Model based Adaptive PID Controller with Parallel Feedforward Compensator for Steam Turbine Speed Control

Model based Adaptive PID Controller with Parallel Feedforward Compensator for Steam Turbine Speed Control

Prof. Dr. Selahattin Ozcelik

Department of Mechanical and Industrial Engineering Texas A&M University-Kingsville

Kingsville, TX USA

Hakan Celep

Department of Mechanical and Industrial Engineering Texas A&M University-Kingsville

Kingsville, TX USA

Abstract In this paper, a new design method of model based adaptive PID controller with parallel feedforward compensator is proposed and improvement of steam turbine control is investigated. The method can be applied to plants, which satisfy the Almost Strict Positive Realness (ASPR) conditions. In order to satisfy ASPR conditions, Parallel Feedforward Compensator (PFC) is introduced. Also, practical applications of the method are discussed and the proposed method is applied to steam turbine speed control system through numerical simulations.

Keywords PID, PFC, Steam Turbine Speed Control, Adaptive PID Control, ASPR.

1. INTRODUCTION

   Apower plant comprises of boiler, gas turbine and steam turbines as major components. Frequency stabilizing is very important for power system security. Because of that, there have been many signs of progress and interests in power systems control in recent years. The new control algorithms and techniques are applied to the power system to obtain the best performance.

   Thus, adaptive control seems one good solution to this problem and has a growing interest in the industrial equipment.

   Most common automatic tuning method is proposed by J.G. Ziegler and N. B. Nichols in 1942 [1]. The method determines Kp, Ki, and Kd gains of PID by using rules, which are developed by Ziegler and Nichols. This method is still used in industrial equipment and commercial sectors. However, it has limited applications on systems, since the rules are based on open-loop responses and cannot perform on closed loop environment.

   Since Ziegler-Nichols tuning method has restrictions on closed-loop processes and therefore other tuning methods have been developed. K. Aström and T. Hägglund proposed Phase Margin Method [2]. However, the drawback of this method is that it may not perform well on systems with big time delays causing oscillations in closed loop systems.

   On the other hand, Popov proposed hyperstability in [3] and introduced the notion of strict positive realness for which Linear Time Invariant (LTI) system is hyper stable asymptotically if the transfer function of the system is

   Compared to gas turbines, steam turbines have been commonly used as for either process equipment in various industries or electricity producer in power plants since the industry began evolving during the 1880s. Because it is reliable and no matter which sector is used for, steam has an important role for various applications.

   There are different types of steam turbines, which are dependent on operational conditions. Steam enters to steam chest and moves to High Pressure (HP) section. Depending on turbine type, steam passes through the Intermediate Pressure (IP) section or leaves the turbine via Exhaust section. The speed of turbine shaft is dependent on inlet steam specifications and load. This process has uncertainties and disturbs turbine speed. Large speed deviations may cause mechanical damages and dont ensure system security. Speed is controlled by the governor, most of which are digital and use conventional PID controller. PID controllers should be tuned under changing operating conditions so that system reliability can be satisfied.

   strictly positive real. M. Hakimi and H. Khaloozadeh in [4] had studied this concept and approached the problem in the frequency domain by using Taylor Series Expansion and Maximum Modulus Principle and showed strict positive realness in the frequency domain.

   A robust optimal LQG controller was proposed for non- minimum phase plant by H. Zargarzadeh and M. M. Arefi in [14]. Robustness of the proposed controller was investigated and for experimental purposes, the controller was later reduced to a PI controller. As compared to loop-shaping H control, proposed LQG controller showed improvement in energy consumption and time responses.

   S. Ozcelik and H. Kaufman presented Direct Model Reference Adaptive Control (DMRAC) algorithm for Single Input Single Output (SISO) plants in [5], where Kharitonovs theorem was used for feed forward compensator design to satisfy Almost Strictly Positive Real (ASPR) conditions. This algorithm was later extended to Multi-Input Multi-Output (MIMO) systems in [6,12].

   Z. Iwai, I. Mizumoto and Y. Nakashima developed a parallel feedforward compensator for (ASPR) systems and developed tuning rules for MIMO systems in [7, 10], where

   1 sFhpTrh

   (Tch s 1)(Trhs 1)

   (2.2)

   an approximate model of the system was used in order keep the stability of the MIMO plant since ASPR conditions were not satisfied in real applications. The drawback of the algorithm is that it has practical and theoretical restrictions and may not be applied to broad range systems due to its control structure.

   Hence, a new model based adaptive PID controller with parallel feedforward compensator is proposed and applied to steam turbine speed control system. Comparisons between proposed algorithm and conventional PID controller are explained and numerical simulations are pursued.

2. MODELING OF STEAM TURBINE SPEED CONTROL SYSTEM

   1. Modeling of Steam Turbine

      Steam enters from main stop valve and flows to governor valve. Governor controls the High Pressure section pressure and then it moves to reheat. Reheated steam passes to

      Droop represented by R is the ratio of the steady state frequency to load or valve gate change [15].

      Percent R= (percent frequency change / percent power output change) x100 (2.3)

   2. Speed Governing Mechanism and Modeling

      The Speed Governor, a type of transducer senses error in speed and converts shaft speed to position output. Control valves conduct input to the turbine and are activated by the governor. Speed control mechanism has other electrical and mechanical components such as linkages, servomotors, levers, and amplifiers. Generic speed governing system is shown in Figure 2 below. There are two valve nonlinearities; Rate limit represents the saturation of the servomotor position, and integrator limits the position of the valve. Limits are shown by per unit (p.u). A new model based adaptive PID controller with parallel feedforward compensator algorithm is applied to steam turbine speed control system.

      Intermediate Pressure section through the intermediate control

      valve and piping. Finally, steam leaves turbine from exhaust

      Load Linearizing Cam

      Rate Limit

      Position Limit

      section through the crossover.

      The simplified block diagram of steam turbine is shown in

      + 1

      KG 1 +

      CV

      + 1 1

      Figure 1 below,

      Speed

      Governor

      –

      LC 0

      Servomotor

      Flow Area

      Turbin

      + + Pg

      + +

      FHP FIP FLP

      Fig.2. Generic Speed Governing System Model Presentation [5]

3. MODEL BASED ADAPTIVE PID CONTROLLER WITH PARALLEL FEEDFORWARD COMPENSATOR

   In this section, the Model Based Adaptive PID Control algorithm is proposed. The algorithm proposed by Iwai [10] has practical, theoretical restrictions and cant be applied to broad range systems due to its control structure. Hence, the

   E 1

   1 +

   1

   1 +

   1

   1 +

   new ModelBased Adaptive PID Controller with Parallel Feedforward Compensator is proposed. If the steady-state analysis of algorithm proposed by Iwai is studied, there are

   Fig.1. Block Diagram of Steam Turbine [8]

   FHP, FIP and FLP being power fractions, determine power, which comes from High Pressure, Intermediate Pressure and Low Pressure sides. Sum of the fractions is equal to 1 (FHP + FIP + FLP = 1). E is valve power. Pg is mechanical power, which is equal to torque. Tch, Trh, and Tco are the time constants of steam turbine chest, re-heater, and crossover piping, respectively. It is assumed that Tco is negligible as compared to Trh. Thus, simplified transfer function of the turbine system, which is the ratio of mechanical power to gate position is [9]

   conditions, which need to be satisfied firstly for zero steady state response. These are,

   — GPFC = 0 (3.1)

   — G (0) 0 and ki 0 (3.2)

   where GPFC is the transfer function of feedforward compensator and ki is the integral gain of adaptive PID. GPFC

   (s) is obtained from GPFC (s) = GASPR(s) G*(s), where

   GASPR(s) is an ASPR transfer function and G*(s) is the nominal transfer function. Since GPFC (0) should be zero and is calculated using G*(s), all physical systems cannot satisfy the above conditions. Hence, steady state error will remain for the controlled plant. To this effect, the new adaptive PID

   Pg

   E

   Fhp

   T s 1 (T

   1 Fhp

   s 1)(T

   s 1)

   (2.1)

   controller is developed to improve the algorithm and to eliminate the error in the system so that adaptive PID can be

   ch ch rh applied to sophisticated systems.

   1. Problem Setup

      Lets consider SISO nth order plant:

      Then defining,

      (s) Gp Gnom

      (3.8)

      xp (t) Apxp (t) Bpup (t)

      yp (t) Cpxp (t)

      (3.3)

      It is clear that the uncertainty is a function of plant parameters varying within the ranges. Hence, a parallel feedforward compensator that satisfies the worst case uncertainty should be designed. An optimization procedure

      where xp is the state vector, up is the control vector, yp is the

      output vector and Ap and Bp are matrix dimensions. The plant can be expressed by transfer function of

      N p

      below will be used to get the worst case uncertainty at each frequency.

      Define a vector whose elements are plant parameters in equation (3.6),

      Gp (s) Cp

      (sI Ap ) D

      p

      (3.4)

      V [CmCm1…C0 Bn Bn1…B0 ]

      Theorem 1 [11]

      Equation (3.4) is ASPR if it satisfies the conditions below,

      • Np(s) is Routh-Hurwitz polynomial.

        Maximize ( j at each

        V

      • Relative degree of the transfer function is 0 or 1.

      • Leading coefficient is positive

      Since most of the plants do not satisfy the ASPR conditions, Parallel Feedforward Compensator is introduced.

      C m j Cm j Cm j , j 0,1,…,m

      Subject to:

      B n j Bn j Bn j , j 0,1,…,n

      (3.9)

   2. Parallel Feedforward Compensator Design

      Nominal plant parameters are known and uncertainties are represented as additive perturbations. Plant uncertainties will

      where (j) is perturbation. Given the worst-case uncertainty for each frequency, it is supposed that there is a rational function which is W(s) RH such that

      be presented by transfer functions. The aim is to develop parallel feedforward compensator so that augmented plant satisfies ASPR conditions and desired responses are achieved despite changes in parameters.

      W ( j) max ( j) ,

      The following assumptions are made for the plant:

      (3.10)

      Now consider a non-ASPR SISO plant in the form of Gp,

      Cmsm Cm1sm1 … C0

      — Nominal plant (Gnom) is known, minimum phase and

      stable.

      — Actual plant is stable.

      G p

      Bnsn

      • Bn1s

      n1

      .. B0

      (3.5)

      — (s) satisfies (3.10).

      If the following augmented plant with parallel feedforward

      Where coefficients are Bn-j and Cm-j may take values within the bounds below,

      compensator is considered as,

      Ga (s) Gp (s) G

      (s)

      (3.11)

      Cm j Cm j Cm j , j 0,1,…,m

      (3.6)

      PFC

      The following theorem gives the design conditions of parallel

      Bn j Bn j Bn j , j 0,1,…,n

      (3.7)

      If the nominal plant parameters are known and Gnom (s) is constructed, uncertainty can be shown as frequency dependent additive perturbation. Hence, actual plant becomes Gp(s) = Gnom (s) + (s), where (s) is additive perturbation.

      (s)

      Actual Plant

      (s)

      Actual Plant

      Gnom(s)

      Gnom(s)

      +

      +

      +

      feedforward compensator. Also, augmented plant satisfies ASPR conditions in the presence of plant uncertainty [12].

      Theorem 1:

      If GPFC(s) is designed as conditions below, the augmented plant Ga(s) = Gp(s) + GPFC(s) will be ASPR.

      — GPFC(s) is stable and relative degree one.

      — The augmented nominal plant Gnom(s) + GPFC(s) is ASPR.

      — (s) RH and (s) < 1

      Where

      +

      +

      +

      GPFC(s)

      Fig.3. Additive Perturbation and Actual Plant Representation.

      ~(s) (Gnom (s) GPFC (s)) 1W (s)

      (s) is the uncertainty of the augmented plant. (Proof of theorem)

      (3.12)

      Consider the closed loop system comprised of the nominal plant Gnom(s) and additive perturbation which is (s) and controller C(s) as shown in Figure 4, the actual plant is

      Suppose the parallel feedforward compensator has the form of Figure 4, then conditions below have to be satisfied,

      — ki > 0

      — Gp (0) 0

      Gp (s) Gnom (s) (s)

      The transfer function from w to z from can be shown as

      w (1 C(s)Gnom (s)) 1C(s).z

      (3.13)

      (3.14)

      — R(s) is step input

      Then, limt e( t ) 0

      Define,

      By using small gain theorem, the closed loop system is

      ym Gm( s )R( s ), y f

      ucGPFC ( s ), yp ucGp( s )

      (3.24)

      stable if and only if

      P 1

      (3.15)

      yma ym y f , ypa yp y f ,GPID C( s ) / s (3.25)

      or it can be written as,

      (1 Gnom (s)C(s)) 1

      (3.16)

      The robust stability condition by using parallel feedforward compensator is

      1

      1

      (3.17)

      Fig.4. New Model Based Adaptive PID with Feedforward Control System

      where the controller can be shown as below,

      GPFC (1 Gnom ) G

      PFC

      C( s ) kd

      s 2 k

      p s ki

      (3.26)

      By replacing with W, one can have

      From Figure 4, one can have the following signals,

      ~

      W 1

      Gnom GPFC

      (3.18)

      Ea ( s ) yma ypa ym y f

      • y p y f

      ym y p (3.27)

   3. Adaptive PID Controller Design

   Suppose that augmented system is ASPR. The system can

   By taking (3.24) and (3.25) into (3.27), one can define,

   Ea ( s ) Gm ( s )R( s ) uc Gp ( s )

   be stabilized by using equation below,

   V (t) k(t)g(t)

   (3.19)

   Gm

   ( s )R( s ) Gp

   ( s ) C( s ) E

   s a

   ( s )

   (3.28)

   where,

   and

   C( s )

   k( t ) k p( t ), kd ( t ), ki ( t )

   (3.20)

   Ea ( s )1 Gp ( s ) Gm ( s )R( s )

   (3.29)

   g( t ) y , y a ,w

   (3.21)

   s

   a

   E ( s )

   sGm ( s )R( s )

   w ya

   (3.22)

   a s G

   ( s )C( s )

   (3.30)

   The adaptive gain vector k(t) is tuned by adaptive law below,

   p

   Using final value theorem, one can have,

   k( t ) g( t )ya ( t )

   0

   (3.23)

   e lim sE ( s ) lim

   s sGm ( s )R( s )

   (3.31)

   ss s0 a

   s0

   s Gp ( s )C( s )

   In this case limt e( t ) 0 . Zero steady state error condition

   0.Gm ( 0 )1 / s

   is satisfied. Error Analysis

   ess lims0 s 0 G

   0

   p

   p

   ( 0 )C( 0 )

   (3.32)

4. SIMULATION RESULTS

   In previous chapters, steam turbine system was discussed and design of new adaptive PID control algorithm was introduced. In this section, an adaptive PID control system with parallel feedforward compensator is implemented. The objective is to control the speed of steam turbine under changing operating conditions. Design procedure utilizes transfer functions which are obtained from equations from Figure 1. Steam turbine system has a governor, turbine and load part. Under changing operating conditions of the turbine system, the transfer function of the plant are Gplant(s)= Ggovernor*Gturbine*Gload.. Simulations are performed using the following nominal parameters and their respective ranges as in Table 1.

   TABLE I

   DIFFERENT PLANT PARAMETERS

   Parameter Nominal Range

   Fig.5. Bounding Function Response, w( j )

   b1 b2	1.6 4.497	1.333 to 2 3.124 to 7.028	satisfies equation (3.12), the parallel feedforward
   b3	1.777	1.029 to 3.472	compensator can be found. Depending on the plant structure,

   b1 b2	1.6 4.497	1.333 to 2 3.124 to 7.028	satisfies equation (3.12), the parallel feedforward
   b3	1.777	1.029 to 3.472	compensator can be found. Depending on the plant structure,

   After getting bounding function, parallel feedforward compensator is to be derived. By using Theorem 1 which

   c1 0.4 0.4

   c2 6.753 7.417 to 6.311

   c3 32.54 26.68 to 41.79

   c4 83.73 63.44 to 119

   c5 106.3 73.19 to 168.2

   the relative degree of the compensator can be chosen either 0 or 1. Relative degree in this simulation is chosen 1 since turbine plant structure satisfied ASPR conditions.

   2

   2

   c6 36.88 21.35 to 72.04

   G (s)

   ps p

   PFC

   s 2s 1

   (4.4)

   Nominal plant transfer function Gnom(s) is

   G ( s )

   1.6 s2 4.49s 1.77

   The denominator part of parallel feedforward structure is chosen by designers regarding of the plant type. Since 2nd

   nom 0.4s 5 6.753s 4 32.34s3 83.73s2 106.3s 36.88

   degree polynomial is chosen, numerator part should be first or

   The actual plant Gp(s) can be written as

   (4.1)

   second degree polynomial. If parallel feedforward structure is formed with low degree polynomial, system response will be faster compared to higher order polynomial structure. Parallel feedforward compensator from optimization method using

   b s 2 b s b

   Lagrange Multiplier is obtained as,

   Gp ( s ) 1 2 3

   c1 s 5 c2 s 4 c3 s 3 c4 s 2 c5 s c6

   (4.2)

   0.1703s 0.09219

   The uncertainty is an additive perturbation in transfer function and can be obtained from (3.9). The worst-case

   GPFC (s)

   s 2 2s 1

   (4.5)

   uncertainty was calculated by optimization procedure by equation (3.9) for 200 frequencies. Then that satisfies equation (3.10) is given below

   The augmented plant transfer function becomes as Ga(s) =

   Gp(s) + GPFC(s), where

   0.0681s6 1.187s5 7.763s4 24.85s3 38.19s2 24.13s 5.177

   a 7 6 5 4 3 2

   a 7 6 5 4 3 2

   G ( s )

   0.4s 7.553s 46.45s 155.6s 306.3s 333.2s 180.1s 36.88

   W( s )

   0.7

   (4.3)

   (4.6)

   1.2s2 3s 5

   The magnitude response showing worst case uncertainty and bounding function W(s) is shown in Figure 5,

   s , being weights for adaptive gains must be positive and can be chosen arbitrarily depending on the system. For this simulation, 1=0.04, 2=0.8, 3=0.04 are chosen. Simulations are done for different cases to observe the changing conditions of the steam turbine. Step input is applied for both PID and adaptive PID controllers and results are shown.

   Changes in turbine time constants are dependent on the formula

   Po

   T V (4.7)

   v PQo

   Po= Rated pressure

   Qo= Rated flow out of the vessel

   Time constants depend on the steam pressure, flow and steam density. In real systems, steam coming through and going through operation may not be stable. Changes in pressure and flow will affect the turbine time constants and the system. Changes in plant parameters from the nominal values are assumed to be 25-30 % for simulations. Simulation results for conventional PID controller were obtained for Kp= 1, Ki =0.4, Kd =0.1. Simulations are carried out for 8 different cases

   as shown in Table II below.

   TABLE II

   Cases	Turbine Chest	Turbine Re-heater	Turbine Crossover
   1	0.25	7. 5	0.4
   2	0.3	9	0.48
   3	0.2	6	0.32
   4	0.2	9	0.48
   5	0.3	6	0.32
   6	0.25	6	0.48
   7	0.3	6	0.48
   8	0.3	6	0.4

   Cases	Turbine Chest	Turbine Re-heater	Turbine Crossover
   1	0.25	7. 5	0.4
   2	0.3	9	0.48
   3	0.2	6	0.32
   4	0.2	9	0.48
   5	0.3	6	0.32
   6	0.25	6	0.48
   7	0.3	6	0.48
   8	0.3	6	0.4

   DIFFERENT PLANT CASES FOR SIMULATIONS

   Simulations for the uncontrolled system responses for a unit step input for all the 8 cases are shown in Figure 6. It is important to note that uncontrolled system responses cannot reach the steady-state value of unity.

   Fig.6. Time vs. Unit-Step Response for Uncontrolled System

   In all the simulations, turbine droop assumed as 5 %. If there is any change in droop characteristic in the steam turbine which usually occurs during operation, the controller should compensate this change. During the operation of the turbine, the load can be changed or can be disturbed. Depending on the load, speed or valve change will be affected and power output will be affected as well. In order to compensate the deviation of the speed, droop needs to be adjusted by the governor. Figure 7 shows the block diagram of turbine governor with droop.

   Fig.7. Turbine Governor Representation with Droop

   As seen in Figure 8, 2% percent drop in droop causes increasing settling time in PID controller for all the 8 cases. However, Adaptive PID Controller can compensate this change and performs well in all responses.

   Fig.8. Time vs. Unit-Step Response under 2 % Droop Change

   However, if droop drop increases further to 4%, PID cannot control and three system responses are unacceptable and conidered unstable from a practical point of view as seen in Figure 9. However, the new model based adaptive PID controller can successfully control the system and maintains desired performance for all cased.

   Fig.9. Time vs. Unit-Step Response under 4 % Droop Change

   Figures 10 and 11 show error history for adaptive PID and regular PID, respectively. While errors in all the 8 cases in adaptive PID approach zero, PID controller errors are taking

   too long with three cases showing unacceptable decay.

   Fig.10. Error of Model Based Adaptive PID Controller at 4 % Droop Change

   Fig.11. Error of Conventional PID Controller at 4 % Droop Change

5. CONCLUSION

In this report, model based adaptive PID controller is introduced. This new algorithm is applied to steam turbine speed control system under droop change. From the results obtained during simulations, the new model based adaptive PID control algorithm can cope with changing operational conditions and is a better alternative to conventional PID controller.

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