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Short Papers_______________________________________________________________________________ Design of Incremental Fuzzy PI Controllers for a Gas-Turbine Plant Jong-Wook Kim and Sang Woo Kim Abstract—In this paper, incremental fuzzy proportional integral (PI) speed and temperature controllers for a heavy-duty gas-turbine plant are presented. To improve performance, an analysis of incremental fuzzy PI control is provided, and new fuzzy control rules are proposed. In applying the fuzzy PI control to a gas-turbine plant, all gains are optimized by an adaptive genetic algorithm. We show the performance improvement of the proposed controller compared with conventional PI controller via simulations. Index Terms—Adaptive genetic algorithm, fuzzy proportional integral (PI) control, gain tuning, gas-turbine plant.
I. INTRODUCTION Recently, many gas–steam combined plants, where heat-recovery steam generators (HRSGs), and steam turbines are added to gas turbines for overall efficiency, are constructed worldwide. In controlling the gas-turbine plants in their steady state, stabilizing the turbine rotor speed and exhaust temperature to rated values under a perturbed load torque is very important. The rotor speed is directly related to the quality of generated electric power, and the well-controlled exhaust temperature contributes to the energy efficiency of a connected steam turbine. The frequency-domain gas-turbine models were provided by Rowen. He first proposed simplified models of simple cycle and single-shaft gas turbines with inlet guide vane (IGV) opened [1], and modified the models by adding the influence of axial flow compressor variable IGVs [2]. Hannett et al.[3] reported that the model structure provided by Rowen was found to be adequate by simulation results. From the viewpoint of the control scheme, Rowen and Hannett et al. used conventional fixed-gain proportional integral (PI) controllers for speed, temperature, and acceleration controllers. In [4] and [5], nonlinear lumped parameter mathematical models of gas-turbine plants were described and decoupling control systems, which operate with no interactions between speed and exhaust temperature loops, were proposed. In order to design the decoupling controller, precise dynamic gas-plant mathematical models and accurate transfer functions of the turbo-gas are required, which are difficult to attain without various experiments and deep expertise. In this paper, to overcome the limited performance of the conventional PI control, we design the controllers for rotor speed and exhaust temperature with the incremental fuzzy PI control [6]–[9]. To improve performance, analysis of incremental fuzzy PI control is provided, and new fuzzy PI control rules are established consequently. In applying the fuzzy PI control to the gas-turbine plant, all the gains are optimized by an adaptive genetic algorithm (AGA) [10]. The performance enhanceManuscript received June 4, 2001; revised October 28, 2002. This work was supported in part by Electrical Enginnering and Science Research Institute (EESRI) under Grant 97-082, and in part by the Electrical and Computer Engineering Division, Pohang University of Science and Technology (POSTECH). The authors are with the Electrical and Computer Engineering Division, Pohang University of Science and Technology, Pohang 790-784, Korea (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/TMECH.2003.816858
ment of a gas-turbine plant by the proposed fuzzy PI control is verified via simulations. This paper is organized as follows. In Section II, a simplified block diagram and nonlinear state space equations of a gas-turbine plant are provided with a brief explanation. The analysis and improvement of incremental fuzzy PI controllers are presented in Section III. The simulation results of optimized incremental fuzzy PI controllers are provided in Section IV. The conclusion is presented in Section V II. GAS–TURBINE PLANT MODEL In this paper, GE MS7001EA is chosen as a basic gas-turbine plant model. Fig. 1 shows a simplified block diagram of a gas-turbine plant. The nomenclature concerning the gas-turbine plant is arranged in Table I, and turbine characteristics and constants are shown in Table II. The gas-turbine plant is a multivariable, highly nonlinear, and stiff process, which consists of three subsystems. The fuel-combustor subsystem is transformed from the block diagrams as
x_ 1 = 0 20x1 (t)+20uf (t)N (t); x_ 2 = 20x1 (t) 0 20x2 (t) x_ 3 = 2:5x2 (t) 0 2:5x3 (t) Wf (t) = x3 (t 0 "cr )
0:15
uf (t) 1:0
(1) (2) (3) (4)
where uf is the speed-control output which manipulates fuel flow. The exhaust temperature subsystem is described as
_ 1 = 0 21 (t) + 2uIGV (t); 25 uIGV (t) 84 _ 2 = 0 0:06672 (t) + 0:0133Tx (t) _ 3 = 0:42 (t) 0 0:43 (t) + 0:32Tx (t)
IGV (t) = 1 (t) Tm (t) = 3 (t)
(5) (6) (7) (8)
where (9)–(11), shown at the bottom of the next page, hold and uIGV is the temperature control output which manipulates IGV. The rotor subsystem is written as
0 1cd 1 (t) + 1cd Wf (t) f2 (1 (t); 2 (t)) 0 Dt (t)2 (t)2 _2 = _1
=
N (t) = 2 (t)
i
=
turbine torque =
(13) (14)
where
f2
(12)
1:16(1
0 0:133) :
2
(15)
The control system includes speed control, fuel temperature control, IGV temperature control, and acceleration control. The speed controller is the primary means of gas-turbine control under part load condition. The fuel temperature control limits gas-turbine output at a predetermined firing temperature, independent of variation in ambient temperature. For heat recovery applications, an optimum part load cycle performance is obtained when the turbine exhaust temperature is at maximum. This is achieved with closing the IGV by the IGV temperature controller within the operating ranges in Table II. Acceleration control is used primarily during gas-turbine startup to limit the rate of rotor acceleration prior to reaching minimum governor speed, thus ameliorating the thermal stresses encountered during startup. Among the controllers, speed and IGV temperature controllers dominate the performance in steady state. Therefore, speed and IGV temperature controllers whose outputs are uf and uIGV in (1) and (5), re-
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Fig. 1. Simplified block diagram of a single-shaft mechanical drive gas turbine. TABLE I NOMENCLATURE
spectively, are designed using fuzzy PI control in this paper. In simulation, Ta is fixed at 18 , and sampling time, Ts , is 0.01 second. The Runge-Kutta’s fourth order method is adopted for the computation of every instantaneous state value with the abovementioned state equations.
C
III. INCREMENTAL FUZZY PI CONTROLLER The proportional integral derivative (PID) controller is commonly used in factories because of its simple structure and robust performance. The output of a conventional PID control is written as
( ) = P e(t) + I
ut
t
0
( ) + D dedt(t)
e d
(16)
where P , I , and D are proportional, integral, and derivative gains, respectively. Much research is carried out to enhance the capabilities of PID controllers in various ways, and a PID controller based on fuzzy logic is increasingly applied to many systems with nonlinearity and uncertainty.
The fuzzy PID control can be classified into the direct action (DA) type and the gain scheduling (GS) type. The majority of fuzzy PID controllers belong to the DA type, which replaces a conventional PID controller by a fuzzy logic controller. However, from the viewpoint of applicability to an existing controller, the GS-type fuzzy controller is preferable, which maintains the structure of a conventional PID controller. Incremental fuzzy PID (IFPID) control, a branch of the GS-type fuzzy controller, was provided in [6]. The IFPID control was categorized as a fuzzy supervision of a PID control, where the PID gains are adjusted in an intelligent way by the help of fuzzy rules [7]. The IFPID control was applied to an industrial robot [8] and a solar power plant [9] with a drastic improvement of performance in terms of overshoot and rising time. Specifically, Hong improved the incremental fuzzy PI (IFPI) control by modifying the control matrices of proportional and integral gains [8]. However, the underlying principle of each control matrix is inconsistent, and the controller has limited performance owing to the fact that the only one matrix is employed. In this section, more efficient control matrices are designed by mapping the region of each control matrix to an output response. The proposed IFPI control output can be formulated as
P I
= P0 (1+k CV (e; 1e)) = I0 (1+ k CV (e; 1e)) p
i
( ) = P e(t)+ I
ut
=
(18)
i
t
0
()
e d t
( )+ I0
P0 e t
+
(17)
p
0
()
e d
( 1 ) ( )+ I0 k CV (e; 1e)
P0 kp CVp e; e e t
i
i
= u (t)+ u1 (t) c
(t 0 ); IGV (t)) ( ) = f1 (T ; Tf3; N(N(t()t;)W ; T ; IGV (t)) T 0 371:46(1 0 W (t 0 ))(N 2 0 4:21N + 4:22) + 722(1 0 N ) + 1:94(MaxIGV 0 IGV ) f1 = 1 + 0:005(15 0 T ) 0 257 519
IGV f3 =exhaust flow = N T + 460 MaxIGV a
Tx t
r
f
td
t
0
()
e d (19)
(9)
a
r
f
td
(10)
a
:
a
(11)
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TABLE II TURBINE CHARACTERISTICS AND CONSTANTS
advance. To this end, elements of proportional and integral fuzzy matrices should be changed by the same rule, i.e. from PB to PZ. By contrast, consider the period in which output values increase from C to D regions, where error and error sum change from NZ to NM and PB to PM, respectively. When e t is negative as in this case, care must be taken in designing CVp owing to the fact the two terms are multiplied in (19), while e d are almost always positive. Therefore, elements of proportional and integral fuzzy matrices should be PB and NB, respectively, in order to suppress a current overshoot. In this manner, every element of fuzzy control matrices can be designed, and all the converted real values are summarized in Table IV. The 14 quantization levels can be readjusted according to a desired control resolution.
()
()
(a)
IV. SIMULATION RESULTS The structure of a proposed IFPI speed controller is rewritten as follows:
( ) = P0s (1 + kps CVp (es (t); 1es (t))) ( ) = I0s (1 + kis CVi (es (t); 1es (t)))
Ps t Is t
( ) = Ps (t)es (t) + Is (t)
uf t
t
0
()
es d
(20)
where
( ) =1:0 0 N (t) 1es (t) = es (t) 0 eTss(t 0 Ts ) es t
(b) Fig. 2. Relational diagrams of output regions in a typical step response and their mapped regions in an error and error-derivative coordinate. (a) Step response. (b) Corresponding error space.
where P0 and I0 represent nominal proportional and integral gains, and CVp and CVi denote proportional and incremental fuzzy-control matrices whose elements are fuzzy gains for corresponding error and error derivative values. The fuzzy coefficients kp and ki are per unit values of the nominal gains, which determine ranges of variation. In (19), the control output is divided into two terms; one is of conventional PI control, uc , and the other is of IFPI control, u1 . In this framework, the IFPI control is interpreted to cooperate with the conventional PI control for quickly tracking a set point or rejecting perturbation. Hence, if the fuzzy matrices of CVp and CVi in (19) are properly scheduled by considering this complementary role, the control performance is to be advanced as a result. Fig. 2 shows the relational diagrams of output regions in a step response and corresponding error space whose horizontal axis represents error derivative values for the directional consistency of fuzzy matrices. Considering the tendencies of error and error sum terms for each region and (19), desired incremental output values and fuzzy-matrix elements are categorized in Table III. For example, in the case output values increase from A to B regions, error and error sum change from PB to PM and PZ to PM, respectively. Since output values increase with maximal speed at this period but will pass over a reference value at the next period, u1(t) should be lowered from PB to PZ to decelerate the rise in
and uf controls the fuel combustor subsystem as in (1). Unlike the speed controller, the anti-windup PI control, which is preferable to a system with limited actuators, is employed with the proposed IFPI scheme for a temperature controller as
( ) = I0t (1 + kit CVi (et (t); 1et (t)))
(21)
( ) = P0t et (t) + P0tIt (t)
(22)
It t
uIGV t
t
0
()
et d
where
( ) = Tm (t) 0 Tra :
et t
Note that a proportional gain P0t is fixed as a constant to avoid wrong fuzzy scheduling, because an integral gain and the proportional gain is multiplied in (22). The configuration of the AGA [10] for tuning the gains are summarized in the following: • maximum generation=2000; • population size=50; • selection: Roulette wheel selection; • bit length per variable=10; • coefficients of AGA k1
= 0:85
k2
= 0:5
k3
= 1:0
k4
= 0:05
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TABLE III DESIRED FUZZY RULES FOR REGIONS OF A STEP RESPONSE(P: POSITIVE, N: NEGATIVE, B: BIG, M: MEDIUM, S: SMALL, Z: ZERO)
TABLE IV PROPOSED FUZZY PROPORTIONAL AND INTEGRAL-CONTROL MATRICES (PROPORTIONAL/INTEGRAL, +: POSITIVE CHANGE,-: NEGATIVE CHANGE, 6: EXTRA LARGE, 5: LARGE, 4: BIG, 3: MEDIUM, 2: SMALL, 1: EXTRA SMALL, 0: ZERO)
Fig. 3. Comparison of controlled rotor speed governed by fuzzy PI control (solid) and conventional PI control (dotted).
Fig. 4. Controlled exhaust temperature under fuzzy PI control (solid) and conventional PI control (dotted).
where k1 , k2 , k3 , and k4 are prescribed by the consideration of typical values of crossover mutation rates. In addition, the fitness function is computed by
The best gains of the temperature controller tuned by AGA for a conventional PI control are P0t = 90:6714, I0t = 0:1893, and those for IFPI control are as
Fs;t (t) =
0 je t
1
j
) d
s;t (
:
(23)
The best gains of the speed controller tuned by AGA for a conventional PI control are P0s = 23:6122, I0s = 7:7822, and those for the IFPI control are as P0s = 43:9253
I0s = 58:9526
kps = 0:7748
kis = 0:2838:
The error and error derivative values are quantized equidistantly inside [00.02, 0.02], which is the maximal deviation under conventional PI control. In Fig. 3, profiles of controlled rotor speed are shown, when load torque drops from 1.0 to 0.7 at t = 1:0. The overshoot and settling time of rotor speed are shown to be considerably reduced by the well-tuned IFPI control action.
P0t = 24:0020
I0t = 0:1264
kit = 4:0627:
The error and error derivative are quantized inside [04.75,4.75] and [04.92, 4.92], respectively. In Fig. 4, controlled exhaust temperature is shown, when load torque drops in the same manner. The deviation of exhaust temperature under IFPI control is much reduced by the improved action of IGV. Therefore, the overall simulation results support the effectiveness of the proposed IFPI control. V. CONCLUSION In this paper, an improved IFPI controller is proposed and used to regulate the rotor speed and exhaust temperature of a gas-turbine plant with the manipulation of fuel flow and IGV angle, respectively. The
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IFPI control sums up changing nominal gains of the existing PI controller by referring to fuzzy matrices, i.e. a skillfully designed look-up table. The main advantage of the proposed technique is that it can be implemented quite easily by adding a microprocessor component that carries out the extra computation to the existing hardware PI controllers. To obtain the optimal gains of controllers, the genetic algorithm, a general optimization method, is used, and the resulting overshoot of rotor speed and exhaust temperature under proposed fuzzy PI controller are considerably decreased. As future work, we plan to investigate the design of the fuzzy compensator to improve the performance of the temperature controller. REFERENCES [1] W. I. Rowen, “Simplified mathematical representations of heavy-duty gas turbines,” J. Eng. Power, vol. 105, pp. 865–869, 1983. , “Simplified mathematical representations of single shaft gas tur[2] bines in mechanical drive service,” Turbomachinery Int., pp. 26–32, 1992. [3] L. N. Hannett and A. Khan, “Combustion turbine dynamic model validation from tests,” IEEE Trans. Power Syst., vol. 8, pp. 152–158, Feb. 1993. [4] G. Crosa, G. Ferrari, and A. Trucco, “Modeling and recoupling the control loops in a heavy-duty gas-turbine plant,” presented at the ASME TURBO Expo’95, June 1995, Paper 95-GT-61. [5] S. M. Camporeale, B. Fortunato, and A. Dumas, “Dynamic modeling and control of regenerative gas turbines,” presented at the ASME TURBO Expo’98 , June 1998, Paper 98-GT-172. [6] S. Tzafestas and N. P. Papanikolopoulos, “Incremental fuzzy expert PID control,” IEEE Trans. Ind. Electron., vol. 37, pp. 365–371, Oct. 1990. [7] R. Ketata, D. De Geest, and A. Tilti, “Fuzzy controller: design, evaluation, parallel and hierarachical combination with a PID controller,” Fuzzy Sets Syst., vol. 71, pp. 113–129, 1995. [8] J. H. Hong, “Position/Force Control of Industrial Robots Using the Fuzzy PI Algorithm,” Master dissertation, Univ. Han-Yang, 1991. [9] M. Berenguel, E. F. Camacho, F. R. Rubio, and P. C. K. Luk, “Incremental fuzzy PI control of a solar power plant,” Proc. Inst. Elect. Eng. Control Theory Appl., vol. 144, no. 6, pp. 596–604, 1997. [10] M. Srinivas and L. M. Patnaik, “Adaptive probabilities of crossover and mutation in genetic algorithms,” IEEE Trans. Syst., Man Cybern., vol. 24, pp. 656–667, Apr. 1994.
Application of Fast Haar Transform and Concurrent Learning to Tool-Breakage Detection in Milling H. K. Tönshoff, Xiaoli Li, and C. Lapp
Abstract—In this paper, an effective monitoring approach for manufacturing processing by combining the in-place fast Haar transform and the concurrent learning is described and applied to detect tool flute breakage during end milling by sensing the feed-motor current signatures. The application procedure and the effectiveness of the proposed method have been delineated by case studies; the results indicate that the proposed approach possessed an excellent potential application to tool breakage detection in milling. Index Terms—Concurrent learning (CL), end milling, fast Haar transform (FHT), finite-impulse response (FIR) median hybrid filters, recursive, tool-flute breakage.
I. INTRODUCTION Immediate response to tool failure during end milling may prevent the workpiece and machine tools from excessive damage. The most frequent approach taken to end the milling process monitoring is to attach sensors to the machine or process and then monitor the signals obtained from these sensors. Research to date has presented investigation on cutting force, acoustic emission (AE), vibration/acceleration, and motor current/power [1] to detect tool failure during end milling. In this paper, we proposed a new approach to monitor tool failure, especially focused on tool flute breakage monitoring during end milling. Tool-flute-breakage detection based on cutting force has been done in several research studies [2]. Cutting force is usually measured by using a dynamometer mounted on machining worktable, or mounted on the tool holder. However, the fixation of dynamometer and its cost are two main problems for the application of the method. AE is another important method. It was very successfully applied to tool condition monitoring in signal-point cutting, like turning operations [3]–[5]. More details can be found in [14]. Its application to end milling, however, involves some disadvantages, such as the sensitivity for cutting conditions, the fixation of the AE sensor and the complexity of AE signal processing. Vibration analysis is also a valuable method, which was widely used for tool-condition monitoring, especially for tool-wear monitoring and tool-failure prediction [6]. However, in the context of tool-condition monitoring in end milling, its application is somewhat limited by the nature of an end milling process as well as AE-based method. Spindle or feed-motor current-based tool-breakage monitoring systems have been presented in the end milling operations to overcome the disadvantages of cutting force and AE/vibration-based methods, described in [7]–[10]. To monitor tool failure successfully through the motor current signals, an appropriate signal-processing algorithm is very important because the motor current signals do not indicate more obviously cutting tool condition than cutting force, AE and vibration signals. To meet the need of tool breakage monitoring by using motorcurrent, we apply recursive in-place growing FIR-median hybrid filters, Manuscript received September 12, 2001; revised March 7, 2003. This work was supported by the Alexander von Humboldt Foundation, Germany. H. K. Tönshoff and C. Lapp are with Institute for Production Engineering and Machine Tools, University of Hannover, D-30159 Hannover, Germany. X. Li is with Institute for Production Engineering and Machine Tools, University of Hannover, University of Hannover, D-30159 Hannover, Germany on leave from the Institute of Electric Engineering, Yanshan University, Hebei, China (e-mail:
[email protected]). Digital Object Identifier 10.1109/TMECH.2003.816830
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