Synthesis of Advanced Mill Control

Copyright © IFAC Control of Power Systems and Power Plants, Beijing, China, 1997

SYNTHESIS OF ADVANCED MILL CONTROL P.A. O'Kelly

P.A. O'Kelly fj Associates 46 Holly St, Castle Cove NSW 2069 Australia tel/fax: +61 29417 6710; e-mail [email protected]

Abstract: Using a robust model-based, receding horizon type of control, this paper demonstrates by simulation the coordinated control of a typical power station vertical spindle coal mill using a high order, strongly non-linear model of the mill with three control inputs. Performance of the controller working with different models of the mill demonstrates the controller's good tolerance of modelling errors while achieving and maintaining target setpoints (statiC and changing) with zero or permissable steadystate measurement offsets. Simulation results show operation of the controller during the phases of mill warming, initial loading and load manoeuvering at high rates. The optimising controller takes explicit account of maximum actuator positioning rates. Copyright © 1998 IFAC

Keywords: Power generation; Boilers; Power station control; Control system synthesis; Receding Horizon Control; Model based control

1

INTRODUCTION

The method of control realisation described in this paper is based on an approach first suggested by w. Kipiniak (1961). This technique described an analog computer implementation of a spatial boundary-value problem solver. The method has been dealt with further by Meer (1961) and a O'Kelly (1964). It was re-examined in 1991 and is developed further in an unpublished technical report (O'Kelly, 1991). It is the subject of a recently completed Ph.D. thesis project (Palizban, 1996).

ric model fitted on-line to the plant using some fonn of plant identification scheme. At each update instant a new linear model is presented to the controller for computation of the next control sequence, of which only the first element is directed to the plant. At each update instant the controller looks forward by a fixed interval of time T, hence the term "receding horizon" . Since the control model need be valid only for the duration T , a linear representation can suffice provided the future interval does not exceed the validity interval of the current linearisation of the model. The control model must be sufficiently detailed to give confidence that (a) it can be initialised to a state closely resembling the actual plant at the sampling time and (b) it will provide a reasonable match to the plant over the future interval T.

The technique's distinguishing features are the use of a convolution integral and impulse response kernels for representation of the plant dynamics and the direct manipulation of the perfonnance functional for derivation of an equation from which the plant control input sequence is calculated. Nonlinearities in the plant representation are handled by assuming that , in the short-term, the plant may be represented by a linear model matched to the plant at the particular operating point. The plant model used by the controller (the control model) may be designed from a consideration of the basic physics of the plant or it may be a paramet-

This paper is an extension of (Palizban, O'Kelly and Rees, 1995). It treats the coordinated control of outlet temperature (new), product mass flow and mill differential pressure for an extended range of operations of a typical vertical spindle coal mill.

171

2

DERIVATION OF THE CONTROL EQUATIONS

The sequence uk(k = 1, 2, . . . N -1) which yields a minimum of L over the future time interval (N 1)~ is given as the solution of the equation

Full details of the derivation of the controller equations are presented by (0 ' Kelly, 1991) and (Palizban, 1996). Only those elements of the derivation of immediate assistance to an understanding of this application are present ed here.

given the current rate of change of the state of the plant iO, the desired future output sequence y~ and the non-controllable inputs zk . This is the matrix formulation of the so-called Euler equation derived from a variational analysis of the performance functional subject to side conditions defined by the plant dynamics and measurement equations. The coefficients of the matrices M , X, Z are functions of G u, H u and their calculation and the derivation of equation (5) is described in (O 'Kelly, 1991) .

It is assumed that the plant can be described by the differential system dx

- = G(x ,u , z) dt

(1)

where G (x, u , z) is an arbitrary function of the states x (a vector of dimension m:z:) , the control inputs u (dimension mu) and uncontrolled inputs z (dimension mz). A first-order Euler approximat ion to the time derivative and retention of the linear terms of a Taylor expansion of G (x, u, z) gives the linearised discrete time form of equation (1)

The underlying structure of the controller is revealed by integration of equation (5) , giving u

A set of plant measurements yk are related to the plant states xk and the plant controls u k by the measurement matrices H:z: and Hu as follows .

2.1

(3) where w k is a vector of random measurement errors.

J

(yO -

y~)dt + K: xO

(6)

The State Estimator

(i) completion of state feedback (ii) re-calculation of the coefficents of the controller equation. A discrete time Kahnan Filter has been selected, being consistent with the state matrix formulation of the control model equations. The algorithm used here is that described by Lewis (1986).

A performance measure summated over the future interval N ~ is defined as N-l

~F(yk, u k , y~)

K;

Estimation of the state vector is required for two purposes

In the general case the matrices H:z:, H u , G:z:, G u , G z are non-linear functions of the state variables and controls. By evaluating the individual terms of the matrices using the state and control vectors known or estimated at time step k , the plant will be treated as a constant coefficient system for the future time interval T.

L

=

where K; and K! are time-varying gain matrices. yO is the vector of actual plant measurements , y~ the vector of desired values (setpoints) corresponding to yO and xO is the vector of current controller plant model states.

(2)

L=

k

2.2

(4)

Controller Computation Sequence

k=l

The composite plant/controller configuration is shown by fig 1. The internal computations performed by the controller are organised into three main groups

where F is a scalar function of the measurable quantities yk , y~ and u k . Define an augmented state vector X

• estimation of the current controller model state vector i: using current measurements y and control inputs u • controller model coefficient update and impulse response kernel calculation using i: and u • calculation of the controller equation coefficient matrices (M , X , Z) and its solution for iLk.

and a rate of control action u. Then the augmented state equation is

dX dt

[~

] [xl + [

~:

] [u] 172

External interfacing is organised into the two functions of (i) plant measurement sampling to yield Y and (ii) generation of actuator drive pulses. To these can be added the prediction of values or trends of future desired outputs Yd.

3

+ "~Pj

k

n

_._J_

U j=l (Uj,max)

}

with

= measured ground coal flow from the mill = measured differential pressure across the mill Y3 = mill outlet air temperature Y1 ,d = fuel flow demand set point Y2,d = mill differential pressure set point Y3,d = mill outlet temperature set point u 1 = rate of change of raw coal flow to the mill u2 = rate of change of hot air flow to the mill u3 = rate of change of cold air flow to the mill n is an even integer :» 2 (set to 6) k is the time step index w y ; , Wuj, Pj are weighting coefficients.

Y1

All control computations are to be completed within one plant sampling interval. A complete computation sequence will include the following.

Y2

1. Sample the plant measurements Y and actual control inputs u 2. Calculate the state estimate x using the current version of the controller plant model 3. Calculate the controller model coefficient matrices G x , G u , G z , H x , Hu using i and u 4. Calculate the controller model impulse response kernels f/J from the updated G x 5. Calculate the controller equation coefficient matrices M, X, Z using the now updated versions of Gx,Gu , Gz,Hx,Hu and f/J. The assumption is now made implicitly that these constant coefficient matrices can adequately represent the plant for the duration of the look-ahead interval. 6. Solve the controller equation (5) for the first element of the control sequence uk, using the current control error (y - Yd). 7. Test for lukl > u for all k and all elements of uk. If true for any k, add penalty loading to the relevant elements of the M matrix and recalculate uk. Iterate until either all iLk are within limits or until the maximum iteration count has been reached. 8. The controller computes the sequence iLk from which only the first element of the sequence iLl is applied to the plant. Integration to give u 1 is performed by driving individual actuators open or closed at a rate proportional to the magni tude of the associated element of u1 .

The set point terms Y1,d, Y1,d and Y3,d are timedependent. All flows were constrained within maximum and minimum limits. The mathematical model of the mill (the "simulation" model) is basically that described in Cao and Rees (1995) with the addition of one state for the mass of ground coal of size category 2 on the table and three states to include thermal effects, giving an 11th order plant model.

max

3

.

The priority of control set by weighting coefficients in the performance cost function was

1. pulverised fuel (= P F) flow (w u1 = 0.4) 2. mill outlet temperature (W u 2 = 0.02) 3. mill differential pressure(wu3 = 0.001) The sequence of events portrayed by each set of attached trend curves is as follows:

1.

t

+ 0:

mill warming commences:

(a) the mill outlet temperature setpoint ramps from an initial value of 20C to reach 70C after 20 minutes

COAL MILL CONTROL

(b) rotation of the table grinds out the residual coal which is removed from the mill by the airflow (the graphs show a small and short lived flow of coal from the mill and a reduction of coal masses in the mill to zero);

The computation results presented by the accompanying figures show the operation of the controller under operating conditions typical of a real mill during startup, initial loading and subsequent load manoeuvering. The controller used 6 terms in the impulse response series and a 4 second discrete time interval (~ = 4 seconds) giving a look-ahead horizon T of 24 seconds. The controller took the maximum rates of change of all actuators into account using high order penalty functions in the cost function L:

(c) the hot air flow increases to follow the mill outlet temperature set point ramp

2. to + 20: the feeder is started; raw coal flows to the mill at a minimum preset rate; the mill outlet temperature setpoint gradient is changed to achieve 85C after a further 13 minutes. The flow of ground coal product (PF flow) from the mill increases as the airflow , responding to temperature control action, picks up the finer particles.

L =

173

3. to + 30: the mill has achieved its normal operating temperature (85C) using hot air alone (inlet temperature 160C); the internal coal masses have reached equilibrium; mill loading now commences at around 10% rated flow per minute to 10 kg/s; the mill differential pressure (mill DP) setpoint is set to a fixed 3.5 kpa. The increase in internal coal masses together with the high airflow leads to a rapid increase in mill DP ; the feeder reduces throughput.

by the controller reproduces the essential input/output associations of the plant, qualitatively in terms of directions of influence of the controlled inputs and quantitatively, with reasonable accuracy, in terms of magnitudes and rates of change. A model or series of models to do this could be relatively simple and easily adaptive, certainly less complex than a rigorous high order non-linear physical model. (iv) Criticism of linearised models on the basis of their limited validity is justified. However, if the operation of the model-based control is limited to the current validity range of the model upon which it is based, as is the case here, and the model is regularly rematched to its reference plant , also the case here, the benefits of model-based predictive control should continue to be available at all times during which the model is an adequate representation of the plant .

+ 90: the mill output was varied in a series of fast ramp changes.

4. to

Simulation results are presented for three different control models: 1. same as the simulation plant model (fig. 2)

2. the plant model but with coefficient errors of up to 2000% (fig. 3) 3. reduced order, no higher order non-linearity (G x and G u linear or n-linear in the states and controls) ; coefficients adjusted to give good control performance rather than good plant modelling match (fig. 4)

4

REFERENCES 1. Kipiniak W. (1961) Dynamic Optimization and Control - A Variational Approach MIT Press Research Monograph

2. Meer A. (1961) Synthesis of an Optimum Non-linear Controller as a Spatial Analog, MS thesis , MIT

CONCLUSIONS

Some general conclusions on the control method may be drawn from the numerical results presented here and from numerous other numerical studies, reported in Palizban (1996) for other types of process.

3. O'Kelly P.A. (1964) The Use of Analog Circuits in the Determination of Optimal Control Functions, MEngSc thesis , University of NSW , Sydney, Australia

(i) The performance achieved by this controller in these simulation studies is superior to that achievable using current mill control techniques, notwithstanding the idealised conditions of simulated as opposed to real plant.

4. O' Kelly P.A. (1991) A General Method of Optimal Control Computation Using Direct Manipulation of the Performance Functional, Dept of Systems and Control, University of NSW , Sydney, Australia 5. Palizban H.A., O 'Kelly P.A. and Rees N.W. (1995) Practical Optimal Predictive Control of Power Plant Coal Mills IFAC Symposium on Control of Power Plants and Power Systems , Cancun Mexico, December 1995

(ii) The technique produces a controller which is very tolerant of modelling errors. The behaviour of the mill simulation model is extremely sensitive to selection of its coefficient values. The coefficients used for the control model in case (ii) above rendered the model unusable as a simulation model of a mill but control was more than adequate. The oscillation observed during the startup phase is indicative of controller instability arising from properties of the control model (poor conditioning of the M matrix) rather than from plant/model mismatch.

6. Palizban H.A. (1996) Design and Implementation of a Practical Receding Hori zon Control System PhD thesis , University of NSW, Sydney, Australia 7. Lewis F .L. (1986) Optimal Estimation (book) Wiley Interscience 8. Cao S.C. and Rees N.W. (1995) Fuzzy Logic Control of Vertical Spindle Mills IFAC Symposium on Control of Power Plants and Power Systems, Cancun Mexico, December 1995

(iii) The author speculates that the controller design method described here will be successful using any plant model which, for the intended range of operations to be handled

174

Plant Measurements y

Plant Model CONTROLLER Kalman State Estimator

1x x= u1

~ IUrncu:1

u k +1 = u k +!:l.u

uk

Mu

'1

[G."Gu ,G.] [H."Hu ]

=X [ ~ ] +y

[

~

]

f---

1

[Yd " .

N-lj f-- Yd

I---

+Z [ z ]

~

ACTUATORS

x=[;]

Control Model

G:r(x, u)x + Gu(x, u)u + G.z y = H:r:(x, u)x + Hu(x, u)u

Apply penalty loading

Figure 1: Composite Plant/controller Layout

C1:I

c..

15

600

~

lj)

ci

.:,c

0

(f)

-

'E (f) -.. 0) ~

~

10

~ co

400

(f)

E

5

/DP

~ 200 Q)

c

0

= u...

a... 0

0

50

100

0

150

time in min

(I)

Air

"Metal

~ 20

""'-Coal

~

"§ 60

0

=

Q)

~ C 10

0-

40 20

150

(f) --. lj)

\

80

~

I-

100

30 Air setpoint

~

E Q)

50

time in min

100 u

0

0

U

0

50

100

150

50

100

150

time in min

time in min

Figure 2: Mill operation case (i) - Control Model identical to Simulation Model

175

15

~

Q. .Y:

800 Cl

cL ::: 10

~

(/) 600

0

Q) (/) (/)

E vi

E400

--

Cl

.x

~

5

~

......... DP

c::

03200

0

'E

~

u....

Cl..

0

0

50

100

0

150

0

time in min

80

--~20 (/)

Cl

~

Metal

~

~ IV

Coal

5: 0

&l

~

Q.

E Q)

]c:: 10

40

I-

20

150

30 Air set point

(/)

100

time in min

100 <...>

50

0

<...>

0

50

100

150

50

time in min

100

150

time in min

Figure 3: Mill operation case (ii) - Control Model has major errors

~

Q.

15

800

.x

0>

ri

~

(/) 600

0

Q) (/)

::: 10 E

(/)

E400

vi

--

Cl

~

.x

c::

~

CD 200

0

'E

~

u.... Cl..

0

0

50

100

0

150

0

100 (f)

--~ 20 Cl

~r --Metal

~

'Coal

5: 0

~ 60

~

IV

:gc:: 10

Q.

E

I-

150

(J)

"-

80

~

IV

100

30 Air set point

<...>

50

time in min

time in min

40

0

U

20

0

50

100

150 time in min

time in min

Figure 4: Mill operation case (iii) - Simplified Control Model adjusted to good controller performance

176