- Email: [email protected]
Final Comments by the Authors
119
Discussion on: “Nuclear Reactor Modeling and Identification”
Here, N ∗ and Tm∗ are desired set-points for N and Tm , respectively; τ is some arbitrary time constant that may serve as a tuning parameter for the controller. Inserting the feedback laws into the equations (15) and (18) of the state space model leads to the following equations of the closed loop system: 1 ∗ dN = N −N dt τ dTm 1 ∗ Tm − Tm = dt τ This kind of control decouples the outputs N and Tm from the rest of the system. It can only be successful, if the remaining internal system states, i.e. the states C, Tf , I, and X, form a stable system. In the case of the nuclear reactor, the zero dynamics [2] is a linear time-variant system with a system matrix A given as ⎞ ⎛ 0 0 0 −λC ⎟ ⎜ ⎟ ⎜ 0 −A1 0 0 ⎟ ⎜ ⎟ ⎜ A=⎜ ⎟ 0 −λI 0 ⎟ ⎜ 0 ⎠ ⎝ N − λX + σX N 0 0 0 λI 0
If N is at the set-point, then the zero dynamics is obviously stable, ∗witheigenvalues −λC , −A1 , −λI , and − λX + σX N N0 0 . This result suggests minimum phase behaviour of the system, which is of course convenient for controller design.
Figure 2 contains a simulation of the nuclear reactor with input output linearisation, where τ is set to 20 min. The response to a step change of the set-point N ∗ is shown. One can see from the top diagram that N has the desired linear behaviour and reaches the new steady state after about 2 hours. However, the internal states are far from stationary at that time, as is illustrated by the lowest diagram (note the different time scales in the diagrams). The zero dynamics is stable but slow. It needs a very long time to settle after a step change. This may be explained by the large time constants 1/λI and 1/λX of the iodine and the xenon concentration, which are between 10 h and 13 h.
4. Conclusions The nuclear reactor model proposed by Gábor et al. shows a rich and interesting dynamic behaviour. The system possesses ignited steady states that only exist for a subset of possible operation conditions. Stable ignited steady states coexist with unstable ones. The reactor model has very slow internal dynamics. Apparently, disturbances affect the system for many hours. In summary, the presented model seems to be a good basis for future control studies.
References 1. Mangold M, Kienle A, Mohl KD, Gilles ED. Nonlinear computation in DIVA – methods and applications. Chem Eng Sci, 55:441–454, 2000. 2. Isidori A. Nonlinear Control Systems. Springer-Verlag, Berlin, 1989.
Final Comments by the Authors A. Gábor, C. Fazekas∗, G. Szederkényi, K.M. Hangos Process Control Research Group, Computer and Automation Research Institute, Hungarian Academy of Sciences, Budapest, Hungary
1. Introduction The discussion by M. Mangold on our paper “Modeling and Identification of a Nuclear Reactor with Temperature Effects and Xenon Poisoning” provides an interesting control-oriented dynamic analysis of our nonlinear statespace model, that nicely complements our understanding of the dynamics based on first engineering principles, and could truly serve as a starting point for future controller design. ∗ E-mail: [email protected]
At the same time, the physical properties of the investigated nuclear reactor itself and that of the entire nuclear power plant put substantial constraints on the scope and validity domain of the dynamic analysis, and determine the possibilities for future controller design.
2. The Validity Domain of the Model and Other Important Operating Conditions The investigated nuclear reactor is part of the primary circuit of an operating block at the Paks Nuclear Power Plant
120
Discussion on: “Nuclear Reactor Modeling and Identification” PSG
Valve positiont Pressurizer: Pressure controller
Steam 46 bar, 260˚C 450 t/h, 0,25%
Pressurizer, PR 123bar 325˚C
Heating power
lSG Steam generator SG
lPR
mSG,SW
mSG,SW
Base signal
Steam generator: Level controller
PPR Valve position
297˚C
N TPC,HL
Reactor, R
Main hidraulic pump
v Reactorpower controller
TPC,CL
Inlet secundary water 222˚C
220 – 230˚C
Steam generator (SG) Level controller. mSG,SV : Outlet sec.steam mass flow lSG : Water level in SG mSG,SW : Inlet sec.water mass flow Pressurizer (PR) Pressure controller PPR: Steam pressure in PR
Pressurizer: Level controller
Base Correction signal signal
Pressurizer (PR )Level controller
Preheater
TPC,HL : Temperature in hotleg TPC,CL : Temperature in coldleg lPR : Water level in pressurizer Reactor(R) power controller N: Neutronflux PSG: Pressure in steam generator v: Rod position
Fig. 1. The flowsheet of the primary circuit.
in Hungary, the flowsheet of which is depicted in Fig. 1. Besides of the nuclear reactor, one has a pressurizer, that keeps the high pressure of this pressurized water nuclear power plant, and a steam generator, that transfers the energy to the turbines as the main operating units in this plant. The system is operating under closed loop conditions, the most important control loops are also shown by using double rectangles. In our earlier work (see [2, 1]) a simple dynamic model of the whole primary circuit has been developed and its parameters have been identified using measured data of normal load transients in the range of 80%–100% nominal power. The nuclear reactor model with temperature effects and xenon poisoning is an improved version of the reactor model there, but its validity domain is the same. Because of physical reasons, the model cannot describe the nonlinear dynamic behavior outside this range, thus neither the “trivial extinguished state” nor the unstable steady state can be reliably described by our model. Because of the operating conditions of the primary circuit, only one of the possible input variables, the
rod position z is really manipulable (the reactor power controller actually really manipulates this) but the cold leg temperature Tin is determined by the amount of the energy transferred to the turbines through the steam generator, therefore it is not a real input variable available for controller design.
3. Comments on the Steady State and Zero Dynamics Analysis Steady State Analysis. As we have noted before, the unstable ignited non-trivial steady-states correspond to less than 80% neutron flux (or power) value, thus these states do not belong to the validity domain of the reactor. In addition, this reactor is the power source of a power plant, so it makes little sense to stabilize a low power unstable steady state anyway. We also note, that the operating values of the inlet temperature are in fact Tin ≈ 265◦ C. Furthermore, the effect of the rod on the reactivity is not described really accurately with our quadratic function above 0.1 m, this also
121
Discussion on: “Nuclear Reactor Modeling and Identification” Rod position 0.5 0.4 0.3
Position [m]
0.2 0.1 0
0
50
100
150 time [min]
200
250
300
200
250
300
250
300
Neutron Flux 105
100
Flux [%]
95
90
85
80
75 0
50
100
150 time [min]
Average temperature of the moderator 281 280
Temperature [˚C]
279 278 277 276 275 274 0
50
Fig. 2. The measured signals during load transient.
100
150 time [min]
200
122
may explain the mismatch between the measured data and the steady state analysis. Zero Dynamics Analysis. Because of the relation of the reactor with the primary circuit, one needs to consider N and Tm as output variables, but only z is available as input variable, Tin can be regarded as measurable disturbance. Because of the two orders of magnitude difference in the time constants in the model, the control system should focus on the dynamics of the neutron flux N that is in the order of seconds and on the time constants of the temperatures, that are in the range of minutes. The effect of xenon causes a slow dynamic component that is taken as a slowly varying disturbance by the controller(s). Therefore, the controller that is proposed in the discussion paper is not realistic from the viewpoint of real operations. Fig. 2 shows the measured data of a load transient, that is roughly a unit step change in power from 80% to 100% under closed loop conditions using the reactor
Discussion on: “Nuclear Reactor Modeling and Identification”
power controller and the turbine controller of the plant. It can be seen that the whole transient takes approximately 30 minutes. The slow drift after the change is caused by the poisoning that is also compensated slowly by the power controller. It is important to note that the load changes between the day and night operating conditions occur roughly in every 12 hours, so there is no possibility to change the rod position so slowly that a steady state in the xenon concentration is achieved.
References 1. Fazekas C, Szederkényi G, Hangos KM. Parameter estimation of a simple primary circuit model of a VVER plant. IEEE Trans Nucl Sci, 55(5):2643–2653, 2008. 2. Fazekas C, Szederkényi G, Hangos KM. A simple dynamic model of the primary circuit in VVER plants for controller design purposes. Nucl Eng Design, 237:1071–1087, 2007.
Discussion on: “Nuclear Reactor Modeling and Identification”
Here, N ∗ and Tm∗ are desired set-points for N and Tm , respectively; τ is some arbitrary time constant that may serve as a tuning parameter for the controller. Inserting the feedback laws into the equations (15) and (18) of the state space model leads to the following equations of the closed loop system: 1 ∗ dN = N −N dt τ dTm 1 ∗ Tm − Tm = dt τ This kind of control decouples the outputs N and Tm from the rest of the system. It can only be successful, if the remaining internal system states, i.e. the states C, Tf , I, and X, form a stable system. In the case of the nuclear reactor, the zero dynamics [2] is a linear time-variant system with a system matrix A given as ⎞ ⎛ 0 0 0 −λC ⎟ ⎜ ⎟ ⎜ 0 −A1 0 0 ⎟ ⎜ ⎟ ⎜ A=⎜ ⎟ 0 −λI 0 ⎟ ⎜ 0 ⎠ ⎝ N − λX + σX N 0 0 0 λI 0
If N is at the set-point, then the zero dynamics is obviously stable, ∗witheigenvalues −λC , −A1 , −λI , and − λX + σX N N0 0 . This result suggests minimum phase behaviour of the system, which is of course convenient for controller design.
Figure 2 contains a simulation of the nuclear reactor with input output linearisation, where τ is set to 20 min. The response to a step change of the set-point N ∗ is shown. One can see from the top diagram that N has the desired linear behaviour and reaches the new steady state after about 2 hours. However, the internal states are far from stationary at that time, as is illustrated by the lowest diagram (note the different time scales in the diagrams). The zero dynamics is stable but slow. It needs a very long time to settle after a step change. This may be explained by the large time constants 1/λI and 1/λX of the iodine and the xenon concentration, which are between 10 h and 13 h.
4. Conclusions The nuclear reactor model proposed by Gábor et al. shows a rich and interesting dynamic behaviour. The system possesses ignited steady states that only exist for a subset of possible operation conditions. Stable ignited steady states coexist with unstable ones. The reactor model has very slow internal dynamics. Apparently, disturbances affect the system for many hours. In summary, the presented model seems to be a good basis for future control studies.
References 1. Mangold M, Kienle A, Mohl KD, Gilles ED. Nonlinear computation in DIVA – methods and applications. Chem Eng Sci, 55:441–454, 2000. 2. Isidori A. Nonlinear Control Systems. Springer-Verlag, Berlin, 1989.
Final Comments by the Authors A. Gábor, C. Fazekas∗, G. Szederkényi, K.M. Hangos Process Control Research Group, Computer and Automation Research Institute, Hungarian Academy of Sciences, Budapest, Hungary
1. Introduction The discussion by M. Mangold on our paper “Modeling and Identification of a Nuclear Reactor with Temperature Effects and Xenon Poisoning” provides an interesting control-oriented dynamic analysis of our nonlinear statespace model, that nicely complements our understanding of the dynamics based on first engineering principles, and could truly serve as a starting point for future controller design. ∗ E-mail: [email protected]
At the same time, the physical properties of the investigated nuclear reactor itself and that of the entire nuclear power plant put substantial constraints on the scope and validity domain of the dynamic analysis, and determine the possibilities for future controller design.
2. The Validity Domain of the Model and Other Important Operating Conditions The investigated nuclear reactor is part of the primary circuit of an operating block at the Paks Nuclear Power Plant
120
Discussion on: “Nuclear Reactor Modeling and Identification” PSG
Valve positiont Pressurizer: Pressure controller
Steam 46 bar, 260˚C 450 t/h, 0,25%
Pressurizer, PR 123bar 325˚C
Heating power
lSG Steam generator SG
lPR
mSG,SW
mSG,SW
Base signal
Steam generator: Level controller
PPR Valve position
297˚C
N TPC,HL
Reactor, R
Main hidraulic pump
v Reactorpower controller
TPC,CL
Inlet secundary water 222˚C
220 – 230˚C
Steam generator (SG) Level controller. mSG,SV : Outlet sec.steam mass flow lSG : Water level in SG mSG,SW : Inlet sec.water mass flow Pressurizer (PR) Pressure controller PPR: Steam pressure in PR
Pressurizer: Level controller
Base Correction signal signal
Pressurizer (PR )Level controller
Preheater
TPC,HL : Temperature in hotleg TPC,CL : Temperature in coldleg lPR : Water level in pressurizer Reactor(R) power controller N: Neutronflux PSG: Pressure in steam generator v: Rod position
Fig. 1. The flowsheet of the primary circuit.
in Hungary, the flowsheet of which is depicted in Fig. 1. Besides of the nuclear reactor, one has a pressurizer, that keeps the high pressure of this pressurized water nuclear power plant, and a steam generator, that transfers the energy to the turbines as the main operating units in this plant. The system is operating under closed loop conditions, the most important control loops are also shown by using double rectangles. In our earlier work (see [2, 1]) a simple dynamic model of the whole primary circuit has been developed and its parameters have been identified using measured data of normal load transients in the range of 80%–100% nominal power. The nuclear reactor model with temperature effects and xenon poisoning is an improved version of the reactor model there, but its validity domain is the same. Because of physical reasons, the model cannot describe the nonlinear dynamic behavior outside this range, thus neither the “trivial extinguished state” nor the unstable steady state can be reliably described by our model. Because of the operating conditions of the primary circuit, only one of the possible input variables, the
rod position z is really manipulable (the reactor power controller actually really manipulates this) but the cold leg temperature Tin is determined by the amount of the energy transferred to the turbines through the steam generator, therefore it is not a real input variable available for controller design.
3. Comments on the Steady State and Zero Dynamics Analysis Steady State Analysis. As we have noted before, the unstable ignited non-trivial steady-states correspond to less than 80% neutron flux (or power) value, thus these states do not belong to the validity domain of the reactor. In addition, this reactor is the power source of a power plant, so it makes little sense to stabilize a low power unstable steady state anyway. We also note, that the operating values of the inlet temperature are in fact Tin ≈ 265◦ C. Furthermore, the effect of the rod on the reactivity is not described really accurately with our quadratic function above 0.1 m, this also
121
Discussion on: “Nuclear Reactor Modeling and Identification” Rod position 0.5 0.4 0.3
Position [m]
0.2 0.1 0
0
50
100
150 time [min]
200
250
300
200
250
300
250
300
Neutron Flux 105
100
Flux [%]
95
90
85
80
75 0
50
100
150 time [min]
Average temperature of the moderator 281 280
Temperature [˚C]
279 278 277 276 275 274 0
50
Fig. 2. The measured signals during load transient.
100
150 time [min]
200
122
may explain the mismatch between the measured data and the steady state analysis. Zero Dynamics Analysis. Because of the relation of the reactor with the primary circuit, one needs to consider N and Tm as output variables, but only z is available as input variable, Tin can be regarded as measurable disturbance. Because of the two orders of magnitude difference in the time constants in the model, the control system should focus on the dynamics of the neutron flux N that is in the order of seconds and on the time constants of the temperatures, that are in the range of minutes. The effect of xenon causes a slow dynamic component that is taken as a slowly varying disturbance by the controller(s). Therefore, the controller that is proposed in the discussion paper is not realistic from the viewpoint of real operations. Fig. 2 shows the measured data of a load transient, that is roughly a unit step change in power from 80% to 100% under closed loop conditions using the reactor
Discussion on: “Nuclear Reactor Modeling and Identification”
power controller and the turbine controller of the plant. It can be seen that the whole transient takes approximately 30 minutes. The slow drift after the change is caused by the poisoning that is also compensated slowly by the power controller. It is important to note that the load changes between the day and night operating conditions occur roughly in every 12 hours, so there is no possibility to change the rod position so slowly that a steady state in the xenon concentration is achieved.
References 1. Fazekas C, Szederkényi G, Hangos KM. Parameter estimation of a simple primary circuit model of a VVER plant. IEEE Trans Nucl Sci, 55(5):2643–2653, 2008. 2. Fazekas C, Szederkényi G, Hangos KM. A simple dynamic model of the primary circuit in VVER plants for controller design purposes. Nucl Eng Design, 237:1071–1087, 2007.