Article pubs.acs.org/IECR
Early Warning System for Chemical Processes with Time Delay and Limited Actuator Capacity Mohammad Aminul Islam Khan, Syed Ahmad Imtiaz,* and Faisal Khan Safety and Risk Engineering Group of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, Newfoundland and Labrador A1B 3X5, Canada ABSTRACT: A new model-based predictive technique is proposed for generating early warnings to operators. The proposed method uses an open-loop process and disturbance model along with process measurements to predict the future state of a process. The method successfully monitors a process vulnerability (i) during the time-delay period and (ii) when the actuator does not have sufficient capacity. On the basis of the predicted process system, an alarm is generated when predictions exceed the threshold of safety limits. A moving horizon predictor combined with bias corrections predicts the dynamic output for the timedelay period, and a linear programming algorithm checks the feasibility of the solution using system gain and constraint information. The proposed method is applied to a single-input-single-output system as well as a multiple-input-multiple- output continuous stirred tank heater system. The method demonstrated performance in issuing early warnings that was superior to that of existing methods and was robust under small disturbances in the process.
1. INTRODUCTION Early detection of an abnormal situation is absolutely critical to avoid both human injuries and equipment damage. A welldesigned alarm system should be able to issue an early warning to operators with minimal number of alarms, i.e., keeping false alarms to a minimum. In order to minimize false alarms, filtering, dead-band, and delay are typically used. However, the implementation of these methodologies causes a detection delay to the fault. Significant research has been focused on optimizing these parameters to keep detection delay to a minimum.1−3 The other focus of research has been to keep the number of alarms to a minimum. Because of the large number of sensors, multiple alarms can be triggered from a single abnormal cause in the process. Elimination of nuisance alarms is important because an operator’s capacity to deal with number of alarms under a stressed condition is limited. Multivariate statistical methods have been used effectively to compress a large set of variables to a small number of pseudo variables and successfully reduced the number of alarms. Since the pioneering work by Kresta et al., multivariate statistical methods such as principal component analysis (PCA) and partial leastsquares (PLS) have been used extensively to monitor chemical processes.4−6 In multivariate analysis, fault is defined as the breakdown of the correlation between variables, which often gives an indication of fault earlier than univariate methods. In alarm system design, emphasis is placed on robustness. Process plants rarely use alarms based on prediction signals. Predictive monitoring can be an efficient tool for the successful forecasting of an abnormal situation. Significant work on predictive alarm systems include a Kalman predictor-based alarm system developed by Juricek et al.7 Zamanizadeh et al. developed an extended Kalman filter-based approach for dealing with process nonlinearity.8 Fernandez et al. proposed a supervisory method for predicting an abnormal situation. When a process variable reaches critical value, the monitoring system starts trending the input and output data using a neural network (NN) and generates alarms based on the trend of the system.9 These © 2014 American Chemical Society
methods had varying degrees of success. One of the main shortcomings of these methods is they use closed-loop prediction for alarm generation; as a result, any change in controller tuning or structure will affect the alarm systems, requiring frequent updating. Also, these methods do not use the full prediction horizon (i.e., prediction up to the steady state) for monitoring; for example, a Kalman filter is typically used for predicting system trend for few time steps, which may not be sufficiently early for corrective action. The objective of this study is to develop a simple, maintenance-free predictive alarm system that will use some of the preexisting resources in a process industry. The main contributions of this paper are listed below: • A framework for systems monitoring using open-loop model during the delay period and for actuator limitations is proposed. • Detailed methodology to generate alarms for any immediate effect of disturbances (called dynamic alarm generation) and long-term effect of disturbances (called steady-state alarm generation) is developed. • The proposed method explicitly separates controller from model prediction. It uses open-loop model (e.g., step response model) typically available from model predictive controller (MPC). • The monitoring system does not use any controller information, as such it remains valid under any controller tuning changes.
2. PROPOSED PREDICTIVE ALARM SYSTEM In many plants where model predictive controller (MPC) applications control the process, the process model and Received: Revised: Accepted: Published: 4763
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models are often used in process industry to tune controllers; also, DMC contains step response models. A step response model can be either converted from other parametric models or estimated directly from the step response.10 The given step response coeffecients between input u and output y are ak, and step response coefficients between the disturbance ud and output y are dk. Also, assuming that no unmeasured disturbance has affected the process, the following convolution model can be used to describe the system:
disturbance model are often available. The proposed early warning system uses such models for early alarm generation. These are models without any controller knowledge. There are several advantages for using open-loop predictions: (i) an openloop model may be already available from an MPC application; (ii) typically, the controller tuning parameters are changed frequently in a process; therefore, closed-loop models need to be updated regularly. Without any controller information, open-loop models remain valid for a longer period. Considering the above facts, open-loop predictions are used for alarm generation. However, it is understood that an open-loop prediction based upon past inputs, called free response in dynamic matrix controller (DMC) terminology, may show that a process variable will violate the threshold, but the future controller action may actually keep the process within the normal operation limit. Therefore, the effect of the controller must be taken into account. In this paper, we state two limiting conditions where the controller does not affect the free response prediction. Under these conditions, free response predictions can be used for alarm generation. Condition 1: In a system with time delay, any controller regulatory action will be reflected on outputs only after the time-delay period has elapsed. Thus, the free response predictions will closely match the measured output within the time-delay periods. Condition 2: At steady state, the ability of a controller to bring a process variable within the control limit will depend entirely on the available actuator capacity and steady-state gain of the process. On the basis of the two conditions above, we develop two alarm generation protocols for the process system which are presented in the following subsections. 2.1. Dynamic Alarm Generation. Condition 1 states that, within the time-delay period, the free response predictions and the actual measurement are the same. This provides a window in which the open-loop prediction based upon the past inputs can be used to monitor the process. This is defined as the monitoring horizon. Within the monitoring horizon, an alarm will be generated if a free response prediction exceeds the alarm threshold. At each execution cycle t, using the output from the controller and measured disturbances to the process, the models predict the process output for the entire monitoring horizon, tp. As new measurements become available, the predicted values are compared with the measurements; if a bias is observed, the predicted values will be corrected for the bias. A Kalman filter type update (i.e., use of a fraction of bias based on adaptive gain), rather than simple bias correction, can be used to update the predictions. However, for most practical purpose, first-order filter would be sufficient to filter the noisy measurement. Subsequently filtered measurements can be used for bias correction. The detailed steps of the methodology are described below. Consider the following dynamic system with a single control input u and measured disturbance ud y(s) = G(s)u + D(s)ud + e
∞
yt =
∞
∑ akΔut− k+ ∑ dkΔutd− k k=1
(2)
k=1
where Δut and Δutd are changes in control input and disturbance at time t. The step response model can be conveniently separated into the following four terms. h
yt =
h
∞
∑ akΔut− k+ ∑ k=1 ∞
k=1
∑
+
dk Δutd− k+
∑
ak Δut − k
k=h+1
dk Δutd− k
(3)
k=h+1
where h is the number of past inputs that affect the current outputs, which is given by the settling time of a finite impulse response model. The last two terms of eq 3 represent the accumulated response due to control action and measured disturbances that affected the system from infinite past all the way to time t−h−1. By denoting these two terms as Zt, eq 3 can be rewritten as h
yt =
h
∑ akΔut− k + ∑ dkΔutd− k + Zt k=1
(4)
k=1
For monitoring purposes, our objective is to predict the system output for at least the next p samples such that monitoring horizon tp ≥ system time delay. We can use the step response model for prediction purposes by time shifting the model appropriately as h
yt +̂ l =
h
∑ akΔut + l− k +
∑ dkΔutd+ l− k + Zt + l
k=1
k=1
(5)
where l = [1, 2, 3,..., p]. The step response model can be further segregated into terms containing future input changes and terms containing past input changes l
yt +̂ l =
l
k=1 h
+
h
∑ akΔut + l− k + ∑ dkΔutd+ l− k ∑
∑
k=1
ak Δut + l − k
k=l+1
dk Δutd+ l − k + Zt + l
(6)
k=l+1
where the first two terms contain the future input changes and the rest contain past input changes. At time t, the future input changes are still not available; therefore, for dynamic alarm generation we use only past inputs and disturbance changes to predict the future output, which in DMC terminology is known as free response and is given by
(1)
where G(s) is the transfer function between control input u and output y, D(s) the transfer function between disturbance ud and output y, and e a lumped parameter consisting of measurement noise and unmeasured disturbances. The first step is to predict the future output for the monitoring horizon tp, i.e, next p samples. To recursively predict the future output, we use a step response model to describe process dynamics. Step response
h
yt*+ l =
h
∑
ak Δut + l − k +
k=l+1
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∑ k=l+1
dk Δutd+ l − k + Zt + l
(7)
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where Zt+l comes from past measured output. For multipleinput-multiple-output(MIMO) system with inputs u(m×1), outputs y(n×1), and disturbance ud(r×1) we can generalize eq 7 for output yi,t+l as m
yi*, t + l =
h
∑ ∑ j=1 k=l+1
+ Zi , t + l
r
aij , k Δuj , t − k +
will bring the system back to the original state or the system will attain a new state that may or may not be within the safety limits. Steady-state conditions are checked to identify whether the controller can bring the variable within the safe operation limits using the available actuator capacity. If it appears that the actuators do not have sufficient capacity to bring the process within the safety limit, an alarm is generated. An alarm generation requires a prediction of the open-loop steady-state values due to the disturbance, calculation of the capacity of the actuators, and maximum possible control action on the variables. The open-loop steady-state value of a variable is predicted by adding the change in the process variable due to the disturbances with the current steady-state value. Consider a step type disturbance udt entering the system at time t. Assuming the system is at steady state, the final value for the ith output is given by
h
∑ ∑
dij , k Δujd, t − k
j=1 k=l+1
(8)
where aij are the step response coefficients between jth input and ith output and dij are the step response coefficients between jth disturbance and ith output. More detailed derivation of the recursive prediction can be found in refs 10−12 described in the context of DMC. In step 2, online output measurements are used to correct the predicted values. At every execution cycle, the output is corrected by comparing the current measurement with the predicted value from the model. However, in the case of the process having measurement noise, online measurement cannot be used directly for bias correction. So the online measurements are passed through a filter, and filtered data can be used for bias correction. The difference between filtered outputs with the predicted values from the model give the bias errors. The bias error at time t can be calculated using eq 9: bt = yt filt − yt*
h
yiss = yi , t +
k=1
(11)
Because it is a step type disturbance, all future changes in disturbance Δudt+1 = Δudt+2 = Δudt+3 = ... = Δudt+h−1 = 0 and eq 11 reduces to yiss = yi , t + dhΔutd
(12)
For implementation purposes, we used the filtered value yfilt i,t instead of yi,t. The minimum requirement from a controller is to make changes in the actuators such that the output remains within the control limits. Assuming that the high and low limits for the ith output are yi,low and yi,high, respectively, the controller must satisfy the following condition:
(9)
where y*t is the one-step-ahead prediction at time t − 1 and yfilt t is the filtered output at time t. On the basis of the calculated error at time t, bias correction is done on all future predictions, as given in eq 10: yt +̂ l = yt*+ l + bt
∑ dkΔutd+ h − k
(10)
yi ,low ≤ yiss + Δyiss ≤ yi ,high
where l = 1, 2,..., p. The updated predictions show the effect of disturbances earlier than the process measurements because they are based on both process and disturbance models. Step 3 is alarm generation. An alarm limit is set for each variable based on process knowledge. At each instant, the predicted values are checked against the limits. If the prediction exceeds the limit within the monitoring horizon, an alarm will be issued to alert the operator. Step 4 improves the robustness of alarm. A single value can sometimes exceed the limit because of measurement noise. To make the alarm robust and avoid nuisance alarms, a further heuristic rule is applied. If three consecutive predicted values cross the limit only at that point, an alarm will be issued. However, this rule can be adjusted depending on the severity of the consequences associated with the variable. 2.2. Steady-State Alarm Generation. The steady-state alarm generation algorithm is developed based on condition 2, which was described earlier. Suppose that a process is at steady state and a disturbance affects the system; if the system is openloop and there is no controller acting on the system, the steady state of the system will be disturbed and the system will eventually become steady at a new state. However, in a closedloop system, the controller will take corrective action and will try to bring the system back to its original state. Assuming the controller is perfect or efficient, the ability of the controller to bring the system back to the original state is limited by the available actuator capacity of the controller. Therefore, depending on the available actuator capacity, either a controller
(13)
where Δyssi is the steady-state change in the ith output due to the input changes made by the controller. For multiple inputs, at steady state the input and output changes are related by the process gain as given by m
Δyiss =
∑ aijssΔuj (14)
j=1
assij
where is the step response coefficient at steady state, which is equivalent to the process gain; eqs 13 and 14 are combined to express the desired condition in terms of the input variable. m
yi ,low −
yiss
≤
∑ aijssΔuj ≤ yi ,high − yiss j=1
(15)
The capacity of an actuator is given by the difference between the current steady-state value of the actuator (e.g., valve) and high and low limits known from the actuator range, which can be written using the input constraint uj ,low − uj , t ≤ Δuj ≤ uj ,high − uj , t
(16)
where uj,low and uj,high are the low and high limit values of the actuator, respectively. The controller will be able to bring all the process variables within the desired limits only if eqs 15 and 16 are satisfied simultaneously. Therefore, eqs 15 and 16 give the desired condition for steady-state alarm generation. If these two equations cannot be satisfied simultaneously, an alarm will be 4765
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Figure 1. Implementation steps of proposed alarm system.
within the time-delay period, an alarm will be generated. Otherwise, the algorithm will proceed to the next time step.
issued. Linear programming (LP) algorithms are widely used to calculate a target and check the feasibility of a target, typically in the upper layer of MPC.13 Linear programs can be used to check the existence of a feasible solution for the output constraints arising from eq 15 and input constraints arising from eq 16. For example, for a system with m inputs and n outputs, there will be m input constraints and n output constraints. An alarm is issued if there is no feasible solution that satisfies all (m + n) constraints. 2.3. Implementation of the Alarm System. The alarm system discussed in this section has two integral parts that monitor the process in two different situations. However, these two algorithms will work in harmony with each other. The implementation steps of the overall alarm system are shown in Figure 1. The steady-state alarm generation method is the top layer of the alarm system. It monitors for the long-term effect of the disturbance. If the LP calculation does not find a feasible solution, an alarm will be issued. On the other hand, if there is a feasible solution, the algorithm will proceed to the dynamic alarm calculation step. The dynamic alarm generation algorithm will use the dynamic model to predict open-loop outputs. The outputs will be updated with the measured values using a bias correction method. If the predicted outputs cross the threshold
3. CASE STUDIES To explain the implementation steps clearly, the proposed methodology is first applied to a single-input-single-output (SISO) system. Then the methodology is applied to a multipleinput-multiple-output (MIMO) continuous stirred tank heater (CSTH) system. 3.1. A Simple SISO Example. Consider a simple SISO system with a disturbance input, as described in eqs 17a−17c: y = G(s)u + D(s)ud + e
(17a)
G (s ) =
e−14.7 s 21.3 s + 1
(17b)
D(s) =
1 25 s + 1
(17c)
where y is an output variable, u an input to the process, and ud a measured disturbance. The system is controlled using a feedback controller, the most common industrial scenario. There is a system time delay of 14.7 s. 4766
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3.1.1. Dynamic Alarm Generation. The purpose of dynamic alarm generation is to monitor the process during the timedelay period or monitoring horizon when the controller does not have any influence on the system. In this case, 16 s is chosen as the monitoring horizon, which is slightly greater than the system time delay. The output over the monitoring horizon is predicted through finite step response models identified through system identification. The sampling interval for the system is 2 s; therefore, at each instant, predictions are made for the next eight samples. At time t = 200, a disturbance ud of step size 5 is applied to the process. In this case, we considered the safe operating limit of the system to be 3. The predicted responses of the system at different times are shown in Figure 2a. At 201 s, the predicted
was generated based solely on the process measurement, the earliest an alarm can be issued is at 213 s. The proposed scheme generated alarms 9 s earlier than with conventional alarm generation. 3.1.2. Steady-State Alarm Generation. Given the current steady-state conditions, process gain, and safety limits, the proposed scheme checks whether the controller has enough capacity to keep the process within the safety limits. At steady state, the following relationship exists between the input and output:
Δy ss = Δu
(18)
where Δyss is the change in measured output at steady state and Δu is the maximum available capacity of the input; the steadystate gain is 1 for this process. High and low limit values for the output are 2 and −2, respectively, whereas for the input variable, capacity varies from −7 to 7. Thus, inequality constraints for this process can be rewritten in input space as −2 − yiss ≤ Δy ss ≤ 2 − yiss
(19)
−7 − uj , t ≤ Δu ≤ 7 − uj , t
(20)
Two different disturbance scenarios were simulated to check the steady-state alarm conditions. In the first scenario, a disturbance of step size 10 is introduced to the system at t = 200. Given that the steady-state values for input and output at t = 200 are u200 = 1 and y200 = 1, the steady-state value for the process at any instant can be predicted using eq 12, which gives the open-loop steady-state value for output, yiss =11. Substituting these values in eqs 19 and 20, we obtain the following inequality constraints arising from output and input limitations: −13 ≤ Δu ≤ −9
(21)
−8 ≤ Δu ≤ 6
(22)
which are plotted in Figure 3. It shows clearly that there is no common space between these two inequalities (A ∩ B = ⌀).
Figure 3. Constraint inequalities for the first scenario. Figure 2. Predictions over horizon at the time of alarm generation and process measurement.
Therefore, a feasible solution does not exist. It is easy to graphically plot and visualize the feasibility in a simple system; however, for complex systems, it is not always possible to graphically represent the inequalities. In such a case, LP can be used to seek a feasible solution. For example, in this case the LP algorithm could not find a feasible solution either, confirming that there is not enough capacity in the actuator to bring the output within the limit. Therefore, an alarm will be issued at t = 200.
response first showed that the output will exceed the threshold at 213 s; however, in order to increase confidence, the alarm was issued at 204 s when three predicted responses exceeded the safe-operating limit. In Figure 2b, the closed-loop process measurement validates the predicted system response. The measured output exceeded the threshold at 213 s. If an alarm 4767
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For the second scenario, a disturbance of step size of −5 is applied to the system at t = 600. Measured output and input at t = 600 are y600= 3 and u600 = 1, respectively. The predicted steady-state output yssi = −4. The inequality constraints for output and input after expressing in the input space are plotted in Figure 4, which shows that there is a feasible solution A ∩ B ≠ ⌀. Therefore, no alarm will be issued in this instance.
2 ≤ Δu ≤ 6
(23)
−6 ≤ Δu ≤ 6.
(24)
Figure 4. Constraint inequalities for the second scenario.
These results are verified in Figure 5a,b, which shows the closed-loop process responses for these two scenarios. Figure 5a shows that the output measurement remains outside the limit at steady state. The measured signal crosses the threshold at 208 s. Therefore, on the basis of the conventional method, an alarm will be issued at 208 s, whereas using the predictive approach the alarm will be issued at t = 200 s. Conversely, Figure 5b shows that the method is robust to false alarm; it does not issue an alarm when the controller is able to nullify small disturbance effects. 3.2. A MIMO Example. The proposed predictive alarm protocol is applied to a continuous stirred tank heater (CSTH) presented by Thornhill et al.14 The study comprises a dynamic model as well as experimental data of a pilot plant located in the Department of Chemical and Materials Engineering at the University of Alberta. For fault diagnosis purposes an equivalent simulink model for the plant is used in the present study. Although this is a simulated model, it is very realistic, as it uses noise obtained from actual sensors. Figure 6 shows the schematic diagram of the CSTH plant. Steam and hot water heat the cold water in a tank. The process dynamics of the plant are discussed in detail in the contribution of Thornhill et al.14 The flow of steam, cold water, and hot water can be manipulated using control valves. The system can be represented by the following equation: ⎡ y1 ⎤ ⎡G11(s) 0 ⎤⎡ u1 ⎤ ⎡ D1(s) ⎤ d ⎥⎢ ⎥ + ⎢ ⎥u ⎢ ⎥=⎢ ⎣ y2 ⎦ ⎢⎣G21(s) G22(s)⎥⎦⎣ u 2 ⎦ ⎢⎣ D2(s)⎥⎦
Figure 5. Simulated results of process variable measurement with limit values for steady state.
stated in Table 1. Identified models for these operating points are mentioned in Thornhill et al.14 The literature also states that the temperature measurement delay for the process is 8 s. The system is controlled using a DMC. The water level and water temperature in the tank are two measured outputs. The DMC manipulates the steam valve and cold water valve to control the water level and temperature of the tank. For this study, the hot water valve position is considered to be a measured disturbance to the system. Units of the different variables (e.g., temperature, level) in the prior work14 are in milliamperes. In our work, we converted the units to conventional units (e.g., temperature in degrees Celsius and level in centimeters). 3.2.1. Dynamic Alarm Generation. Several disturbance scenarios were simulated through changes in the hot water valve position. The monitoring system tracked the process for these abnormal conditions. Here, we report two such scenarios. At time t = 600 s, the hot water valve is opened from 7% to 8%. Both output variables start to increase from their set point with the introduction of the disturbance. As the process time delay is 8 s, the disturbance starts to affect the process measurements at t = 608 s. However, level and temperature predictions are continuously assessed over the monitoring horizon, which in this case are the next eight samples at every time interval. The predicting model used for the alarm generation contains the process model as well as the disturbance model relating hot water flow to level and
(25)
where y1 is the level, y2 the temperature, u1 the cold water valve position, u2 the steam valve position, and ud the hot water valve position. Standard operating points for the simulink model are 4768
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Figure 6. Schematic diagram of the CSTH plant.
Table 1. Operating Points of CSTH for Predictive Monitoring variable
operating point
level (cm) temperature (°C) CW valve (%) steam valve (%) HW valve (%)
20.50 42.50 17.64 9.77 7.14
temperature. The predicted response is also corrected for bias at every second based on actual measurements. Because the monitoring scheme contains the disturbance model, as soon as the disturbance entered the system, the prediction showed the effect of the disturbance on the output variables. In this case, we considered the alarm threshold to be 43.2 °C. The predicted responses are shown at different time instants in Figure 7a. At 625 s the prediction showed that the output will exceed the threshold at 633 s. However, for increased reliability, the alarm was issued at 627 s when three predicted values exceeded the threshold. In Figure 7b, the closed-loop process measurement shows that the measured output exceeded the threshold at 634 s. If an alarm was generated based solely on the process measurement, the earliest it could be issued is at 634 s. In this case, the proposed alarm system issues the alarm 7 s earlier than an alarm system based on a measured signal, which gives the operator time to take corrective action. 3.2.2. Steady-State Alarm Generation. The outputs, level y1 and temperature y2, and inputs, steam valve position u1 and cold water valve position u2, give rise to four constraints. The output constraints involve the safe operation of the process system, and the input constraints are due to the limited capacities of the valves. In addition to these constraints, there also exist the input−output relationships arising from the steady-state process gain. Equations 26a and 26b give the input−output relationship for the CSTH system at steady state: Δy1ss = 2.766Δu1
Figure 7. Predictions at different times and process measurement in the dynamic state.
Δy2ss = −0.293Δu1 + 0.369Δu 2
(26b)
(26a) 4769
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Figure 8. Constraint inequalities for the first scenario.
where Δy1ss is the change in level, Δy2ss the change in temperature, Δu1 the change in cold water valve position, and Δu2 the change in steam valve position. The high and low limit values for the level are defined as 25.2 and 15.8 cm, respectively. For temperature, high and low limits are 43.2 and 39.2 °C, respectively. For both the cold water valve position and steam valve position, high and low limit values are selected as 100% and 0%, respectively. Using these values, four inequality constraints for the system can be written as in eqs 27a−27d: 15.8 − y1ss ≤ Δy1ss ≤ 25.2 − y1ss
(27a)
39.2 − y2ss ≤ Δy2ss ≤ 43.2 − y2ss
(27b)
0 − u1, t ≤ Δu1 ≤ 100 − u1, t
(27c)
0 − u 2, t ≤ Δu 2 ≤ 100 − u 2, t
(27d)
a feasible solution; therefore, no alarm is issued. Constraints are also depicted in Figure 8, where the feasible region that satisfies all four constraints simultaneously is shown by the hatched area. Therefore, the process will be at no alarm state at t = 600 s, despite the disturbance being present. This result is also supported by the actual closed-loop measurements, which show that level and temperature do not exceed the alarm limits for the above disturbance scenario (Figure 9). The second
For the first disturbance scenario, the hot water valve position is changed from 7.1% to 7.6% at t = 600 s. This change of hot water valve position causes a rise in both the level and temperature of the water from their nominal values of 20.5 cm and 42.5 °C, respectively. The steady-state values of the process variables are predicted using eq 12. For this scenario, the predicted open-loop steady-state value of the process variables are yss1 = 50 cm and yss2 = 42.85 °C. Also, the input values at t = 600 s are u1,600 = 17.95% and u2,600 = 9.79%. Substituting these values in eqs 27a−27d, we obtain the following output and input constraints: −34.2 ≤ Δy1ss ≤ − 24.81
(28a)
Δy2ss
(28b)
−3.65 ≤
≤ 0.36
−17.95 ≤ Δu1 ≤ 82.05
(28c)
−9.79 ≤ Δu 2 ≤ 90.2.
(28d)
These inequalities, together with the steady-state input− output relationships described in eqs 26a and 26b, are used to check for feasibility for Δu1 and Δu2. At every instant, the LP algorithm checks whether there is a feasible solution for these constraints. For this disturbance scenario, the LP is able to find
Figure 9. Simulated results of level and temperature measurement with limit value for scenario 1. 4770
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Figure 10. Constraint inequalities for the second scenario.
disturbance scenario is similar to scenario 1, except that a bigger step size was considered. The hot water valve position is changed from 7.1% percent to 9.5% percent at t = 800. The consequence of the introduction of this disturbance is the same as that in the previous scenario, but with a greater intensity because the disturbance size is larger. For this scenario, the predicted open-loop steady-state values of the process variables are yss1 = 50 cm and yss2 = 44.21 °C, and the input values at 800 s are u1,800 = 17.95% and u2,800 = 9.79%. Using these values, output and input constraints for the process can be written as −34.2 ≤ Δy1ss ≤ − 24.81
(29a)
Δy2ss
≤ − 1.01
(29b)
−17.95 ≤ Δu1 ≤ 82.05
(29c)
−9.79 ≤ Δu 2 ≤ 90.21.
(29d)
−5.01 ≤
For the given conditions, a feasible solution that meets all the constraints does not exist. This is also depicted in Figure 10, which shows there is no feasible region for the given conditions. Therefore, an alarm is issued at the time the disturbance is measured (at t = 800 s). Closed-loop process measurements for this particular scenario are presented in Figure 11, which confirms that the tank temperature exceeds the alarm limit at 825 s. Therefore, the proposed system was able to provide a warning to the operator 25 s earlier than the system that would generate an alarm based on process measurement.
4. CONCLUSIONS A new model-based predictive alarm generation technique is proposed. The proposed methodology uses open-loop process and disturbance models to predict the system responses. These open-loop models are developed using system identification around the operating point. These models are chosen because chemical processes are typically operated near the constraints; therefore, models built for control purposes are actually valid near the constraints. For highly nonlinear systems, the identified linear model may not remain valid. In that case, nonlinear models must be used; however, the methodology remains the same. The open-loop responses are bias-corrected
Figure 11. Simulated results of level and temperature measurement with limit value for Scenario 2.
using the available measurements to nullify the mild nonlinearity. Alarms are generated based on the bias-corrected predictions. Two limiting conditions arising from controller and actuator limitations were identified, and on the basis of these two conditions, alarm generation methodologies were developed. The dynamic alarm generation procedure looks at the immediate effect of the disturbance, and the steady-state 4771
dx.doi.org/10.1021/ie402101x | Ind. Eng. Chem. Res. 2014, 53, 4763−4772
Industrial & Engineering Chemistry Research
Article
(13) Zou, T.; Li, H.; Zhang, X.; Gu, Y.; Su; H. Y. Feasibility and soft constraint of steady state target calculation layer in LP-MPC and QPMPC cascade control systems. In Proceedings of 2011 International Symposium on Advanced Control of Industrial Processes (ADCONIP), Hangshou, China, May 23−262011. (14) Thornhill, N. F.; Patwardhan, S. C.; Shah, S. L. A Continuous Stirred Tank Heater Simulation Model with Applications. J. Process Control 2008, 18, 347−360.
alarm generation procedure monitors the process for long-term effects of a disturbance. The method has several advantages: (i) it provides an alarm earlier than any conventional alarm generation method; (ii) once developed, the system does not require frequent updating, as it uses open-loop models for predictions; and (iii) the technique is robust because it exploits the fundamental limitations of controller and actuator for alarm generation. The methodology has been applied to a SISO system and a more complex MIMO system, where the technique was used to monitor the system for different disturbance scenarios. In both examples, the methods generated alarms in a consistent manner and demonstrated robustness against false alarms.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank Research and Development Corporation (RDC) and Natural Sciences and Engineering Research Council (NSERC) for financial support.
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REFERENCES
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