Multivariable Control for Regulating High Pressure ... - Semantic Scholar
2014 IEEE International Conference on Control Applications (CCA) Part of 2014 IEEE Multi-conference on Systems and Control October 8-10, 2014. Antibes, France
Multivariable Control for Regulating High Pressure Centrifugal Compressor with Variable Speed and IGV Toufik Bentaleba , Alessio Cacittib , Sergio De Franciscisb , Andrea Garullia Inlet guide vane
Abstract— The objective of this paper is to develop a multivariable control system for a class of centrifugal compressors, which exploit as control signals both the rotational speed and the Inlet Guide Vane (IGV). Linear Quadratic Gaussian control with Integral action (LQGI) and Model Predictive Control (MPC) are investigated. The LQGI and MPC controllers are compared to a standard proportional integral (PI) controller, to regulate the discharge pressure of the compressor. The control algorithms are simulated and compared in different operating scenarios. Results demonstrate that the proposed multivariabe control schemes provide better performance than the singleloop PI controller, thus motivating the use of IGV for control purposes. Index Terms— Multivariable control, Optimal control, Model predictive control, Linear quadratic regulator, Power plant.
Gas Turbine N
Ps Pd
T1 P1
DV
UV V2
V1
T2 P2
Gas Outlet
Gas Inlet
I. I NTRODUCTION
Fig. 1. Compression system: gas turbine-driven centrifugal compressor, equipped with IGV, and two volumes at suction and discharge.
High-pressure multistage centrifugal compressors are an essential part of the process machinery in the oil and gas industry across a wide variety of applications. Centrifugal compressors in a connected process system are very sensitive to changes in the inlet conditions [1], such as the suction pressure, the temperature, as well as the inlet gas density. This type of gas compressors requires quick response and reliable control systems to increase their capacity. Process regulation (for example discharge pressure control) is usually performed by acting on the shaft speed. More recently, a variable Inlet Guide Vane (IGV) has been used for control purposes. The IGV system allows wide capacity control of the centrifugal compressor with reduced energy losses. In centrifugal compressors, IGV is used to control the mass flow rate with negligible change in compressor ratio and shaft speed [2]. When both the turbine speed and the IGV are used to regulate the process, a multivariable control system is necessary. In this paper two types of multivariable control, namely a linear quadratic regulator (LQR) and a model predictive controller (MPC), are presented, and compared to a standard PI controller based on speed regulation. The LQR is widely and effectively used in many industrial applications [3], [4], including control of the compressor stations for natural gas pipelines [5], [6]. Since complete state information is not available, it is necessary to use a LQG control scheme, which includes Kalman filter estimation of the state vector. The LQG scheme includes an integral action to compensate the reference tracking error. In recent years, MPC has confirmed
itself as a successful approach to multivariable control due to its advantages over traditional controllers [7]. In particular, it is widely employed in the oil and gas industry to deal with power plant control [8], [9], [10]. Although there is an increasing research interest in nonlinear MPC [11], [12], [13] most of this literature is dedicated to systems described by analytical models, while the model of the reference application considered in this paper contains non analytical parts (e.g. look-up tables). Hence, in this study linear MPC is employed, based on a linearized model of the plant. The paper is structured as follows: in Section II, the power plant model is presented. Section III deals with the PI, LQGI, and MPC control schemes and the details about the implementation of each controller. Simulation results are presented in Section IV to compare the performances of the PI, LQGI and MPC controllers, while some concluding remarks are given in Section V. II. M ODELING This section describes the dynamic model of a compression system. The plant model, depicted in Figure 1, includes a gas turbine, a centrifugal compressor, driven by the turbine, equipped with an adjustable inlet guide vane (IGV), two volumes, and two actuator valves. The two volumes are placed one in suction and the other in discharge of the compressor. The valves are placed upstream the first volume, in compressor suction, and downstream the second volume. Model parameters are shown in Table I for a specific operating point considered in the simulations.
a Dipartimento di Ingegneria dell’Informazione, Universit` a di Siena, Italy. E-mail: [email protected]; [email protected]. b Nuovo Pignone Ge Oil & Gas, Florence, Italy. E-mail: [email protected]; [email protected].
978-1-4799-7408-5/14/$31.00 ©2014 IEEE
Centrifugal Compressor
Shaft Speed
486
TABLE I P LANT PARAMETERS AT DESIGN CONDITION Value 47.926 90.441 103.1 70 71.578 121.177 6053.264 0.00 460.386 275.65 18.591 18126 15 340 80 80
IGV=-69.55 [deg] Hp[kJ/kg]
Parameter Upstream valve U V (%) Downstream valve DV (%) Inlet pressure of the plant p1 (bar-a) Outlet pressure of the plant p2 (bar-a) Compressor suction pressure ps (bar-a) Compressor discharge pressure pd (bar-a) Rotational speed N (rpm) Inlet guide vane IGV (deg) Flow rate W c (kg/s) Compressor inlet temp. Ts (K) Molecular weight M W (g/mol) Power P w (kW) Ambient temperature Tamb (◦ C) Speed of sound a (m/s) Volume V1 (m3 ) Volume V2 (m3 )
IGV=0 [deg]
SCL A 6659 6741
SLL
6054 6128
B 4290
4238
Speed lines Choke line Qv[m3 /s]
Fig. 3.
A. Gas Turbine Model
Characteristic map of variable geometry compressor.
representation is often called the compressor performance map (or compressor map). The polytropic head is defined as the amount of reversible work required to compress a unit mass of gas. Figure 3 shows the compressor map for one of the simulations performed in the present study. The usable region of the map is limited by the resistance lines (Surge Limit Line SLL, Surge Control Line SCL, and Choke line) and the maximum and minimum rotational speed of the compressor. The region bounded by dashed lines is the compressor map when IGV = 0 [deg], while that bounded by solid lines corresponds to IGV = −69.55 [deg]. It is clear from the figure that the operating region of the compressor is significantly enlarged when IGV is used as a control variable.
Gas turbine modeling has been addressed in many studies [14], [15], [16], [17], [18]. In [19], a detailed review of the typical gas turbine performance maps is provided. In this paper, the gas turbine steady-state operating maps are used to determine the engine heat rate at a given ambient temperature, as a function of compressor power and turbine speed. Data of the power gas turbine PGT25 DLE are used. Figure 2 shows the considered heat rate map. The nominal speed of the gas turbine is Nref = 6100 [rpm]. The rotational speed can be varied from 70% to 105% of the nominal turbine speed. The ambient temperature is assumed to be 15◦ C.
Heat rate
C. Linearization of plant model The overall dynamic model of the plant in Fig. 1 includes the mass flow dynamics, the models of speed and IGV actuators and the dynamics of the suction and discharge volumes (the equations are standard and are omitted due to lack of space). Model-based control techniques usually require an analytical model of the plant to be controlled: this is not the case for the considered plant, which contains lookup tables for the turbine and compressor maps. The standard approach used in these cases is to generate an approximate model which is used only for the purpose of designing the controller. The model does not have to be accurate enough to reproduce the behaviour of the system: it just has to capture the dominant dynamics which are relevant to control design. Most of the times, linearized models around the considered operating point are employed. This greatly simplifies the control design procedure. In order to get a linear approximation of the plant nonlinear model, the function linearize of the Matlab/Simulink Control Toolbox has been applied to the Simulink model of the plant. The input, state and output variables of the system are as follows: • Input variables: – u1 (t): commanded IGV [deg]
B
Sp ee d A
Fig. 2.
er Pow
Steady-state heat rate map of the PGT25 DLE gas turbine.
B. Centrifugal Compressor Model Usually the performance curves of the compressor are given as compressor ratio versus inlet flow rate (Rc-Qv coordinates) at certain suction conditions. Other coordinates can be used for the analysis of the compressor: for example, the polytropic head (Hp) versus volumetric flow (Qv) representation (Hp-Qv plot) presents a number of advantages. This 487
122.8
– u2 (t): commanded rotational speed [rpm] State variables: – – – – – •
x1 (t): x2 (t): x3 (t): x4 (t): x5 (t):
122.6
mass flow rate [kg/s] actual rotational speed [rpm] actual IGV [deg] suction pressure [bara] discharge pressure [bara]
122.4 Discharge Pressure (bara)
•
Output variable: – y(t): discharge pressure [bara]
122.2
122
121.8
121.6
The values of the above variables at the nominal operating point are denoted by uss , xss , y ss , respectively. The resulting linearized model is given by
121.4
121.2 0
δ x(t) ˙ δy(t)
= =
Aδx(t) + Bδu(t) Cδx(t)
B= C=
0 0 1 0 0
0 0 0
0 1 0 0 0
Fig. 4.
20
30
40
50 Time (s)
60
70
80
90
100
Comparison between plant model and linearized model.
III. C ONTROL SYSTEM DESIGN A. Anti-windup proportional-integral (PI) control PI control is a standard control technique, which is commonly used to track a reference signal. The basic discretetime PI controller is described by: Ts e(k) (2) u(k) = Kp + Ki q−1 where u is the control signal, e = ysp − y is the tracking error, y is the measured process variable, ysp is the set point and q is the forward shift operator. The controller parameters are the proportional gain Kp and the integral gain Ki . Kp
e
0 1
10
(1)
where the notation δ(.) = (.) − (.)ss represents the variation of the considered variable with respect to the nominal value. The input values at the operating point considered in Table I are uss = 0 [deg] and uss = 6053.264 1 2 [rpm]. The corresponding state vector is equal to xss = [454.8877 6053.264 0 74.1093 121.27]T and the system discharge pressure is y ss = 121.27 [bara]. The system matrices A, B and C turn out to be: −15.7 2.6327 17.028 270.9 −96.8 0 −1.00 0 0 0 0 0 −1.00 0 0 A= −0.026 0 0 −0.18 0 0.023 0.0003 0.0017 0.01 −0.13
Nonlinear Linearized Reference
Ki
+
R
()
+ +
Actuator limits u′ u
− −
+
1/Taw
Fig. 5.
The eigenvalues λ of system matrix A are found to be
PI controller with back-calculation anti-windup scheme.
To prevent actuators saturation (i.e., gas turbine speed at minimum or maximum value) the back-calculation antiwindup scheme in Figure 5 has been implemented [20]. A single-input single-output PI controller for the rotational speed u2 (k) has been used to regulate the discharge pressure y(k) to its reference value, while keeping the IGV input constant to u1 (k) = 5 [deg] (it has been observed that the plant achieves the minimum fuel consumption with this value of IGV). Results with this standard approach have been used as a baseline solution, for comparison with multi-input multioutput (MIMO) control techniques exploiting also the IGV as a control variable. The PI controller parameters are reported in Table II. The proportional and integral gains and the antiwindup back calculation coefficient have been obtained via manual tuning.
λ = {−15.0729, − 0.7986, − 0.1379, − 1, − 1}. Figure 4 compares the behaviour of the nonlinear plant and of the corresponding linearized model, when they are inserted in a feedback loop with an MPC control scheme (see section III-C). To this purpose, the linearized model is discretized with sampling time Ts = 1 second. In the considered test, the control has to regulate the discharge pressure to the reference value 122.48, starting from the steady state initial condition y ss . It can be observed that the output behaviours are very similar. This suggests that the linearized model captures the essential dynamics of the plant, at least for small variations with respect to the considered operating point. 488
TABLE II
The classic LQ control theory provides the way to compute the state feedback matrix K by solving a Riccati equation. When the state vector is not fully measurable, δx(k) is replaced by the corresponding estimate δ x ˆ(k) provided by the Kalman filter. The augmented state is hence ξ(k) = [δ x ˆ(k) xi (k)]T . The presence of the integral action ensures that the output y(k) tracks the reference command ysp (k) asymptotically. The tuning parameters of the LQ controller are the matrices Q and R in the cost function (7). In the simulations presented in this paper the following values have been used
PARAMETERS FOR PI CONTROLLER Parameter Proportional gain Kp Integral gain Ki Taw Initial value of the integral Falling slew rate Rising slew rate Upper saturation limit Lower saturation limit
Value 220 20 1 6053.264 -15 15 6405 4270
Q R
B. LQG control with integral action In this section, LQ optimal control for multivariable feedback design is considered. In order to perform output tracking of a non-zero reference signal ysp , an integral action is added to classic LQ optimal control, by suitable augmenting the original state-space system. Since complete state information is not available, it is necessary to use a LQG control scheme, which includes Kalman filter estimation of the state vector. The complete control scheme is depicted in Figure 6. Controller R xi
e
y sp +
K
-
+
Plant
Actuator limits +
+
δˆ x Fig. 6.
C. Model predictive control In this study, linear MPC is employed, based on the linearized model derived as explained in Section II-C. The main idea behind MPC is that at each time instant k, the controller solves an optimal control problem over a finite prediction horizon [k, k + m]. To limit the number of optimization variables, the control input may be allowed to change over a shorter control horizon [k, k + p], with p < m, and then kept constant from k + p + 1 to k + m. Then, the first element of the optimal input sequence is applied to the plant. A new optimal control problem is solved at time k + 1 and so on. Assume the system is described by the discrete-time model (3) (to simplify notation, hereafter the symbol δ before the variable names is omitted). The objective is to find at each time k the solution of the optimization problem ( m−1 X min wi∆u1 ∆u1 (k + i|k)2 +
y
+
Kalman Filter -
y ss
Block diagram of closed loop system with LQGI controller.
Assume that around the operating point (xss , uss ), the system is described by the discretized version of system (1), δx(k + 1) δy(k)
= =
Ad δx(k) + Bd δu(k) Cd δx(k).
(3)
u(k|k),...,u(k+p−1|k)
Define the output tracking error as e(k) = ysp (k) − y(k) = δysp (k) − δy(k) and the error sum as xi (k + 1)
=
xi (k) + Ts e(k).
(4)
umin ≤ uh (k + i|k) ≤ umax h h ∆umin ≤ ∆uh (k + i|k) ≤ ∆umax h h ∆uh (k + j|k) = 0 for h = 1, 2, i = 0, . . . , m − 1, j = p, . . . , m − 1 In (8), y(k + i|k) denotes the predicted output value at time (k + i), based on the information available at time k, while u(k + i|k) is the input value at time k + i, for the input sequence starting at time k, and ∆u(k + i|k) = u(k + i + 1|k) − u(k + i|k). It can be observed that the above formulation is able to cope with several types of hard process constraints, involving both input variables and their rates. The tuning parameters of the MPC scheme are:
(6)
minimizing the quadratic cost function ∞ X
δξ(k)T Qδξ(k) + δu(k)T Rδu(k).
(8)
subject to the constraints
The aim is to define a state feedback control law
J=
i=0
wi∆u2 ∆u2 (k + i|k)2 + !) 2 y wi y(k + i + 1|k) − ysp (k + i + 1)
By defining the augmented state as ξ(k) = [δx(k) xi (k)]T , one gets the augmented system equations " # " #" # δx(k + 1) Ad 0 δx(k) = + xi (k + 1) −Ts Cd 1 xi (k) # " # " 0 Bd δu(k) + δysp (k). (5) 0 Ts
δu(k) = −Kξ(k)
diag([0 0 0 0 30 15]) diag([10 0.1]),
where diag(v) denotes a diagonal matrix, with diagonal v. Notice that by acting on the matrix R one can tune the relative influence on the control action of compressor speed and IGV opening.
uss δu +
= =
(7)
k=0
489
the prediction horizon length m; the control horizon length p; ∆u • the input increment weights wi h , h = 1, 2; y • the output weights wi . All weights have been set to constant values over the prediction horizon, i.e. wiy = wy , and wi∆uh = w∆uh for all i = 1, ..., m. The values of the tuning parameters adopted in the simulations are reported in Table III. The cost function is minimized by using the quadratic programming solver provided by the Model Predictive Control Toolbox for Matlab [21].
only speed is used as a control variable (PI controller). This demonstrates that the variation of inlet guide vane can improve the regulation of the discharge pressure, compared to controlling only the rotational speed. The improvement provided by the MIMO controllers with respect to the PI one is much more evident when the change in the downstream valve is fast. It is also observed that when fast decrease of the flow rate occurs, the IGV tends to be driven to the lowest admissible value in order to achieve a faster regulation of the output signal. Then, at 150 [s] both valves undergo a slow increase: the downstream valve returns to its initial position at 300 [s], while the upstream valve goes from 47.93% to 65% and then remains constant. From 150 [s] to 300 [s], all controllers manage to keep the discharge pressure at the desired reference value, due to the fact that in this case the valves variation is slow. At 300 [s] the upstream valve undergoes a step change going from 65% to 48% and at 350 [s] the downstream valve undergoes the same step change as at 15 [s]. Once again, thanks to the combined use of speed and IGV, the LQGI and MPC control schemes show a much better control performance than the PI one. In all the simulation tests, the performance of the MPC and LQGI controllers turned out to be similar, the main advantage of the MPC being the ability to incorporate input constraints into the control design, while in the LQGI one can only act indirectly on the input ranges by suitably tuning the R matrix in the cost function (7).
•
•
TABLE III PARAMETERS FOR MPC CONTROLLER Parameter [p; m] [w∆u1 ; w∆u2 ; wy ] [∆umin ; ∆umax ; ∆umin ; ∆umax ] 1 1 2 2 max ; umin ; umax ] [umin ; u 1 1 2 2
IV.
Value [2; 5] [5; .1; 30] [−4; 4; −15; 15] [−70; 10; 4270; 6405]
SIMULATION RESULTS
In this section, the MIMO controllers (MPC and LQGI) are evaluated and compared to the PI one, within a typical scenario in which the output pressure changes due to external disturbances occurring at the suction and discharge volumes. A number of different tests have been performed: in the one presented here the valves of the two volumes are opened and closed several times in order to test the controllers performance in different operating conditions. In the simulation reported in Figure 7, the opening of the upstream and downstream valves follows the profiles shown in Figure 7-a, where the solid curve represents the opening (%) of the upstream valve (UV), while the dashed curve shows that of the downstream valve (DV). The remaining plots in Figure 7 report the results for the MPC controller (solid), the LQGI controller (dashed) and the PI controller (dash-dotted). Figure 7-b depicts the mass flow rate. Figure 7-c shows the discharge pressure, while Figures 7-d and 7-e report the IGV and rotational speed, respectively. Notice that for the PI controller, the IGV is kept constant at 5 [deg] during the whole experiment. In Figure 3 the trajectory of the operating point in the compressor map, corresponding to the MPC simulation, is reported. The same trajectory is shown in Figure 2, on the heat rate map of the power gas turbine. In both figures, A denotes the starting operating point while B is the final one: it can be noticed that during the simulation the operating point covers different paths from A to B and backwards, depending on the different variation profiles of the valves. The simulation starts by acting on the downstream valve which undergoes a step change (from 90% to 65% at 15 [s]), thus causing a quick move of the operating point. It can be observed that the combined use of IGV and rotational speed allows one to regulate the discharge pressure to the desired reference value in a much faster way and with much more limited transients, with respect to the case in which
V. C ONCLUSIONS This paper studied the problem of the pressure regulation in an industrial gas centrifugal compressor. Two multivariable feedback control schemes have been proposed and tested, based on a linearized model of the plant. This study demonstrates that the variation of IGV can improve the regulation of the discharge pressure, compared to controlling only the rotational speed. Future work will include the use of anti-surge control action and the investigation of linear parameter varying (LPV) control techniques. R EFERENCES [1] S. Golden, S. Fulton, and D. Hanson. Understanding centrifugal compressor performance in a connected process system. In Petroleum Technology Quarterly,, Spring 2002. [2] A. Mohseni, R. A. Van den Braembussche, J. R. Seume, and E. Goldhahn. Novel IGV designs for centrifugal compressors and their interaction with the impeller. Journal of Turbomachinery, 134(2), June 2011. [3] S. Varigonda, J. Eborn, and S.A. Bortoff. Multivariable control design for the water gas shift reactor in a fuel processor. In Proceedings of the 2004 American Control Conference, volume 1, pages 840–844, June 2004. [4] T. Goya, E. Omine, T. Senjyu, M. Tokudome, A. Yona, N. Urasaki, T. Funabashi, and Chul-Hwan Kim. Torsional torque suppression of decentralized generators using LQR observer with parameter identification. In IEEE International Symposium on Industrial Electronics, pages 2109–2114, July 2009. [5] R. E. Stillwagon. Economic aspects of electrically driven compressor stations for natural gas pipelines. IEEE Transactions on Industry Applications, IA-11(2):240–245, March 1975. [6] N. Uddin and J.T. Gravdahl. Piston-actuated active surge control of centrifugal compressor including integral action. In 11th International Conference on Control, Automation and Systems (ICCAS), pages 991– 996, October 2011.
490
90 80
50
70 60 450
b)
350 140
c) 130 120 0
d)
−20 −40
Speed [rpm]
−60 6000
e)
Inlet Guide Vane [deg.]
Discharge Pressure [bara]
400
DV [%]
60
Flow Rate [kg/s]
UV [%]
100
a)
5500 5000 4500 0
50
100
150
200
250 Time [s]
300
350
400
450
500
Fig. 7. Simulation results for MPC (solid), LQGI (dashed) and PI (dash-dotted) controllers: a) upstream (solid) and downstream (dashed) valve opening; b) flow rate; c) discharge pressure; d) IGV; e) rotational speed.
IEEE Transactions on Power Systems, 9(3):1698–1708, Aug 1994. [15] M. Nagpal, A. Moshref, G. K. Morison, and P. Kundur. Experience with testing and modeling of gas turbines. In IEEE Power Engineering Society Winter Meeting, volume 2, pages 652–656, 2001. [16] K. Kunitomi, A. Kurita, H. Okamoto, Y. Tada, S. Ihara, P. Pourbeik, W. W. Price, A.B. Leirbukt, and J. J. Sanchez-Gasca. Modeling frequency dependency of gas turbine output. In IEEE Power Engineering Society Winter Meeting, volume 2, pages 678–683, 2001. [17] H. Iki, M. Komatsu, Y. Uriu, M. Watanabe, and Y. Mitani. The development of a model of the industrial gas turbine generator on the analysis tool for the power system dynamics. In 42nd International Universities Power Engineering Conference, UPEC 2007, pages 148– 151, September 2007. [18] S. Massucco, A. Pitto, and F. Silvestro. A gas turbine model for studies on distributed generation penetration into distribution networks. IEEE Transactions on Power Systems, 26(3):992–999, 2011. [19] R. Kurz and K. Brun. Gas turbine performance - What makes the map. In 29th Texas A&M Turbomachinery Symposium, pages 247– 262, Houston, TX, USA, September 2000. [20] X-L. Li, J-G. Park, and H-B. Shin. Comparison and evaluation of anti-windup PI controllers. Journal of Power Electronics, 11(1):45– 50, January 2011. [21] A. Bemporad, M. Morari, and R. N. Lawrence. Model Predictive Control Toolbox for Use with Matlab. Mathworks, 2005.
[7] S. J. Qin and T. A. Badgwell. A survey of industrial model predictive control technology. Control Engineering Practice, pages 733–764, 2003. [8] J. P. M. Smeulers, W. J. Bouman, and H. A. Van Essen. Model predictive control of compressor installations. In Proceedings of the Conference on Compressors and their Systems, I Mech E, London, 1999. [9] H. Ghorbani, A. Ghaffari, and M. Rahnama. Constrained model predictive control implementation for a heavy-duty gas turbine power plant. WSEAS Transactions on System and Control, 2008. [10] A. Cortinovis, D. Pareschi, M. Mercangez, and Th. Besselmann. Model predictive anti-surge control of centrifugal compressors with variable-speed drives. In IFAC Workshop on Automatic Control in Offshore Oil and Gas Production, pages 251–256, Norway, May 31 June 1 2012. [11] B. Chaudhuri, R. Majumder, and B.C. Pal. Application of multiplemodel adaptive control strategy for robust damping of interarea oscillations in power system. IEEE Transactions on Control Systems Technology, 12(5):727–736, 2004. [12] R.K. Al Seyab and Y. Cao. Nonlinear model predictive control for the ALSTOM gasifier. Journal of Process Control, 16(8):795 – 808, 2006. [13] J. Prakash and R. Senthil. Design of observer based nonlinear model predictive controller for a continuous stirred tank reactor. Journal of Process Control, 18(5):504 – 514, 2008. [14] F. P. De Mello, D. J. Ahner, P. M. Anderson, J. H. Doudna, P. Kundur, L. M. Richardson, G. Tandy, C. W. Taylor, and F. Van De Meulebroeke. Dynamic models for combined cycle plants in power system studies.
491
Multivariable Control for Regulating High Pressure Centrifugal Compressor with Variable Speed and IGV Toufik Bentaleba , Alessio Cacittib , Sergio De Franciscisb , Andrea Garullia Inlet guide vane
Abstract— The objective of this paper is to develop a multivariable control system for a class of centrifugal compressors, which exploit as control signals both the rotational speed and the Inlet Guide Vane (IGV). Linear Quadratic Gaussian control with Integral action (LQGI) and Model Predictive Control (MPC) are investigated. The LQGI and MPC controllers are compared to a standard proportional integral (PI) controller, to regulate the discharge pressure of the compressor. The control algorithms are simulated and compared in different operating scenarios. Results demonstrate that the proposed multivariabe control schemes provide better performance than the singleloop PI controller, thus motivating the use of IGV for control purposes. Index Terms— Multivariable control, Optimal control, Model predictive control, Linear quadratic regulator, Power plant.
Gas Turbine N
Ps Pd
T1 P1
DV
UV V2
V1
T2 P2
Gas Outlet
Gas Inlet
I. I NTRODUCTION
Fig. 1. Compression system: gas turbine-driven centrifugal compressor, equipped with IGV, and two volumes at suction and discharge.
High-pressure multistage centrifugal compressors are an essential part of the process machinery in the oil and gas industry across a wide variety of applications. Centrifugal compressors in a connected process system are very sensitive to changes in the inlet conditions [1], such as the suction pressure, the temperature, as well as the inlet gas density. This type of gas compressors requires quick response and reliable control systems to increase their capacity. Process regulation (for example discharge pressure control) is usually performed by acting on the shaft speed. More recently, a variable Inlet Guide Vane (IGV) has been used for control purposes. The IGV system allows wide capacity control of the centrifugal compressor with reduced energy losses. In centrifugal compressors, IGV is used to control the mass flow rate with negligible change in compressor ratio and shaft speed [2]. When both the turbine speed and the IGV are used to regulate the process, a multivariable control system is necessary. In this paper two types of multivariable control, namely a linear quadratic regulator (LQR) and a model predictive controller (MPC), are presented, and compared to a standard PI controller based on speed regulation. The LQR is widely and effectively used in many industrial applications [3], [4], including control of the compressor stations for natural gas pipelines [5], [6]. Since complete state information is not available, it is necessary to use a LQG control scheme, which includes Kalman filter estimation of the state vector. The LQG scheme includes an integral action to compensate the reference tracking error. In recent years, MPC has confirmed
itself as a successful approach to multivariable control due to its advantages over traditional controllers [7]. In particular, it is widely employed in the oil and gas industry to deal with power plant control [8], [9], [10]. Although there is an increasing research interest in nonlinear MPC [11], [12], [13] most of this literature is dedicated to systems described by analytical models, while the model of the reference application considered in this paper contains non analytical parts (e.g. look-up tables). Hence, in this study linear MPC is employed, based on a linearized model of the plant. The paper is structured as follows: in Section II, the power plant model is presented. Section III deals with the PI, LQGI, and MPC control schemes and the details about the implementation of each controller. Simulation results are presented in Section IV to compare the performances of the PI, LQGI and MPC controllers, while some concluding remarks are given in Section V. II. M ODELING This section describes the dynamic model of a compression system. The plant model, depicted in Figure 1, includes a gas turbine, a centrifugal compressor, driven by the turbine, equipped with an adjustable inlet guide vane (IGV), two volumes, and two actuator valves. The two volumes are placed one in suction and the other in discharge of the compressor. The valves are placed upstream the first volume, in compressor suction, and downstream the second volume. Model parameters are shown in Table I for a specific operating point considered in the simulations.
a Dipartimento di Ingegneria dell’Informazione, Universit` a di Siena, Italy. E-mail: [email protected]; [email protected]. b Nuovo Pignone Ge Oil & Gas, Florence, Italy. E-mail: [email protected]; [email protected].
978-1-4799-7408-5/14/$31.00 ©2014 IEEE
Centrifugal Compressor
Shaft Speed
486
TABLE I P LANT PARAMETERS AT DESIGN CONDITION Value 47.926 90.441 103.1 70 71.578 121.177 6053.264 0.00 460.386 275.65 18.591 18126 15 340 80 80
IGV=-69.55 [deg] Hp[kJ/kg]
Parameter Upstream valve U V (%) Downstream valve DV (%) Inlet pressure of the plant p1 (bar-a) Outlet pressure of the plant p2 (bar-a) Compressor suction pressure ps (bar-a) Compressor discharge pressure pd (bar-a) Rotational speed N (rpm) Inlet guide vane IGV (deg) Flow rate W c (kg/s) Compressor inlet temp. Ts (K) Molecular weight M W (g/mol) Power P w (kW) Ambient temperature Tamb (◦ C) Speed of sound a (m/s) Volume V1 (m3 ) Volume V2 (m3 )
IGV=0 [deg]
SCL A 6659 6741
SLL
6054 6128
B 4290
4238
Speed lines Choke line Qv[m3 /s]
Fig. 3.
A. Gas Turbine Model
Characteristic map of variable geometry compressor.
representation is often called the compressor performance map (or compressor map). The polytropic head is defined as the amount of reversible work required to compress a unit mass of gas. Figure 3 shows the compressor map for one of the simulations performed in the present study. The usable region of the map is limited by the resistance lines (Surge Limit Line SLL, Surge Control Line SCL, and Choke line) and the maximum and minimum rotational speed of the compressor. The region bounded by dashed lines is the compressor map when IGV = 0 [deg], while that bounded by solid lines corresponds to IGV = −69.55 [deg]. It is clear from the figure that the operating region of the compressor is significantly enlarged when IGV is used as a control variable.
Gas turbine modeling has been addressed in many studies [14], [15], [16], [17], [18]. In [19], a detailed review of the typical gas turbine performance maps is provided. In this paper, the gas turbine steady-state operating maps are used to determine the engine heat rate at a given ambient temperature, as a function of compressor power and turbine speed. Data of the power gas turbine PGT25 DLE are used. Figure 2 shows the considered heat rate map. The nominal speed of the gas turbine is Nref = 6100 [rpm]. The rotational speed can be varied from 70% to 105% of the nominal turbine speed. The ambient temperature is assumed to be 15◦ C.
Heat rate
C. Linearization of plant model The overall dynamic model of the plant in Fig. 1 includes the mass flow dynamics, the models of speed and IGV actuators and the dynamics of the suction and discharge volumes (the equations are standard and are omitted due to lack of space). Model-based control techniques usually require an analytical model of the plant to be controlled: this is not the case for the considered plant, which contains lookup tables for the turbine and compressor maps. The standard approach used in these cases is to generate an approximate model which is used only for the purpose of designing the controller. The model does not have to be accurate enough to reproduce the behaviour of the system: it just has to capture the dominant dynamics which are relevant to control design. Most of the times, linearized models around the considered operating point are employed. This greatly simplifies the control design procedure. In order to get a linear approximation of the plant nonlinear model, the function linearize of the Matlab/Simulink Control Toolbox has been applied to the Simulink model of the plant. The input, state and output variables of the system are as follows: • Input variables: – u1 (t): commanded IGV [deg]
B
Sp ee d A
Fig. 2.
er Pow
Steady-state heat rate map of the PGT25 DLE gas turbine.
B. Centrifugal Compressor Model Usually the performance curves of the compressor are given as compressor ratio versus inlet flow rate (Rc-Qv coordinates) at certain suction conditions. Other coordinates can be used for the analysis of the compressor: for example, the polytropic head (Hp) versus volumetric flow (Qv) representation (Hp-Qv plot) presents a number of advantages. This 487
122.8
– u2 (t): commanded rotational speed [rpm] State variables: – – – – – •
x1 (t): x2 (t): x3 (t): x4 (t): x5 (t):
122.6
mass flow rate [kg/s] actual rotational speed [rpm] actual IGV [deg] suction pressure [bara] discharge pressure [bara]
122.4 Discharge Pressure (bara)
•
Output variable: – y(t): discharge pressure [bara]
122.2
122
121.8
121.6
The values of the above variables at the nominal operating point are denoted by uss , xss , y ss , respectively. The resulting linearized model is given by
121.4
121.2 0
δ x(t) ˙ δy(t)
= =
Aδx(t) + Bδu(t) Cδx(t)
B= C=
0 0 1 0 0
0 0 0
0 1 0 0 0
Fig. 4.
20
30
40
50 Time (s)
60
70
80
90
100
Comparison between plant model and linearized model.
III. C ONTROL SYSTEM DESIGN A. Anti-windup proportional-integral (PI) control PI control is a standard control technique, which is commonly used to track a reference signal. The basic discretetime PI controller is described by: Ts e(k) (2) u(k) = Kp + Ki q−1 where u is the control signal, e = ysp − y is the tracking error, y is the measured process variable, ysp is the set point and q is the forward shift operator. The controller parameters are the proportional gain Kp and the integral gain Ki . Kp
e
0 1
10
(1)
where the notation δ(.) = (.) − (.)ss represents the variation of the considered variable with respect to the nominal value. The input values at the operating point considered in Table I are uss = 0 [deg] and uss = 6053.264 1 2 [rpm]. The corresponding state vector is equal to xss = [454.8877 6053.264 0 74.1093 121.27]T and the system discharge pressure is y ss = 121.27 [bara]. The system matrices A, B and C turn out to be: −15.7 2.6327 17.028 270.9 −96.8 0 −1.00 0 0 0 0 0 −1.00 0 0 A= −0.026 0 0 −0.18 0 0.023 0.0003 0.0017 0.01 −0.13
Nonlinear Linearized Reference
Ki
+
R
()
+ +
Actuator limits u′ u
− −
+
1/Taw
Fig. 5.
The eigenvalues λ of system matrix A are found to be
PI controller with back-calculation anti-windup scheme.
To prevent actuators saturation (i.e., gas turbine speed at minimum or maximum value) the back-calculation antiwindup scheme in Figure 5 has been implemented [20]. A single-input single-output PI controller for the rotational speed u2 (k) has been used to regulate the discharge pressure y(k) to its reference value, while keeping the IGV input constant to u1 (k) = 5 [deg] (it has been observed that the plant achieves the minimum fuel consumption with this value of IGV). Results with this standard approach have been used as a baseline solution, for comparison with multi-input multioutput (MIMO) control techniques exploiting also the IGV as a control variable. The PI controller parameters are reported in Table II. The proportional and integral gains and the antiwindup back calculation coefficient have been obtained via manual tuning.
λ = {−15.0729, − 0.7986, − 0.1379, − 1, − 1}. Figure 4 compares the behaviour of the nonlinear plant and of the corresponding linearized model, when they are inserted in a feedback loop with an MPC control scheme (see section III-C). To this purpose, the linearized model is discretized with sampling time Ts = 1 second. In the considered test, the control has to regulate the discharge pressure to the reference value 122.48, starting from the steady state initial condition y ss . It can be observed that the output behaviours are very similar. This suggests that the linearized model captures the essential dynamics of the plant, at least for small variations with respect to the considered operating point. 488
TABLE II
The classic LQ control theory provides the way to compute the state feedback matrix K by solving a Riccati equation. When the state vector is not fully measurable, δx(k) is replaced by the corresponding estimate δ x ˆ(k) provided by the Kalman filter. The augmented state is hence ξ(k) = [δ x ˆ(k) xi (k)]T . The presence of the integral action ensures that the output y(k) tracks the reference command ysp (k) asymptotically. The tuning parameters of the LQ controller are the matrices Q and R in the cost function (7). In the simulations presented in this paper the following values have been used
PARAMETERS FOR PI CONTROLLER Parameter Proportional gain Kp Integral gain Ki Taw Initial value of the integral Falling slew rate Rising slew rate Upper saturation limit Lower saturation limit
Value 220 20 1 6053.264 -15 15 6405 4270
Q R
B. LQG control with integral action In this section, LQ optimal control for multivariable feedback design is considered. In order to perform output tracking of a non-zero reference signal ysp , an integral action is added to classic LQ optimal control, by suitable augmenting the original state-space system. Since complete state information is not available, it is necessary to use a LQG control scheme, which includes Kalman filter estimation of the state vector. The complete control scheme is depicted in Figure 6. Controller R xi
e
y sp +
K
-
+
Plant
Actuator limits +
+
δˆ x Fig. 6.
C. Model predictive control In this study, linear MPC is employed, based on the linearized model derived as explained in Section II-C. The main idea behind MPC is that at each time instant k, the controller solves an optimal control problem over a finite prediction horizon [k, k + m]. To limit the number of optimization variables, the control input may be allowed to change over a shorter control horizon [k, k + p], with p < m, and then kept constant from k + p + 1 to k + m. Then, the first element of the optimal input sequence is applied to the plant. A new optimal control problem is solved at time k + 1 and so on. Assume the system is described by the discrete-time model (3) (to simplify notation, hereafter the symbol δ before the variable names is omitted). The objective is to find at each time k the solution of the optimization problem ( m−1 X min wi∆u1 ∆u1 (k + i|k)2 +
y
+
Kalman Filter -
y ss
Block diagram of closed loop system with LQGI controller.
Assume that around the operating point (xss , uss ), the system is described by the discretized version of system (1), δx(k + 1) δy(k)
= =
Ad δx(k) + Bd δu(k) Cd δx(k).
(3)
u(k|k),...,u(k+p−1|k)
Define the output tracking error as e(k) = ysp (k) − y(k) = δysp (k) − δy(k) and the error sum as xi (k + 1)
=
xi (k) + Ts e(k).
(4)
umin ≤ uh (k + i|k) ≤ umax h h ∆umin ≤ ∆uh (k + i|k) ≤ ∆umax h h ∆uh (k + j|k) = 0 for h = 1, 2, i = 0, . . . , m − 1, j = p, . . . , m − 1 In (8), y(k + i|k) denotes the predicted output value at time (k + i), based on the information available at time k, while u(k + i|k) is the input value at time k + i, for the input sequence starting at time k, and ∆u(k + i|k) = u(k + i + 1|k) − u(k + i|k). It can be observed that the above formulation is able to cope with several types of hard process constraints, involving both input variables and their rates. The tuning parameters of the MPC scheme are:
(6)
minimizing the quadratic cost function ∞ X
δξ(k)T Qδξ(k) + δu(k)T Rδu(k).
(8)
subject to the constraints
The aim is to define a state feedback control law
J=
i=0
wi∆u2 ∆u2 (k + i|k)2 + !) 2 y wi y(k + i + 1|k) − ysp (k + i + 1)
By defining the augmented state as ξ(k) = [δx(k) xi (k)]T , one gets the augmented system equations " # " #" # δx(k + 1) Ad 0 δx(k) = + xi (k + 1) −Ts Cd 1 xi (k) # " # " 0 Bd δu(k) + δysp (k). (5) 0 Ts
δu(k) = −Kξ(k)
diag([0 0 0 0 30 15]) diag([10 0.1]),
where diag(v) denotes a diagonal matrix, with diagonal v. Notice that by acting on the matrix R one can tune the relative influence on the control action of compressor speed and IGV opening.
uss δu +
= =
(7)
k=0
489
the prediction horizon length m; the control horizon length p; ∆u • the input increment weights wi h , h = 1, 2; y • the output weights wi . All weights have been set to constant values over the prediction horizon, i.e. wiy = wy , and wi∆uh = w∆uh for all i = 1, ..., m. The values of the tuning parameters adopted in the simulations are reported in Table III. The cost function is minimized by using the quadratic programming solver provided by the Model Predictive Control Toolbox for Matlab [21].
only speed is used as a control variable (PI controller). This demonstrates that the variation of inlet guide vane can improve the regulation of the discharge pressure, compared to controlling only the rotational speed. The improvement provided by the MIMO controllers with respect to the PI one is much more evident when the change in the downstream valve is fast. It is also observed that when fast decrease of the flow rate occurs, the IGV tends to be driven to the lowest admissible value in order to achieve a faster regulation of the output signal. Then, at 150 [s] both valves undergo a slow increase: the downstream valve returns to its initial position at 300 [s], while the upstream valve goes from 47.93% to 65% and then remains constant. From 150 [s] to 300 [s], all controllers manage to keep the discharge pressure at the desired reference value, due to the fact that in this case the valves variation is slow. At 300 [s] the upstream valve undergoes a step change going from 65% to 48% and at 350 [s] the downstream valve undergoes the same step change as at 15 [s]. Once again, thanks to the combined use of speed and IGV, the LQGI and MPC control schemes show a much better control performance than the PI one. In all the simulation tests, the performance of the MPC and LQGI controllers turned out to be similar, the main advantage of the MPC being the ability to incorporate input constraints into the control design, while in the LQGI one can only act indirectly on the input ranges by suitably tuning the R matrix in the cost function (7).
•
•
TABLE III PARAMETERS FOR MPC CONTROLLER Parameter [p; m] [w∆u1 ; w∆u2 ; wy ] [∆umin ; ∆umax ; ∆umin ; ∆umax ] 1 1 2 2 max ; umin ; umax ] [umin ; u 1 1 2 2
IV.
Value [2; 5] [5; .1; 30] [−4; 4; −15; 15] [−70; 10; 4270; 6405]
SIMULATION RESULTS
In this section, the MIMO controllers (MPC and LQGI) are evaluated and compared to the PI one, within a typical scenario in which the output pressure changes due to external disturbances occurring at the suction and discharge volumes. A number of different tests have been performed: in the one presented here the valves of the two volumes are opened and closed several times in order to test the controllers performance in different operating conditions. In the simulation reported in Figure 7, the opening of the upstream and downstream valves follows the profiles shown in Figure 7-a, where the solid curve represents the opening (%) of the upstream valve (UV), while the dashed curve shows that of the downstream valve (DV). The remaining plots in Figure 7 report the results for the MPC controller (solid), the LQGI controller (dashed) and the PI controller (dash-dotted). Figure 7-b depicts the mass flow rate. Figure 7-c shows the discharge pressure, while Figures 7-d and 7-e report the IGV and rotational speed, respectively. Notice that for the PI controller, the IGV is kept constant at 5 [deg] during the whole experiment. In Figure 3 the trajectory of the operating point in the compressor map, corresponding to the MPC simulation, is reported. The same trajectory is shown in Figure 2, on the heat rate map of the power gas turbine. In both figures, A denotes the starting operating point while B is the final one: it can be noticed that during the simulation the operating point covers different paths from A to B and backwards, depending on the different variation profiles of the valves. The simulation starts by acting on the downstream valve which undergoes a step change (from 90% to 65% at 15 [s]), thus causing a quick move of the operating point. It can be observed that the combined use of IGV and rotational speed allows one to regulate the discharge pressure to the desired reference value in a much faster way and with much more limited transients, with respect to the case in which
V. C ONCLUSIONS This paper studied the problem of the pressure regulation in an industrial gas centrifugal compressor. Two multivariable feedback control schemes have been proposed and tested, based on a linearized model of the plant. This study demonstrates that the variation of IGV can improve the regulation of the discharge pressure, compared to controlling only the rotational speed. Future work will include the use of anti-surge control action and the investigation of linear parameter varying (LPV) control techniques. R EFERENCES [1] S. Golden, S. Fulton, and D. Hanson. Understanding centrifugal compressor performance in a connected process system. In Petroleum Technology Quarterly,, Spring 2002. [2] A. Mohseni, R. A. Van den Braembussche, J. R. Seume, and E. Goldhahn. Novel IGV designs for centrifugal compressors and their interaction with the impeller. Journal of Turbomachinery, 134(2), June 2011. [3] S. Varigonda, J. Eborn, and S.A. Bortoff. Multivariable control design for the water gas shift reactor in a fuel processor. In Proceedings of the 2004 American Control Conference, volume 1, pages 840–844, June 2004. [4] T. Goya, E. Omine, T. Senjyu, M. Tokudome, A. Yona, N. Urasaki, T. Funabashi, and Chul-Hwan Kim. Torsional torque suppression of decentralized generators using LQR observer with parameter identification. In IEEE International Symposium on Industrial Electronics, pages 2109–2114, July 2009. [5] R. E. Stillwagon. Economic aspects of electrically driven compressor stations for natural gas pipelines. IEEE Transactions on Industry Applications, IA-11(2):240–245, March 1975. [6] N. Uddin and J.T. Gravdahl. Piston-actuated active surge control of centrifugal compressor including integral action. In 11th International Conference on Control, Automation and Systems (ICCAS), pages 991– 996, October 2011.
490
90 80
50
70 60 450
b)
350 140
c) 130 120 0
d)
−20 −40
Speed [rpm]
−60 6000
e)
Inlet Guide Vane [deg.]
Discharge Pressure [bara]
400
DV [%]
60
Flow Rate [kg/s]
UV [%]
100
a)
5500 5000 4500 0
50
100
150
200
250 Time [s]
300
350
400
450
500
Fig. 7. Simulation results for MPC (solid), LQGI (dashed) and PI (dash-dotted) controllers: a) upstream (solid) and downstream (dashed) valve opening; b) flow rate; c) discharge pressure; d) IGV; e) rotational speed.
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