Table of Contents
ISRN Electronics
Volume 2012, Article ID 134580, 8 pages
http://dx.doi.org/10.5402/2012/134580
Research Article

Using LQG/LTR Optimal Control Method to Improve Stability and Performance of Industrial Gas Turbine System

School of Electrical Engineering, Shiraz University, Shiraz, Fars, Iran

Received 27 April 2012; Accepted 3 July 2012

Academic Editors: C. W. Chiou and M. Hopkinson

Copyright © 2012 Fereidoon Shabaninia and Kazem Jafari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The gas turbine is a power plant, which produces a great amount of energy for its size and weight. Its compactness, low weigh, and multiple fuels make it a natural power plant for various industries such as power generation or oil and gas process plants. In any of these applications, the performance and stability of the gas turbines are the end products that strongly influence the profitability of the business that employs them. Control and analyses of gas turbines for achieving stability and good performance are important so that they have to operate for prolong period. Effective control system design usually benefits from an accurate dynamic model of the plant. Characteristic component parts of the system are considered in this model. Gas turbine system is described by specified thermodynamic equations that can be used for defining its model. This paper introduces an optimal LQG/LTR control method for a gas turbine. Analysing the gas turbine dynamic in time and frequency domain by using proposed control compared to PID controller is followed. Applying this optimal control method can provide good performance and stability for the component parts of system.

1. Introduction

The object of a control system is to make the outputs behave in a desired way by manipulating the plant inputs [1]. So the control systems are being required to deliver more accurate and better overall performance in the face of difficult and changing operating conditions [2]. Many complex engineering systems are equipped with several actuators that may influence their static and dynamic behavior. Systems with more than one actuating control input and more than one sensor output may be considered as multivariable systems or multi-input multioutput (MIMO). The control objective for multivariable systems is to obtain a desirable behavior of several output variables by simultaneously manipulating several input channels [3]. The feedback control systems are widely used in design controller. Optimal control design is one choice for many MIMO systems. Linear Quadratic Gaussian and Loop Transfer Recovery (LQG/LTR) is an optimal control problem whose name is derived from the fact that it assumes a linear system, quadratic cost function, and Gaussian noise.

In some practical circumstances, the dynamics of controlled plant may not be exactly modeled, and there may be system disturbances and measurement noises in the plant. The LQG/LTR controller can provide good performance and guaranteed stability in the face of such noises [3, 4]. More details about this method can be consulted in [1, 3, 57]. Effective control system design usually benefits from an accurate model of the plants.

The system to be investigated is a gas turbine. Various approaches such as Fuzzy [8], MPC [9], and Neural Network [10] are used for control of different types of this system. The aim of the present paper is to design an optimal LQG/LTR controller for this system. Gas turbines are widely used as source of power generation. Some features such as ease of installation and maintenance, high reliability, and quick response have made it an attractive means of producing mechanical energy [11]. There are literature on the modeling of gas turbine that described the system by mathematical dynamic equations [12, 13] or transfer function block diagram [14, 15]. Gas turbine model dynamic equations in [9] are used for control object. In this model, all states, inputs, and outputs are related by nonlinear mathematical equations.

Next sections are concerned with a general description of the gas turbine system, definition of its model, LQG/LTR method, and introducing state space model and simulation results. Matlab mfile commands are used for this idea. PID (Proportional Integrated Derivative) control [1] is applied to this system to compare results.

2. System Description

Gas turbines are designed for many different purposes. In the petroleum industry, they are commonly used to drive compressors for transporting gas through pipelines or generators that produce electrical power. The gas turbine is a constant flow cycle with a constant addition of heat energy. It is commonly referred to as the Brayton cycle [16, 17]. Gas turbine consists basically of a compressor, combustion chamber, and turbine. The compressor draws air from the ambient and increases the pressure of the incoming air. The pressurized and hot air then used as combustion air in combustion chamber, in which a fuel is injected and combusted continuously. The hot pressurized gases are then expanded slightly above atmospheric in turbine and cause to produce rotational speed at output. When compressing the air, power is required and when expanding the hot gas through the turbine, power is generated [11]. The gas turbine is the best suited prime mover when the needs at hand such as capital cost, time from planning to completion, maintenance costs and fuel cost, are considered [18].

The design of any gas turbine must meet essential criteria based on operational considerations such as high efficiency, high reliability and thus high availability, ease of service, ease of installation and commission, flexibility to meet various service and fuel needs, conformance with environmental standards, and auxiliary and control systems. The two factors, which most effect high turbine efficiencies, are pressure ratios and temperature. High-pressure ratios and turbine inlet temperatures improve efficiencies on the simple cycle gas turbine. It should also be noted that the very high-pressure ratios tend to reduce the operating range of the turbine compressor [18].

Auxiliary systems and control systems must be designed carefully, since they are often responsible for the downtime in many units. Control systems provide acceleration time and temperature time controls for startups as well as control various antisurge valves. At operating speeds they must regulate fuel supply and monitor vibrations, temperatures, and pressures throughout the entire range [18].

3. Gas Turbine Model

Gas turbine system is described by nonlinear mathematical equations that used thermodynamic principle rules such as conservation of total mass, energy, and ideal gas equation [2, 12, 17, 19]. The investigation of steady state behavior of gas turbine in terms of static turbine characteristic is a traditional area in engineering [12].

Nonlinear state equations of gas turbine system in [9, 20] describe the gas turbine model and are used for control objects in this paper. This model is based upon the characteristics of the component parts of system such as pressure and temperature in compressor, plenum, turbine, and rotational speed.

The compressor pressure (𝑝comp) is relative to air mass flow rates in compressor (̇𝑚comp), plenum (̇𝑚bl) blow off valve (̇𝑚bl) inlet pressure ambient temperature, and compressor temperature according to nonlinear dynamic equation as follows [9, 20]: 𝑑𝑝comp=𝑑𝑡𝛾𝑅𝑉comṗ𝑚comp𝑇cpouṫ𝑚pl̇𝑚bl𝑇comp,(1) where 𝑇cpout=𝑇in𝑝comp𝑝in(𝛾1)/𝛾𝜂comp,̇𝑚pl=2𝑝comp𝐴2comp𝑝comp𝑝pl𝜉𝑅𝑇comp.(2) Nonlinear dynamic equation of turbine temperature is 𝑑𝑇tbin=𝑅𝑑𝑡𝑇𝑇tbin𝑝tbin𝑉cc𝛾𝑇̇𝑚thr𝑇ccin𝑐pcomp𝑐pṪ𝑚tb𝑇tbin+power𝑐pT𝑇tbiṅ𝑚thṙ𝑚tb,(3) where 𝑇ccin=𝑇pl𝑃tbin𝑝pl(𝛾1)/𝛾.(4) Rotational speed described by 𝑑𝑁=𝑑𝑡̇𝑚tb𝑐pT𝑇tbin𝑇tbouṫ𝑚comp𝑐pcomp𝑇cpout𝑇in1,𝑁𝐼(5) where 𝛾=𝑐𝑝/𝑐𝑣 is specific heat capacity ratio, 𝑅 is gas constant (Nm/kg K), 𝐼 is inertia (Kg m2), 𝜉 is friction factor, and 𝜂 is efficiency.

Other parameters state by specific nonlinear dynamic equations same as these at [9, 20].

The state space model of gas turbine is defined with 5 inputs as 𝑈=̇𝑚comṗ𝑚bl̇𝑚thṙ𝑚tbpower.(6) And 7 states as 𝑝𝑋=comp𝑇comp𝑝pl𝑇pl𝑝𝑡bin𝑇tbin𝑁.(7) Parameters of this model are defined as bellows:̇𝑚comp,̇𝑚bl,̇𝑚thr,̇𝑚tb: Mass flows through compressor, blow off, throttle, and turbine, respectively (kg/s).𝑝comp,𝑝pl,𝑝tbin: Pressure at compressor, plenum, and turbine, respectively (pa).𝑇comp,𝑇pl,𝑇tbin: Temperature at compressor, plenum, and turbine, respectively (k).

The system dynamic equations can be expressed as follows 𝑋=𝑓(𝑋,𝑈).(8)

4. LQG/LTR Method

Linear quadratic Gaussian or LQG problem is a method based on optimal control theory. The main results of this theory are stated in this section. But fuller details can be referred to [1, 3, 57]. This method is one which allows the designer to shape the principle gains of the return ratio at either the input or the output of the plant, to achieve required performance or robustness specifications. Stability is obtained automatically [3].

Suppose the plant is generally described by the dynamic equations in the form of state-space representation as follows: 𝑦̇𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+Γ𝑤(𝑡),(𝑡)=𝐶𝑥(𝑡)+𝑣(𝑡),(9) where 𝑥(𝑡)𝑅𝑛,𝑢(𝑡)𝑅𝑚,and𝑦(𝑡)𝑅𝑞 are the states, inputs, and outputs vectors, respectively, 𝐴𝑅𝑛×𝑛,𝐵𝑅𝑛×𝑚,Γ𝑅𝑛×𝑝,𝐶𝑅𝑞×𝑛 are the system states, inputs of plant, inputs of disturbance, and outputs matrices, respectively. The system disturbance 𝑤(𝑡) and the measurement noise 𝑣(𝑡) are 𝑝- and 𝑞-dimensional uncorrelated Gaussian white noise processes with zero-mean, and the associated covariance matrices are defined as 𝐸𝑤𝑤𝑇=𝑊0,𝐸𝑣𝑣𝑇𝐸=𝑉0,𝑤𝑣𝑇=0,(10) where 𝐸{} is an expectation function operator, and 𝑊 and 𝑉 are the system disturbance and measurement noise covariance matrices, respectively. The problem is then to devise a feedback control low which minimizes the cost: 𝐽=lim𝑇𝐸𝑇0𝑧𝑇𝑄𝑧+𝑢𝑇,𝑅𝑢𝑑𝑡(11) where 𝑍=𝑀𝑥 is some linear combination of the states, and 𝑅=𝑅𝑇0,𝑄=𝑄𝑇0 are weighting matrices.

The solution to the LQG problem is prescribed by the seperation principle, which states that the optimal result is achieved by adopting the following procedure. First obtain an optimal estimate ̂𝑥 of the state 𝑥, optimal in the sense that 𝐸{(𝑥̂𝑥)𝑇(𝑥̂𝑥)} is minimized, and then use this estimate as if it was an exact measurement of the state to solve the deterministic linear quadratic control problem. The point of this procedure is that it reduces the problem to two sub problems, the solutions to which are known.

The solution to the first subproblem that of estimating the state is given by Kalman filter theory. Figure 1 shows the block diagram of a Kalman filter, which is seen to have the structure of a state observer; it is distinguished from other observers by the choice of gain matrix 𝑘𝑓. Note that the inputs to the Kalman filter are the plant input and output vectors, 𝑢 and 𝑦, and that its output is the state estimate vector ̂𝑥.

134580.fig.001
Figure 1: The Kalman filter.

The second subproblem is to find the control signal which will minimize the cost: 𝐽=𝑇0𝑧𝑇𝑄𝑧+𝑢𝑇𝑅𝑢𝑑𝑡.(12) On the assumption that 𝑥=𝐴𝑥+𝐵𝑢(13) the solution to this is to let the control signal 𝑢 be a linear function of the state: 𝑢=𝐾𝑐𝑥,(14) where 𝐾𝑐 is the state feedback matrix gain.

𝐾𝑐 and 𝑘𝑓 are obtained from solution of algebraic Riccati equations [3, 5].

By substitute (14) in to (13): 𝑥=𝐴𝐵𝐾𝑐𝑥,(15) as the equation of the close loop system.

From Figure 1, The state equation of the Kalman filter is to be 𝑑𝑑𝑡̂𝑥=𝐴𝐾𝑓𝐶̂𝑥+𝐵𝑢+𝐾𝑓𝑦.(16) It is shown that the kalman filter and optimal state feedback are asymptotically stable [3].

The equation of the combined Kalman filter-optimal state feedback scheme is 𝑑𝑥=𝑑𝑡̂𝑥𝐴𝐵𝐾𝑐𝐾𝑓𝐶𝐴𝐾𝑓𝐶𝐵𝐾𝑐𝑥+𝐾̂𝑥Γ𝑤𝑓𝑣.(17) If 𝜀 define as state estimation error, 𝜀=𝑥̂𝑥, then above equations can be rewritten as 𝑑𝑥𝜀=𝑑𝑡𝐴𝐵𝐾𝑐𝐵𝐾𝑐0𝐴𝐾𝑓𝐶𝑥𝜀+Γ𝑤Γ𝑤𝐾𝑓𝑣,(18) which shows that the close loop eigenvalues of the LQG compensated plant are just the union of eigenvalues of the optimal state feedback scheme with those of the Kalman filter. The overall scheme is therefore internally stable under the stated assumption [3].

5. Loop Transfer Recovery (LTR)

Since both the optimal state feedback regulator and the Kalman filter have such good properties, it might be expected that LQG compensator would generally yield good robustness and performance. Unfortunately, this is not the case and the LQG designs can exhibit arbitrarily poor stability margins [3].

The LQG loop transfer function can be made to approach filter transfer function with its guaranteed stability margins if 𝐾𝑐, state feedback gain, is designed to be large by specific procedure [1, 3, 5, 6]. It is necessary to assume that the plant model is minimum phase and that it has at least as many inputs as outputs. Alternatively, the LQG loop transfer function can be made to approach state feedback transfer function by designing 𝑘𝑓 in the Kalman filter to be large using specific procedure. Again, it is necessary to assume that the plant model is minimum phase and that it has at least as many outputs as inputs [1, 3, 5, 6].

6. State Space Description

The state space detailed model of gas turbine at a given operation point helps to understand the dynamic behavior of system. A nonlinear model can capture the dynamic behavior of gas turbine system [12]. By using nonlinear equations described at [9, 20], a linear time invariant, LTI, model of the gas turbine is developed for control purposes. LTI state space model derived by using Jacobian method [1] at operation condition for MS5002D General Electric gas turbine system. Matlab and Control toolbox used for creating this model. Validation of this model is done by measurement data with Matlab mfile commands [21] and so at [9].

The MS5002D mechanical drive gas turbine is used to drive a centrifugal load compressor to compress treated gas and deliver export gas at 90 bar pressure for export via pipeline. The gas turbine is that part of the mechanical drive gas turbine, exclusive of control and protection devices, in which fuel and air are processed to produce shaft horsepower. Gas turbine axial air compressor has 17 stages with 10.75 : 1 pressure ratio. The output shaft speed is 4670 rpm with 32.5 MW output power. Combustion section has 12 multiple combustors with reverse flow type.

Control of the gas turbine in providing the shaft horse power required by the operation or process is accomplished using parameters such as fuel flow, compressor inlet pressure, compressor discharge pressure, shaft speed, compressor inlet temperature, and turbine inlet or exhaust temperature [19].

The SPPEDTRONIC MARK VI turbine control is the current state of the art control for General Electric turbine. It contains a number of control protection and sequencing systems designed for reliable and safe operation of the gas turbine [22].

Control of the turbine is done mainly by startup, speed, acceleration, synchronization, and temperature control. Figure 2 illustrates the three control modes and the means of fuel control in relation to the fuel command signal. Sensors monitor the turbine speed, temperature, and compressor discharge pressure to determine the operating conditions of the unit. When it is necessary for the turbine control to alter the turbine operating conditions because of changes in load or ambient conditions, it is accomplished by modulating the flow of fuel to the turbine. For example, if the exhaust temperature starts to exceed its permitted value for a given operating condition, the temperature control circuit will cause a reduction in the fuel supplied to the turbine and thereby limit the exhaust temperature.

134580.fig.002
Figure 2: Simplified control schematic.

Operating conditions of the turbine are sensed and utilized as feedback signals to the SPEEDTRONIC control system. There are three major control loops—startup, speed, and temperature, which may be in control during turbine operation. These loops command fuel stroke reference (FSR), the command signal for fuel. The outputs of these control loops are connected to a minimum value select algorithm as shown in Figure 2. The minimum value select algorithm selects the lowest FSR called for by the control loops and passes the result to the FSR controller. The action of this circuit is similar to a low-voltage selector. The lowest voltage output of the control loops is allowed to pass the gate to the fuel control system as the controlling FSR voltage. Using this method of FSR selection, switching between the control modes of speed, temperature, and startup control takes place without any discontinuity. The controlling FSR will establish the fuel input to the turbine at the value required by the system which is in control. Displays on the turbine control panel CRT indicate which of the control systems is controlling FSR [22].

7. Simulation Results

Matlab mfile commands used for simulating gas turbine system [21]. The value of inlet temperature is equal to 321 k that can be changed. The state weighting matrix 𝑄 and the control weighting matrix 𝑅 are chosen by use Bryson’s rule [5]. Often, this choice is just the starting point for a trial and error iterative design procedure aimed at obtaining desirable properties for the close loop system [5].

The gain matrix of state feedback controller 𝐾𝑐 is obtained as:𝐾𝑐=1.148120551.32193967.80.00164421.71171.3466e-0071.16482057.91.34354025.50.00201212.02591.578e-0071.16532076.71.34544041.40.07483645.2081.4127e-0070.0008739319.7410.001105821.9690.1212260.7762.0258e-007𝐴𝐵𝐶𝐷𝐸𝐹𝐺1𝑒6,(19)

where 𝐴: 1.463𝑒010,  𝐵: 7.8539𝑒007,  𝐶: 8.3254𝑒010,  𝐷: 8.4805𝑒006,  𝐸: 4.6762𝑒008,  𝐹: 1.3999𝑒00=   and  𝐺: 4.3014𝑒013.

Analysis the gas turbine dynamic in time and frequency domain by using LQG/LTR control method and with compare to PID controller is flowed. Step response, impulse response and principal gain consider for simulation. Figure 3 shows the unit step response of the compressor pressure, turbine temperature and rotational speed respect to compressor mass flow rate for LQR, LQG and LQG/LTR controller design of proposed gas turbine. So that to compare the results, unit step response of these parameters for LQG/LTR and PID control is shown in Figure 4.

fig3
Figure 3: Step response of the compressor pressure (a), turbine temperature (b), and rotational speed (c) respect to compressor mass flow rate.
fig4
Figure 4: Step response of the compressor pressure (a), turbine temperature (b), and rotational speed (c) respect to compressor mass flow rate for LQG/LTR and PID control.

Simulation results for rotational speed and turbine temperature by using impulse function as input to power depict in Figure 5 for LQR and LQG/LTR controller design. Also Figure 6 shows impulse response of the turbine temperature respect to mass flow rate of compressor.

fig5
Figure 5: Impulse response of the rotational speed (a) and turbine temperature (b) respect to power.
134580.fig.006
Figure 6: Impulse response of the turbine temperature respect to mass flow rate of compressor.

The structure singular value can be used to analyze the ability of a given design on achieves robust performance [1, 3]. Singular value or principal gain plot for main system and controlled system by LQG/LTR are shown in Figure 7.

fig7
Figure 7: Singular value diagram of main system (a) and return ratio at input for LQG/LTR controlled system.

8. Conclusion

This paper introduces the gas turbine systems. Control and satisfactory performances are important concept for this system. By using thermodynamic equations, a nonlinear dynamic model is specified to study control objects. A gas turbine type is considered and its control system is described. Analyses the system is followed by making LTI state space model obtained of linearization nonlinear dynamic model and apply suggested LQG/LTR optimal control method. By LQG/LTR controller, and set parameters of this control by specified procedure, simulation results in time and frequency domain are achieved. To compare results, we use PID control method for described system model.

System analysis shows response of uncontrolled dynamics and effect of LQR, LQG, and LQG/LTR controller designed and PID control. LQG/LTR controller improved principal gain diagram to attain good performance. With LTR method a satisfactory return ratio is recovered at the plant input.

Settling times of system characteristic component parts by using LQG/LTR controller are shorter, and stability of system makes it better by applying this optimal control method.

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