pulping and recovery sections. This process is based on an ... recovery boiler, smelt dissolving tank, clarifiers, slaker, causticizers and lime kiln. Only the pulp ...
REAL-TIME OPTIMIZATION OF A BLEACH PLANT USING AN IMC-BASED OPTIMIZATION ALGORITHM Charles Vanbrugghe, Department of Chemical Engineering, École Polytechnique de Montréal.
Michel Perrier, Department of Chemical Engineering, École Polytechnique de Montréal. André Desbiens, Department of Electrical and Computer Engineering, Université Laval. Paul Stuart, Department of Chemical Engineering, École Polytechnique de Montréal. ABSTRACT Real-time optimization was applied to the kraft pulp mill benchmark [1]. The optimum was obtained with a modified version of an IMC-based optimization method. This technique uses simplified static models to accelerate the convergence. To improve the performance, the model parameters were estimated on-line and the total cost of the bleaching section was reduced by 10.6 %. INTRODUCTION Papermakers evolve in a very competitive market where cost reduction is a priority. Real-time optimization (RTO) is an efficient way to address this cost reduction problem while maintaining product quality and respecting operating and safety constraints. Unfortunately, few RTO applications exist in the pulp and paper industry [2]. Experienced operators dictate the operating set-points, but as the process slowly drifts away from the optimum, due to changes in the raw material properties, chemical prices, energy prices and equipment degradation, adjustments are required to maintain optimal operating conditions. The optimization of first principle models with large number of possibly non-linear equations is time consuming and requires significant computing power. Real-time optimization
in these conditions is a very challenging task to apply in practice. A possible solution to this problem is given by an IMC-based optimization algorithm [3]. This method was used to optimize the kraft pulp mill benchmark developed by Castro and Doyle [1]. A simplified version of the IMC-optimization was applied to the bleaching section of this process. The objective function to be minimized is the total cost of the bleach plant and the error on the desired production rate and quality of the pulp. The main constraints that must be respected are the production rate, the kappa number and the brightness of the pulp. Prior to the optimization, SVD and RGA analysis were performed to generate a controllability index of the plant. An adaptation algorithm was also introduced to minimise model mismatch and improve the optimum solution. The model parameters were therefore estimated on-line to compensate for process non-linearities. KRAFT PULP MILL BENCHMARK The kraft pulp mill benchmark is a dynamic simulator of the pulping and recovery sections. This process is based on an actual mill. The mill model has 114 outputs, 82 manipulated variables and 58 disturbances. The major units of operation are: a double vessel Kamyr digester, pulp washers, oxygen tower, storage vessels, bleaching towers, evaporators, recovery boiler, smelt dissolving tank, clarifiers, slaker, causticizers and lime kiln. Only the pulp washers are modeled by algebraic equations. All other units are modeled by ordinary differential equations (ODE) or partial differential equations (PDE). This gives a total of 8200 states. Every unit is programmed with C code and the complete simulation runs under the MATLAB/SIMULINK environment. The entire source code is open and fully available for the academic community. This simulator is intended to have multiple utilities. It can be used for modeling, state estimation, process control or optimization.
D0
D0
D1
Figure 1 Bleach plant ODED sequence. Squares (output variables), hexagons (process connections), valves (manipulated variables).
BLEACH PLANT PROCESS The bleaching process (Figure 1) is a crucial operation because of its major influence on the final product quality. Chemicals, energy and water are required in this process. The pulp from the brown stock washers is first sent to the oxygen tower (O). It is mixed with white liquor, oxygen, steam and NaOH. The bleaching of the pulp begins and kappa number is reduced. The oxygen tower is the section of the bleaching plant that exhibits the highest level of interaction (see interaction analysis, Figure 2). The exiting pulp passes through a post-oxygen washer and a fraction of the wash water (extract) is recirculated to control the consistency of the pulp before the washer (Figure 1). The rest of the wash water is sent as dilution liquor to the brown stock washers and to the digester. The pulp is then sent to a storage tank for a few hours. Afterwards, it goes to the D0 tower where it reacts with ClO2. After being washed, the pulp goes to the E tower where lignin is solubilized with NaOH and removed by washing. The dilution water for E washer is the extract water from the D1 tower. Finally, the pulp goes to the D1 tower where most of the brightening is done with the addition of ClO2. In the bleaching process, it is important to control the delignification profile. The first tower uses oxygen because its cost is low and it has poor selectivity. At the O tower, if the kappa is too high, the cost in the later stage will increase. If the kappa is too small there will be a loss of pulp yield. Also, the kappa target at the E tower is important in reducing the use of the more expensive ClO2 in the D1 tower. In this optimization project, the production rate, the kappa and brightness targets must be maintained. Therefore, the objective function consists of minimizing the cost of the bleaching section while maintaining the pulp quality and production targets.
RGA values
INTERACTION ANALYSIS Determining the level of multivariable interactions in the bleaching process was performed before the optimization. This is done with the RGA (relative gain array) and SVD (singular values decomposition) methods.
The analysis shows that the RGA is close to the identity matrix (Figure 2), which validates the choice of loop paring. As for the SVD analysis, it indicates that few variables are prone to saturation [4] or over sensitivity. Since the bleach plant can be adequately controlled, optimization can be performed. IMC-BASED OPTIMIZATION ALGORITHM This algorithm is based on the internal model controller design. The IMC-based optimization uses an approximate model of the process to simplify and accelerate the minimization. In the original algorithm [1], H and P are accurate static models of the plant; H generates the outputs required for the objective function calculation while the outputs of P are the states to be constrained. When too complex, these models are not appropriate for real-time optimization because of convergence problems. Therefore, the objective function is instead based on reduced models HM and PM. An IMC structure estimates the model mismatches and the objective minimization is recalculated taking into account the mismatches. The optimization results are low-pass filtered hence acting as a variable step method. This procedure is repeated until convergence: the optimization is then terminated and the optimal set points can then be applied to the process. At each sampling time, the cost function based on the reduced models must be minimized several times (referred to as optimization steps) before applying the results to the plant. The filter time constant is the only parameter required to modify the trade-off between robustness and performance of the algorithm. The procedure shown in Figure 3 is a modified version of the IMC-based optimization and can be seen as an IMCoptimizing controller. In this case, H and P represent the actual plant. At each sampling time, only one optimization step is performed. The minimization is still based on the reduced models HM and PM. The true values from H and P are only used to compute the prediction errors. Hence, at each iteration step, a move is taken toward the optimum. The filter determines the length of the step. The adaptation algorithm is optional and its role will be discussed later on. The problem solved is the following: min Jc(U ' , β i)
1
s.t
(1)
U'
0.5 0
5
y- CV
10
16 1415 1213 1011 9 7 8 56 3 4 2 1
u - MV
15
Figure 2 Relative gain array from bleach plant.
g(U ' , β i) ≥ 0
(2)
U min ≤ U ' ≤ Umax
(3)
Vmin ≤ V ≤ Vmax
(4)
where JC is the total operating cost of the bleach plant, U’ are the degrees of freedom (set point values) and βi are the model parameters. There is a total of 56 constraints out of which 24 are constraints on the operating points and 32 on the actuators’ range.
βi
z-1
β i0
U'(t)
min JC(t) U'(t)
structure. The precision of the reduced model will determine how close the solution is in regards to the optimum.
Adaptation Algorithm z-1
U(t)
Filter
H
HM
εy(t-1)
RESULTS The bleach plant has 16 degrees of freedom and obtaining a quadratic model requires enormous effort even with a fractional experimental design. It was decided to generate a simple linear model:
Y(t)
YM(t)
+ -
z-1
P
y =
V(t)
16
∑ bu i =1
εv(t-1)
-
-
z-1
+
Vmax Vmin
Figure 3 Modified IMC-optimization structure. The required models are obtained through experimental design and their general form is: y =
n
∑bu i =1
i
i
+
n
∑∑b uu i =1 j ≥ i
ij
i
j
+
factorial
n
∑∑∑b i =1 j ≥ i k ≥ j
ijk
ui u j u k
(5)
where y is the output of the model, ui are the degrees of freedom and βi are the parameters of the model. The degree of freedom selection is made once all the control loops concerned with security, production rate and product quality have been assigned and closed. The production rate, bleach pulp yield, O Kappa, E Kappa and D2 brightness are therefore not available for optimization. Of the 16 degrees of freedom, 12 are controlled variables and 4 are manipulated variables. In the model and in the simulator, the values are scaled [-1,1]. In most cases, it also corresponds to the values of the constraints. The constraint ranges have been tighten by 5%. This prevents actuator saturation and limits the process to an operable region in the event of disturbances. As a result the constraint range for optimization is [-0.95; 0.95]. Table 1 AVAILABLE DEGREES OF FREEDOM. O D1 E E D2 O D1 E D2 O O O
Controlled variable set point temperature temperature temperature washer [OH-] temperature washer DF washer DF washer DF washer DF washer effluent T washer inlet consistency pressure
O D1 E D2
= b1u1 + b2u 2 + ... + b16 u16
(6)
During the experiments, the operating conditions were varied only over a 5% span because of stability and convergence issues. This gave a model that only represents the process and the constraints near the nominal operating points. Since optimization often brings the process near the constraints, the model parameter and the true parameters will probably be different due to the non-linearities.
+
VM(t)
Manipulated variable caustic flow water flow back flush flow caustic flow
The main disadvantage with IMC-optimization is its suboptimal convergence given that it relies on approximate models, although corrections are calculated through the IMC
The mill simulator was run for 2x104 minutes (Figure 4) with no noise signal. The optimizer receives the process values and computes the new optimal values every 400 min (6.6hr) and all constraints are respected. The cost of the bleaching is reduced by 8.9% even if an important difference exists between the model and the actual process. 22 Model Process
21 Cost (US$)
PM
i i
20 19 18
0
0.5
1
1.5
2 4
x 10
Figure 4 Bleach plant total cost in chemicals, energy and utilities. During the optimization process, the objective function was minimized and many constraints were reached yet not crossed. It remains impossible to determine whether this is the true optimum or if another optimum exists. Since there is a significant error between the process and the model prediction (Figure 4), the model will be corrected on-line through parameter adaptation. RESULT WITH ADAPTATION A least-squares adaptation method was chosen [5] which consist in minimising the error ε(t) over a certain horizon: t
∑ε i =1
2
(i − 1) = min J a(t) = ˆ θ(t)
∑ [y(i) − θˆ (t)φ(i − 1) ] t
i =1
T
2
(7)
θˆ(t + 1) = θˆ(t) + F(t)φ(t)ε(t + 1)
(8)
with θˆ(t + 1) the new parameter estimate and F(t) the adaptation gain matrix. A constant forgetting factor method is used to adjust the adaptation gain. This enables the adaptation algorithm to put more importance on the new measurements. This is appropriate in the presence of a system with slow time varying parameters. The procedure starts by running the adaptation algorithm alone for 5x103 minutes with an excitation signal to improve the model estimate around the original operating point. Afterwards, only the five most relevant parameters were adapted [6] and additional coefficients were added to obtain a quadratic model for the bleach plant yˆ =
16
∑ bu i =1
i i
+
5
∑bu j =1
j
2 j
(9)
The parameters from the non-adapted simulation are used as initial parameter estimates. The initial estimates for the quadratic terms are zero. This is a reasonable estimate since the higher order terms are smaller than the primary effects determined with the experimental design. After the initial adaptation, the excitation signal is reduced by half and the optimization is switched on for 1,5x105 minutes (Figure 5). The minimized objective function reaches a new stable value near t=6x104 minutes. 21.5
Model Process
Cost (US$)
21 20.5 20 19.5 19
0
5
10
15 4
x 10
Figure 5 Bleach plant total cost with adapted parameters. The new optimal operating conditions reduce the total cost by 10.6%. This is a notable improvement (1.7%) from the first simulation. During the first 5x103 minutes, the initial adaptation occurred and no optimization took place. From this point on, a rapid decrease of the objective function takes place, followed by a slower change while the model parameters are being adapted. Eventually, as the calculated optimum is approached, the operating conditions stabilize
(Figure 6), the adapted parameters become constant and the optimal operating conditions are assigned. The adaptation is adequate since the model follows the process curve closely (Figure 5). An increase in performance through the use of a quadratic model instead of a linear model is noticed. 1 Operating points
with θˆ(t) the vector of parameters, φ(t) the measurement vector (regressor) and y(t) the actual output. The parameters are adjusted recursively such that:
0.5 0 -0.5 -1
0
5
10
15 4
x 10
Figure 6 Optimal set points trajectories. CONCLUSION As a result of the limited level of multivariable interactions and the high consumption of chemicals, the bleach plant is an appropriate candidate for RTO. Although the objective function was initially reduced by 8.9%, the use of model adaptation is advantageous. An additional 1.7% reduction of cost was attained. ACKNOWLEDGEMENT The authors are grateful to NSERC strategic grant program and to the Environmental Design Engineering chair at École Polytechnique for their financial support. REFERENCES [1] CASTRO, J.J. and F.J. DOYLE III, ‘A Pulp Mill Benchmark Problem for Control: Problem Description’, Journal of Process Control, 14(1), pp. 17-29 (2003). [2] FLISBERG P., RÖNNQVIST M., Optimized Control of the Bleaching Process at Pulp Mills, Control Systems 2002, Stockholm, Sweden, pp 210-214 (2002). [3] DESBIENS A. and A.A. SHOOK, ‘IMC-Optimization of a Direct Reduced Iron Phenomenological Simulator’, 4th International Conference on Control and Automation, Montréal, Canada, 446-450 (2003). [4] MCAVOY T., BRAATZ R.D., ‘Controllability of Processes with Large Singular Values’, Indus. and Engi. Chemistry Research, 42(24), p 6155-6165 (2003). [5] LANDAU I.D., LOZANO R., M’SAAD M., ‘Adaptive Control’, London, Springer-Verlag (1998). [6] KRISHNAN S., BARTON G.W., PERKINS J.D., ‘Robust Parameter Estimation In On-Line Optimization—Part I. Methodology and Simulated Case Study’, Computers Chem. Eng., 16(6), pp 545-562 (1992).
