INTEGRATED DESIGN OF WASTEWATER TREATMENT PROCESSES USING MODEL

PREDICTIVE CONTROL

Mario Francisco, Pastora Vega

University of Salamanca – Spain

European Control Conference. (Kos, July 2007)

2

Index

1. Introduction and objectives

1.1 Classical Design

1.2 Integrated Design

1.3 Objectives

2. Description of the activated sludge process

3. Optimal automatic tuning of model predictive controller

4. Integrated Design problem

5. Conclusions and future work

3

Introduction: Classical design

Selection of the process units and interconnection

Calculation of plant parameters and steady state

Control system selection and tuning

All this minimizing construction and operational costs

Process engineer

Control engineer

Sequential procedure

4

Introduction: Integrated Design

Structure selection( PLANT + MPC CONTROL )

Definition of the optimization problem (Costs, controllability indexes, model,

constraints)

Calculation of the optimum design parameters (plant, controllers, steady

state point)

Plant and controller are designed at

the same time

5

Objectives

Develop a method for optimal automatic tuning of Model Based Predictive Controllers (MPC) using dynamic and norm based performance indexes.

Develop Integrated Design techniques that use this new automatic tuning method.

Apply this methodology to the activated sludge process in a wastewater treatment plant, in order to obtain optimal plants that minimize substrate variations at the process output, considering typical process disturbances at the input.

Introduce some benchmark plant characteristics for better interpretation of results (disturbances, indexes)

6

1. Introduction and objectives2. Description of the activated sludge process

2.1 Process2.2 Disturbances2.3 Closed loop configuration

3. Optimal automatic tuning of model predictive controller4. Integrated Design problem5. Conclusions and future work

Index

7

Description of the process

EffluentSettlerBioreactorInfluent

Recycling

Benchmark configuration

(control of substrate, oxygen,

nitrogen)

Substrate and oxygen control

problem

waste

EFFLUENTINFFLUENT

Bioreactors

Unaerated aerated

Settler

Recycling sludge

Nitrate internal recycling

8

Process disturbances: input flow and substrate

Substrate concentration at

the plant input (si)

Flow rate at the plant input (qi)

Real data from a wastewater plant Benchmark disturbances

9

General MPC controller structure

s1,c1 controlled x1 constrained

qr1,fk1,qp manipulated variables

12 2

0

ˆ ˆmin ( ) ( ( | ) ( | )) ( ( | ))Hp Hu

y uui Hw i

V k W y k i k r k i k W u k i k

Standard linear multivariable MPC controller, using state space model for prediction (MPC Toolbox MATLAB)

MPC controller

index

Ref. s1

Ref. c1 c1

fk1

Controller

qr1 s1 x1

PROCESS qp

1

1

1

20 150

400 3000

1 8

s

x

c

1

1

0 3500

0 1

0 200p

qr

fk

q

1

1

0 1000

0 0.2

0 100p

qr

fk

q

MPC constraints

10

1. Introduction and objectives2. Description of the activated sludge process3. Optimal automatic tuning of model predictive controller

3.1 Optimization problem3.2 Tuning parameters3.3 Algorithm description3.4 Tuning results

4. Integrated Design problem5. Conclusions and future work

Index

11

Optimal automatic tuning of MPC

The optimal automatic tuning problem is stated as a non-linear mixed integer constrained optimization problem (MINLP)

Penalty factor added when controller is infeasible2min ( ) nc

f c W INDEX

. . 1; 2s t PE b AE b c = tuning parameters

Performance indexes

Integral square error for both outputs

max max2 2

1 1 1 1

0 0

( ) ( )T T

r r

t t

ISE s s dt c c

Index based on the norm of the error signals

1 1 1 110r rDEV s s c c

max ( ( ))w

S S jw

H norm of the closed loop

disturbances transfer function

Indexes for disturbance

rejection

Pumping energy

max22

1 1max 0

24(0.4032 ( ) 7.8408 ( ))

T

t

AE Kla fk t Kla fk t dtT

max

1max 0

0.04 24( ( ) ( ))

T

p

t

PE qr t q t dtT

Aeration energy

Benchmark indexes for operational

costs

12

Optimal automatic tuning of MPC

TUNING PARAMETERS

Hp : Maximum prediction horizonHw : Minimum prediction horizonHc : Control horizonWu: Weights of the changes of manipulated variablesTref: Time constants of the exponential reference trajectories

Integer parameters (Hp, Hc, Hw)

Real parameters (Wu, Tref, s1ref)

Modified random search method for all variables

S1ref: Optimal reference for substrate

13

Optimization algorithm description

Algorithm steps

Modified random search method for tuning MPC parameters

2. A random vector ξ(k) of Gaussian distribution is generated, with

integer and real elements.

1. An initial point for controller parameters, variances and centre of gaussians (for

random numbers generation) is chosen.

3. Two new points are obtained by adding and removing this vector to

the current point.

4. Cost function is evaluated at the original point and at new points, and the algorithm chooses the point with

smallest cost.

5. If some convergence criteria is satisfied, stop the algorithm, otherwise return to step

2. Variances are decreased.

1( ) , , , , ,p c w u ref rc k H H H W T s

( ) ( )c k k

( ) ( )c k k

14

Tuning results (I)

Results considering ISE as performance index

With constraints over PE, AE (solid lines) and without these constraints (dashed-dotted lines)

TABLE I

INDEX ISE+PE,AE ISE

Wu(1) 0.0015 0

Wu(2) 0 0

Wu(3) 8.3033 10

Tref 0.356 0

Hp 25 12

Hc 10 10

Hw 3 1

f2 =Wn * ISE 0.1737 0.09030

PE 190.15 276.84

AE 470.36 502.83

0.0654 0.0032

S1r 73.55 60.50

Computational time (min)

39.9 129.34

Control variable: qr1

Output variable: s1

S

Control effors are smaller in the

first case

Fixed plantV1=7668

A=2970.88

15

Tuning results (II)

Results considering as performance index

Results considering (solid lines) compared with ISE (dashed-dotted lines)

TABLE I

INDEX ISE

Wu(1) 0 0

Wu(2) 0 0

Wu(3) 10 9.8648

Tref 0 0

Hp 12 5

Hc 10 5

Hw 1 1

f2 =Wn * ISE 0.09030 0.1045

PE 276.84 299.94

AE 502.83 511.47

0.0032 5.9191e-6

S1r 60.50 58.45

Computational time (min) 129.34 2.22

Control variable: qr1

Output variable: s1

S

S

S

S

Results are similar but computa-

tional time is smaller

16

1. Introduction and objectives

2. Description of the activated sludge process

3. Optimal automatic tuning of model predictive controller

4. Integrated Design problem

4.1 Two steps approach

4.2 Optimization problem

4.3 Integrated Design results

5. Conclusions and future work

Index

17

Integrated Design problem

Integrated Design of plant and MPC: Two iterative steps approach

Step 2:Controller parameters fixed,

plant design (NLP/DAE problem)

Step 1:Optimal MPC tuning previously explained

(MINLP problem)

Plant dimensions steady state point

Controller parameters

CONTROLLER TUNING (optimization of f2)

Random search algorithm

PLANT DESIGN (optimization of f1)

SQP algorithm

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Optimization problem

1 1 1 2 3 1 4 2( ) df x w V w A w fk w q

Optimization problem: non-linear constrained problem (NLP /DAE). Solved using SQP algorithm

1 1 1 1 1 1 1 1min ( , , , , , , , , , , )d b r pxf x x V A fk x s c x x x qr q

Construction costs (reactor volume and settler area)

Operational costs (reactor aeration and pumps)

Objective function:

The weights wi (i = 1,…,4) are selected from CAPDET model (benchmark)

w1=1; w2=3.1454 w3=1; w4=1

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Optimization problem

Constraints on the non-linear differential equations of the plant :

Controllability constraints:

Residence time1

12

4 8v

q

1 1

1 1

0.001 0.06i i rq s q s

v x

Mass loads

1 13 1024d r r

p r

v x A l x

q x

Sludge age

111

121

1

21

1

111 xxirv

qxk

s

xk

sk

xsy

dt

dxcd

s

111

121

1

21

1

111 ssirv

qxkf

s

xkf

sk

xs

dt

dsckddkd

s

Process constraints:

20.1 0.9i

q

q Relationships between flows

INDEX 1; 2PE b AE b

where INDEX= performance index (ISE, norms, etc.)

2

0.03 0.18pq

q

20

Control variable qr1

Output s1

TABLE IIIINTEGRATED DESIGN RESULTS WITH ISE

INDEX ISE+ PE,AE ISE

Wu [10 0 0.28] [0.0001 0.0001 10]

Tref 3.88 0

Hp, Hc , Hw 5,2,1 5,2,1

V1 8786 7153

A 1800 2062

S1r 70 70

f2 =Wn * ISE 0.9628 0.1175

Plant cost= f1 (x) 1.2684 1.5499

PE 120.40 180.72

AE 217.08 280.59

Control variable qp

Results considering ISE as performance index

Substrate variations are much smaller when constraints over

PE,AE are not considered

With constraints over PE, AE (solid lines) and without these constraints (dashed-dotted lines)

Integrated Design results (I)

21

Control variable qr1

Output s1

Results considering as performance index S

TABLE IVINTEGRATED DESIGN RESULTS WITH

INDEX + PE,AE

Wu [0 7.34 9.93] [0 0 6.23]

Tref 1.97 0

Hp, Hc , Hw 7, 7, 1 5, 5, 1

V1 9217 7100

A 1800 1800

f2 =Wn * ISE 1.3707 0.1110

Plant cost= f1 (x) 1.2647 1.0393

0.0136 3.3958e-5

PE 126.19 194.48

AE 226.89 290.32

S

S

S

S

Integrated Design results (II)

With constraints over PE, AE (solid lines) and without these constraints (dashed-dotted lines)

Results are similar but computational time is smaller

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Integrated Design results (III)

Integrated Design: (plant + MPC)

Results only for MPC tuning:Results with

Integrated Design Improvement

Results with

Improvement in operational and construction costs

S

S

V1=7668A=2970.88PE=299.9AE=511.5

V1=7099.9A=1800PE=194.4AE=290.3

A comparison between automatic tuning and Integrated Design results

Dashed-dotted lines

Dashed-dotted lines

Output s1

Output s1

23

1. Introduction and objectives

2. Description of the activated sludge process

3. Optimal automatic tuning of model predictive controller

4. Integrated Design problem

5. Conclusions and future work

Index

24

Conclusions and future work

For optimal automatic MPC tuning:– A new algorithm for tuning horizons and weights has been

developed, considering dynamic and norm based indexes– It has been tried in the activated sludge process, with good results.

For Integrated Design of plant and MPC:– The design procedure produces better controllable plants than the

classical procedure.– The designed plant satisfies all basic working requirements, is

optimum cost (optimum units), and furthermore it attenuates the substrate load disturbances.

Future work:– Consider different norm based performance indexes (mixed

sensitivity problems based on H and l1 norms of sensitivity transfer functions)

– Include some robust stability and performace indexes.