross sections (from SERPENT) for each coarse
zone and slice
β’ Calculations of the integrals
Ξ£ π, π =π
π0Ξ£0 + πΌ β πππ
π
π0
πΏπΏ π π, π = π β β πΏπΏ π, π ππππΊ =
= π· ππ§, ππ§ π»π β β π»ππ ππΊπ§ + πΎπ π
β β ππππ
ππΊπ
+ πΎπ π β β ππππ +
ππΊππ§
+π΄π ππ§, ππ§ π β β ππ ππΊπ§ + π΄π ππ§, ππ§ π
β β ππ ππΊπ§
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, βTf1= βTf2 = βTf3= βTl=-50 K; 2. Temperature enhancement in a single pin and in the 5th axial slice, i.e.,
βTf1=+400 K, βTf2=+300 K, βTf3=+200 K, βTl=+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 β Increase of the inlet lead temperature (20Β°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 (
Β°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).
β’ ODEs system
der(N) = (A1() + Pert(T))*N + A2()*C;
der(C) = A3()*N + A4()*C;
β’ Thermal feedback
Pert=Coeff_T_F(T_fuel,T_fuel_ref)*A1FT()+Coeff_T_L(T_lead,T_lead_ref)*A1PLT()
β’ 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]
β’ 6 energy groups, 8 precursors group, 3
pin, 4 radial coarse zones, 10 axial slices
β’ 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
βto provide insight and understandingβ 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 β 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 ODEβs. 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.
ALFRED and LEADER . Alemberti, A. et al. 2013. The Lead fast reactor demonstrator (ALFRED) and ELFR design. Proceedings of the International Conference on Fast Reactors and Related Fuel Cycles: Safe Technologies and Sustainable Scenarios (FR 13), Paris, March 4-7, 2013. . http://www.leader-fp7.eu
Object-oriented and Modelica
Control: RGA, controller tuning, Petri-net . Bristol, E.H., 1966. On a new measure of interaction of multivariable process control. IEEE Trans. Auto. Cont. 11, 133e134. . Γ strΓΆm, K.J., HΓ€gglund, T., 1995. PID Controllers: Theory, Design and Tuning. Instrument Society of America, Research Triangle Park, NC, USA.
. F. Li, et al.βHarmonics synthesis method for core flux distribution reconstructionβ. Prog. Nucl. Energ., 31 (4), 369β372, 2001. . R. Miro, et al. βA nodal modal method for the neutron diffusion equation. Application to BWR instabilities analysisβ. Ann. Nucl. Energy, 29 (10), 1171β1194, 2002. . L. Xia, et al. βPerformance evaluation of a 3-D kinetic model for CANDU reactors in a closed-loop environmentβ. Nucl. Eng. Des., 76β 86, 2012. . G. Berkooz, et al. βThe proper orthogonal decomposition in the analysis of turbolent flowsβ. Annu. Rev. Fluid. Mech., 25, 539β575, 1993. . Chatterjee, A., 2000. An introduction to the proper orthogonal decomposition. Current Science 78, 808β817.
Modal Method and POD
Control: RGA, controller tuning, Petri-net . Skogestad, S., Postlethwaite, I., 2005. Multivariable Feedback Control: Analysis and Design. John Wiley and Sons, New York, USA. . Bernard, J.A., 1999. Light water reactor control systems. In:Webster, J.G. (Ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. New York, NY, USA. . Cassandras, C.G., Lafortune, S., 2010. Introduction to discrete event systems. Springer. . Petri, C., 1962. Kommunikation mit Automaten. Ph.D. Thesis, Darmstadt University of Technology, Germany.
The example of Slide 15 is taken from the slides of the course βTecniche e strumenti di simulazione heldβ by Prof. G. Ferretti, Politecnico di Milano. Slide 19 and 24 are taken from the slides of the presentation βModelica Overviewβ , available at www.modelica.org. Copyright Β© 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 βModelica tutorialβ, ver 1.4, available at www.modelica.org.