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This document must be cited according to its fjnal version which is published in a conference as: V.Grelet, P. Dufour, M. Nadri, V.Lemort, T. Reiche, "Explicit multi model predictive control

• f a waste heat Rankine based system for heavy duty trucks",

54rd IEEE Conference on Decision and Control (CDC), Osaka, Japan, pp. 179-184, december 15-18, 2015. You downloaded this document from the CNRS open archives server, on the webpages of Pascal Dufour: http://hal.archives-ouvertes.fr/DUFOUR-PASCAL-C-3926-2008

Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix

EXPLICIT MULTI-MODEL PREDICTIVE CONTROL OF A WASTE HEAT RANKINE BASED SYSTEM FOR HEAVY DUTY TRUCKS

Vincent GRELET1,2,3, Pascal DUFOUR2, Madiha NADRI2, Vincent LEMORT 3 and Thomas REICHE1

1Volvo Group Trucks Technology Advanced Technology and Research, 1 avenue Henri Germain, 69800 Saint Priest, France 2Universit´ e de Lyon, Lyon F-69003, Universit´ e Lyon 1, CNRS UMR 5007, Laboratory of Process Control and Chemical Engineering (LAGEP), Villeurbanne 69100, France 3LABOTHAP, University of Liege, Campus du Sart Tilman Bat. B49 B4000 Liege, Belgium

54th IEEE Conference on Decision and Control (CDC 2015) 15-18 December, Osaka, Japan

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix

Table of contents

1

Context and motivations

2

Rankine cycle based heat recovery system Rankine process Studied system and controller objective

3

Nonlinear evaporator detailed model

4

Controller development Identification Piecewise linear approach MMPC strategy

5

Simulation results

6

Conclusion and next steps

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix

Context and motivations

In nowadays heavy duty engines, a major part of the chemical energy contained in the fuel is released to the ambient through heat.

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix

Context and motivations

In nowadays heavy duty engines, a major part of the chemical energy contained in the fuel is released to the ambient through heat. Waste heat recovery based on the Rankine cycle is a promising technique to increase fuel efficiency.

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix

Context and motivations

In nowadays heavy duty engines, a major part of the chemical energy contained in the fuel is released to the ambient through heat. Waste heat recovery based on the Rankine cycle is a promising technique to increase fuel efficiency. Dynamic models needed for concept

• ptimization, fuel economy evaluation

and control algorithm development.

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix Rankine process Studied system and controller objective

Rankine process

Liquid compression (1 → 2) from condensing to evaporating pressure by means of the pump power ˙ Win. Preheating (2 → 3a), vaporization (3a → 3b) and superheating (3b → 3c) by means of the input heat power ˙ Qin. Vapor expansion (3c → 4) from evaporating to condensing pressure creating power ˙ Wout on the expander shaft. Condensation (4 → 1) releasing heat ˙ Qout in the heat sink. 12 3a 3b 3c 4 ˙ Qin ˙ Qout T s ˙ Wout ˙ Win

Figure: Temperature-entropy diagram of the Rankine cycle

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix Rankine process Studied system and controller objective

Studied system and controller objective

Recover heat from both EGR and exhaust in a serial configuration. Working fluid: water ethanol mixture. Focus on the control of the working fluid superheat at the expansion machine inlet. Even more critical when using a kinetic expander. Control issue: Reduce the deviation of the superheat around its set point to have safe and efficient operation.

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix

Nonlinear evaporator detailed model

Model representation ˙ xi = fi (xi, u) , (1) uT = ˙ mf0 Pf0 hf0 ˙ mgL TgL

• , xT

i

=

• ˙

mfi hfi Twinti Tgi Twexti

• (2)

fi(xi, u) =                    

˙ mfi−1

hfi−1 ρfi−1 ∂ρfi−1 ∂hfi−1

+

1 ρfi−1 ∂ρfi−1 ∂hfi−1

αfi Aexchintf

• Tfi −Twinti
• 1−

hfi ρfi ∂ρfi ∂hfi

− ˙ mfi

• ˙

mfi−1 hfi−1 − ˙ mfi hfi

• −αfi Aexchintf
• Tfi −Twinti
• ρfi Vf

αfi Aexchintf

• Tfi −Twinti
• +αg Aexchintg
• Tgi −Twinti
• ρwint Vwint

˙ mg cpg (Tgi )

• Tgi−1 −Tgi
• −αg
• Aexchintg
• T ∗

gi −Twinti

• −Aexchextg
• Tgi −Twexti
• ρgi Vg cpg (Tgi )

αambAexchextamb

• Tamb−Twexti
• +αg Aexchextg
• Tgi −Twexti
• ρwext Vwext

                   

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix Identification Piecewise linear approach MMPC strategy

Identification

Dynamic relation between u (working fluid mass flow) and y (working fluid superheat) can be described around an

• perating point by a first order plus time

delay (FOPTD) model: y(s) u(s) = G 1 + τs e−Ls, (3) High variation in FOPTD parameters shows high nonlinearity. Linear time invariant controller will hardly achieve the control objective with good performance under transient driving cycle.

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix Identification Piecewise linear approach MMPC strategy

Piecewise linear approach

Multi linear model approach consists into identifying a bank of N linear models and combine them by means of a weighting scheme. Global model output is (at time tk): yk =

N

• i=1

yi,kWi,k (4) Key design issues are : 1/ the selection of the good model(s) in the bank. 2/ linear models mixing. Modeling error of the ith model at the current time tk is defined by: ǫi,k = yp,k − yi,k. (5)

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix Identification Piecewise linear approach MMPC strategy

Weighting scheme

Bayesian recursive scheme

pi,k = exp(− 1

2 ǫi,kKǫT i,kpi,k−1) N

• m=1

(exp(− 1

2 ǫm,kKǫT m,kpm,k−1)

(6) Wi,k =     

pi,k N

• m=1

pm,k

for pi,k > δ for pi,k < δ (7) where K is a vector and δ a scalar.

New proposed scheme

˜ ǫi,k = ǫ2

i,k N

• m=1

ǫ2

m,k

(8) Xi,k = (1 − ˜ ǫi,k)

j=N

• j=i,j=1

˜ ǫj,k (9) ˜ Xi,k = Xi,k

N

• m=1

Xm,k (10) Wi,k = 1 1 + Ts ˜ Xi,k (11) where T is a scalar.

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix Identification Piecewise linear approach MMPC strategy

Optimization problem

Model Predictive Control cost function for set-point tracking

     min

uinf ≤uk ≤usup

J(uk) =

tk +tp

• tk

(yp(t) − y sp)2 + wu∆u2

k dt,

∆uk = uk − uk−1, (12)

where wu is a scaling factor and a penalty weight. Modeling error

ek = yp,k − yk. (13)

Output prediction yp(t) in (12) can be written based on the N models and feedback:

yp(t) = y(t) + ek. (14)

Output response of a FOPTD model

yi(t) = yp,ke

−(t−tk ) τi

+ t

tk

(e

−(t−s) τi

Gi τi u(s − Li))ds. (15)

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix Identification Piecewise linear approach MMPC strategy

MMPC framework

Integration intervals A model response (15) can be developed as:

yi(t) = yp,ke

−(t−tk ) τi

+ Gi τi e

−t τi

t

tk

(e

s τi u(s − Li))ds,

(16)

Based on the time delay Li, let us define:

• λi

= max(ai ∈ N|ai ≤

Li Ts )

∆Li = Li − λiTs, ∈ R+. (17)

Integration in (16) is done by parts, where the λi + 2 time intervals are:

s s − Li u(s − Li) tk → tk + ∆Li tk − Li → tk−λi u(tk−λi −1) tk + ∆Li → tk + ∆Li + Ts tk−λi → tk−λi +1 u(tk−λi ) . . . . . . . . . tk + ∆Li + (λi − j)Ts → tk + ∆Li + (λi − j + 1)Ts tk−j → tk−j+1 u(tk−j) . . . . . . . . . tk + ∆Li + (λi − 1)Ts → tk + Li tk−1 → tk u(tk−1) tk + Li → t tk → t − Li u(tk) = uk

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix Identification Piecewise linear approach MMPC strategy

MMPC framework

Explicit MMPC formulation Once integrated, (16) is a linear expression in the optimization argument uk: yi(t) = yp,kf1i(τi, tk, t) + f2i(Ts, Gi, τi, ∆Li, λi, tk, t, u(past)) +ukf3i(Gi, τi, Li, tk, t), (18) where the f. may be explicitly defined offline and updated online at each time tk. Hence the initial cost function is: J(uk) =

tk +tp

• tk

 

• N
• i=1
• wi,kyi(t)
• + ek − ysp

k

2 + wu∆u2

k

  dt (19) where the prediction horizon tp = max(tpi) ∀i may be tuned as:    tpi = γp ∗ τi + Li; γp ∈ R+, e.g.: γp = 1 (63% of the dynamics is predicted)

• r γp

= 3 (95% of the dynamics is predicted). (20)

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix Identification Piecewise linear approach MMPC strategy

MMPC framework

Explicit MMPC formulation Based on the step response series (16) of the N linear FOPTD models:

J(uk) = β2,k(N, Gi, τi, Li, tp, wu, wi,k)u2

k

+β1,k(N, Ts, Gi, τi, Li, tp, ∆Li, λi, wu, yp,k, y sp

k , ek, u(past), wi,k)uk

+β0,k(N, Ts, Gi, τi, Li, tp, ∆Li, λi, wu, yp,k, y sp

k , ek, u(past), wi,k),

(21)

Minimization of (21) obtained with the first order optimality at each tk:

∂J ∂uk = 0 at uk = umin

k

. (22)

Calculation of umin

k

is then straightforward:

umin

k

=

−β1,k 2β2,k

(23)

which leads to the explicit formulation of the solution u⋆

k :

   if uinf ≤ umin

k

≤ usup : u⋆

k = umin k

if umin

k

≤ uinf : u⋆

k = uinf

if usup ≤ umin

k

: u⋆

k = usup.

(24)

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix

Input disturbances

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix

Tracking error and manipulated variable

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix

Conclusion and next steps

Conclusion New modeling weighting scheme based on a piecewise linear approach has been developed and validated. New scheme has less tuning parameters than the Bayesian scheme. Explicit MMPC strategy for Rankine cycle based heat recovery system is presented. MMPC is compliant with classical automotive integration constraints (i.e., basic CPU and fast sampling time). Next steps Experimental validation. Robustness study.

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix

Contacts and discussion

Authors

Vincent GRELET: greletv@live.com Pascal DUFOUR: dufour@lagep.univ-lyon1.fr Madiha NADRI: nadri@lagep.univ-lyon1.fr Vincent LEMORT: vincent.lemort@ulg.ac.be Thomas REICHE: thomas.reiche@volvo.com

Acknowledgement

This PhD thesis is collaboration between UCBL1, ULg and Volvo Trucks which is gratefully acknowledged for the

• funding. The French ministry of higher education and research for the financial support of the CIFRE PhD thesis

2012/549 is also acknowledged.

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix βi expressions Experimental results

βi expressions

β1,k =

N

• i=1
• β

′

1,k + β

′′

1,k + β

′′′

1,k + β

′′′′

1,k

• − 2wuuk−1tp,

(25) β

′

1,k

= −Gi τi wi,k 2 yp,k

• e

Li τi + 2 e

− tp

τi − e Li −2 tp τi

− 2

• (26)

β

′′

1,k

= −Gi 2 τi u(tk−λi −1) wi,k 2 e

− 2 tp

τi

• e

∆Li τi

− 1 e

tp τi − 1

• . . .
• e

Li τi − 2 e tp τi + e Li +tp τi

• (27)

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix βi expressions Experimental results

βi expressions

β

′′′

1,k

=

λi

• j=1

Gi 2 τi u(tk−j) wi,k 2 e

∆Li −tp−j Ts +λi Ts τi

• e

Ts τi − 1

e

Li −tp τi

− 2

• . . .

−Gi 2 τi u(tk−j) wi,k 2 e

∆Li −j Ts +λi Ts τi

• e

Li τi − 2

e

Ts τi − 1

• (28)

β

′′′′

1,k

= 2 Gi wi,k

• ek − ySP

tp + τi e

Li −tp τi

− τi e

Li τi

• (29)

β2,k = Gi 2 wi,k 2

• 2 tp + 4 τi e

Li −tp τi

− τi e

2 Li −2 tp τi

− 4 τi e

Li τi + τi e 2 Li τi

• 2

+ wutp (30)

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix βi expressions Experimental results

Input disturbances

Engine speed and torque

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Context and motivations Rankine cycle based heat recovery system Nonlinear evaporator detailed model Controller development Simulation results Conclusion and next steps Contacts and discussion Appendix βi expressions Experimental results

Tracking error and manipulated variable

Developed weighting scheme

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