Baydali, S.A.

Manenti, F.

Mikhalevich, S.S.

Rossi, F.

Mikhalevich, S. S.

Baydali, S. A.

MANENTI, FLAVIO

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 CHEMICAL ENGINEERING TRANSACTIONS   VOL. 43, 2015 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Chief Editors: Sauro Pierucci, Jiří J. Klemeš Copyright © 2015, AIDIC Servizi S.r.l., ISBN 978-88-95608-34-1; ISSN 2283-9216                                                                                Robust PI/PID Controller Design for the Reliable Control of Plug Flow Reactor Sergei S. Mikhalevicha,b, Francesco Rossib, Flavio Manenti*b, Sergei A Baydalia a National Research Tomsk Polytechnic University, Department of Electronics and Automation of Nuclear Plants, Lenin Street 50, 634050 Tomsk, Russian Federation b Politecnico di Milano, Dip. Chimica, Materiali e Ingegneria Chimica “Giulio Natta”, Piazza Leonardo da Vinci 32, 20133 Milano, Italy flavio.manenti@polimi.it The article presents mathematical description of gas-solid flow reactor and its solution with C++ BzzMath library. This work is illustrating the way of developing the control system of flow reactor with PI/PID controller. Quasi linear regions of output depend on input and disturbance signals were found and several conventional and modern tuning techniques were applied to provide the robustness and performance trade-offs. The system stability was checked for robustness in different regimes (quasi linear and non-linear). 1. Introduction Nowadays heterogeneous catalysis rules are used in almost industrial processes, especially in the form of heterogeneous gas-solid flow reactors. This equipment has been deeply investigated in chemical engineering history and several modelling strategies have been developed (Manenti et al., 2013). However, these reactors are typically managed by conventional (Proportional-Integral-Derivative, PID) controls and the definition of a smart, effective and robust methodology to tune the control loops may still be a challenge, especially when very exothermic reactions are carried out in these reactors. PID based control systems are those that are commonly employed in the industrial practice because, if they are properly tuned, they can ensure both good performances in servomechanism and regulation problems and the controlled system stability. However, the definition of a proper set of tuning rules, based on a balance between the need for performance and that for stability, is nontrivial. A novel robust PID tuning method, which accounts for both control performance and control stability, has been described elsewhere (Mikhalevich et al., 2015). The goal of the present work is to apply such a technique to tune the PID control system of a catalytic flow reactor (Figure 1). There used notifications: ρi(G) is the mass concentration of the i-th component in the reactor, Acs is the reactor vessel cross-sectional area, z is the space position in the axial reactor length, Q is the volumetric flowrate inside the reactor, T is the temperature in the reactor, Aexc is the reactor heat exchange area, Tc is the coolant temperature, P is the pressure in the reactor, zF is the maximum value of z, QIN is the inlet volumetric flowrate, TIN is the temperature of the reactor feed, PIN is the pressure of the reactor feed, ρi(G,IN) is the mass concentration of the i-th component in the reactor feed.  Figure 1: Layout of the catalytic PFR                                                                                          DOI: 10.3303/CET1543255                                                             Please cite this article as: Mikhalevich S., Rossi F., Manenti F., Baydali S., 2015, Robust pi/pid controller design for the reliable control of plug flow reactor, Chemical Engineering Transactions, 43, 1525-1530  DOI: 10.3303/CET154325515252. Dynamic model of the plug flow reactor The plug flow (PFR) reactor, shown in Figure 1, is modeled by means of a first principles approach using the following assumptions to simplify the modeling issues. The reacting mixture is supposed to behave as a perfect gas and its thermodynamic properties (constant pressure and constant volume specific heat, enthalpy of reaction and so on) are considered constant with temperature. Moreover, the gas transport properties (axial dispersion and thermal conductivity, viscosity, global heat transfer coefficient and so on) are assumed to be constant with both temperature and composition. All these assumptions are reasonable for a wide number of different cases. The additional assumptions, introduced for the catalytic phase, include isothermal catalyst and no diffusive limitations inside the catalyst (the efficiency factor equals one). Finally, the reactor temperature is assumed to be controlled with a phase changing fluid, thus there is no variation in the coolant temperature profile with respect to the space. By exploiting all the reported assumptions and neglecting the gas – catalytic phase gradients, the reactor model is constituted by the component material balances, the energy balance and two additional equations used to predict the pressure losses and the volumetric flow variation. In the end, a pseudo homogeneous model is developed. The material balance for the i-th component is described in Eq(1) while the energy balance can be found in Eq(2): ( )( ) 2 ( )( ) ( )211 1 RNG GG eff ri ii i i ij jjcsQ D PM Rt A z zρ ρ ερ νε ε=∂ ∂ ∂ −= − + +∂ ∂ ∂   (1) ( ) ( )( )2( ) ( )21 1( )(P)1( )1 ( )(P)1 1(1 )1 1(1 )1C RjCCC RN NG r effi i R ji jcsNGexcc iNiG cs cs ii i c c N Ni riij ji jiQ T TCp H R kA z zT U dA PT T Qt A dz A zCv CpPMP Rρ ερρε ρ ε ρε νρ= ==== = ∂ ∂− − − Δ + + ∂ ∂  ∂ ∂ = + − − +∂ ∂ + −   + −     (2) where t  is the time, ε  is the catalytic bed void fraction, effiD  is the effective axial dispersion of the i-th component, iPM  is the molecular mass of the i-th component, RN  is the number of chemical reactions, ijν  is the stoichiometric coefficient of the i-th component in the j-th reaction, ( )rjR  is the rate of the j-th reaction, iCv  is the i-th component constant volume specific heat inside the reactor, cρ is the catalyst intrinsic density, cCp  is the catalyst constant pressure specific heat, iCp is the i-th component constant pressure specific heat inside the reactor, jRHΔ  is the enthalpy of reaction for the j-th reaction, effk  is the effective axial thermal conductivity, U  is the global heat transfer coefficient, (P)iρ  is the density of the i-th pure component. A Langmuir-Hinshelwood rate equation is chosen (its general functional structure is shown in Eq(3)). Such a rate equation is chosen since the majority of the catalytic reaction kinetics are reported under this formalism in literature: 0( ) 1,011CjjijeqC jiNEaRTj ir ijN Eaeq RTji iik e PRk e Pγψ===  +   ∏ (3) In Eq. (3), 0jk  is the pre-exponential factor of the j-th reaction rate constant, jEa  is the activation energy of the j-th reaction, CN  is the number of components in the reacting mixture, iP  is the partial pressure of the i-th component inside the reactor, jiγ  is the reaction order of the i-th component in the j-th reaction, ,0eqjk  is the pseudo pre-exponential factor of the equilibrium constant of the i-th component in the j-th reaction, eqjiEa  is the pseudo activation energy of the equilibrium constant of the i-th component in the j-th reaction, RT  is the temperature T  multiplied by the universal gas constant. The mathematical expression that can be used to connect the pressure losses to the reactor operating conditions should be derived from the momentum balance. Nevertheless, since the pressure dynamics are much faster than the material/thermal dynamics and of little interest, the dynamic momentum balance can be replaced with the simpler Ergun equation (this equation comes from the momentum balance in steady-state conditions). Ergun equation is stated as follows: 1526( ) ( )3( )11 1.75 150 1CGmixNG cc iiP G Gz DPDPε μεε ρ=  ∂ −= − + −   ∂     (4) where G  is the mass flowrate per unit cross-sectional area inside the reactor, cDP  is the catalyst particles diameter, ( )Gmixμ  is the reacting mixture viscosity. Finally, a mathematical formula for the evaluation of the volumetric flow is needed. In this model, a simplified expression (Eq(5)), which comes from the perfect gases equation of state, is used. ININ mixININ mixT P PMQ QT P PM    =             (5) where mixPM  is the molecular mass of the reacting mixture inside the reactor, INmixPM  is the molecular mass of the inlet mixture. All the model equations have been reported and described but the PDEs require proper initial and boundary conditions to be solved. The initial conditions for Eq. (1-2) consist of the spatial profiles of the state variables at a certain initial time (see Eq(6)) while no initial condition is required for Eq(4). The boundary conditions for Eq. (1-2) have to be imposed both in 0z =  and in Fz z=  and consist of a set of material/energy balances on the reactor inlet interface (Eq(7)) and a zero gradient condition (Eq(8)), respectively. Once again, Eq(4) is an exception because only the boundary condition in 0z = , which comes from a momentum balance on the reactor inlet interface, must be added (Eq(7)). 00( ) ( ,0)t t0t t( ) 0T ( ) 0G Gi i zT zρ ρ==− =− = (6) ( )( ) ( )( ) ( )( ) ( )( )( , ) ( )00( , )01 0000( ) 0CGG IN G eff iIN i i cs izzNG IN IN effIN IN i i cszi zINzQ t Q A DzTQ t T t T t Cp A kzP P tρρ ρ ερ==== == ∂− + = ∂ ∂− + = ∂− =  (7) ( )00FFGiz zz zzTzρ==∂= ∂∂= ∂ (8) In Eq(6-8), 0t  is the initial value of t , ( ,0)Giρ is the mass concentration of the i-th component inside the reactor at 0t t= , 0T  is temperature inside the reactor at 0t t= , INiCp  is the i-th component constant pressure specific heat in the reactor feed. All the equations and the proper initial/boundary conditions belonging to the reactor model have been shown but these equations must be solved with a suitable numerical method, thus some information on the model solution approach must be added. Here, the finite difference method, based on backward derivatives approximation, is employed. In detail, all the reported equations are discretized in space in order to reduce the PDAE system to an initial value DAE system. Then, the DAE system is solved by the BzzMath DAE integrator (Buzzi-Ferraris and Manenti, 2012), which is based on improved Gear multi-value methods. In order to ensure an elevated efficiency in the discretized PFR model solution, it is mandatory to exploit the system Jacobian structure (Figure 2). The BzzMath DAE integrator can be provided with the Jacobian matrix sparsity pattern, thus it is suitable for this application. 1527 Figure 2: Sparsity pattern of the Jacobian of the discretized PFR reactor model 3. Development and tuning of the PFR control system The proportional-integral (PI) and proportional-integral-derivative (PID) controllers are widely used in industry (Visioli, 2012). This type of control is the most popular because a wide class of industrial processes can be in control of it. According to this remarks, the authors have decided to use the control system presented in Figure 3. Nowadays there are a lot of techniques (Aström and Hägglund, 2005) for the optimal calculation of the parameters of PI/PID controllers and ones modifications (Shamsuzzoha, Al-Mutairi, 2011). Each of them has advantages and disadvantages. Therefore, the authors have chosen several modern and conventional PI/PID tuning methods and have applied them to the proposed control system in and compare their performance. An identification procedure must be completed before PID tuning can be obtained. For this purpose the step response of the PFR model is computed and, subsequently, used to generate a simplified linearized model, based on a transfer function representation. Note that the identification is performed by using eT  as the only manipulated variable and outT  as the only controlled variable of the PFR. The transfer function resulting from the identification is presented in Eq(9).  Figure 3: PFR control system where spT  is the PFR hot spot temperature set-point, inT  is the temperature of the PFR inlet flowrate, outT  is the PFR hot spot temperature 0.477( )89 1PFR ss=+ (9)  The transfer function of PFR is true for linear bounds shown on Figure 4. These bounds required for the functionality of the control system must be coupled with the physical constraints. It must be ensured that  eT  < 525 K and inT  ϵ [425;485] K (according to the technical requirement). After combining theoretical and practical bounds, the final allowed intervals for eT  and inT  are: eT ϵ [395;525) and inT  ϵ [450;485]. Notice that the PI/PID controller in the proposed control system must consider the saturation for the control signal according to the eT  bounds, described above. As an alternative, we may suppose that our reactor is an interval system and, subsequently, tune the PI/PID controller as for an interval system (i.e., each parameter of the transfer function is not constant). In this latter option, we need to check the behavior of the control system in the case of nonlinear response of the target process. 1528  Figure 4: Identification of the quasi-linear operation regions In order to tune the PI/PID controller in proposed control system, the authors have used several methods. Magnitude Optimum (MO) (Aström and Hägglund, 2005) provides the maximum bandwidth of system. Mikhalevich et. al. (Mikhalevich et al., 2015) provides minimal overshoot for desired phase margin and crossover frequency. Good Gain Method (Haugen, 2012) is an alternative of Ziegler-Nichols (Ziegler and Nichols, 1942) that provides better stability robustness with Ziegler-Nichols method.. Also, several control quality and robustness indexes have been evaluated: maximum sensitivity (Ms), phase margin (φm), and settling time (ts). The results of the servomechanism and disturbance attenuation, in the region where the reactor response is almost linear, are presented in Figure 5. The control systems tuned by means of the Good gain method is not shown because the PFR becomes unstable. The control system performance measures and robustness indices are shown in Table 1. According to Figure 4, the PFR reactor is a nonlinear system. Therefore, we need to check the performance of the control system, tuned with different methods, also based on a servomechanism problem performed in the region where the reactor response is nonlinear. The results of this servomechanism problem are presented in Figure 6. According to Figures 5, Figure 6 and Table 1, the most appropriate control system for a PFR appears to be a PID, tuned by the Mikhalevich et. al. method. It provides minimum overshoot and settling time along with reasonably large robust indices. The proposed PID controller also allows both to exclude the saturation block from the control system and to control the reactor over the entire range of feasible operation. Table 1. Control system performance and robustness when different tuning methods are employed Method Settling time, s Maximum sensitivity phase margin, ° MO-PI 179 1 78.49 MO-PID 175 0.67 105.3 Mikhalevich et. al. method 165 0.81 104   Figure 5: PFR transient in a servomechanism and disturbance attenuation problem (the reactor response is quasi-linear) 1529 Figure 6: PFR transient in a servomechanism problem (the reactor response is nonlinear) 4. Conclusions This paper has outlined a mathematical model of a catalytic flow reactor, made of partial differential equations, algebraic equations and boundary conditions. Then, the PFR dynamic behavior has been approximated with a first order transfer function (FOTF) without delay time and the regions of linear dynamic response have been identified as a function of the manipulated variable and the external perturbation. Despite the widespread belief that the best controller for chemical plants is a PI, both PI and PID controllers have been tested. Their tuning procedures have been addressed with several methods. The performances of the PI/PID controllers, all tuned with different strategies, have been compared on several servomechanism and disturbance attenuation problems in the region where the PFR exhibits both a quasi-linear and a nonlinear behavior. It results that PID appears superior to PI, in contrast to the standard beliefs, and the Mikhalevich et al. (2015) method ensures the best performances. Finally, a modern PID tuning method has been used to develop a reliable and efficient control system for PFR reactors. References Aström K. J., Hägglund T., 2005. Advanced PID Control in Research Triangle Park, NC 27709, ISA - Instrumentation, Systems, and Automation Society, Durham, North Carolina, USA. Buzzi-Ferraris G., Manenti F., 2012. BzzMath: Library Overview and Recent Advances in Numerical Methods. Computer-Aided Chemical Engineering, 30, 1312-1316. Haugen F., 2012. The Good Gain method for simple experimental tuning of PI controllers. Modeling, Identification and Control, 33, 4, 141-152. Manenti F., Papasidero D., Frassoldati A., Bozzano G., Pierucci S., Ranzi E., 2013. Multi-scale modeling of Claus thermal furnace and waste heat boiler using detailed kinetics. Computers & Chemical Engineering, 59, 219-225. Mikhalevich S. S., Baydali S. A., Manenti F., 2015, Development of a tunable method for PID controller to achieve the desired phase margin. Journal of Process Control, 25, 28-34. Visioli A., 2012. Research trends for PID controllers. Acta Polytechnica, 52, 6, 144-150. Shamsuzzoha M., Al-Mutairi E., 2011. A New Approach for PID Controller Tuning from Closed-loop Setpoint Experiment. Chemical Engineering Transactions, 24, 3, 445-450. Ziegler J. G., Nichols N. B., 1942. Optimum Settings for Automatic Controllers. Transactions of the A.S.M.E., 115, 759-768.  1530

Robust PI/PID controller design for the reliable control of plug flow reactor

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Robust PI/PID controller design for the reliable control of plug flow reactor

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