Optim Eng (2013) 14:547–563 DOI 10.1007/s11081-013-9237-3
Economic oriented stochastic optimization in process control using Taguchi’s method András Király · László Dobos · János Abonyi
Received: 5 March 2011 / Accepted: 6 September 2013 / Published online: 24 October 2013 © Springer Science+Business Media New York 2013
Abstract The optimal operating region of complex production systems is situated close to process constraints related to quality or safety requirements. Higher profit can be realized only by assuring a relatively low frequency of violation of these constraints. We defined a Taguchi-type loss function to aggregate these constraints, target values, and desired ranges of product quality. We evaluate this loss function by Monte-Carlo simulation to handle the stochastic nature of the process and apply the gradient-free Mesh Adaptive Direct Search algorithm to optimize the resulted robust cost function. This optimization scheme is applied to determine the optimal set-point values of control loops with respect to pre-determined risk levels, uncertainties and costs of violation of process constraints. The concept is illustrated by a well-known benchmark problem related to the control of a linear dynamical system and the model predictive control of a more complex nonlinear polymerization process. The application examples illustrate that the loss function of Taguchi is an ideal tool to represent performance requirements of control loops and the proposed Monte-Carlo simulation based optimization scheme is effective to find the optimal operating regions of controlled processes. Keywords Monte-Carlo simulation · Model predictive control · Economic assessment · Stochastic optimization 1 Introduction Due to the dynamic and significant changes of the economic environment, performance assessment of process control is a highlighted area of chemical engineering (HoLee et al. 2010). Our aim is to develop an optimization framework based
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A. Király · L. Dobos · J. Abonyi ( ) Department of Process Engineering, University of Pannonia, Egyetem street 10, Veszprém, 8200, Hungary e-mail:
[email protected]
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on Taguchi’s method (Taguchi et al. 1989) designed to determine optimal operating regimes of chemical processes by taking process constraints, desired maximum number (frequency) of constraint violations and process uncertainties into consideration. Variance in the closed control loop caused by unmeasured disturbances and badly designed controllers might cause variations in the product quality. In case of increasing variance of process variables the probability and frequency of violation of quality requirements is increasing that might lead to increase the amount of less valuable offset products. Typical examples when reduced number of violations of the predetermined process constraints is acceptable can be found in the field of statistical process control (SPC; Oakland 2007). In SPC statistical tools are applied to monitor the performance of the production process and detect significant deviations that may later result in offset products. We follow a more sophisticated model based approach during this work. Modern process analysis, monitoring, control and optimization tools are mainly based on some kind of process model. It is obvious to utilize these process models also in the economic assessment and optimization. Usually the output of cost-benefit analysis is cost reduction or profit increment expressed by a cost function. These functions incorporate the costs of the operation, raw materials, current prices of products (Bauer and Craig 2008), and risks of malfunctions. In our economic oriented optimization strategy the aim is to find steady state operation points (controller set points) where profit is realized. This task is fulfilled at the supervisory control level (Chen et al. 2012). The general approach for economic performance evaluation comprises the following steps: reduce the variance of the controlled variable and shift the set points (process mean) closer to the operation limits (Lee et al. 2008) without increasing the frequency of the violation. This operation is referred to the improved control (Zhao et al. 2009). The variance reduction might mean to re-tune the existing controllers, or, in more radical cases, change the whole control strategy. To handle uncertainty and effects of measurement noise, a novel Monte-Carlo (MC) simulation based approach is proposed. MC simulation is frequently applied in various areas (Rubinstein and Kroese 2008). This tool has also been proven its efficiency in risk related optimization of chemical processes, e.g. it is applied in optimizing maintenance strategies of operating processes (Borgonovo et al. 2000). There is a common characteristic in these solutions: the stochastic nature of the studied system has to be modeled. In the applied methodology this simulation is related to the modeling of the unmeasured disturbances of the control loops. To handle this random effect, MC simulation is applied with the characterized noise. An economic cost function is calculated in each case to measure the economic efficiency of the process. We use Taguchi’s quality loss function to adjust the economic benefit, which is also based on MC simulation. Using this function, soft constraints can be applied for the optimization. Taguchi’s loss function is a widely used method for quality control (Taguchi et al. 1989). Di Mascio and Barton proposed a novel technique for measuring dynamic control quality within the Taguchi framework to compare several controllers based on performance and stability (Mascio and Barton 2001). Maghsoodloo et al. (2004) give a comprehensive analysis of Taguchi’s contribution to the field of quality from a statistical (presenting Statistical Quality Control) and an engineering
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viewpoint. Using Taguchi’s loss function in robust stochastic optimization together with MC simulation can be often found in the literature. Zang et al. (2005) give a review of robust design based on these concepts in dynamics, while Yang and Chou propose a hybrid Taguchi method to solve a multi-response simulation-optimization problem (Yang and Chou 2005). Rezaie et al. present a robust tool to handle uncertainty permutations by MC simulation (Rezaie et al. 2007). Recent methods can be found in Subbaraj et al. (2011), where Taguchi’s function is used to develop a self-adaptive real-coded genetic algorithm, while Yang and Peng (2012) propose an improvement of Taguchi method by the integration of particle swarm optimization. Integrating this benefit analysis tool into the mesh adaptive direct search optimization algorithm—where the task is to find the most beneficial steady state operation point—resulted the proposed economic oriented optimization framework. In the proposed multilayer optimization framework, the application of gradient based methodologies for maximizing the economic throughput is not possible, due to the stochastic characteristics caused by the closed loop variance. Therefore, the utilization of direct search methods are necessary. Mesh Adaptive Direct Search (MADS) (Audet and Dennis 2006) class of algorithms is a relatively new set of direct search methods for nonlinear optimization, i.e. these algorithms are capable of calculate the extremums of a non-smooth functions, like our economic objective function. Since the steady state operation points are mainly determined by the variance of the controlled variables, incorporating this effect into the model is inevitable. The created optimization framework functions as an industrial Advanced Process Control system (Ray 1981; Bauer and Craig 2008). The paper is organized as follows: in Sect. 2 the economic cost function based multilayer optimization framework is introduced. In Sect. 3 the applied methodology is explained in detail. In Sect. 4, the efficiency of the proposed methodology is illustrated throughout a linear benchmark control problem. The economic performance measure has been formalized as a basis for optimizing the set point signal. In this simple case of the benchmark example the process is controlled with a PI controller. To determine optimal set point of the controller, Taguchi’s quality loss function is applied to maximize the economic benefit. As a second example a non-linear process controlled by a linear model based predictive controller MPC (MPC; García et al. 1989) is considered, since this combination highly applicable for variance-reduction purposes is widely applied in chemical process industry. In this case study the process variance is caused by an unmeasured disturbance, model mismatch, and noise added to the controlled variable. In this example the effect of the unmeasured disturbance with different amplitude is examined in detail. These examples show the realistic benefits of the proposed methodology.
2 Taguchi loss function based multilayer optimization The proposed optimization framework could be considered as a multilayer optimization problem, as depicted in Fig. 1. This framework is rather similar to the Advanced Process Control (APC) systems applied in process industry for online profit optimization (Bauer and Craig 2008).
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Fig. 1 The layers of an economic optimization of an operating technology
In the following sections the main aspects and tasks are introduced, which have to be taken into consideration in different optimization levels. 2.1 Applying Taguchi’s loss function at the supervisory control level The main task at the supervisor level is to maximize the economic throughput by varying the steady state set point signal. In general the economically optimal set point is close to the operation limits of the process. For that reason the reduction of the closed loop variance is necessary. Due to the process variance—caused by disturbances, noise, etc.—there is a risk of process constraint violation which has to be taken when the new set point is determined. The essence of this economic optimization approach is depicted in Fig. 2. The aim of the economic oriented process optimization framework could be formulated as minimizing a quality cost function. As we discussed earlier, the economic loss can be expressed by the Taguchi loss function which is formulated mathematically in basic case as follows: (y − t)2 LSL ≤ y ≤ USL L(y) = (1) otherwise Cr where L(y) is the loss associated with the value of quality characteristic y, t is the target value. Cr is the cost of rejection while LSL and USL are the lower and upper specification limits respectively. Based on the equation above, Kapur and Cho (1996) developed a multivariate loss function for the multivariate quality characteristic y1 , y2 , . . . , yn : p i j =1 Kij (yi − ti )(yj − tj ) LSLi ≤ yi ≤ USLi i=1 L(y1 , y2 , . . . , yn ) = (2) Cr (y1 , y2 , . . . , yn ) otherwise
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Fig. 2 Approach to economic benefit estimation with variance reduction Fig. 3 Traditional rectangular loss quality function
where tj is the target, Kij is the loss coefficient of the j th quality characteristic. In many situations, target value is defined as a range instead of a number, and within this range the same loss or profit is realized. The traditional loss function in this case is depicted in Fig. 3. In our optimization problem target is defined as a range, however, using classical function is not appropriate because of its sharp boundaries. Therefore we have defined steps in quality function as sigmoid (see Fig. 7 and Fig. 9) and the loss function is formed as follows. ⎧ a LSL ≤ y ≤ USL ⎪ ⎪ ⎨ 1+e y−b c a LSL < y ≤ USL + δ (3) L(y) = y−b) ⎪ 1+e−( c ⎪ ⎩C otherwise r
where a, b, c are the parameters of the sigmoid functions and δ is a necessary small -LSL . number, e.g. USL100 In our case the optimization can be expressed as a minimization problem according to the set points of local controllers of operative control level w = [w1 , . . . , wp ]T . p is the number of the controlled variables, denoted by yi . The task at the operative control level can be summarized as yi should be as close to wi as possible: min L(y1 , y2 , . . . , yn ) w
(4)
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During development of process control in economic point of view, the main task is to find the set point signal with the possible highest economic performance value. In steady state operation the value of (4) is often decreasing by shifting the steady state operation point closer to the constraints of the process. To reach this goal, the reduction in variance of the key process variables is necessary with e.g. re-tuning the controller or even re-design the existing control strategy. If the variance of the key process variables is reduced, the extra profit will be realized as the difference of the income in the old steady state operation point and the income in the improved operation point. Figure 2 illustrates this approach of increasing the economic performance. The final goal in multilayer optimization is to maximize the accessible profit, determined by the cost function mentioned before in Eq. (4), by finding the optimal steady state operation point with respect to the process constraints. This optimization problem represents the supervisory control layer. 2.2 The operative control level In optimization and control of complex production processes, the role of Model Predictive Controllers (MPCs) is increasing. The more and more widespread application is reasonable, thanks to the good variance reduction ability. MPC is a model based control algorithm where models are used to predict the behavior of process outputs of a dynamical system with respect to changes in the process inputs. The MPC uses the models and current plant measurements to calculate future moves in the manipulated variables, that will result in operation that honors all input and output variables’ constraints. Predictive control uses the receding horizon principle. This means that after the computation of the optimal control sequence, only the first control action will be implemented, subsequently, the horizon is shifted one sampling period and the optimization is restarted with new information about the measurements. That is why MPCs do not optimize the operation on the time horizon of the whole steady state operation, but consider just the horizon, implemented in the controller and solve the optimization problem iteratively. By the help of Fig. 4 the essence of the model predictive control is easily understandable. Utilizing this control strategy in the operative control level the previously proposed multilayer optimization framework can be obtained (depicted in Fig. 1), thanks to the control rule of MPCs, expressed as: min
Δu(k+j )
Hp p i=1 j =1
(wi,k+j − yi,k+j ) + 2
m i=1
λi
Hc j =1
Δu2i,k+j −1
(5)
where p and m are the numbers of the controlled variables and manipulated variables, respectively. In MPC control strategy the different number of controlled and manipulated variables is acceptable, since the interconnection between the different manipulated and controlled is considered in the process model, which is applied in the MPC. The tuning parameters of the controller are: Hp and Hc are the length of prediction and control horizon, λ is the factor for penalizing the change of the control signal. In addition there is one more tuning parameter of the MPC, which is the
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Fig. 4 Illustration of essence of MPC in control of yi by manipulating uj
length of the model horizon, Hm (see Fig. 4). This horizon considers the effect of the previous control actions on the current state. MPCs formulate an objective function, which is for finding the optimal input signal sequence to eliminate the difference of the controlled variable and the set point. Since MPCs with the formalized objective function, Eq. (5), are for ensuring the smooth and stable operation, it is not ensuring the maximization of the economic performance, formalized in Eq. (4). This random phenomenon has to be taken into account during the economic performance optimization, so in this paper we suggest the application of Monte Carlo simulation by applying the augmented process model (see Fig. 1). The result of the Monte Carlo simulation is an aggregated economic performance (e.g. mean of the economic performance of the individual runs). Integrating this Monte Carlo simulation based economic performance assessment tool into an optimization algorithm, the previously proposed multilayer framework could be obtained. The optimization algorithm should be very effective computationally since the gradient of economic cost function is difficult to calculate, because of the random phenomena of process variance. That is why a direct search method should be chosen for this purpose. We suggest the Mesh Adaptive Direct Search methodology, since its’ efficiency has already been proven. In the following section the multilayer optimization framework and its’ two main constituents—Monte Carlo simulation and Mesh Adaptive Direct Search methodology—are going to be introduced in details.
3 Maximization of economic benefit by stochastic modeling and direct search methodology Taking the process variance into account the previously proposed economic oriented objective function, Eq. (4), has a stochastic characteristic, since the process variation is considered as a random phenomenon. To handle these uncertainties, Monte Carlo
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simulation is applied. The Monte Carlo method is applied frequently in the solution of stochastic optimization problems, in stochastic linear programming (Prékopa 1995; Marti et al. 2004). Kjellström (1969) was the first using Monte Carlo estimators for the iterative improvement of convergence behavior in nonlinear stochastic optimization. In the proposed multilayer optimization framework, the application of gradient based methodologies for maximizing the economic throughput is not possible, due to the stochastic characteristics caused by the closed loop variance. Integrating the simulation based economic performance assessment methodology into a direct search optimization algorithm an effective optimization framework is obtained. Mesh Adaptive Direct Search (MADS) (Audet and Dennis 2006) class of algorithms is a relatively new set of direct search methods for nonlinear optimization, i.e. these algorithms are capable of calculate the extremums of a non-smooth objective functions, like our economic objective function. Our methodology is the following: – Economic performance assessment of the considered steady state operation point. It means applying a set point (w) and calculating the value of the economic cost function, Eq. (4), with respect to the process constraints. Because of considering the process variance as random phenomena, Monte Carlo simulation with multiple runs of augmented process simulator is applied to aggregate the effect of random variances in a final economic cost function. – Integrate the economic performance evaluation tool into the MADS optimization algorithm to find the economically optimal steady state operation point. The previously applied economic cost function (the quality loss function) has to be minimized with respect to the proposed constraints with varying set point signal (w). This algorithm can handle constraint limits of process variables at certain confidence levels. In the following section way of application of Monte Carlo simulation and MADS optimization algorithm is introduced briefly. 3.1 Monte Carlo simulation Monte Carlo Simulation (MCS) methods are widely applied in mathematical modeling problems, where some kind of random phenomena must be handled. In the proposed multilayer optimization framework, process variance caused by noise and unmeasured disturbances is considered as random phenomena. The Monte Carlo method consists of the following steps: – Define the domain of possible inputs. – Generate inputs from this domain randomly using a specified probability distribution. – Execute deterministic computation using the inputs. – Aggregate the results of the computations into the final result. In engineering practice, the normal distribution is considered as an adequate assumption for characterizing many uncertainties. In the modeling of the considered
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process, the next steps are followed: at first the mathematical model of the process is created. Then noise signals with different variance have been added to input and output variables to approximate the real process variance. By setting the added noise signals randomly using a specified probability distribution, different result can be obtained in each individual simulation. In case of economic oriented optimization, the result is obviously the value of the economic objective function, Eq. (4). Aggregating the economic performances of the individual runs into a final value the statistical economic performance can be obtained with respect to the examined steady state operation point. Since the complex production processes are mostly characterized with non-linear process models, the economic assessment and optimization need an optimization algorithm which is able to handle the non-linear optimization problem with respect to the process constraints. In addition the economic cost function is nonsmooth, caused by the closed loop variance. 3.2 The mesh adaptive direct search methodology Since the evaluation of gradient of the economic objective function respect to the steady state operation points is highly CPU time intensive, there is a need to utilize any gradient free optimization method. Mesh Adaptive Direct Search (MADS) (Audet and Dennis 2006) class of algorithms is a relatively new set of direct search methods for nonlinear optimization, i.e. these algorithms are capable of minimize a non-smooth function, like our economic cost function (Eq. (4)) under the proposed constraints. According to Audet and Dennis (2006), Abramson et al. (2009), MADS can be interpreted as a generalization of Generalized Pattern Search (GPS) (Torczon 1997) algorithms, with the restriction to finitely many pool direction removed. MADS is an iterative algorithm, where at each iteration a finite number of test points are generated. At a beginning of an iteration, the infeasible test points are filtered (discarded), i.e. infinite objective value is assigned to them (f (x) = +∞). Thereafter the feasible test points are evaluated by the objective function, and compared with the current best objective function value found so far. Each of these test point lies on the current mesh, which is constructed from a finite set of nD directions n D ∈ Rn and scaled by the mesh size parameter Δm k ∈ R . If we find a point with lower objective value than the current best one, this test point is a so-called improved mesh point and the iteration is a successful iteration. Each iteration consist of two steps, the so-called SEARCH step and POLL step. SEARCH step can return any point of the underlying mesh, it is trying to find an unfiltered point. If it fails to generate an improved mesh point, then the second step, the POLL is invoked. POLL step consists of a local exploration around the current best solution, and the test points are generated in some directions scaled by the mesh size parameter. MADS are novel in the number of usable directions, since in GPS, POLL directions belong to a finite set, while POLL direction in MADS belongs to a much larger set, in fact if the iteration number k goes to infinity, the union of the normalized POLL directions over all k become dense in the unit sphere. According to Audet and Dennis (2006), this algorithmic construction allows faster convergence. Another imp portant difference between MADS and GPS is the so called poll size parameter, Δk . This parameter determines the size of the frame where the POLL step can operate.
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Fig. 5 Example of GPS frames (above) and MADS frames (bottom) Pk = {xk + Δm k d : d ∈ Dk } = p = Δ . In all six figures, the mesh M is the intersection of all {p1 , p2 , p3 } for different values of Δm k k k lines (Audet and Dennis 2006) p
In case of GPS, mesh size and poll size are equal (Δm k = Δk ), while in MADS these two parameters can differ. This difference is depicted in Fig. 5. Additional pieces of information like convergence analysis or practical implementations can be found in Abramson et al. (2009). In the economic oriented multilayer optimization framework (see Fig. 1) MADS is applied in the supervisory level to maximize the economic performance formalized as Eq. (4). The optimization problem is solved with respect to the process constraints, with varying set point signal (w). Since MADS needs a reduced number of runs of the augmented process simulator, the optimal value of the set point signals can be quickly obtained. The low number of iteration during optimization is necessary, since Monte Carlo simulation of the operative control level (augmented process simulator) is applied, which is highly computation demanding process. In the following section the effective application of the proposed framework is going to be examined throughout the case studies of a benchmark, linear process and a MPC controlled highly non-linear technology.
4 Application examples In this section, two application examples are presented to demonstrate the applicability of the proposed framework for enhancing the economic benefit of the operating technologies. The calculations for both examples are based on closed loop-data, generated using MATLAB-Simulink. To ensure reproducibility the MATLAB files are downloadable from the website of the authors at www.abonyilab.com.
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Fig. 6 Block diagram of the SISO closed loop system
4.1 Optimization of a constrained single-input process Consider a single-input single-output (SISO) process, characterized by Gp shown in Fig. 6 subject to disturbance dynamics Gd described by: yk = Gp uk + Gd αk =
0.6299z−1 1 − 0.8z−1 u + αk k−2 1 − 0.8899z−1 1 − 0.8899z−1
k = 1, . . . , q (6)
where αt is a normally distributed white noise sequence of mean 0 and variance 1. q denotes the last time step of the considered simulation. The objective in the supervisory control level is to minimize the loss function, L(y)—mean of the Taguchi’s loss function value on the considered time horizon, respect to the process constraints. The optimization problem can be formalized as: ⎧ 50 LSL ≤ y ≤ USL ⎪ ⎪ ⎨ 1+ey−0.5 50 min 2L(y) where L(y) = (7) LSL < y ≤ USL + δ 1+e−(y−0.5) w ⎪ ⎪ ⎩ 50 otherwise where the loss function of the outputs is depicted in Fig. 7 with δ = 2. The applied Taguchi functions have a larger is better shape. The specification limits for inputs and outputs are the following: LSL = −10 ≤ yk ≤ 10 = USL;
−5 ≤ uk ≤ 5;
where k = 1, . . . , q
(8)
In the base case a PI controller has been chosen, and the controller parameters are Kc = 1.926, TI = 0.6. As previously remarked, the probability of not crossing the limits of the output means a non-linear constraint for the optimization problem. During the examinations this confidence level is assigned as 95 % and 90 %. In the literature (Zhao et al. 2009) the same SISO process is applied as an example, with the same probabilities. The means of the output are y¯ = 1.4844 and y¯ = 2.4531 at confidence level of 95 % and 90 %, respectively. The output data in 95 % confidence level is depicted in Fig. 8.
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Fig. 7 Taguchi loss function for outputs in the SISO process
Fig. 8 Base case operation with probability constraint level of 95 %
As we mentioned earlier, the number of individual economic performance evaluations in the Monte Carlo simulation has been set to 100. There has been an attempt to apply quadratic programming as optimization algorithm (utilizing MATLAB, Optimization Toolbox), but the computational demand was extremely high, almost one hour even in this simple example, but with the same result. By applying MADS, the computation demand has been significantly decreased into 2 minutes. It can be possible, since the computation of the gradient of the economic cost function is not necessary. The initial set point for the optimizer was set equal to the upper constraint of the output variable, w0 = 10. Thanks to the application of Taguchi’s loss function, soft constraints and quality evaluation can be easily realized. Utilizing the previously introduced Monte Carlo simulation based optimization methodology new steady state operation points have been determined with multiplied economic performance respect to the defined confidence level. 4.2 The polymerization process The process under consideration is a polymerization process controlled by a linear MPC, the previously mentioned DMC (Ricker 1988). The controlled system possesses all those difficulties which exist in an operating polymerization process.
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The reactor what has been studied is a CSTR where a free radical polymerization reaction of methyl-methacrylate is considered using azobisisobutironitil (AIBN) as initiator, and toluene as solvent. The aim of the process is to produce different kinds of product grades. The number-average molecular weight is used for qualifying the product and process state. The polymerization process can be described by the following model equations (Silva-Beard and Flores-Tlacuahuac 1999): dCm dt dCI dt dD0 dt dD1 dt
F (Cmin − Cm ) V FI CI in − F CI = −kI CI + V F D0 = (0.5ktc + ktd )P02 + kf m Cm P0 − V F D1 = Mm (kp + kf m )Cm P0 − V = −(kp + kf m )Cm P0 +
where
P0 =
2f ∗ CI kI ktd + ktc
(9) (10) (11) (12)
(13)
The actual values of parameters used in the equations can be seen in Table 1. The number average molecular weight (NAMW) is defined by the ratio of D1 /D0 . By assuming an isotherm operation model the process model consists of four states, represented by four differential equations Eq. (9)–Eq. (12) (Maner and Doyle 1997). During the simulations Ts = 0.03 h is applied as sample time. The qualification of the product and process operation is based on the number average molecular weight. Due to the non-linear model equations, the development economic performance turns into a highly non-linear optimization problem. The control objective on the supervisory control level is to maximize the economic performance of the process. The objective function is formalized as ⎧ 100 LSL ≤ y ≤ USL ⎪ ⎪ ⎨ 1+e y−25000 100 100 (14) min 2L(y) where L(y) = LSL < y ≤ USL + δ −(y−25000) w ⎪ 100 ⎪ 1+e ⎩ 100 otherwise where the loss functions of the inputs and outputs are depicted in Fig. 9 with δ = 20. The applied Taguchi functions have a larger is better shape. The specification limits are the following: LSL = 24000 ≤ yk ≤ 26000 = USL
(15)
An important characteristic of the process is the increasing of product quantity when shifting the steady state operation closer to the lower limit, hence the optimal steady state operation point is expected near to the lower limit. The maximum probability of violating the process constraints is 1 %, so the mentioned confidence level is 99 %. The number of Monte Carlo simulations is 100, similarly to the previous
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Fig. 9 Taguchi loss function for outputs in the polymerization process
Fig. 10 Improved case, optimal steady state operation point of PMMA reactor
case. The closed loop variance of the process is caused by the noise added to the inlet monomer flowrate (F) with mean of 0 and σ¯ = 0.014. Other source of the closed loop variance is the noise added to the controlled variable with the mean of 0 and σ¯ = 143. On the operative control level a linear MPC, a Dynamic Matrix Controller (DMC) is installed (Ricker 1988). The DMC applies the linear convolution model of the process for predicting the effects of the considered manipulated variable sequence. The manipulated variable in the control strategy of the reactor is the initiator inlet flow rate. The tuning parameters of the applied DMC are Hm = 30, Hp = 3, Hc = 3. The value of λ is chosen as 4 · 1012 . As base case the safest steady state operation point has been chosen which is in the middle of the specified operation range (w = 25000). As the result of the economic performance optimization the optimal steady state set point is w = 25632. Thanks to this set point modification the quantity of the produced polymer has been increased with 5 % and throughout this the economic performance also increased with 5 %. The result of the closed loop simulation with the optimal setpoint is depicted in Fig. 10.
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As it is mentioned the number of individual economic performance evaluations in the Monte Carlo simulation has been set to 100. In this case study there has been an attempt to apply quadratic programming as optimization algorithm. Using this tool, the same result has been obtained but the computational demand was extremely high, almost 10 hours. By applying MADS, the computation demand has been decreased into half an hour. As it can be seen, the quadratic programming might be applicable but its’ computation demand is exaggerated. The initial set point for the optimizer was set equal to the lower constraint of the output variable, w0 = 24000. As Fig. 10 shows the frequency of constraint violation is conspicuously low, the process constraint has been violated only once. It means 99.85 % probability of not violating the limits, in contrast to the previously determined confidence level, which was 99 %. It may happen since the off-specification product (products which does not fulfill the requirements) means extra outgoings in the economic objective function. Accordingly it is not worthwhile to produce even just 1 % off-specification product, however this amount can be accepted technologically. The results confirmed the assumption that the economically optimal operation is close to the process constraints. However 99 % was set as confidence level of limit violation, the way of formulating the economic cost function does not allow such a low quantity of off-specification product, since it causes extra outgoings during operation.
5 Conclusion We introduced an economic oriented optimization framework for the optimization of operating regimes of controlled processes. We applied Taguchi’s quality loss function to aggregate process constraints, target values, and desired ranges of product quality. Due to unmeasured disturbances and noise caused process variance, the determination of the optimal operating region (setpoints of the local controllers) is rather difficult, since shifting the operation point closer to the process limits means a risk of violation of constraints and reducing product quality. Economic performance and risk level of the operation can be measured as quality loss. Monte Carlo simulation is applied with multiple runs of process model (augmented with the model of the control system) with economic performance assessment to handle the random phenomena of process variance and to provide robust estimate of the loss function. We showed the stochastic nature of the cost function is effectively handled by utilized gradient-free Mesh Adaptive Direct Search algorithm. The application of MADS reduced the time demand of optimization by one order of magnitude compared to applying quadratic programming. The resulted optimization scheme is applied to determine the optimal set-point values of control loops with respect to pre-determined risk levels, uncertainties and costs of violation of process constraints. The efficiency of the proposed framework is demonstrated throughout an example of a benchmark, linear system and a nonlinear process. In the first benchmark problem, significant economic benefit could be realized, the profit could be increased substantially by finding the optimal set point signal after variance reduction.
562 Table 1 Design parameters for MMA polymerization reactor
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F
3 1.0 mh
Cmin
6.4678 kmol 3
CI in
8 kmol 3
V
0.1 m3
Mm
100.12 kmol
f∗
0.58
R
kJ 8.314 kmol·K
kp
m 2.4952 · 106 kmol·h
kI
1.0224 · 10−1 1h
kf m
m 2.4522 · 103 kmol·h
ktc
m 1.3281 · 1010 kmol·h
ktd
m 1.0930 · 1011 kmol·h
m
m
kg
3
3
3 3
These application examples illustrate that Taguchi’s loss function is an ideal tool to represent performance requirements of control loops. This is important result since the proposed methodology helps to develop and improve control performance assessment tools and integrate hard to difficult requirements of process engineers into optimization algorithms. With this tool risk management and optimal set point calculation can be handled together. In this paper we demonstrated the application of our framework in a optimal control problem, however it can also be applied in other optimization and risk management problems where uncertainties should be handled and the user can express the requirements by tuning the shapes of the loss function. Acknowledgements This publication/research has been supported by the European Union and the Hungarian Republic through the projects GOP-1.1.1-11-2011-0045 and TÁMOP-4.2.2.C-11/1/KONV-20120004—National Research Center for Development and Market Introduction of Advanced Information and Communication Technologies. The research of Janos Abonyi was realized in the frames of TÁMOP 4.2.4. A/2-11-1-2012-0001 “National Excellence Program—Elaborating and operating an inland student and researcher personal support system”. The project was subsidized by the European Union and co-financed by the European Social Fund.
References Abramson M, Audet C, Chrissis J, Walston J (2009) Mesh adaptive direct search algorithms for mixed variable optimization. Optim Lett 3:35–47 Audet C, Dennis JJE (2006) Mesh adaptive direct search algorithms for constrained optimization. SIAM J Control Optim 17:188–217 Bauer M, Craig IK (2008) Economic assessment of advanced process control—a survey and framework. J Process Control 18(1):2–18 Borgonovo E, Marseguerra M, Zio E (2000) A Monte Carlo methodological approach to plant availability modeling with maintenance, aging and obsolescence. Reliab Eng Syst Saf 67:61–73 Chen X, Heidarinejad M, Liu J, Christofides PD (2012) Distributed economic mpc: application to a nonlinear chemical process network. J Process Control 22:689–699 García CE, Prett DM, Morari M (1989) Model predictive control: theory and practice–a survey. Automatica 25(3):335–348
Economic oriented stochastic optimization in process control
563
HoLee K, Tamayo EC, Huang B (2010) Industrial implementation of controller performance analysis technology. Control Eng Pract 18:147–158 Kapur KC, Cho BR (1996) Economic design of the specification region for multiple quality characteristics. IIE Trans 28(3):237–248 Kjellström G (1969) Network optimization by random variation of component values. Ericsson Tech. 25(3):133–151 Lee KH, Huang B, Tamayo EC (2008) Sensitivity analysis for selective constraint and variability tuning in performance assessment of industrial mpc. Control Eng Pract 16:1195–1215 Maghsoodloo S, Ozdemir G, Jordan V, Huang CH (2004) Strengths and limitations of Taguchi’s contributions to quality, manufacturing, and process engineering. J Manuf Syst 23(2):73–126 Maner B, Doyle F (1997) Polymerization reactor control using autoregressive Volterra-based mpc. AIChE J 43:1763–1784 Marti K, Ermoliev Y, Pflug G (2004) Dynamic stochastic optimization. Springer, Berlin Mascio RD, Barton GW (2001) The economic assessment of process control quality using a Taguchi-based method. J Process Control 11(1):81–88 Oakland J (2007) Statistical process control. Butterworth/Heineman, Oxford Prékopa A (1995) Stochastic programming. Springer, Berlin Ray WH (1981) Advanced process control. McGraw-Hill, New York Rezaie K, Amalnik MS, Gereie A, Ostadi B, Shakhseniaee M (2007) Using extended Monte Carlo simulation method for the improvement of risk management: consideration of relationships between uncertainties. Appl Math Comput 190(2):1492–1501 Ricker N (1988) The use of biased least—squares estimators for parameters in discrete—time pulse response model. Ind Eng Chem Res 27:343–350 Rubinstein R, Kroese D (2008) Simulation and the Monte Carlo method. Wiley-Interscience, New York Silva-Beard A, Flores-Tlacuahuac A (1999) Effect of process design/operation on steady-state operability of a methyl-metacrylate polymerization reactor. Ind Eng Chem Res 38:4790–4804 Subbaraj P, Rengaraj R, Salivahanan S (2011) Enhancement of self-adaptive real-coded genetic algorithm using Taguchi method for economic dispatch problem. Appl Soft Comput 11(1):83–92 Taguchi G, Elsayed EA, Hsiang TC (1989) Quality engineering in production systems. McGraw-Hill, New York Torczon V (1997) On the convergence of pattern search algorithms. SIAM J Control Optim 7(1):1–25 Yang HT, Peng PC (2012) Improved Taguchi method based contract capacity optimization for industrial consumer with self-owned generating units. Energy Convers Manag 53(1):282–290 Yang T, Chou P (2005) Solving a multiresponse simulation-optimization problem with discrete variables using a multiple-attribute decision-making method. Math Comput Simul 68(1):9–21 Zang C, Friswell MI, Mottershead JE (2005) A review of robust optimal design and its application in dynamics. Comput Struct 83(4a5):315–326 Zhao C, Zhao Y, Su H, Huang B (2009) Economic performance assessment of advanced process control with lqg benchmarking. J Process Control 19(4):557–569
