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Steam Generator control in Nuclear Power Plants by water mass inventory
Available online at www.sciencedirect.com
Nuclear Engineering and Design 238 (2008) 859–871
Steam Generator control in Nuclear Power Plants by water mass inventory Wei Dong, J. Michael Doster ∗ , Charles W. Mayo North Carolina State University, Department of Nuclear Engineering, Box 7909, Raleigh, NC 27695-7909, United States Received 15 February 2007; received in revised form 5 September 2007; accepted 10 September 2007
Abstract Control of water mass inventory in Nuclear Steam Generators is important to insure sufficient cooling of the nuclear reactor. Since downcomer water level is measurable, and a reasonable indication of water mass inventory near steady-state, conventional feedwater control system designs attempt to maintain downcomer water level within a relatively narrow operational band. However, downcomer water level can temporarily react in a reverse manner to water mass inventory changes, commonly known as shrink and swell effects. These complications are accentuated during start-up or low power conditions. As a result, automatic or manual control of water level is difficult and can lead to high reactor trip rates. This paper introduces a new feedwater control strategy for Nuclear Steam Generators. The new method directly controls water mass inventory instead of downcomer water level, eliminating complications from shrink and swell all together. However, water mass inventory is not measurable, requiring an online estimator to provide a mass inventory signal based on measurable plant parameters. Since the thermal-hydraulic response of a Steam Generator is highly nonlinear, a linear state-observer is not feasible. In addition, difficulties in obtaining flow regime and density information within the Steam Generator make an estimator based on analytical methods impractical at this time. This work employs a water mass estimator based on feedforward neural networks. By properly choosing and training the neural network, mass signals can be obtained which are suitable for stable, closed-loop water mass inventory control. Theoretical analysis and simulation results show that water mass control can significantly improve the operation and safety of Nuclear Steam Generators. © 2007 Elsevier B.V. All rights reserved.
1. Introduction Control of water mass inventory in Nuclear Steam Generators is important to insure sufficient cooling of the nuclear reactor. Since downcomer water level is a reasonable indicator of water mass inventory at steady-state, or near steady-state conditions, conventional control system designs attempt to maintain downcomer water level between narrow limits in order to provide for sufficient cooling of the reactor, to provide for good performance of the steam separators and dryers, and also to reduce risks of hydrodynamic instability. Failure to maintain the water level within reasonable bounds typically results in a reactor trip. As a result, a good control system for the water level can be a major factor in overall plant availability (Masche, 1971). The water level system in Nuclear Steam Generators is a complex nonlinear system with inverse dynamics. When piecewise
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0029-5493/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2007.09.001
linearized, the system is a MIMO (multi-input–multi-output), variable parameter, non-minimum phase system and good controller designs are difficult to achieve (Choi et al., 1989; Irving and Bihoreaux, 1980). During plant transients, level control is complicated by hydrodynamic effects known as “shrink and swell”. Due to the presence of steam bubbles in the tube bundle region, the water level measured in the downcomer can temporarily react in a reverse manner to water mass inventory changes. This behavior is accentuated during start-up and low power operation. Under these conditions, the only true indication of water mass inventory change is the difference between steam flow and feed water flow. However, at low power levels, flow measurement uncertainty can be too large for these parameters to be used for effective control. It is well known that traditional control schemes do not permit satisfactory automatic level control during start-up/low power conditions. Unsatisfactory performance of automatic level control may either produce a reactor trip or require the operator to initiate manual control. Even for a skilled operator, it can be difficult to react properly to the reverse level indications resulting from shrink and swell
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effects. It has been observed that reactor trips are caused by the operator’s overreaction while restoring water level (Choi et al., 1989). Digital feedwater control systems (Carmichael et al., 1984; SAIC, 1987; Popovic and Hammer, 1987; Zapotocky et al., 1988; Singh et al., 1989) have successfully mitigated some of the stability problems associated with analog control systems, and have been applied to the low power range (Popovic and Hammer, 1987; Zapotocky et al., 1988; Singh et al., 1989). However, these control systems still suffer from the inverse dynamics of shrink and swell and are not minimum phase. Many studies have contributed to level controller design (Choi et al., 1989; Irving and Bihoreaux, 1980; Kuan et al., 1992; Kothare et al., 1996). Conclusions that may be drawn from these previous studies include: • Good controller designs for Nuclear Steam Generators are difficult, although significant progress has been achieved in the period since 1975. With the help of high-speed computer simulation, almost every modern control strategy has been applied to the Nuclear Steam Generator level control problem. • Since the inverse system is non-minimum phase and has unstable dynamics, it is difficult to train a neural network controller via the back-propagation of the system output error (Park et al., 1995). For this reason, artificial neural networks are precluded from the design when water level in Nuclear Steam Generators is chosen as the controlled variable. • Most controller designs treat the plant as a single-input/singleoutput system even though the Steam Generator is in fact a multi-input–multi-output system. Feedwater flow rate is typically taken as the system input, narrow range downcomer water level as the system output, and steam flow rate is treated as a disturbance. The impact of other factors such as feedwater temperature, primary side water temperature and primary side water flow rate on the reverse dynamics are included in one lumped parameter—reactor power level (Irving and Bihoreaux, 1980; Kothare et al., 1996). • At low plant power levels, efforts to develop a good controller are still hindered by the lack of accuracy in steam flow and/or feed flow measurements. It is noted that controllers with less dependence on these flow measurements may be desirable (Irving and Bihoreaux, 1980; Kothare et al., 1996). • A good Steam Generator model (well validated against plant data) is essential for good controller design (Choi et al., 1989; Kuan et al., 1992; Kothare et al., 1996; Dong et al., 2000). Simplified piecewise linearized transfer function models (Irving and Bihoreaux, 1980) are widely used by designers familiar with control theory to develop linear control strategies; while physical models are used by designers familiar with thermodynamics and for fuzzy controller design (Kuan et al., 1992). By changing the controlled variable in Nuclear Steam Generators from water level to water mass inventory, the controlled system becomes a linear system with minimum phase, which simplifies both the control system design as well as manual operation. At the same time, the water mass inventory is a true
measure of the instantaneous cooling capacity of the Nuclear Steam Generator. Combined with a conventional level signal, water mass control provides the opportunity for improved plant operation (Dong, 2001). Unlike the conventional water level control system, the water mass inventory within the Steam Generator is not measurable, and must be inferred from measurable plant variables. Since the thermal-hydraulic response of a Steam Generator is highly nonlinear, a linear state-observer is not feasible. In addition, difficulties in obtaining flow regime and density information within the Steam Generator make an estimator based on analytical methods impractical at this time. In this paper, a water mass estimator based on feedforward neural networks is developed, and shown to be capable of estimating water mass inventory over a broad range of plant operating conditions. The neural network water mass estimator is then embedded in a close-loop water mass control system. Simulation results indicate the mass controller can maintain both Nuclear Steam Generator water mass and water level within operating limits, even when the system is subjected to large load disturbances at low plant power levels. In the following sections, the Steam Generator dynamic model is described, and evaluation of the model is carried out. The details of the new controller design are described and theoretical analysis of the new water mass control system is presented. A water mass estimator based on neural networks is developed and embedded into a Nuclear Power Plant simulation model. A conventional PI controller uses the estimated water mass as input to form a close-loop control system where water mass is the controlled variable, and the performance of the system is evaluated. 2. Steam Generator model A computational model of the Steam Generator is necessary to provide insight into the Steam Generator dynamics and replace actual plant data for developing and evaluating the controller. The dynamic behavior of U-Tube Steam Generators is a function of not only the thermal-hydraulic conditions within the Steam Generator, but also the forcing functions or boundary conditions imposed on the Steam Generator by the remainder of the plant. This implies that a full plant simulation is required to adequately characterize Steam Generator behavior over the full range of operating conditions. A full plant engineering simulation code has been developed at North Carolina State University for simulating the dynamic response of pressurized water reactor systems during normal operational transients as well as design basis events. The model is capable of simulating multi-loop PWR systems with U-Tube Steam Generators. Both primary and secondary sides are represented including automatic control systems, reactor protection systems and engineered safety features. The Steam Generator model is based on a four-equation drift flux model that allows for non-equilibrium treatment of the liquid phase. Level tracking logic is included to allow for simulation of vertical stratified flows and accurate predictions of Steam Generator liquid levels
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Fig. 1. Simulation of “shrink ” following a 10% step reduction in load from 80% power.
(Kim and Doster, 1991). Details of the numerical solution can be found elsewhere (Kim and Doster, 1991, 1995). In addition to this work, this software has served as the simulation platform to develop methods for fault diagnosis and isolation (March-Leuba et al., 2001; Upadhyaya et al., 2001, 2002a,b) and automatic controller optimization (March-Leuba et al., 2001, 2002; Mullen et al., 2002). The plant simulation model is capable of simulating the shrink and swell phenomena as illustrated in Fig. 1 for a step reduction in load (shrink) and Fig. 2 for a step increase in load (swell). As the turbine control valve closes following load reduction, steam pressure increases. This results in bubble collapse in the bundle region and a corresponding draw down of the liquid level in the downcomer. In a conventional level control system, the feedwater control valve will open to increase the feedwater flow rate as the indicated water level decreases, even though the water mass inventory is increasing. Similarly, when the turbine control valves open following an increase in load, the steam pressure decreases resulting in bubble expansion in the bundle region. This expansion forces water back up into the downcomer, increasing downcomer level, even though the mass inventory is decreasing. As shown in Figs. 1 and 2, the water mass in the Steam Generator shows no reverse dynamics. Water mass
increases when the feedwater flow rate is greater than the steam flow rate and decreases when the feedwater flow rate is less than the steam flow rate. 3. Water mass inventory controller 3.1. Conventional water level control system Conventional level control strategies are based on threeelement, proportional-integral (PI) controllers, where the controller inputs are feedwater flow rate, steam flow rate, and downcomer water level. At low plant power levels, flow measurement uncertainties can render the flow measurements unreliable, and a single-element PI controller based on the level error signal is often used. The control action of a three-element proportional-integral controller is described by Eq. (1) in terms of the primary control variables: feedwater flow rate Wfw ; steam flow rate Ws ; and downcomer water level Lw ; (1) Wfw = kp (Aw ew + Al el ) + ki (Aw ew + Al el ) dt where kp = proportional gain; ki = integral gain; ew = (Ws − Wfw ) = flow error (kg/s); el = (Lref −Lw ) = level error (m);
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Fig. 2. Simulation of “swell” following a 10% step increasing in load from 80% power.
Lref = reference (programmed) water level (m); Aw = flow error weight; Al = level error weight. An installed controller may differ in the details but uses the same three variables (Wfw , Ws , and Lw ) to calculate a control error signal. In addition, feedwater flow is often controlled by a combination of pump speed and control valve position using more complex control logic with cascaded PI controllers. The objective of this control strategy is to eliminate the steam generator water level error which is affected by the shrink and swell phenomena. As steam flow decreases because of turbine load reduction, the rate of steam removal from the Steam Generator drops below the rate of steam generation. In a saturated system, this imbalance results in a pressure rise and subsequent collapse of steam bubbles that exist in the tube bundle region. The volume taken up by the liquid/vapor mixture decreases with the collapse of the steam bubbles, causing the indicted downcomer water level to “shrink”. However, the fluid mass in the Steam Generator is actually increasing (feed flow > steam flow). It normally takes several seconds for the water mass in the downcomer to move into the bundle region and to reestablish the force balance between friction and buoyancy forces. When steam flow increases, excess
steam removal results in a decrease in Steam Generator pressure. The decrease in pressure causes expansion of the liquid/vapor mixture in the tube bundle region. This forces water back up into the downcomer, causing the indicted water level to “swell”. However, the fluid mass in the Steam Generator is actually decreasing (feed flow < steam flow). Shrink and swell effects are also observed with increases and decreases in feedwater flow. As the flow rate of relatively cold feedwater increases, the inlet enthalpy into the tube bundle region decreases so that the steam bubbles collapse causing the indicated downcomer water level to “shrink”. As the feedwater flow decreases, the inlet enthalpy into the tube bundle increases, leading to expansion of the steam bubbles, causing the indicated downcomer water level to “swell”. Changes in feedwater temperature, primary side water temperature and primary side flow rate can also cause “shrink” and “swell”. Shrink and swell effects within the downcomer give a reverse level indication that complicates the choice of the proper control action. If these reverse effects are combined with high controller gains in a level controlled system, the system may become unstable. Since shrink and swell do not affect the actual mass inventory in the Steam Generator, it is postulated that
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Fig. 3. Schematic of water mass control system.
direct control mass of inventory would have improved performance. 3.2. Water mass control system A schematic of the water mass control system is given in Fig. 3. A PI type controller is still assumed, allowing continued use of the feed water controllers installed in existing plants. Only the controller inputs would be changed and the controller retuned. This would minimize modification costs. Also, it would be possible to return to the original level controller during the initial proving phase of the mass inventory controller. As shown in Fig. 3, following a load change, the position of the turbine control valve will change resulting in a steam flow rate change. The steam flow rate change is a disturbance on the system, and will change the states of the Steam Generator water mass, water level, steam pressure and so on. After the water mass estimator provides a water mass signal to the comparator, the comparator calculates the water mass error by subtracting the estimated water mass from the reference water mass. Then, the PI controller uses the water mass error signal to generate a control signal for the feed control valve which modifies the feedwater flow rate to compensate for the disturbance in the steam flow. To perform a theoretical analysis of the water mass control system, the water mass system is represented as a simple linear integrator, with the turbine control valve and feed control valves taken as first order systems with specified characteristic time constants. This system is shown in Fig. 4.
3.3. Stability of the water mass control system To analyze the stability of the system, the transfer function from the change of plant load to the Steam Generator water mass inventory is given in Eq. (2), assuming the true water mass inventory is fed back to calculate the error signal for the PI controller. By applying the Routh–Hurwitz stability criteria to Eq. (2), we can prove that the system is stable when ki > 0 and kp > τ fcv ki . If there is no other restriction on these stability criteria, satisfying these conditions implies the system will be stable throughout the entire power range. H(s) =
water mass load
=−
(τfcv
s3
+ s2
ktcv s(τfcv s + 1) + kfcv kp s + kfcv ki )(τtcv s + 1)
(2)
where kfcv = feed control valve gain; ktcv = turbine control valve gain; τ tcv = characteristic time constant of the turbine control valve; τ fcv = characteristic time constant of the feed control valve; kp = proportional gain of the PI controller; ki = integral gain of the PI controller. 3.4. Robustness of the water mass control system Over the life of a Nuclear Steam Generator, component characteristics may change. In addition, the water mass signals that are sent to the feedwater controller could have errors. These factors will affect the performance of the water mass control
Fig. 4. Schematic of simplified water mass control system.
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Fig. 5. Error channel schematic in water mass control.
system. From Eq. (2), the turbine control valve gain will not affect the stability of the system since it is not contained in the poles of the transfer function. The characteristic time constant of the feed control valve will affect the stability of the system. If the feed control valve degrades such that its response time τ fcv increases, the stability criteria Kp > τ fcv Ki may be violated and lead this system into regions of instability. This analysis provides guidance in setting control parameters in the PI controller and implies the proportional gain should be much larger than the integral gain to guarantee stability of the water mass control system. Since we cannot measure the water mass inventory in the Steam Generator directly, by necessity mass inventory must be approximated using some form of estimator. As no estimator will be perfect, it is important to know how estimation errors will affect the behavior of the mass control system. From the water mass controller in Fig. 4, if water mass estimation error is taken as input and the true water mass inventory as output, the resulting system is illustrated in Fig. 5, with the transfer function given as Eq. (3). He (s) =
kfcv (kp s + ki ) true mass =− 3 mass est · error τfcv s + s2 + kfcv kp s + kfcv ki (3)
Eq. (3) has three of the four poles in Eq. (2), and therefore the system is guaranteed to be stable if the water mass control system is stable. The system represented by Eq. (3) is a tracking system in which the output will follow the input in the opposite direction. According to the final value theorem, the difference between the true water mass and the reference water mass will converge to zero for an impulse error input. em |t=∞ = (m ˆ − mref )|t=∞ = lims→0 sHe (s) = 0
satisfies the following two conditions, the water mass control system will be stable and the true water mass inventory will eventually converge to the reference water mass: • At steady-state, the water mass estimation error be close to zero; • Under transient conditions, large water mass estimation errors should be short lived. If the estimated water mass meets these conditions, the true water mass will eventually converge to the reference water mass, even if the estimated water mass does not follow the reference water mass during rapid transients. 3.5. Programmable reference water mass In conventional water level control system designs, the reference water level is programmed as a function of the plant power level. While specific level programs may vary from vendor to vendor and plant to plant, in general, the reference level increases linearly with increasing reactor power. A representative level control program is illustrated in Fig. 6. The corresponding steady-state steam generator liquid mass is given in Fig. 7. A reference mass program was selected to approximately span the mass range indicated in Fig. 7. For simplicity, the reference water mass was set as the linearly decreasing function of power shown in Fig. 8, though in principle any reference mass program could be employed. The corresponding steady-state Steam Generator narrow range water level is shown in Fig. 9. The normalized
(4)
When there is a unit step water mass estimation error, the final mass error will converge to ˆ − mref )|t=∞ = lims→0 He (s) = −1 em |t=∞ = (m
(5)
In conclusion, when the system reaches steady-state, the true water mass will not converge to the reference water mass if the water mass estimation error is not zero. The difference between the reference water mass and the true water mass inventory will be the mass estimation error. This analysis implies the water mass estimation must be close to the true water mass, particularly as the system approaches steady-state. In practice, the water mass estimation is unlikely to be equal to the true water mass inventory. Under this condition, if the water mass estimation
Fig. 6. Reference water level as a function of plant power level.
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4. Neural network water mass inventory estimator
Fig. 7. Steady-state steam generator liquid mass under level control.
An online estimator is required to provide a mass inventory signal based on measurable plant parameters. Since the thermalhydraulic response of a Steam Generator is highly nonlinear, a linear state-observer is not feasible. In addition, difficulties in obtaining flow regime and density information within the Steam Generator make an estimator based on analytical methods impractical at this time (Dong et al., 2000). Neural networks are well known for their ability to handle nonlinear systems. For this work, a feedforward neural network is chosen because it is guaranteed to be stable. Since the power plant instrumentation system cannot provide water mass signals and there is no analytical model available to calculate the water mass in the steam generator, the neural network was trained off-line using mass inventory and system information produced by a full plant engineering simulation code developed at North Carolina State University (Dong, 2001). 4.1. A macroscopic view of the Steam Generator In pressurized water reactors, the process instrumentation system provides the following ten measurable parameters which are directly related to the thermal-hydraulic behavior of the Steam Generator: • Narrow and wide range water level; • Feedwater temperature, primary side hot leg temperature, primary side cold leg temperature and steam temperature; • Steam flow rate, feedwater flow rate and primary side water flow rate; • Steam pressure.
Fig. 8. Reference water mass as a function of plant power level.
steady-state narrow range level is between 30% and 70%. Thus, water mass control with the linear reference water mass in Fig. 8 produces steady-state narrow range water levels in the range of traditional level controllers.
To provide insight into the parameters governing Steam Generator behavior, a simplified space average view of the Steam Generator was developed. Steam Generator water mass is the sum of the mixture mass in the bundle and riser sections, and the liquid mass in the downcomer. In terms of density, m = ρm V + ρdc Ax L
(6)
where Ax = cross-section flow area of downcomer; L = downcomer water level; m = water mass inventory in the SG; ρm = bundle and riser section mixture water density; ρdc = downcomer water density; V = bundle and riser section volume. Since the density is not measurable, and cannot be derived from other measurable parameters, water mass cannot be obtained directly from Eq. (6). Taking the derivative of Eq. (6) and rearranging, we get ∂L ∂m ∂ρm ∂ρdc m = f L, , , ρm , , ρdc , (7) ∂t ∂t ∂t ∂t
Fig. 9. Steady-state narrow range water level under the linear reference water mass controller.
If ρ = ρ(u, P), then ∂ρ/∂t = f(u, ∂u/∂t, P, ∂P/∂t), where these functions can be highly nonlinear. Since the internal energy u for single phase liquids and saturated mixtures can be expressed as u = f(T), or, u = f(αg , P) where αg is the void fraction. Then, ∂u/∂t = f(T, ∂T/∂t)) or ∂u/∂t = f(αg , ∂αg /∂t, P, ∂P/∂t).
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From eq. (7), the mass within the Steam Generator can be expressed as ∂αg ∂P ∂L ∂m ∂T , T, αg , , P, (8) m = f L, ∂t ∂t ∂t ∂t ∂t where ∂m/∂t can be approximated by the difference between feedwater flow rate and steam flow rate, P can be taken as the average steam pressure and T is the average single-phase liquid temperature. The average void fraction αg and ∂T/∂t are related to the energy input and output of the Steam Generator, i.e. ∂αg ∂T ∂E , ⇒ = f (W1 , T1hot , T1cold , Wfw , Tfw ) (9) αg , ∂t ∂t ∂t Combining Eqs. (8) and (9), and assuming the vapor mass is small, the liquid mass inventory becomes, ∂L ∂P , Ws , W1 , Wfw , T1cold , T1hot , Tfw , P, ml = f L, ∂t ∂t (10) where L = downcomer water level; Ws = steam flow rate; W1 = primary side water flow rate; Wfw = feedwater flow rate; T1hot = primary side hot leg temperature; T1cold = primary side cold leg temperature; Tfw = feedwater temperature; P = steam pressure. The variables on the right hand side of Eq. (10) are all measurable and available from current plant instrumentation. This equation can be utilized to design the mass inventory estimator. 4.2. Neural network water mass estimator Finding Steam Generator water mass from measurable plant parameters is a system identification problem. Mathematically, system identification is a functional approximation task for developing the plant model that is the most consistent with the measurable or simulation data. Since the water mass inventory is not measurable, only off-line training can be utilized to train the neural network given the data set collected from the Steam Generator model. Theoretically, all the measurable variables on the right hand side of Eq. (10) should be used as neural network inputs. But for ease of implementation and to improve the generality of the neural network, the following factors are further considered in picking the inputs for the water mass estimator: 1. Narrow range level and wide range level are both provided by the plant instrumentation system and are highly correlated. As a result, only narrow range level is used as an input for the neural network. 2. The primary side water flow rate is nearly constant, therefore W1 can be neglected as an input to the neural networks. 3. Heat transfer across the tube bundle is directly related to the average primary side temperature within the bundle. This can be approximated as the average between the hot leg and cold leg temperatures. Average temperature is chosen to replace T1hot and T1cold in training of the neural network.
In conclusion, the following nine measurable parameters are chosen as the inputs for the neural network to perform the water mass inventory estimation: • L = Steam Generator downcomer narrow range water level. • dL/dt = the first-order derivative of the downcomer water level. • Ws = steam flow rate. • Wfw = the feedwater flow rate. • Tave = the primary side average temperature. • Tfw = the feedwater temperature. • P = the steam pressure. • dP/dt = the first-order derivative of the steam pressure. • dm/dt = the first-order derivative of the water mass in the Steam Generator, which is approximated by the difference between feedwater flow rate and steam flow rate. The basic structure of the neural network developed to estimate the Steam Generator water mass is the two-layer neural network illustrated in Fig. 10. The input layer is connected to the plant instrumentation system. The hidden layer takes a summation of the weighted neural network inputs and evaluates them before they are passed to the output layer. The neural network output layer produces the predicted water mass signal for the water mass controller. Sigmoid nonlinearity is used in the hidden layer neurons. The output layer neuron uses pure linear functions to sum the weighted outputs from hidden neurons and bias the summation (Principe et al., 2000). The two most widely used sigmoid functions in neural networks are “logsig” and “tansig”. The difference being that the “logsig” function has an output between 0 and 1, while the “tansig” function has an output between −1 and 1. From computational experiments, neither “logsig” nor “tansig” functions showed significant differences in approximating the water mass inventory, and the “tansig” function was chosen for our design. The “tansig” transfer function is: f (net) = tanh(net)
Fig. 10. Two-layer neural network for water mass estimation.
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Table 1 Training and validating set for the neural networks Initial nominal plant power level (%) 20 40 60
Increase 10%, decrease 5%, 10%, 15%, 20% Increase 10%, decrease 10%, 20%, 30%, 40% Increase 10%, decrease 10%, 20%, 40%, 50%, 60% Increase 10%, decrease 10%, 20%, 40%, 60%, 70%, 80% Decrease 10%, 20%, 40%, 60%, 80%, 90%, 100%
80 100
Fig. 11. Training and validation error for different numbers of hidden neurons.
where net =
wi xi + b
i
w = neuron input weights; x = neuron inputs; b = neuron bias. One of the central issues in neural-computation is to appropriately set the number of hidden neurons. Too many hidden neurons may cause over-fitting and accumulate noise in the training signals; too few hidden neurons will have insufficient freedom to map the input/output function and the error will stabilize at a high value. Guidelines have been established for setting the number of hidden neurons (Baum and Haussler, 1989; Hrycej, 1997; Kolmogorov, 1969). In this research, the number and type of simulation data sets used for training and validation were frequently changed based upon results from the computational experiments, such that setting the number of hidden neurons analytically or by semi-analytical computational methods is not feasible. As a result, the number of hidden neurons used in this work is chosen empirically. Fig. 11 shows the training and validation results as a function of the number of hidden neurons. The neural network with 15 hidden neurons has the lowest mean square error in estimating the water mass inventory, and was selected for the remainder of the neural network development. 4.3. Training the neural network for water mass estimation The generality of the neural network over the state space of interest is affected by the data sets chosen for network training. From our previous analysis, the neural network will perform a 10-dimensional function approximation corresponding to the nine inputs and one output. With such a high dimensionality, the state space is large. To make the water mass estimator feasible, the choice of training and testing data should satisfy these two conditions: • The data must cover the whole state space, or at least the region of significance. • The distribution of training and testing data in the state space should not be heavily biased.
Step change (change in load)
If the neural network is applied to a region of state space that is far from where it is trained, the outputs of the neural networks are unpredictable and cannot be guaranteed to match the physical system. If the data densities in some regions are much higher than the data density in others, the neural network will be over trained in these regions and will not generalize well. To adequately represent the state space, it is necessary to collect as many simulation data sets as possible. Data generated when the plant is subjected to large transients are considered more important in training the neural network since transients of this type cover a wider region of state space. The transients included in Table 1 were used in developing the training and validating data sets for the neural network. The gains of the PI controller were set to very large values (kp = 2, ki = 0.2) compared with the nominal gains of the PI level controller (kp = 0.07, ki = 0.07) in our plant model. Step change responses were selected as opposed to ramp change responses due to their increased coverage of the state space. Since large load decreases are more frequent in real plant operation than load increases, the training set included more load decrease responses than load increase responses. For example, there can be a reactor trip that brings the plant from full power to zero power, but the plant is unlikely to experience a step increase in power level from zero to full power. In addition, more low power transients are selected because the “shrink and swell” effect at low power is more severe than it is at high-power level. To evenly distribute the training data sets in state space, data sets collected when the plant is at steady-state should be removed. During steady-state, the data repeat rate is very high and the neural network will be over-trained in these regions if these data are included in the training data set. Even after removing steady-state data, the data distribution in the state space is still not balanced. At the beginning of a step response, the plant states change so quickly that data is scarce in those regions. To balance the data distribution, the data sets collected in those regions can be repeated several times (bootstrapped). This is similar to giving those data higher weights. The data sets are duplicated according to their validation error in ratio to the mean square error of the whole training set after early stop training. nk =
|ek | N
(1/N)
i=1 |ei |
(11)
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Table 2 Range of training and validation data set Variable name
Minimum
Maximum
Mass (kg) Narrow range level (%) Steam pressure (Pa) Steam flow rate (kg/s) Feed-water flow rate (kg/s) Feed-water temperature (K) Tave (K) dP/dt (Pa/s) dL/dt (%/s) dm/dt (kg/s)
44,500 0 6.2e6 (900 psia) 0 0 305 (90 ◦ F) 555 (540 ◦ F) −9.0e4 (−13 psia/s) −0.5 −500
60,000 100 8.3e6 (1200 psia) 580 580 505 (450 ◦ F) 590 (600 ◦ F ) 9.0e4 (13 psia/s) 0.5 500
nk is how many times the kth data is duplicated; |ek | is the magnitude of the kth data validation error; N is the number of data in training data sets. To improve noise immunization of the neural network, 1% measurement noise in uniform distribution was added to each of the measurable inputs to the neural networks training sets. One practical issue in implementing the learning algorithm is the avoidance of neural network saturation. The output of the sigmoid activation function used to represent the hidden neuron will be close to either 1 or −1 when the weighted input net >2 or net <−2. This will stall the neuron weight updates and is a typical “retarded” learning example. One technique for solving this problem is to scale the neural network input values to be within (−1, 1) by normalizing the input and output vectors. In practice, sensor outputs are all normalized to a fixed range of voltage or current, such that the normalization here should not only make the network training converge more quickly, but also make future implementation easier. By simulating extreme cases of plant load changes, the minimum and maximum of the measurable variables were determined and are listed in Table 2. First-order derivative inputs are normalized within (−1, 1) to capture the changing direction of the parameters, while non-derivative parameters are normalized on (0, 1). The water mass estimator was trained by the second-order Levenberg–Marquardt algorithm (Park et al., 1995). Training results are used to bootstrap the data in fast transient regions,
Fig. 12. Distribution of water mass estimation error.
with the neural network retrained after each bootstrap. Early stopping is employed to prevent over-training. Simulation results show that the water mass estimator generalized well after training. The probability distribution function for the water mass estimation error in our validating data set is shown in Fig. 12. These results indicate that the water mass estimation error is sufficiently small to provide the water mass signal for the water mass controller. 4.4. Close-loop water mass control system using the water mass estimator The water mass estimator was embedded in the water mass control system and implemented in the plant simulator as indicated in Fig. 13. In this scheme, x are measurable plant parameters such as downcomer water level, steam pressure, etc. and dx/dt are first-order derivatives of selected parameters. A dead-zone nonlinear block was added after the derivative to prevent noise in the measurable parameters from giving an unreasonable parameter change rate and lead the water mass estimator into an untrained region of state space. The plant parameters x and their derivatives dx/dt are fed into the neural network water mass estimator. The output of the
Fig. 13. Water mass control system with neural network water mass estimator.
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Fig. 14. System performance under ramp load change.
Fig. 15. System performance with step load change at low power.
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Fig. 16. System performance for a step change in load at low power with large controller gains.
neural network is the estimated steam generator water mass. The estimated water mass is fed back to the control system, compared to a reference mass and a water mass error signal is generated. The error signal is fed to the PI controller and the PI controller generates a control signal to drive the feed control valve. Plant safety functions (i.e. reactor trips) are assumed to be based on conventional downcomer level signals, and therefore independent of the mass controller. The new control system was tested under different plant load changes. The gains of the PI controllers were varied to evaluate their effects on the system performance. Sensor noise was also included in the measurable parameters to see its effect on the system behavior. Fig. 14 illustrates controller performance for a 5%/min ramp load decrease from 60% power. The ramp load change lasted 2 min resulting in a final power of 50%. The PI controller gains were Kp = 1.0, Ki = 0.1. In this simulation, the water mass estimator gives a very good estimation of the true water mass inventory. The water mass controller brings the water mass and water level to steady-state within 200 s after the disturbance. The water mass controller also shows good performance for step changes in load at low power levels. Fig. 15 shows controller performance for a 10% step increase in load from 20% initial power. The proportional gain was set to Kp = 0.5. Integral gain was set to Ki = 0.07. The water mass estimator provides reasonable water mass inventory estimation for the duration of the transient and the water mass controller brings the system to steady-state smoothly.
The water mass control system provides acceptable performance at low power levels with large PI controller gains. In Fig. 16, the plant was subjected to a 10% step increase in load from an initial power level of 20%, resulting in a final power of 30%. The PI controller gains were Kp = 4.0 and Ki = 0.07. [The nominal gains of the PI controller are Kp = 0.07, Ki = 0.07 in the water level controller in our plant model.] The estimated water mass at the beginning of the transient has an error of about 7% normalized mass. After 50 s however, the controller brings the system into a region of state space where the water mass estimator provides a good estimation of the water mass inventory. The true water mass reaches the reference water mass approximately 200 s after the step change in load. Once the water mass reaches steady-state, the large PI controller gains cause the feedwater control valve to “chatter” resulting in changes in the feed flow, leading to oscillations in the estimated water mass. However, the true water mass inventory is not affected because of the integral effect of the PI controller. The narrow range water level also remains within the normal operational range. 5. Conclusion A new control strategy is presented that changes the controlled variable in Nuclear Steam Generators from downcomer level to steam generator water mass inventory. Compared with a conventional water level control system, the water mass control approach has been shown to have the following advantages:
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• The controlled system is a linear system with minimum phase, which simplifies both the control system design as well as manual operation; • The water mass inventory is a true measure of the instantaneous cooling capacity of the Steam Generator. Combined with a conventional level signal, safety could be enhanced. • Control of water mass inventory can increase margin to set points and relax constraints on system output resulting in a more robust system. • The controller design can accommodate “shrink and swell” caused not only by steam and water flow change, but also by the change of feedwater temperature, the change of Steam Generator primary side water temperature and the change of primary side water flow rate. • If reactor trip is still based on downcomer water level signals, safety functions are unchanged. For example, if the water mass estimator fails or the sensors providing inputs for the water mass estimator fail, normal safety systems will trigger a water level trip. In summary, the presented water mass controller is an improvement over the conventional water level control system in almost every aspect of control system design. Unlike most water level control systems, the water mass inventory within the Steam Generator is not measurable and must be inferred from measurable plant variables. A water mass estimator based on neural networks has demonstrated that a water mass control system is feasible, and simulation results indicate good controller performance over a wide range of operating conditions. References Baum, E.B., Haussler, D., 1989. What size net gives valid generalization? Neural Comput. 1, 151–160. Carmichael, L.A., Toye, G., Ipakchi, A. Sievers, M.W., 1984. Digital feedwater controller for a BWR: a conceptual design study, EPRI NP-3323. Choi, J.I., et al., 1989. Automatic controller for steam generator water level during low power operation. Nucl. Eng. Des. 117, 263–274. Dong, W., 2001. Water mass control system based on artificial neural networks for the steam generator in a pressurized water reactor, Ph.D. Thesis, Department of Nuclear Engineering, North Carolina State University. Dong, W., Mayo, C.W., Doster, J.M., 2000. A neural network based mass inventory observer for UTSG in PWR. In: International Topical Meeting on Nuclear Plant I&C, and Human–Machine Interface Tech, NPIC & HMIT 2000, Washington, DC.
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Hrycej, T., 1997. Neural Control: Towards an Industrial Control Methodology. John Wiley & Sons, Inc. Irving, E., Bihoreaux, C., 1980. Adaptive control of non-minimum phase system: application to the PWR SG water level control. In: Proceedings of the 19th IEEE Conference Control and Decision, Albuquerque, NM. Kim, K., Doster, J.M., 1991. Application of mixture drift flux equations to vertical separating flows. Nucl. Technol. 95, 103. Kim, K., Doster, J.M., 1995. Implementation of a courant violating scheme for mixture drift-flux equations. Nucl. Sci. Eng. 119, 18–33. Kolmogorov, A.N., 1969. On the representation of continuous functions of many variables by superposition of continuos functions of one variable and addition. Am. Math. Soc. 2, 55–59. Kothare, M.V., et al., 1996. Level control in the steam generator of a nuclear power plant. In: Proceedings of the 35th Conference on Decision and Control, Kobe, Japan. Kuan, C.C., Lin, C., et al., 1992. Fuzzy logic control of steam generator water level in pressurized water reactors. Nucl. Technol. 100, 125–134. March-Leuba, J., Mullen, J.A., Wood, R.T., Brittain, C.R., 2001. Demonstration of a methodology to capture performance requirements in reactor control system design. Trans. Am. Nucl. Soc. 85, 204. March-Leuba, J., Mullens, J.A., Wood, R.T., Upadyahya, B.R., Doster, J.M., Mayo, C.W., 2002. NERI Project 99-119. A New paradigm for Automatic Development of Highly Reliable Control Architectures for Nuclear Power Plants. Final Report, ORNL/TM-2002/193. Masche, G., 1971. Systems Summary of a Westinghouse Pressurized Water Reactor Nuclear Power Plant. Westinghouse Electric Corporation. Mullen, J.A., March-Leuba, J., Wood, R.T., Brittain, C.R., 2002. A computing environment integrating diagnostics and control for reactor systems. Nucl. Plant J.. Park, S., Park, L. J. Park, C. H., 1995. A neuro-genetic controller for nonminimum phase systems’. IEEE Trans. Neural Networks 6. Popovic, J.R., Hammer, M.F., 1987. Testing and installation of a BWR digital feedwater control system, EPRI NP-5524. Principe, Jose, C., et al., 2000. Neural and Adaptive Systems. John Wiley & Sons, Inc. SAIC, 1987. Requirement and design specifications of a BWR digital feedwater control system, EPRI NP-5502. Singh, G. Tsiouris, M.G., Pearce, L.W., Burton, C., 1989. Design and operation of a digital low-power feedwater control system for PWRs, EPRI NP-6149. Upadhyaya, B.R., Zhao, K., Lu, B., Doster, J.M., 2001. Fault detection and isolation of sensors and actuators in a nuclear plant steam generator. Trans. Am. Nucl. Soc. 85, 350. Upadhyaya, B.R., Lu, B., Zhao, K., Doster, J.M., 2002a. Equipment monitoring during process transients and multiple fault conditions. In: MARCON 2002, Knoxville, TN. Upadhyaya, B.R., Zhao, K., Lu, B., Doster, J.M., Na, M.G., Sim, Y.R., Park, K.H., 2002b. Nuclear plant system monitoring under process transients and multiple fault conditions. Trans. Am. Nucl. Soc. 86, 182. Zapotocky, A., Popovic, J.R. Fournier, R.D., 1988. Implementation of a digital feedwater control system at dresden nuclear power plant units 2 and 3, EPRI NP-6143.
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Steam Generator control in Nuclear Power Plants by water mass inventory Wei Dong, J. Michael Doster ∗ , Charles W. Mayo North Carolina State University, Department of Nuclear Engineering, Box 7909, Raleigh, NC 27695-7909, United States Received 15 February 2007; received in revised form 5 September 2007; accepted 10 September 2007
Abstract Control of water mass inventory in Nuclear Steam Generators is important to insure sufficient cooling of the nuclear reactor. Since downcomer water level is measurable, and a reasonable indication of water mass inventory near steady-state, conventional feedwater control system designs attempt to maintain downcomer water level within a relatively narrow operational band. However, downcomer water level can temporarily react in a reverse manner to water mass inventory changes, commonly known as shrink and swell effects. These complications are accentuated during start-up or low power conditions. As a result, automatic or manual control of water level is difficult and can lead to high reactor trip rates. This paper introduces a new feedwater control strategy for Nuclear Steam Generators. The new method directly controls water mass inventory instead of downcomer water level, eliminating complications from shrink and swell all together. However, water mass inventory is not measurable, requiring an online estimator to provide a mass inventory signal based on measurable plant parameters. Since the thermal-hydraulic response of a Steam Generator is highly nonlinear, a linear state-observer is not feasible. In addition, difficulties in obtaining flow regime and density information within the Steam Generator make an estimator based on analytical methods impractical at this time. This work employs a water mass estimator based on feedforward neural networks. By properly choosing and training the neural network, mass signals can be obtained which are suitable for stable, closed-loop water mass inventory control. Theoretical analysis and simulation results show that water mass control can significantly improve the operation and safety of Nuclear Steam Generators. © 2007 Elsevier B.V. All rights reserved.
1. Introduction Control of water mass inventory in Nuclear Steam Generators is important to insure sufficient cooling of the nuclear reactor. Since downcomer water level is a reasonable indicator of water mass inventory at steady-state, or near steady-state conditions, conventional control system designs attempt to maintain downcomer water level between narrow limits in order to provide for sufficient cooling of the reactor, to provide for good performance of the steam separators and dryers, and also to reduce risks of hydrodynamic instability. Failure to maintain the water level within reasonable bounds typically results in a reactor trip. As a result, a good control system for the water level can be a major factor in overall plant availability (Masche, 1971). The water level system in Nuclear Steam Generators is a complex nonlinear system with inverse dynamics. When piecewise
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0029-5493/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2007.09.001
linearized, the system is a MIMO (multi-input–multi-output), variable parameter, non-minimum phase system and good controller designs are difficult to achieve (Choi et al., 1989; Irving and Bihoreaux, 1980). During plant transients, level control is complicated by hydrodynamic effects known as “shrink and swell”. Due to the presence of steam bubbles in the tube bundle region, the water level measured in the downcomer can temporarily react in a reverse manner to water mass inventory changes. This behavior is accentuated during start-up and low power operation. Under these conditions, the only true indication of water mass inventory change is the difference between steam flow and feed water flow. However, at low power levels, flow measurement uncertainty can be too large for these parameters to be used for effective control. It is well known that traditional control schemes do not permit satisfactory automatic level control during start-up/low power conditions. Unsatisfactory performance of automatic level control may either produce a reactor trip or require the operator to initiate manual control. Even for a skilled operator, it can be difficult to react properly to the reverse level indications resulting from shrink and swell
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effects. It has been observed that reactor trips are caused by the operator’s overreaction while restoring water level (Choi et al., 1989). Digital feedwater control systems (Carmichael et al., 1984; SAIC, 1987; Popovic and Hammer, 1987; Zapotocky et al., 1988; Singh et al., 1989) have successfully mitigated some of the stability problems associated with analog control systems, and have been applied to the low power range (Popovic and Hammer, 1987; Zapotocky et al., 1988; Singh et al., 1989). However, these control systems still suffer from the inverse dynamics of shrink and swell and are not minimum phase. Many studies have contributed to level controller design (Choi et al., 1989; Irving and Bihoreaux, 1980; Kuan et al., 1992; Kothare et al., 1996). Conclusions that may be drawn from these previous studies include: • Good controller designs for Nuclear Steam Generators are difficult, although significant progress has been achieved in the period since 1975. With the help of high-speed computer simulation, almost every modern control strategy has been applied to the Nuclear Steam Generator level control problem. • Since the inverse system is non-minimum phase and has unstable dynamics, it is difficult to train a neural network controller via the back-propagation of the system output error (Park et al., 1995). For this reason, artificial neural networks are precluded from the design when water level in Nuclear Steam Generators is chosen as the controlled variable. • Most controller designs treat the plant as a single-input/singleoutput system even though the Steam Generator is in fact a multi-input–multi-output system. Feedwater flow rate is typically taken as the system input, narrow range downcomer water level as the system output, and steam flow rate is treated as a disturbance. The impact of other factors such as feedwater temperature, primary side water temperature and primary side water flow rate on the reverse dynamics are included in one lumped parameter—reactor power level (Irving and Bihoreaux, 1980; Kothare et al., 1996). • At low plant power levels, efforts to develop a good controller are still hindered by the lack of accuracy in steam flow and/or feed flow measurements. It is noted that controllers with less dependence on these flow measurements may be desirable (Irving and Bihoreaux, 1980; Kothare et al., 1996). • A good Steam Generator model (well validated against plant data) is essential for good controller design (Choi et al., 1989; Kuan et al., 1992; Kothare et al., 1996; Dong et al., 2000). Simplified piecewise linearized transfer function models (Irving and Bihoreaux, 1980) are widely used by designers familiar with control theory to develop linear control strategies; while physical models are used by designers familiar with thermodynamics and for fuzzy controller design (Kuan et al., 1992). By changing the controlled variable in Nuclear Steam Generators from water level to water mass inventory, the controlled system becomes a linear system with minimum phase, which simplifies both the control system design as well as manual operation. At the same time, the water mass inventory is a true
measure of the instantaneous cooling capacity of the Nuclear Steam Generator. Combined with a conventional level signal, water mass control provides the opportunity for improved plant operation (Dong, 2001). Unlike the conventional water level control system, the water mass inventory within the Steam Generator is not measurable, and must be inferred from measurable plant variables. Since the thermal-hydraulic response of a Steam Generator is highly nonlinear, a linear state-observer is not feasible. In addition, difficulties in obtaining flow regime and density information within the Steam Generator make an estimator based on analytical methods impractical at this time. In this paper, a water mass estimator based on feedforward neural networks is developed, and shown to be capable of estimating water mass inventory over a broad range of plant operating conditions. The neural network water mass estimator is then embedded in a close-loop water mass control system. Simulation results indicate the mass controller can maintain both Nuclear Steam Generator water mass and water level within operating limits, even when the system is subjected to large load disturbances at low plant power levels. In the following sections, the Steam Generator dynamic model is described, and evaluation of the model is carried out. The details of the new controller design are described and theoretical analysis of the new water mass control system is presented. A water mass estimator based on neural networks is developed and embedded into a Nuclear Power Plant simulation model. A conventional PI controller uses the estimated water mass as input to form a close-loop control system where water mass is the controlled variable, and the performance of the system is evaluated. 2. Steam Generator model A computational model of the Steam Generator is necessary to provide insight into the Steam Generator dynamics and replace actual plant data for developing and evaluating the controller. The dynamic behavior of U-Tube Steam Generators is a function of not only the thermal-hydraulic conditions within the Steam Generator, but also the forcing functions or boundary conditions imposed on the Steam Generator by the remainder of the plant. This implies that a full plant simulation is required to adequately characterize Steam Generator behavior over the full range of operating conditions. A full plant engineering simulation code has been developed at North Carolina State University for simulating the dynamic response of pressurized water reactor systems during normal operational transients as well as design basis events. The model is capable of simulating multi-loop PWR systems with U-Tube Steam Generators. Both primary and secondary sides are represented including automatic control systems, reactor protection systems and engineered safety features. The Steam Generator model is based on a four-equation drift flux model that allows for non-equilibrium treatment of the liquid phase. Level tracking logic is included to allow for simulation of vertical stratified flows and accurate predictions of Steam Generator liquid levels
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Fig. 1. Simulation of “shrink ” following a 10% step reduction in load from 80% power.
(Kim and Doster, 1991). Details of the numerical solution can be found elsewhere (Kim and Doster, 1991, 1995). In addition to this work, this software has served as the simulation platform to develop methods for fault diagnosis and isolation (March-Leuba et al., 2001; Upadhyaya et al., 2001, 2002a,b) and automatic controller optimization (March-Leuba et al., 2001, 2002; Mullen et al., 2002). The plant simulation model is capable of simulating the shrink and swell phenomena as illustrated in Fig. 1 for a step reduction in load (shrink) and Fig. 2 for a step increase in load (swell). As the turbine control valve closes following load reduction, steam pressure increases. This results in bubble collapse in the bundle region and a corresponding draw down of the liquid level in the downcomer. In a conventional level control system, the feedwater control valve will open to increase the feedwater flow rate as the indicated water level decreases, even though the water mass inventory is increasing. Similarly, when the turbine control valves open following an increase in load, the steam pressure decreases resulting in bubble expansion in the bundle region. This expansion forces water back up into the downcomer, increasing downcomer level, even though the mass inventory is decreasing. As shown in Figs. 1 and 2, the water mass in the Steam Generator shows no reverse dynamics. Water mass
increases when the feedwater flow rate is greater than the steam flow rate and decreases when the feedwater flow rate is less than the steam flow rate. 3. Water mass inventory controller 3.1. Conventional water level control system Conventional level control strategies are based on threeelement, proportional-integral (PI) controllers, where the controller inputs are feedwater flow rate, steam flow rate, and downcomer water level. At low plant power levels, flow measurement uncertainties can render the flow measurements unreliable, and a single-element PI controller based on the level error signal is often used. The control action of a three-element proportional-integral controller is described by Eq. (1) in terms of the primary control variables: feedwater flow rate Wfw ; steam flow rate Ws ; and downcomer water level Lw ; (1) Wfw = kp (Aw ew + Al el ) + ki (Aw ew + Al el ) dt where kp = proportional gain; ki = integral gain; ew = (Ws − Wfw ) = flow error (kg/s); el = (Lref −Lw ) = level error (m);
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Fig. 2. Simulation of “swell” following a 10% step increasing in load from 80% power.
Lref = reference (programmed) water level (m); Aw = flow error weight; Al = level error weight. An installed controller may differ in the details but uses the same three variables (Wfw , Ws , and Lw ) to calculate a control error signal. In addition, feedwater flow is often controlled by a combination of pump speed and control valve position using more complex control logic with cascaded PI controllers. The objective of this control strategy is to eliminate the steam generator water level error which is affected by the shrink and swell phenomena. As steam flow decreases because of turbine load reduction, the rate of steam removal from the Steam Generator drops below the rate of steam generation. In a saturated system, this imbalance results in a pressure rise and subsequent collapse of steam bubbles that exist in the tube bundle region. The volume taken up by the liquid/vapor mixture decreases with the collapse of the steam bubbles, causing the indicted downcomer water level to “shrink”. However, the fluid mass in the Steam Generator is actually increasing (feed flow > steam flow). It normally takes several seconds for the water mass in the downcomer to move into the bundle region and to reestablish the force balance between friction and buoyancy forces. When steam flow increases, excess
steam removal results in a decrease in Steam Generator pressure. The decrease in pressure causes expansion of the liquid/vapor mixture in the tube bundle region. This forces water back up into the downcomer, causing the indicted water level to “swell”. However, the fluid mass in the Steam Generator is actually decreasing (feed flow < steam flow). Shrink and swell effects are also observed with increases and decreases in feedwater flow. As the flow rate of relatively cold feedwater increases, the inlet enthalpy into the tube bundle region decreases so that the steam bubbles collapse causing the indicated downcomer water level to “shrink”. As the feedwater flow decreases, the inlet enthalpy into the tube bundle increases, leading to expansion of the steam bubbles, causing the indicated downcomer water level to “swell”. Changes in feedwater temperature, primary side water temperature and primary side flow rate can also cause “shrink” and “swell”. Shrink and swell effects within the downcomer give a reverse level indication that complicates the choice of the proper control action. If these reverse effects are combined with high controller gains in a level controlled system, the system may become unstable. Since shrink and swell do not affect the actual mass inventory in the Steam Generator, it is postulated that
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Fig. 3. Schematic of water mass control system.
direct control mass of inventory would have improved performance. 3.2. Water mass control system A schematic of the water mass control system is given in Fig. 3. A PI type controller is still assumed, allowing continued use of the feed water controllers installed in existing plants. Only the controller inputs would be changed and the controller retuned. This would minimize modification costs. Also, it would be possible to return to the original level controller during the initial proving phase of the mass inventory controller. As shown in Fig. 3, following a load change, the position of the turbine control valve will change resulting in a steam flow rate change. The steam flow rate change is a disturbance on the system, and will change the states of the Steam Generator water mass, water level, steam pressure and so on. After the water mass estimator provides a water mass signal to the comparator, the comparator calculates the water mass error by subtracting the estimated water mass from the reference water mass. Then, the PI controller uses the water mass error signal to generate a control signal for the feed control valve which modifies the feedwater flow rate to compensate for the disturbance in the steam flow. To perform a theoretical analysis of the water mass control system, the water mass system is represented as a simple linear integrator, with the turbine control valve and feed control valves taken as first order systems with specified characteristic time constants. This system is shown in Fig. 4.
3.3. Stability of the water mass control system To analyze the stability of the system, the transfer function from the change of plant load to the Steam Generator water mass inventory is given in Eq. (2), assuming the true water mass inventory is fed back to calculate the error signal for the PI controller. By applying the Routh–Hurwitz stability criteria to Eq. (2), we can prove that the system is stable when ki > 0 and kp > τ fcv ki . If there is no other restriction on these stability criteria, satisfying these conditions implies the system will be stable throughout the entire power range. H(s) =
water mass load
=−
(τfcv
s3
+ s2
ktcv s(τfcv s + 1) + kfcv kp s + kfcv ki )(τtcv s + 1)
(2)
where kfcv = feed control valve gain; ktcv = turbine control valve gain; τ tcv = characteristic time constant of the turbine control valve; τ fcv = characteristic time constant of the feed control valve; kp = proportional gain of the PI controller; ki = integral gain of the PI controller. 3.4. Robustness of the water mass control system Over the life of a Nuclear Steam Generator, component characteristics may change. In addition, the water mass signals that are sent to the feedwater controller could have errors. These factors will affect the performance of the water mass control
Fig. 4. Schematic of simplified water mass control system.
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Fig. 5. Error channel schematic in water mass control.
system. From Eq. (2), the turbine control valve gain will not affect the stability of the system since it is not contained in the poles of the transfer function. The characteristic time constant of the feed control valve will affect the stability of the system. If the feed control valve degrades such that its response time τ fcv increases, the stability criteria Kp > τ fcv Ki may be violated and lead this system into regions of instability. This analysis provides guidance in setting control parameters in the PI controller and implies the proportional gain should be much larger than the integral gain to guarantee stability of the water mass control system. Since we cannot measure the water mass inventory in the Steam Generator directly, by necessity mass inventory must be approximated using some form of estimator. As no estimator will be perfect, it is important to know how estimation errors will affect the behavior of the mass control system. From the water mass controller in Fig. 4, if water mass estimation error is taken as input and the true water mass inventory as output, the resulting system is illustrated in Fig. 5, with the transfer function given as Eq. (3). He (s) =
kfcv (kp s + ki ) true mass =− 3 mass est · error τfcv s + s2 + kfcv kp s + kfcv ki (3)
Eq. (3) has three of the four poles in Eq. (2), and therefore the system is guaranteed to be stable if the water mass control system is stable. The system represented by Eq. (3) is a tracking system in which the output will follow the input in the opposite direction. According to the final value theorem, the difference between the true water mass and the reference water mass will converge to zero for an impulse error input. em |t=∞ = (m ˆ − mref )|t=∞ = lims→0 sHe (s) = 0
satisfies the following two conditions, the water mass control system will be stable and the true water mass inventory will eventually converge to the reference water mass: • At steady-state, the water mass estimation error be close to zero; • Under transient conditions, large water mass estimation errors should be short lived. If the estimated water mass meets these conditions, the true water mass will eventually converge to the reference water mass, even if the estimated water mass does not follow the reference water mass during rapid transients. 3.5. Programmable reference water mass In conventional water level control system designs, the reference water level is programmed as a function of the plant power level. While specific level programs may vary from vendor to vendor and plant to plant, in general, the reference level increases linearly with increasing reactor power. A representative level control program is illustrated in Fig. 6. The corresponding steady-state steam generator liquid mass is given in Fig. 7. A reference mass program was selected to approximately span the mass range indicated in Fig. 7. For simplicity, the reference water mass was set as the linearly decreasing function of power shown in Fig. 8, though in principle any reference mass program could be employed. The corresponding steady-state Steam Generator narrow range water level is shown in Fig. 9. The normalized
(4)
When there is a unit step water mass estimation error, the final mass error will converge to ˆ − mref )|t=∞ = lims→0 He (s) = −1 em |t=∞ = (m
(5)
In conclusion, when the system reaches steady-state, the true water mass will not converge to the reference water mass if the water mass estimation error is not zero. The difference between the reference water mass and the true water mass inventory will be the mass estimation error. This analysis implies the water mass estimation must be close to the true water mass, particularly as the system approaches steady-state. In practice, the water mass estimation is unlikely to be equal to the true water mass inventory. Under this condition, if the water mass estimation
Fig. 6. Reference water level as a function of plant power level.
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4. Neural network water mass inventory estimator
Fig. 7. Steady-state steam generator liquid mass under level control.
An online estimator is required to provide a mass inventory signal based on measurable plant parameters. Since the thermalhydraulic response of a Steam Generator is highly nonlinear, a linear state-observer is not feasible. In addition, difficulties in obtaining flow regime and density information within the Steam Generator make an estimator based on analytical methods impractical at this time (Dong et al., 2000). Neural networks are well known for their ability to handle nonlinear systems. For this work, a feedforward neural network is chosen because it is guaranteed to be stable. Since the power plant instrumentation system cannot provide water mass signals and there is no analytical model available to calculate the water mass in the steam generator, the neural network was trained off-line using mass inventory and system information produced by a full plant engineering simulation code developed at North Carolina State University (Dong, 2001). 4.1. A macroscopic view of the Steam Generator In pressurized water reactors, the process instrumentation system provides the following ten measurable parameters which are directly related to the thermal-hydraulic behavior of the Steam Generator: • Narrow and wide range water level; • Feedwater temperature, primary side hot leg temperature, primary side cold leg temperature and steam temperature; • Steam flow rate, feedwater flow rate and primary side water flow rate; • Steam pressure.
Fig. 8. Reference water mass as a function of plant power level.
steady-state narrow range level is between 30% and 70%. Thus, water mass control with the linear reference water mass in Fig. 8 produces steady-state narrow range water levels in the range of traditional level controllers.
To provide insight into the parameters governing Steam Generator behavior, a simplified space average view of the Steam Generator was developed. Steam Generator water mass is the sum of the mixture mass in the bundle and riser sections, and the liquid mass in the downcomer. In terms of density, m = ρm V + ρdc Ax L
(6)
where Ax = cross-section flow area of downcomer; L = downcomer water level; m = water mass inventory in the SG; ρm = bundle and riser section mixture water density; ρdc = downcomer water density; V = bundle and riser section volume. Since the density is not measurable, and cannot be derived from other measurable parameters, water mass cannot be obtained directly from Eq. (6). Taking the derivative of Eq. (6) and rearranging, we get ∂L ∂m ∂ρm ∂ρdc m = f L, , , ρm , , ρdc , (7) ∂t ∂t ∂t ∂t
Fig. 9. Steady-state narrow range water level under the linear reference water mass controller.
If ρ = ρ(u, P), then ∂ρ/∂t = f(u, ∂u/∂t, P, ∂P/∂t), where these functions can be highly nonlinear. Since the internal energy u for single phase liquids and saturated mixtures can be expressed as u = f(T), or, u = f(αg , P) where αg is the void fraction. Then, ∂u/∂t = f(T, ∂T/∂t)) or ∂u/∂t = f(αg , ∂αg /∂t, P, ∂P/∂t).
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From eq. (7), the mass within the Steam Generator can be expressed as ∂αg ∂P ∂L ∂m ∂T , T, αg , , P, (8) m = f L, ∂t ∂t ∂t ∂t ∂t where ∂m/∂t can be approximated by the difference between feedwater flow rate and steam flow rate, P can be taken as the average steam pressure and T is the average single-phase liquid temperature. The average void fraction αg and ∂T/∂t are related to the energy input and output of the Steam Generator, i.e. ∂αg ∂T ∂E , ⇒ = f (W1 , T1hot , T1cold , Wfw , Tfw ) (9) αg , ∂t ∂t ∂t Combining Eqs. (8) and (9), and assuming the vapor mass is small, the liquid mass inventory becomes, ∂L ∂P , Ws , W1 , Wfw , T1cold , T1hot , Tfw , P, ml = f L, ∂t ∂t (10) where L = downcomer water level; Ws = steam flow rate; W1 = primary side water flow rate; Wfw = feedwater flow rate; T1hot = primary side hot leg temperature; T1cold = primary side cold leg temperature; Tfw = feedwater temperature; P = steam pressure. The variables on the right hand side of Eq. (10) are all measurable and available from current plant instrumentation. This equation can be utilized to design the mass inventory estimator. 4.2. Neural network water mass estimator Finding Steam Generator water mass from measurable plant parameters is a system identification problem. Mathematically, system identification is a functional approximation task for developing the plant model that is the most consistent with the measurable or simulation data. Since the water mass inventory is not measurable, only off-line training can be utilized to train the neural network given the data set collected from the Steam Generator model. Theoretically, all the measurable variables on the right hand side of Eq. (10) should be used as neural network inputs. But for ease of implementation and to improve the generality of the neural network, the following factors are further considered in picking the inputs for the water mass estimator: 1. Narrow range level and wide range level are both provided by the plant instrumentation system and are highly correlated. As a result, only narrow range level is used as an input for the neural network. 2. The primary side water flow rate is nearly constant, therefore W1 can be neglected as an input to the neural networks. 3. Heat transfer across the tube bundle is directly related to the average primary side temperature within the bundle. This can be approximated as the average between the hot leg and cold leg temperatures. Average temperature is chosen to replace T1hot and T1cold in training of the neural network.
In conclusion, the following nine measurable parameters are chosen as the inputs for the neural network to perform the water mass inventory estimation: • L = Steam Generator downcomer narrow range water level. • dL/dt = the first-order derivative of the downcomer water level. • Ws = steam flow rate. • Wfw = the feedwater flow rate. • Tave = the primary side average temperature. • Tfw = the feedwater temperature. • P = the steam pressure. • dP/dt = the first-order derivative of the steam pressure. • dm/dt = the first-order derivative of the water mass in the Steam Generator, which is approximated by the difference between feedwater flow rate and steam flow rate. The basic structure of the neural network developed to estimate the Steam Generator water mass is the two-layer neural network illustrated in Fig. 10. The input layer is connected to the plant instrumentation system. The hidden layer takes a summation of the weighted neural network inputs and evaluates them before they are passed to the output layer. The neural network output layer produces the predicted water mass signal for the water mass controller. Sigmoid nonlinearity is used in the hidden layer neurons. The output layer neuron uses pure linear functions to sum the weighted outputs from hidden neurons and bias the summation (Principe et al., 2000). The two most widely used sigmoid functions in neural networks are “logsig” and “tansig”. The difference being that the “logsig” function has an output between 0 and 1, while the “tansig” function has an output between −1 and 1. From computational experiments, neither “logsig” nor “tansig” functions showed significant differences in approximating the water mass inventory, and the “tansig” function was chosen for our design. The “tansig” transfer function is: f (net) = tanh(net)
Fig. 10. Two-layer neural network for water mass estimation.
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Table 1 Training and validating set for the neural networks Initial nominal plant power level (%) 20 40 60
Increase 10%, decrease 5%, 10%, 15%, 20% Increase 10%, decrease 10%, 20%, 30%, 40% Increase 10%, decrease 10%, 20%, 40%, 50%, 60% Increase 10%, decrease 10%, 20%, 40%, 60%, 70%, 80% Decrease 10%, 20%, 40%, 60%, 80%, 90%, 100%
80 100
Fig. 11. Training and validation error for different numbers of hidden neurons.
where net =
wi xi + b
i
w = neuron input weights; x = neuron inputs; b = neuron bias. One of the central issues in neural-computation is to appropriately set the number of hidden neurons. Too many hidden neurons may cause over-fitting and accumulate noise in the training signals; too few hidden neurons will have insufficient freedom to map the input/output function and the error will stabilize at a high value. Guidelines have been established for setting the number of hidden neurons (Baum and Haussler, 1989; Hrycej, 1997; Kolmogorov, 1969). In this research, the number and type of simulation data sets used for training and validation were frequently changed based upon results from the computational experiments, such that setting the number of hidden neurons analytically or by semi-analytical computational methods is not feasible. As a result, the number of hidden neurons used in this work is chosen empirically. Fig. 11 shows the training and validation results as a function of the number of hidden neurons. The neural network with 15 hidden neurons has the lowest mean square error in estimating the water mass inventory, and was selected for the remainder of the neural network development. 4.3. Training the neural network for water mass estimation The generality of the neural network over the state space of interest is affected by the data sets chosen for network training. From our previous analysis, the neural network will perform a 10-dimensional function approximation corresponding to the nine inputs and one output. With such a high dimensionality, the state space is large. To make the water mass estimator feasible, the choice of training and testing data should satisfy these two conditions: • The data must cover the whole state space, or at least the region of significance. • The distribution of training and testing data in the state space should not be heavily biased.
Step change (change in load)
If the neural network is applied to a region of state space that is far from where it is trained, the outputs of the neural networks are unpredictable and cannot be guaranteed to match the physical system. If the data densities in some regions are much higher than the data density in others, the neural network will be over trained in these regions and will not generalize well. To adequately represent the state space, it is necessary to collect as many simulation data sets as possible. Data generated when the plant is subjected to large transients are considered more important in training the neural network since transients of this type cover a wider region of state space. The transients included in Table 1 were used in developing the training and validating data sets for the neural network. The gains of the PI controller were set to very large values (kp = 2, ki = 0.2) compared with the nominal gains of the PI level controller (kp = 0.07, ki = 0.07) in our plant model. Step change responses were selected as opposed to ramp change responses due to their increased coverage of the state space. Since large load decreases are more frequent in real plant operation than load increases, the training set included more load decrease responses than load increase responses. For example, there can be a reactor trip that brings the plant from full power to zero power, but the plant is unlikely to experience a step increase in power level from zero to full power. In addition, more low power transients are selected because the “shrink and swell” effect at low power is more severe than it is at high-power level. To evenly distribute the training data sets in state space, data sets collected when the plant is at steady-state should be removed. During steady-state, the data repeat rate is very high and the neural network will be over-trained in these regions if these data are included in the training data set. Even after removing steady-state data, the data distribution in the state space is still not balanced. At the beginning of a step response, the plant states change so quickly that data is scarce in those regions. To balance the data distribution, the data sets collected in those regions can be repeated several times (bootstrapped). This is similar to giving those data higher weights. The data sets are duplicated according to their validation error in ratio to the mean square error of the whole training set after early stop training. nk =
|ek | N
(1/N)
i=1 |ei |
(11)
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Table 2 Range of training and validation data set Variable name
Minimum
Maximum
Mass (kg) Narrow range level (%) Steam pressure (Pa) Steam flow rate (kg/s) Feed-water flow rate (kg/s) Feed-water temperature (K) Tave (K) dP/dt (Pa/s) dL/dt (%/s) dm/dt (kg/s)
44,500 0 6.2e6 (900 psia) 0 0 305 (90 ◦ F) 555 (540 ◦ F) −9.0e4 (−13 psia/s) −0.5 −500
60,000 100 8.3e6 (1200 psia) 580 580 505 (450 ◦ F) 590 (600 ◦ F ) 9.0e4 (13 psia/s) 0.5 500
nk is how many times the kth data is duplicated; |ek | is the magnitude of the kth data validation error; N is the number of data in training data sets. To improve noise immunization of the neural network, 1% measurement noise in uniform distribution was added to each of the measurable inputs to the neural networks training sets. One practical issue in implementing the learning algorithm is the avoidance of neural network saturation. The output of the sigmoid activation function used to represent the hidden neuron will be close to either 1 or −1 when the weighted input net >2 or net <−2. This will stall the neuron weight updates and is a typical “retarded” learning example. One technique for solving this problem is to scale the neural network input values to be within (−1, 1) by normalizing the input and output vectors. In practice, sensor outputs are all normalized to a fixed range of voltage or current, such that the normalization here should not only make the network training converge more quickly, but also make future implementation easier. By simulating extreme cases of plant load changes, the minimum and maximum of the measurable variables were determined and are listed in Table 2. First-order derivative inputs are normalized within (−1, 1) to capture the changing direction of the parameters, while non-derivative parameters are normalized on (0, 1). The water mass estimator was trained by the second-order Levenberg–Marquardt algorithm (Park et al., 1995). Training results are used to bootstrap the data in fast transient regions,
Fig. 12. Distribution of water mass estimation error.
with the neural network retrained after each bootstrap. Early stopping is employed to prevent over-training. Simulation results show that the water mass estimator generalized well after training. The probability distribution function for the water mass estimation error in our validating data set is shown in Fig. 12. These results indicate that the water mass estimation error is sufficiently small to provide the water mass signal for the water mass controller. 4.4. Close-loop water mass control system using the water mass estimator The water mass estimator was embedded in the water mass control system and implemented in the plant simulator as indicated in Fig. 13. In this scheme, x are measurable plant parameters such as downcomer water level, steam pressure, etc. and dx/dt are first-order derivatives of selected parameters. A dead-zone nonlinear block was added after the derivative to prevent noise in the measurable parameters from giving an unreasonable parameter change rate and lead the water mass estimator into an untrained region of state space. The plant parameters x and their derivatives dx/dt are fed into the neural network water mass estimator. The output of the
Fig. 13. Water mass control system with neural network water mass estimator.
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Fig. 14. System performance under ramp load change.
Fig. 15. System performance with step load change at low power.
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Fig. 16. System performance for a step change in load at low power with large controller gains.
neural network is the estimated steam generator water mass. The estimated water mass is fed back to the control system, compared to a reference mass and a water mass error signal is generated. The error signal is fed to the PI controller and the PI controller generates a control signal to drive the feed control valve. Plant safety functions (i.e. reactor trips) are assumed to be based on conventional downcomer level signals, and therefore independent of the mass controller. The new control system was tested under different plant load changes. The gains of the PI controllers were varied to evaluate their effects on the system performance. Sensor noise was also included in the measurable parameters to see its effect on the system behavior. Fig. 14 illustrates controller performance for a 5%/min ramp load decrease from 60% power. The ramp load change lasted 2 min resulting in a final power of 50%. The PI controller gains were Kp = 1.0, Ki = 0.1. In this simulation, the water mass estimator gives a very good estimation of the true water mass inventory. The water mass controller brings the water mass and water level to steady-state within 200 s after the disturbance. The water mass controller also shows good performance for step changes in load at low power levels. Fig. 15 shows controller performance for a 10% step increase in load from 20% initial power. The proportional gain was set to Kp = 0.5. Integral gain was set to Ki = 0.07. The water mass estimator provides reasonable water mass inventory estimation for the duration of the transient and the water mass controller brings the system to steady-state smoothly.
The water mass control system provides acceptable performance at low power levels with large PI controller gains. In Fig. 16, the plant was subjected to a 10% step increase in load from an initial power level of 20%, resulting in a final power of 30%. The PI controller gains were Kp = 4.0 and Ki = 0.07. [The nominal gains of the PI controller are Kp = 0.07, Ki = 0.07 in the water level controller in our plant model.] The estimated water mass at the beginning of the transient has an error of about 7% normalized mass. After 50 s however, the controller brings the system into a region of state space where the water mass estimator provides a good estimation of the water mass inventory. The true water mass reaches the reference water mass approximately 200 s after the step change in load. Once the water mass reaches steady-state, the large PI controller gains cause the feedwater control valve to “chatter” resulting in changes in the feed flow, leading to oscillations in the estimated water mass. However, the true water mass inventory is not affected because of the integral effect of the PI controller. The narrow range water level also remains within the normal operational range. 5. Conclusion A new control strategy is presented that changes the controlled variable in Nuclear Steam Generators from downcomer level to steam generator water mass inventory. Compared with a conventional water level control system, the water mass control approach has been shown to have the following advantages:
W. Dong et al. / Nuclear Engineering and Design 238 (2008) 859–871
• The controlled system is a linear system with minimum phase, which simplifies both the control system design as well as manual operation; • The water mass inventory is a true measure of the instantaneous cooling capacity of the Steam Generator. Combined with a conventional level signal, safety could be enhanced. • Control of water mass inventory can increase margin to set points and relax constraints on system output resulting in a more robust system. • The controller design can accommodate “shrink and swell” caused not only by steam and water flow change, but also by the change of feedwater temperature, the change of Steam Generator primary side water temperature and the change of primary side water flow rate. • If reactor trip is still based on downcomer water level signals, safety functions are unchanged. For example, if the water mass estimator fails or the sensors providing inputs for the water mass estimator fail, normal safety systems will trigger a water level trip. In summary, the presented water mass controller is an improvement over the conventional water level control system in almost every aspect of control system design. Unlike most water level control systems, the water mass inventory within the Steam Generator is not measurable and must be inferred from measurable plant variables. A water mass estimator based on neural networks has demonstrated that a water mass control system is feasible, and simulation results indicate good controller performance over a wide range of operating conditions. References Baum, E.B., Haussler, D., 1989. What size net gives valid generalization? Neural Comput. 1, 151–160. Carmichael, L.A., Toye, G., Ipakchi, A. Sievers, M.W., 1984. Digital feedwater controller for a BWR: a conceptual design study, EPRI NP-3323. Choi, J.I., et al., 1989. Automatic controller for steam generator water level during low power operation. Nucl. Eng. Des. 117, 263–274. Dong, W., 2001. Water mass control system based on artificial neural networks for the steam generator in a pressurized water reactor, Ph.D. Thesis, Department of Nuclear Engineering, North Carolina State University. Dong, W., Mayo, C.W., Doster, J.M., 2000. A neural network based mass inventory observer for UTSG in PWR. In: International Topical Meeting on Nuclear Plant I&C, and Human–Machine Interface Tech, NPIC & HMIT 2000, Washington, DC.
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