7. Study of the sweeping phase regulations

7. Study of the sweeping phase regulations

Having defined all the “pieces” of the model (equations that describes the physical phenomena, I&C devices, etc.) it is now possible to build the model on which the study will be carried on.

According to §6.1.1 it is possible to analyze the two regulation separately, so first part of this chapter will deal with the regulation into the RCP volume, and the second part with the regulation of the pressure at the outlet of the phase separator of the RPE vacuum unit (in following RPE pressure regulation for the sake of brevity).

For each regulation the results for the study in the frequency domain and in the time domain will be shown.

7.1.

Reactor Coolant System (RCP) pressure regulation

According to what has been stated in §6.1.1 the system part analyzed in this section is defined in Figure 28. Boundary conditions are going to be provided for the mass extracted from the primary volume (RCP) by the vacuum pump and for pressure upstream the valve CV1 according to Table 5.

Figure 28: RCP pressure regulation

Figure 29 shows the resulting Simulink® model. In green the blocks that contains the equation to model the thermal-hydraulic system and in blue I&C related blocks.

The signal named mass out represents the mass pulled out from the RCP primary volume by means of the vacuum pump and represent a disturbance for the closed loop controller. Various steps have been built by means of a step source block.

The signal named “Set point” is the pressure set point and by substituting the block “Constant” it is possible to drive the model with the required signal.

A “Transport delay” block has been added after the CTRL block to perform sensitivity analysis in function of the delay introduced into the loop by the computational time required by the I&C computers. During all analysis this delay is set to 0 s except when performing the sensitivity study to this parameter.

Table 9 shows main parameters of the I&C used in the Simulink® model. The most relevant are the run time of the valve CV1 and the sampling time of the ADC implemented in the acquisition module because they have an influence on dynamics of the system and an the range of frequencies transmitted to PI controller respectively.

Pressure set-point bara 0.8

Time constant of PT1 s 0.25

Time constant of the ACQ software filter s 1

CV1 total run time s 30

Initial P value [-] 33.4

Initial I value s 3.34

Range of computational delay s 0:1:4; 5:5:30

Sampling period (τ) s 0.4

ω

f

7.1.1. Frequency domain analysis

STABILITY MARGINS

First stage is the evaluation of stability margin for the actual controller configuration.

Following picture shows open and closed loop Bode diagrams together with GM and PM (the yellow segment on the relative Bode diagram) for the reference choice of controller parameters.

Figure 32: RCP Pressure reference configuration – PINK: closed loop Bode from set point to Mes (ACQ Output) – GREEN: Bode plot of PI controller transfer function

From Figure 32 it is possible to note that this set of parameter (P and I) results in a resonance around 5e-3 Hz. This would results in oscillatory behaviour before reaching steady state. Graphs also show the position of poles (x marker) and zeros (o marker).

Table 10 summarizes the results of this analysis.

P 33.4

I 3.34 s-1

Gain margin 51.6 dB at 0.298 Hz Phase margin 16.7 deg

at 5.57e-3 Hz

According to these results it is difficult to satisfy the time requirements of §6.1.2, so new values for P and I have been derived to satisfy frequency domain specification of §6.1.1. Following Bode plots refer to the new optimized configuration.

Figure 34: RCP Pressure optimized configuration – OL Bode plot from set point to Mes (ACQ Output)

Figure 35: RCP Pressure optimized configuration– PINK: CL Bode plot from set point to Mes (ACQ Output) – GREEN: Bode plot of PI controller transfer function

As can be seen from Figure 35 no more resonance is present in the closed loop response of the system and it behaves as a first order system. Table 11 shows the impact of the optimization on stability margins together with the new controller parameters.

P 70

I 0.07 s-1

Gain margin 40.2 dB at 0.0213 Hz Phase margin 85.4 deg

at 0.00386 Hz

Table 11: RCP Pressure optimized configuration - Stability margins

VERIFICATION OF SHANNON’S CONDITION ON SAMPLING FREQUENCY7

Next step is to verify if sample frequency satisfies the condition of the Shannon’s theorem. In fact the control system is a time discrete one and it is important that all the “modes” are correctly sampled without the appearance of aliasing. This can be easily done by the pole – zeros map of the closed loop system. It is evident that this diagram is derived from the continuous system8.

7

Enunciation: a function f(t) with a spectrum bounded by 0 and ωs, that is its Fourier transformed F(jω) is

equal to 0 for ω> ωs, can be rebuilt from the sequence of samples ~f(k)= f(kT)_{, only and only if the sampling} pulsation Ω=2π/T is not smaller than 2ωs

8

The “Control System Tool Box” of the Matlab® R2006b version, does not account for sampling, so also other diagram referring to frequency domain are derived for the continuous system.

Figure 36: RCP Pressure optimized configuration– closed loop pole-zero map

As can be seen in Figure 36 no pole is out of the stripe bounded by ±1.25 Hz (refer to Table 9) and no pole has positive real part and hence the system is stable in a bounded input bounded output sense.

SOME PHYSICAL REMARKS

At this point, concluding the frequency domain analysis of this regulation, it is possible to trace some conclusions about the dynamic behaviour of the TH system involved. It is useful to do so in order to better understand the link between the frequency domain and the time domain but also to better understand what will be the behaviour of the system and what will be its limitations.

The aim of this regulation is to control pressure into a big volume, that one of the RCP, by means of admitting gas trough a DN50 valve (named CV1 in this study). It possible to foreseen that the pressure will rise very slowly compared to percentage of opening of the valve. This finds a counter part in the linearized TH system.

The Bode plot of Figure 37 resembles that one of a first order system with a pole at 2e-6 Hz and so with a time constant of 5e5 sec. In other words the system is very slow due to its geometry.

Figure 37: RCP Pressure - Bode plot of TH system

7.1.2. Time domain analysis

Proposed configuration for controller involved in this closed loop regulation is going to be tested in the time domain to verify the robustness of the control regarding non linearities and parameter variation. Moreover results of the same test for original parameters are reported. Tests involve steps in the following variables at t=5000 s:

̶ -20% of nominal step in mass flow rate to RPE to asses disturbance rejection of the control loop: it is credible that mass flow rate to RPE may only diminish as consequence of operator’s action on TEG admission valve and automatic regulation of pressure downstream the vacuum pump that is at the exit of the vacuum package; ̶ -25% of nominal step in RCP pressure set point to asses dynamic behaviour of the

-20%STEP FROM NOMINAL IN MASS FLOW RATE TO RPE(DISTURBANCE REJECTION)

Figure 38 shows time domain results for this test for the original and optimized configuration. The original configuration shows an oscillatory behaviour and prevents the system from reaching a new steady state condition.

The optimized one satisfies all the requirements shown in Table 12. Even if integral action is present in the controller, steady state error is not zero because of a dead band on process deviation before the input of the controller. This dead band is defined as (MEASURING RANGE)*DEADBAND*0.01 and is modelled with the Simulink® block “backslash”: when input is into the dead band, output is the middle value of the dead band. (See Figure 26).

Figure 38:RCP Pressure (measured by PT1) – step in mass flow rate to RPE

Overshoot 0.89% Ref. Param. Optim. Param Undershoot NA Stable Settling time NA Controlled variable in ±5% SP range Error -0.46% Reach SS

STEP VARIATION FROM 0.8 TO 0.6 BARA IN RCPPRESSURE SET POINT (RESPONSE TO SET

POINT CHANGES)

Figure 39 shows that the original configuration is unstable. With the optimized one the dynamic behaviour is sensibly improved. As soon as the variation in the set point takes place, controller closes the N2 admission valve and consequently pressure starts decreasing only

because of mass extraction by means of the vacuum pump. This phenomenon characterizes the linear part of the response. In the meantime integral section of the controller keeps on integrating the error, “loading its memory”. To bring again the integral error to zero it is necessary that the process deviation change sign and this justifies the undershoot. Exponential trend of last part of the transient is due to the dumping action of the integral part.

Figure 39 also shows the temporal counterpart of Figure 37 confirming the previous analysis.

Overshoot NA Ref. Param.

Optim. Param

Undershoot 22.94% Stable

Settling time 5837 s Controlled variable in ±5% SP range Error steady

state

-0.065%

Reach SS

Table 13: RCP Pressure regulation summary – Test 2

SENSITIVITY ANALYSIS TO COMPUTATIONAL DELAYS

All calculations are performed by a digital processor and data are retrieved/published from/on the plant bus (cfr. §4.3). According to the available bandwidth on the bus and computational load on the processor, a delay can be introduced in the control chain.

At the moment no quantification of potential delays has been performed neither an estimation of their source; it has been assumed that a representative delay exists between the publishing of the order and its reception by the actuator on the valve. In other terms the actuators receive a function that has the following shape:

) ( )

(_t =_{F t}−τ f_{actuator controller}

To have a sound understanding of the performance of the control loop, linearization has been performed for various time delays in order to find GM and PM (see Figure 40). Time domain simulations have been performed for delays of 0; 10, 30 seconds registering how the delays affect system response to a step in mass flow rate to RPE and to a step in the set point.

Figure 40: RCP Pressure - control loop GM and PM for various time delays

Figure 42: RCP Pressure - Tme response to a -25% step in pressure set point for various time delays

Figure 41 and Figure 42 show that the loop can tolerate well delays because of physical system size and inertia. Notwithstanding this a non-oscillatory behaviour is ensured for delays up to 10 seconds.

7.1.3. Conclusions

Proposed regulator parameters allow good disturbance rejection and ensure good dynamics regarding set point changes too, keeping in mind the physical time constant of the TH system. The lack of speed of the system is not a problem: in fact sweeping process is going to be performed at a constant set-point of 0.8 bara which is not going to be changed during the process; on the other hand, no information is provided about time trend of the mass flux pulled out from the vessel. So, for this process, the need of good disturbance rejection prevails on the need of good tracking dynamics.

Regarding delays introduced by I&C processing, results show that a “stable” behaviour (no oscillations) is ensured up to 10 s delay: a threshold that can be easily guaranteed by computer available nowadays.

7.2.

Nuclear Island Vent and Drain (RPE) pressure regulation

According to what has been stated in §6.1.1 the system part analyzed in this section is defined in Figure 28. Boundary conditions are going to be provided for the RCP pressure (that is the pressure downstream the valve BP) and for the mass flow rate discharged to TEG system. This mass flow rate is controlled by the operator from the MCR via the TEG admission valve CV2. Values used for these variables are reported in Table 5.

Figure 43: RPE Pressure regulation

Figure 44 shows the resulting Simulink® model for this section of the global system.

In green the blocks that contains the equation to model the thermal-hydraulic system and in blue I&C related blocks.

The “set-point” signal is the set point of the regulation and can be provided by any source block (in this case it is a step).

The “calculation delay” block is used to simulate the effect on the regulation of computational time; it has been set to zero for the following tests and to a non-zero value to perform sensitivity analysis.

The block named “mTEG” represents the flow rate that is routed toward the TEG and represents a disturbance for the system.

Table 14 shows main parameters of the I&C used in the Simulink® model. The most relevant are the run time of the valve CV1 and the sampling time of the ADC implemented in the acquisition module because they have an influence on dynamics of the system and an the range of frequencies transmitted to PI controller respectively.

Pressure set-point bara 1

Time constant of PT2 s 0.25

Time constant of the ACQ software filter s 1

Sampling period (τ) s 0.4

ω

f

BP total run time s 40

Initial P value [-] 1

Initial I value s 1

Range of the delay s 0:1:4; 5:5:30

7.2.1. Frequency domain analysis

STABILITY MARGINS

Also for this section first stage is the evaluation of stability margin for the actual controller configuration.

Following picture shows open and closed loop Bode diagrams together with GM and PM (the yellow segment on the relative Bode diagram) for the reference choice of controller parameters.

Figure 46: RPE Pressure reference configuration – OL Bode plots from set point to Mes (ACQ Output)

Figure 47: RPE Pressure reference configuration– PINK: CL Bode plot from set point to Mes (ACQ Output) – GREEN: Bode plot of PI controller transfer function

From Figure 47 it is possible to note that this set of parameter (P and I) results in a resonance around 1.5e-1 Hz. This would results in oscillatory behaviour before reaching steady state. Graphs also show the position of poles (x marker) and zeros (o marker).

Table 15 summarizes the results of this analysis.

P 1

I 1 s-1

Gain margin 2.99 dB at 0.151 Hz Phase margin 4.24 deg

at 0.127 Hz

Table 15: RPE Pressure original configuration - Stability margins

According to these results it is difficult to satisfy the criteria exposed at beginning of §6.1.2; moreover poor GM and PM assure no robustness regarding to parameter variation and non linear phenomena (neglected by the linearization tool). So new values for P and I are going to be inspected.

Figure 49: RPE Pressure optimized configuration – OL Bode plots from set point to Mes (ACQ Output)

Figure 50: RPE Pressure optimized configuration– PINK: CL Bode plot from set point to Mes (ACQ Output) – GREEN: Bode plot of PI controller transfer function

As can be seen from Figure 50 no more resonance is present in the closed loop response of the system and it behaves as a first order system. Table 16 shows the impact of the optimization on stability margins together with the new controller parameters.

P 0.2

I 0.06 s-1

Gain margin 30.9 dB at 0.311 Hz Phase margin 68.8 deg

at 0.0268 Hz

Table 16: RPE Pressure optimized configuration - Stability margins

Some considerations can be done about proposed configuration (Figure 48, Figure 49, Figure 50):

̶ Stability margins have been increased;

̶ Bandwidth of the loop has been consequently diminished and so response time increased: system will respond more slowly to perturbations;

̶ System will have a first order like behaviour.

VERIFICATION OF SHANNON’S CONDITION ON SAMPLING FREQUENCY9

For the theoretical needs of this analysis refer to §7.1.1.

As can be seen from Figure 51, even if no pole has positive real part, and hence the system is stable in a bounded input bounded output sense, a pole is out of the stripe bounded by ±1.25 Hz with a pole frequency of 1.61 Hz. In order to make the system stable at all frequencies, the sampling period should be diminished or an anti-aliasing filter (basically a low pass filter) placed before sampling acquired measure. Assuming that the filter cannot be placed in the signal processing chain and the sampling time cannot be increased because of design

9

Enunciation: a function f(t) with a spectrum bounded by 0 and ωs, that is its Fourier transformed F(jω) is equal to 0 for ω> ωs, can be rebuilt from the sequence of samples ( ) ( )

~

kT f k

f = _{, only and only if the sampling} pulsation Ω=2π/T is not smaller than 2ωs

constraints, an other way can be followed: if the system dumps in all operating condition, in spite of parameter variation the frequencies above 1.25 Hz, it would be stable too. This latter condition is summarized by saying that the open loop system should have high gain and phase margins.

Figure 51: RPE Pressure optimized configuration– closed loop pole-zero map

SOME PHYSICAL REMARKS

At the conclusion of this section it is worthy to make the point on the main aspects of this regulation.

The aim of regulation is to control the pressure at the outlet of the phase separator (or in the phase separator according to the assumption of this study) by means of a DN80 control valve, named BP in the present study..

It is possible to envisage that pressure will rise fast due to the dimensions of the separator. This finds a counterpart in the linearized model whose bandwidth reaches frequencies up to 2e-2 Hz. Accordingly, by approximating the response with that one of a first order system, the time constant will be of 50 sec (see Figure 52).

Figure 52: RPE Pressure - Bode plot of TH system

7.2.2. Time domain analysis

Proposed configuration for controller involved in this closed loop regulation is going to be tested in the time domain to verify the robustness of the control regarding non linearities and parameter variation. Moreover results of the same test for original parameters are reported. Tests involve steps in the following variables at t=1500 s:

̶ -20% of nominal step in mass flow rate to TEG to asses disturbance rejection of the control loop: it is credible that mass flow rate to RPE may only diminish as consequence of operator’s action on TEG admission valve;

̶ +20% of nominal step in RPE pressure set point to asses dynamic behaviour of the control loop: it is credible that pressure may only be increased to overcome pressure losses in the pipeline between vacuum package and TEG system.

-20%STEP FROM NOMINAL IN MASS FLOW RATE TO TEG(DISTURBANCE REJECTION)

Figure 53: RPE Pressure (measured by PT2) – step in mass flow rate to TEG

As can be seen from Table 17 both configurations ensure satisfaction of time requirements even if proposed configuration shows a more dumped response (see Figure 53). Pressure does not reach steady state because of the presence of a dead band defined on the process deviation. Overshoot 4.01% Ref. Param. Optim. Param Undershoot NA Stable

Settling time NA Controlled

variable in ±5% SP range Error steady state -0.75% Reach SS

Table 17: RPE Pressure regulation summary – Test 1

STEP VARIATION FROM 1 TO 1.2 BARA IN SEPARATOR PRESSURE SET POINT (RESPONSE TO

As shown by Figure 54 reference configuration is unstable. Moreover to obtain this result the pump has been forced to work in the model by increasing its compression ratio: pump would never have the power to realize such a compression.

The optimized configuration show the typical dumped response of first order system as it was predicted by the CL Bode plot of the system.

Table 18 synthesizes the results of these tests.

Figure 54: RPE Pressure (measured by PT2) – step in pressure set point

Overshoot NA Ref.

Param.

Optim. Param

Undershoot NA Stable

Settling time NA Controlled variable

in ±5% SP range Rise time 32 s (t10%=1504 t90%=1536 s) Reach SS Error steady state 0.62%

Table 18: RPE Pressure regulation summary – Test 2

General aspects of this point already treated in § 7.1.2 also held for the present point.

Since the system is faster and shows little inertia due to its relatively small dimensions, it is more sensible to delays introduced in the loop. This is illustrated in Figure 55 which shows large variation in the GM and PM even for small variation in the delay introduced.

Figure 55: RPE Pressure - control loop GM and PM for various time delays

From Figure 56 together with the aid of Figure 55 it is possible to conclude that the loop is stable for delays up to 5 s. this can be justified by saying that even if PM is lower than 0 deg, there is enough GM to dump the out of phase waves which come back to the controller input. Moreover it is worthy note that with decreasing GM the problem of aliasing is exacerbated.

Figure 56: RPE Pressure - Tme response to a +20% step in pressure set point for various time delays

7.2.3. Conclusions

The optimized configuration assures good set-point tracking performance and good disturbance rejection as well.

Influence of dead band and non-linearities is well borne by the system even if it is relatively fast.

This time margin in computational delay is tighter (always due to the characteristics of the TH system and constraints on the sampling period) in fact “stable” behaviour is ensured for delays up to 5 s. Once again this constraint can be widely satisfied by available processing units.

7.3.

Results for the sweeping by N

As stated in §6.1, it is now possible to re-couple the two sub-systems and inspect the behaviour of the two loops.

This “validation” will be performed in the time domain, testing the system with the same kind of perturbations used in previous chapter to be able to make a comparison.

̶ -25% nominal step in RCP pressure set point Figure 57; ̶ +20% nominal step in RPE pressure set point Figure 58;

̶ -20 nominal step in mass flow rate to TEG Figure 59 and Figure 60.

This time pressure upstream CV1 and mass flow rate to TEG system are the only boundary conditions that must be provided.

Simulation is stopped 10800 seconds after sweeping has started (after 0.8 bara are reached starting from atmospheric pressure).

Figure 57: -25% nominal step in RCP pressure set point – step at t=6000 seconds

Pressure at PT2 Pressure at PT1

Figure 58: +20% nominal step in RPE pressure set point – step at t=8000 seconds

Figure 59: -20% nominal step in mass flow rate to TEG - RCP side

Pressure at PT2 Pressure at PT1

Figure 60: -20% nominal step in mass flow rate to TEG - RPE side

Initial peak in phase separator pressure is due to the temperature of gas coming from RCP: it becomes colder as time passes and moreover it is mixed with gas coming from the phase separator which has been cooled by the pump.

Generally speaking there are no interactions that cause instability. Oscillations at the end of the transient reported in Figure 57 are due to the influence of temperature on pressure into the phase separator.

Moreover an operating procedure can be proposed (it is also the procedure used to simulate the entire process):

̶ CV1 is controlled in manual mode via the HMI of its control block CTRL (see Figure 6);

̶ CV1 is closed;

̶ Vacuum pump is started and BP is controlled in automatic mode;

̶ CV1 control is switched to automatic mode.

This procedure, namely of controlling CV1 first in manual mode and then switching to the automatic, is required because the control loop is designed to operate in a pressure range near 0.8 bara; starting automatic control from atmospheric pressure would cause integrator wind-up. To solve this problem a desaturation algorithm should be implanted in the PI controller.

7.4.

Results for the sweeping by Air

In this phase the entire mass flow rate is routed to EBA as a consequence of the lack of constraints on the incoming mass flow rate for this system. In this condition vacuum unit works under mass equilibrium state (the same amount of gas extracted from the RCP by the pump is discharged to EBA). As a consequence, by pass valve BP is closed and the pressure regulation into RCP is the only closed loop regulation active.

According to this operating condition RCP pressure set point step is the only meaningful test to perform. Step considered in this section is a+25% step: in fact it is reasonable to think that pressure at suction side of the pump needs to be increased in order to overcome pressure losses.

Even if controller has been tuned and tested considering only sweeping by nitrogen, control loop performs well also during sweeping by air: this is a consequence of having provided high stability margins.

From Figure 61 it is possible to note a faster response of the system: this is due to the higher initial pressure with respect to the sweeping by nitrogen and to the higher initial opening of the valve CV1.

References

[1] Areva’s internal documentation