IMC-optimization of a direct reduced iron phenomenological simulator

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1 IMCoptimization of a direct reduced iron phenomenological simulator André Desbiens Andrew A. Shook LOOP (Process Observation and Optimization Laboratory) Department of Electrical and Computer Engineering, Université Laval Pavillon AdrienPouliot, Quebec City (Québec), Canada G1K 7P4 desbiens@gel.ulaval.ca BHP Billiton Minerals Technology Off Vale Street, Shortland, NSW, Australia 237 andrew.a.shook@bhpbilliton.com Abstract This paper deals with the optimization of a direct reduced iron (DRI) phenomenological static simulator. The technique is based on an internal model controller (IMC) designed using a static approximate reduced (even linear) model. The cost function depends only on the reduced model rendering the minimization much easier. Appropriate corrections are calculated by the means of the IMC structure. Even in presence of a large model mismatch, constraints are respected. The convergence is usually much faster than a direct optimization based of the nonlinear process itself. The algorithm is successfully applied to a direct reduced iron phenomenological simulator, allowing to draw some metallurgical conclusions about the guidance of the process. 1 Introduction Several powerful optimization techniques exist. A comprehensive and uptodate description of these methods is detailed in [8]. When the model used to define the behaviour of the process to be optimized is linear, the minimization of the performance index is usually easy, even in presence of constraints. However if the model is nonlinear, more complex algorithms are needed. They are often hard to implement and setting their parameters may sometimes be difficult. The calculation time may also be long. The first aim of this paper is to present a static optimization algorithm which relies on an approximate reduced or even linear model of the nonlinear process. The proposed algorithm is simple to tune and to implement and converges rapidly. The solution may be suboptimal but it remains sufficiently close to the optimum for pratical applications as will be illustrated in this paper and in [1, 2]. Even in presence of a mismatch between the nonlinear process and the linear model, constraints are respected. The convergence is usually faster than a direct optimization based of the nonlinear process. The second objective of the paper is to apply the IMCoptimization to a direct reduced iron (DRI) phenomenological static simulator. The optimization consists in calculating the manipulated variables to obtain the desired metallurgical product properties while respecting production and safety constraints. 2 IMCoptimization The system to be optimized is static and nonlinear. It is represented by an operator H which acts on the inputs U R nu to produce the outputs Y = (HU) R ny. The cost function to be minimized is J = f (Y, U) (1) The optimization is subject to the following constraints V V max W = W eq (2) where V = (P U) R nx and W = (QU) R nw are some states of the static process, which are respectively obtained by applying the static and nonlinear operators P and Q to U. The operators H, P and Q being possibly highly nonlinear, the minimization of (1) subject to (2) may be difficult and time consuming. The IMC structure [3] depicted in Figure 1 facilitates and accelerates the optimization. The solution is obtained by simulating Figure 1 until convergence. The operators H, P and Q are only needed to calculate the corrections ɛ y, ɛ v and ɛ w through the IMC structure. The cost function and the constraints are based on simpler operators H M, P M

2 H M, P M, Q M W eq min J C U V max U e y e v e w (1ρ) 1ρ I U H H M P P M Q Q M Y M V M W M Y V W Under Assumption 1, the optimal solution is then U op = ( H T ΩH Ψ ) 1 ( H T ΩR Λ ) (7) Simulating until convergence Figure 1 using the error weight Ω IMC instead of Ω leads to U IMC = ( H T M Ω IMC H Ψ ) 1 ( H T M Ω IMC R Λ ) (8) The above equations are developed and analyzed only to gain a better understanding of the behaviour of the structure depicted in Figure 1. Comparing (8) to (7) shows that the IMCoptimization leads to an optimal result only if the IMC error weight is selected to respect the following equality Figure 1: IMCoptimization structure H T Ω = H T M Ω IMC (9) and Q M, which may even be linear. The cost function J C at step k of the simulation is J C = f (Y M ɛ y (k 1), U ) (3) and the constraints are V M V max ɛ v (k 1) W M = W eq ɛ w (k 1) (4) The time constant of the lowpass filter is defined by setting the parameter ρ ( ρ < 1). Theorem 1 (Respect of the constraints) If simulating Figure 1 converges to a steadystate when minimizing the objective function (3) subject to the constraints (4), then in the steadystate the constraints (2) are also respected. Proof. The following equations are obtained from Figure 1 ɛ v (k 1) = V (k 1) V M (k 1) ɛ w (k 1) = W (k 1) W M (k 1) Inserting (5) into (4) leads to (2) in steadystate. Assumption 1 The cost function is (5) J =.5 (R HU) T Ω (R HU).5U T ΨU Λ T U (6) where Ω R ny ny and Ψ R nu nu are positive definite weights. The vector Λ R nu is also a weight selected by the user. Also, to help the analysis of Figure 1, it is assumed that it converges to a steadystate, H and H M are matrices of linear gains and there is no constraint (they would be respected if present, see theorem 1). In practice, this is not critical since the weights are sometimes difficult to select and must often be fine tuned by trials and errors. The examples in Section 3.4 illustrate that, even when the process to be optimized is nonlinear, the proposed method leads to results certainly sufficiently close in practice to the optimal solution. Other examples confirming this property are detailed in [1, 2]. The stability (convergence) analysis of the system depicted in Figure 1 is a difficult topic. The tool to analyze the stability of discrete multivariable nonlinear systems is the Tsypkin criterion, which is the discrete analog to the Popov criterion used for analyzing continuous nonlinear systems. However the structural restrictions of even the more recent results [5, 6, 9] make impossible the stability analysis of Figure 1. To apply the Tsypkin criterion and its extensions, it is required to know the nonlinearities sector bounds which is difficult in practice when H consists of phenomenological equations (which is the case for the example presented in this paper). But even worse, H must be diagonal. Consequently, the stability analysis is restricted to the case where H is linear. In that case, the IMC structure is very robust and it can therefore be hoped that nonlinear systems can also be stabilized. For r r linear systems, it was demonstrated [4, 7] that, with an IMC controller, stability can be guaranteed by increasing the filter parameter ρ as long as Re ( [ ]) λ j HH 1 M > j = 1,..., r (1) where λ j [A] denotes the j th eigenvalue of A. Theorem 6.7 in [1] also confirms that a controller with integral action in all channel (which is the case for an IMCstructure) can be made stable even with large model errors by decreasing the performance (increasing ρ), provided that deth/deth M >, i.e. both H and H M have the same sign. Thus, ρ offers a

3 compromise between performance and robustness. To avoid unbounded constraints, it is also required to have detp /detp M > and detq/detq M >. Ore Gas to Recycle Loop When implementing the proposed structure, the simulation can be stopped when U U(k 1) 2 < η n u, where η >. The parameter η is therefore the desired precision for the elements of the solution U, equivalent to the termination tolerance for usual optimization algorithms. R4 R3 R2 R1 Recycle Gas HBI 3 Optimization of the direct reduced iron phenomenological simulator 3.1 Plant and simulator description BHP Billiton s Boodarie Iron direct reduced iron plant is located in Port Hedland, Western Australia. The plant uses FINMET technology to convert beneficiated iron ore fines to metallic iron in a series of pressurized fluidisedbed reactors. The iron ore feed is substantially hematite (Fe 2 O 3 ) particles smaller than 1 mm in diameter. The reactors are arranged in 4 parallel trains, each consisting of four sequential reactors (Figure 2). Within each reactor, the oxygen is removed from the ore by reaction with the gas, which contains large quantities of hydrogen. The gas passes upward through each reactor in the train, accumulating oxygen in the form of gaseous water. The ore passes down through each reactor in series under the influence of gravity. The reactors operate at temperatures between 4 and 8 C and at pressures of MPa. The extent of reduction of the iron ore is determined by a number of factors, including the intrinsic ore reducibility, the reducing gas composition, operating temperature and reactor residence time. The material from the final solids reactor (R1) is continuously formed into briquettes at elevated temperature. The plant is capable of producing 79 tonnes/hr of product from each of the four reactor trains. A fundamentallybased steadystate mathematical simulation of the plant was developed by personnel at BHP Billiton s Newcastle Technology Centre. This model applies laws of conservation of energy and mass together with descriptions of the thermodynamic and kinetic behaviour of the gas and ore to compute the temperatures and compositions occurring within a reactor train. A Gibbs free energy minimization technique is used to compute the gasphase thermodynamic equilibrium. Experimentallydetermined kinetic rates are applied to compute the degree of solids reaction. The mixing within each of the fluidised beds is computed using standard chemical engineering correlations. The composition of the reducing gas (containing H 2, Figure 2: DRI plant H 2 O, CH 4, CO, CO 2, N 2, H 2 S) is such that a large number of interacting gasgas and gassolid reactions are possible. As a result, the simulation is capable of predicting the significant changes to plant operations that occur due to alterations in gasphase equilibria. However, the complexity and nonlinear nature of these calculations mean that repeated simulations consume excessive amounts of computer time, thus rendering the online direct optimization very difficult. 3.2 The variables and the reduced models of the DRI simulator The manipulated variables (U) are the ore flow rate, the pressure at R1 inlet, the residence time of all four reactors, the reducing gas flow rate and temperature and the reformed gas flow rate. Since the recycle loop is closed, the recycle gas flow rate is obtained from a balance with other flow rates and therefore cannot be manipulated. The output of the simulator (Y ) is the percentage of metallization. The constrained variables (V ) are the level of all four reactors and the bed temperature of R1 and R2. In order to have signals with approximately all the same magnitude, they are normalized and the operation points are removed. Since the operators H and P (the DRI simulator itself) are complex and nonlinear, the optimization will rely instead on reduced models H M and P M and the structure depicted in Figure 1. The models were calibrated on data recorded from running the DRI simulator at various operating points. The operator H M predicting the normalized percentage of metallization is as follow y = 9 b i u i i=1 9 b ij u i u j i=1 j i 9 b ijk u i u j u k i=1 j i k j (11) Similar equations define the operator P M used to predict v 1 to v 6. Several bparameters in the reduced models are zero.

4 Percentage of metallization Set point NLop R1 bed temperature x Constraint NLop Percentage of metallization Set point NLop R1 bed temperature Constraint NLop R1 level NLop Ore flow rate NLop R1 level NLop Ore flow rate NLop Figure 3: Example The objective function and the constraints The objective function to be minimized by manipulating u 1 to u 9 is J = (y r) 2 λ 1 u 1 λ 7 u 7 λ 8 u 8 λ 9 u 9 (12) where r is the desired percentage of metallization. The scalars λ 1, λ 7, λ 8 and λ 9 are weights to be selected by the user. Minimizing J therefore corresponds to bringing the percentage of metallization y close to its set point (maintaining the quality of the product) while maximizing the ore flow rate u 1 (maximizing throughput) and minimizing the reducing gas flow rate u 7, the reducing gas temperature u 8 and the reformed gas flow rate u 9 (minimizing production cost). The minimization of J is subject to 9 inequality constraints on U and 1 inequality constraints on V. 3.4 Comparisons with direct optimization The IMCoptimization (IMCop) presented in Section 2 will be compared to two direct optimizations of (1) with respect to (2). The first direct optimization (NLop) uses the same initial guess as the IMCoptimization, which is zero for all variables. The initial guess for the second scheme () is the result of the optimization based on H M and P M, which is in fact the result of the first iteration of the IMCoptimization. To find the optimum, (NLop) and () rely on the Matlab function fmincon with the termination tolerance on U set to The parameter η in the IMCop scheme is set to the same value. No lowpass filter is used, i.e. ρ =. The minimum of (3) with respect to (4) is also found with fmincon. Example 1. The weights are λ 1 =.1 and λ 7 = λ 8 = λ 9 =. Figure 3 shows the stepbystep behaviour of some (normalized with the operating point removed) variables during the IMCoptimization. The lines labelled represent the parameter values of the DRI simulator while those labelled are the re Figure 4: Example 2 duced model states. In all graphs, the first point (iteration ) represent the simulator in openloop at its nominal operating point. The IMC structure starts to be effective only at the next iteration. Note that because all initial conditions are zero, the first iteration consists in doing an optimization based on H M and P M : the percentage of metallization predicted by H M is therefore equal to the set point at that first step. In the following iterations, corrections are calculated through the IMC structure to take into account the model mismatches. Since λ 1 was set to.1, the ore flow rate is increased to its maximum permissible value. To ease comparisons, the results obtained with NLop and are also plotted on the same graphs. The IMCop method needed to run the DRI simulator only 3 times (total calculation time: 28.3 s) while it was respectively run 23 times (191.9 s) and 11 times (625.4 s) for and NLop. All methods give almost equal terminal value of the cost function (12), IMCop being the best by.2% and.3%. All constraints are respected. Example 2. The weights are the same as in Example 1 but the desired percentage of metallization is larger. Again, IMCop needs to run the DRI simulator very few times (IMCop: 6 runs, 33.4 s; : 68 runs, s; : 178 runs, 92.9 s). IMCop leads to the smallest terminal value of the cost function by 1.5% and.4%. Figure 4 depicts some results. 4 Conclusion An interesting characteristic of the proposed IMCoptimization is to show the duality between control and optimization algorithms. The rapidity of convergence for optimization corresponds to the performance for control; divergence of the optimization is similar to unstability of the closedloop system. Hence, the

5 parameter ρ is related to the length of the steps for optimization and to the closedloop time constant for control, therefore allowing a tradeoff between performance and stability. The variable k in Figure 1 can be interpreted either as the optimization step from an optimization point of view or as the time when thinking in terms of control. The main advantages of the proposed IMCoptimization are: It can rely on a simple optimization algorithm and is easy to implement: If H M is linear, the minimum of the cost function can simply be found using quadratic programming. An analytical solution even exists in absence of constraints. If H M is nonlinear but simple, the optimization routine do not have to be robust and complex. For the DRI problem, when using a commercial optimization routine written in C instead of the Matlab function fmincon, the method NLop frequently diverges. Divergence has never occured using the same optimization routine combined with the proposed IMC approach. It is easy to tune: Because of the simplicity of the model appearing in the cost function, the default parameters of the minimization algorithm are usually adequate. The only parameters to tune are η and ρ, both with clear meanings. It converges rapidly: As illustrated in the examples detailed in Section 3.4 and in [1, 2], IMCop usually converges much faster than NLop when they both rely on the same optmization routine. If the behaviour of H differs significantly for various possible operating points, it is recommended to use a different H M for each of them. The IMCoptimization makes it possible to rapidly evaluate different situations according to the weights in the cost function, the desired percentage of metallization and the selected constraints. In most situtations, when the throughput is increased as much as possible, the quality (y) noticeably decreases and the production cost (u 7 u 8 u 9 ) is increased. Trying to minimize the production costs is translated into very large decreases of throughput and product quality. The tests allow to quantify the gains and losses and to determine the best tradeoff. Currently underway, the following step of the project is the online implementation of the IMCoptimization at the Port Hedland plant. Every time when needed, the simulator will be put in a state similar to the plant using online data. Then, the optimization will produce guidance advices to the operators by minimizing a cost function representative of the aimed metallugical product properties and constraints to be respected. Acknowledgements The authors are grateful to CIDAC (Centre for Integrated Dynamics and Control at the University of Newcastle), NSERC (Natural Science and Engineering Research Council of Canada) and BHP Billiton for their financial support and to BHP Billiton for its authorization to publish the results. References [1] D. Pomerleau, A. Desbiens, G.W. Barton, Realtime optimization of an extrusion cooking process using a first principles model, sumitted to CCA 23 Conf. on Control Appl., (Istanbul, Turkey), (23). [2] D. Pomerleau, A. Desbiens and D. Hodouin, Optimization of a simulated ironoxide pellets induration furnace, accepted for publication, 11th Mediterranean Conf. on Control, (Rhodes, Greece), 23. [3] C.E. Garcia and M. Morari, Internal model control 1. A unifying review and some new results, Ind. Engng Chem. Process Des. Dev., vol. 21, pp , [4] C.E. Garcia and M. Morari, Internal model control 2. Design procedure for multivariable systems, Ind. Engng Chem. Process Des. Dev., vol. 24, pp , [5] V. Kapila and W. Haddad, A multivariable extension of the Tsypkin criterion using a Lyapunovfunction approach, IEEE Trans. Autom. Control., vol. 41(1), pp , [6] M. Larsen and P. Kokotović, A brief look at the Tsypkin criterion: from analysis to design, Int. J. Adapt. Control Signal Process., vol. 15, pp , 21. [7] M. Morari, Response to comments on Internal model control 2. Design procedure for multivariable systems, Ind. Engng Chem. Res., vol. 26, pp. 633, [8] J. Nocedal and S.J. Wright, Numerical optimization, Springer, New York, [9] P. Park and S.W. Kim, A revisited Tsypkin criterion for discrete nonlinear Lur e systems with monotonic sectorrestrictions, Automatica, vol. 34(11), pp , [1] S. Skogestad and I. Postlethwaite, Multivariable feedback control: Analysis and design, John Wiley & Sons, England, 1996.