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Jasper van Dessel
(ORC) System for Waste Heat RecoveryMultivariable Optimal Control in Organic Rankine Cycle
Academic year 2013-2014Faculty of Engineering and ArchitectureChairman: Prof. dr. ir. Jan MelkebeekDepartment of Electrical Energy, Systems and Automation
Master of Science in Electromechanical EngineeringMaster's dissertation submitted in order to obtain the academic degree of
Counsellor: Jairo Andres Hernandez NaranjoSupervisor: Prof. dr. ir. Robain De Keyser
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Jasper van Dessel
(ORC) System for Waste Heat RecoveryMultivariable Optimal Control in Organic Rankine Cycle
Academic year 2013-2014Faculty of Engineering and ArchitectureChairman: Prof. dr. ir. Jan MelkebeekDepartment of Electrical Energy, Systems and Automation
Master of Science in Electromechanical EngineeringMaster's dissertation submitted in order to obtain the academic degree of
Counsellor: Jairo Andres Hernandez NaranjoSupervisor: Prof. dr. ir. Robain De Keyser
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Permission & Copyright
The author gives permission to make this master dissertation available for consultation and
to copy parts of this master dissertation for personal use.
In the case of any other use, the limitations of the copyright have to be respected, in particu-
lar with regard to the obligation to state expressly the source when quoting results from this
master dissertation.
Jasper van Dessel
may 31, 2014
i
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Preface
I would like to thank all people that helped me during this thesis work.
First of all ir. Andres Hernandez who was always present for discussions, questions and
support. His helpful attitude and openness helped me to progress on my thesis. I would like
to which him good luck in his future work as a PhD. in the department.
Then I would also like thank the people of the control laboratory and in particular my
promotor prof. dr. ir. Robain De Keyser who made it possible for me to do this interesting
thesis in the field of energy systems in which I have a large interest. I would also like to
thank prof. De Keyser to give me the opportunity to go on erasmus during the previous
academic year where I had the opportunity to experience education in one of the best control
laboratories of Europe.
And last but not least I would like to thank all the people that gave me support, coloured
my days during the hard work over the last year and could entertain me.
ii
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Summary
Multi-variable optimal control of an Organic Rankine Cycle for wasteheat recovery
Jasper van Dessel
This master thesis has the purpose to address the need for optimal regulation for organic
rankine cycles with a focus on waste heat. The organic rankine cycle is a promising player
in the renewable energy field but also for its ability to increase the overall efficiency in power
plants. For both uses the ORC subtracts waste heat that would otherwise not be used.
Due to its low efficiency (±5%) its regulation strongly determines the performance of the
cycle. Because of the high variability of the waste heat which alters the optimal regulation
this not only causes thermodynamic challenges but certainly even important are the control
challenges. Therefore a real-time optimisation (RTO) is needed together with advanced con-
trol.
This thesis is build up in as follows. First the reader is introduced to the concept of an organic
rankine cycle. In chapter 2 the RTO of a subcritical ORC is addressed. In chapters 3, 4 and
5 a first attempt is made to regulate the cycle using through identification obtained models
and the limitations of linear control are addressed.
In the last part of this work a non-linear model of the ORC is developed for use in NMPC.
keywords: organic rankine cycle (ORC), real-time optimisation (RTO), non-linear model
predictive control (NMPC), moving boundary model
iii
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List of Symbols
Symbols
A area
c specific heat capacity
C(s) controller transfer function
D diameter
G(s) plant transfer function
FF filling factor
h enthalpy
L length moving boundary region
m mass flow rate
M mass or mass matrix
N actuator speed or horizon (MPC)
O circumference
p pressure
P Power or error covariance matrix
Q Heat transfer
q heat transfer per unit mass flow rate
R gas constant
rν , rp specific volume ratio, pressure ratio
s entropy
T temperature
∆ T temperature difference or time-constant
TS thermodynamic states vector
V volume
Vs swept volume
W Work
w work per unit mass flow rate
u input vector
U heat transfer coefficient
x state vector
X pump capacity fraction
z−1 time shift operator
ε isentropic efficiency
η efficiency
φ level receeiver
µ condition number
ν specific volume
ω acentric factor or frequency
σ singular value
ρ density
γ mean void fraction
iv
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Subscripts
0 initial
amb ambient
bw bandwidth
c critical
cd condenser
cool, cf cooling water, cold fluid
d delay
exe exergy
exp expander
ev evaporator
g gaseous
gen generator
h constant enthalpy
heat, hf waste heat, hot fluid
hr heat recovery
i inner
in inlet or internal
l liquid or lower
k sample k
mech mechanical
n, nom nominal
net netto (generated power)
o outer
out outlet
p constant pressure
pp pump
sat saturation
sh superheating
ss steady state
su supply
sub subcooling
th thermal
tp two-phase
tot total
u upper
vap vaporisation
wf working fluid
w wall
′ saturated liquid
′′ saturated gas
Abbreviations
DC decoupled
EKF extended kalman filter
ICE internal combustion engine
IWT innovatie door wetenschap en technologie
FE finite element
FMI functional mock-up interface
MB moving boundary
(N)MPC (non-linear) model predictive control
(N)EPSAC (Non-linear) Extended Prediction Self-Adaptive Control
ORC organic rankine cycle
PI proportional integrator controller
RGA relative gain array
WHR waste heat recovery
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Multivariable Optimal Control in Organic Rankine Cycle (ORC) System
for Waste Heat Recovery
Jasper van DesselSupervisors: prof. dr. ir. Robain De Keyser, ir. Jairo Andres Hernandez Naranjo
June 2, 2014
Abstract The optimal control of an organic rankine cycleis addressed for waste heat systems with strongly vary-ing conditions using R245fa as working fluid where linearcontrol fails. A real-time steady state optimiser is imple-mented in the control system such that the system canadapt to the conditions and a non-linear model is devel-oped using moving boundary models. The model is usedin a constrained NMPC controller using the NEPSAC al-gorithm.
Keywords organic rankine cycle (ORC), real-time op-timisation (RTO), non-linear model predictive control(NMPC), moving boundary model
1 Introduction
Expander
Pump
Receiver
Condenser
Evaporator
Figure 1: The ORC cycle
The organic rankine cycle (figure 1) is a thermodynamiccycle that extracts low grade waste heat (typically <200◦C) to produce electricity. It’s overall efficiency fromtotal available waste heat to produced net electrical poweris around 5%:
ηoverall =Wexp −Wpp
mheat(hin − hamb)≈ 5%
Applications are geothermal energy, industrial waste heat,biomass plants, exhaust of an ICE etc. The first oneshave rather stable conditions, whereas exhaust gases havea more aggressive profile. This causes some thermody-namic and control challenges.
2 Optimised conditions
The rather low efficiency of the ORC makes that the pro-duction of electricity is strongly dependant on the oper-ation of the cycle. Optimal conditions of the cycle aredetermined by the waste heat conditions (and thus theapplication) and the working fluid R245fa. Optimal op-eration of the cycle can be summarised to (Quoilin et al.[2011]):
• superheating ∆Tsh as low as possible (but non-zero)
• optimal evaporator saturation temperature (or pres-sure)
• subcooling condenser non-zero
• low condenser pressure
Based on the work of He et al. [2012] the saturation tem-perature is optimised. A theoretical expression of the netgenerated power is derived:
Wnet = ηsmheatcp,heat×(Theat −∆Tcd,out − Tsat)(Tsat − Tcd,out)
Tsat×
(1 +
Cp,wfTsat2∆hvap
ln
(TsatTcd,out
))
It is a function of the waste heat (Theat, mheat), the satu-ration temperature, the evaporator pinch point ∆T1, thecondenser outlet temperature and the latent heat ∆hvap.The latter is a function of the saturation temperaturewhere the following correlation is suggested:
∆hvap = RgTc
[7.08
(1− Tsat
Tc
)0.354
+ 10.95ω
(1− Tsat
Tc
)0.456]
The optimal value is found by solving the following equa-tion using newton-iteration:
dWnet
dTsat+
∂Wnet
∂∆hvap
d∆hvapdTsat
= 0
This then yields an optimal value independent on the typeof waste heat (!), as a function of Theat, Tcd,out and ∆T1.A mapping is plotted is figure 2 for various waste heattemperatures and condenser outlet (for ∆T1 = 2.5).
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100
150
200
0
5050
100
150
Theat
[°C]T
cd,out [°C]
Tsa
t [°C
]
60
70
70
70
80
80
80
90
90
90
100
100
100
110
110
Theat
[°C]
Tcd
,ou
t [°C
]
100 120 140 160 18010
15
20
25
30
35
40
45
50
Figure 2: Mapping optimal conditions waste heat
3 Real-time optimised control
Superheating and evaporator saturation temperature arethe two most important operation parameters. Since thesystem has 2 actuators (pump and expander speed arecontrollable) a two input two output system results.The following MIMO transfer function model was identi-fied around the nominal operating point (Npp = 1800rpm,Nexp = 3000rpm) using PEM-method with multisine sig-nals (86% fitting):
G(s) =
−4.34 e−32.6s
92.8s+1 0.76−17.9s+176.7s+1
0, 64 e−37.9s
83.2s+1 −0.47 145.1s+1
(1)
Because the ORC system has several mechanical andthermodynamic limitations a constrained MPC is prefer-able. An EPSAC controller (De Keyser [2003]) was im-plemented on a simulation model of the ORC and hardconstraints are imposed to the control. The following op-timisation problem results:
min J =
N2∑
k=N1
[Tsat,opt(t+ k|t)− Tsat(t+ k|t)]2 (2)
subjected to:
Y 1 = Y1,base +G1Uopt
Y 2 = Y2,base +G2Uopt (3)
Umin < Uopt < Umax
|Uk − Uk−1| < 500rpm
5 < ∆Tsh < 20 (4)
The input constraints for pump are 1000rpm →3000rpm and for expander 2000rpm → 4000rpm. Anexperiment with strongly varying conditions is executedusing a sampling time of 5s. The result and waste heatprofile is plotted in figure 5. The controller is clearly notable to obey the constraints.
4 Non-linear MPC
Due to the poor results of an MPC controller, a non-linearmodel is developed to be used in an NMPC controller. the
aim is a low order non-linear model that is computationalefficient.Heat exchanger models: The evaporator and condenserare modelled using the moving boundary technique wherethe component is divided in a liquid, two-phase and gasregion. For each part mass balances and energy balancesare obtained yielding a system of differential equations ofthe form:
M(x, u)dx
dt= f(x, u) (5)
yk = h(xk, uk) (6)
The system is modelled as a non-linear state space systemwith 10 states. For superheating and downcooling regionsstatic models are usedActuators: For pump and expander semi-empirical staticmodels are used because of the fast time-scales.State estimation: an extended kalman filter is build to ob-serve the states using 4 measurements.The model is now used with a NEPSAC controller. Theoverall system becomes:
NEPSAC Systemu y
sensorsEKF ym
x
optimiser
d
r
Figure 3: overall closed loop system
The control system is tested with a very aggressivewaste heat stream and the heat profile and result is shownin figure 6 (the model is used as plant with artificial noise
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added). Here the evaporator pressure is tracked (equiv-alent to tracking saturation temperature), the controllerperformes well with aggressive waste heat.
The computation time per sample is shown in figure4. The average computation time is 4.5s. It can be con-cluded that the control system can be implemented on areal set-up.
60 80 100 120 140 160 1804.2
4.4
4.6
4.8
5
5.2
sample
com
puta
tion tim
e [s]
Figure 4: Computation time per sample
5 Conclusion
A real-time optimiser for an organic rankine cycle systemis implemented. Applications with strongly varying condi-tions were considered and it was shown that linear controlcan be insufficient. A non-linear model of the ORC was
developed and an extended kalman filter was designed forstate estimation. A non-linear model predictive controllerwas able to track optimised conditions without violatingconstraints.
6 References
R De Keyser. Model based Predictive Controlfor Linear Systems. In UNESCO Encyclopaediaof Life Support Systems http://www.eolss.net, vol-ume 35 pages. Eolss Publishers Co Ltd, Oxford,2003. Article contribution 6.43.16.1 (available on-line at: http://www.eolss.net/sample-chapters/c18/e6-43-16-01.pdf).
Chao He, Chao Liu, Hong Gao, Hui Xie, Yourong Li,Shuangying Wu, and Jinliang Xu. The optimal evap-oration temperature and working fluids for subcriticalorganic Rankine cycle. Energy, 38:136–143, 2012.
Sylvain Quoilin, Richard Aumann, Andreas Grill, AndreasSchuster, Vincent Lemort, and Hartmut Spliethoff. Dy-namic modeling and optimal control strategy of wasteheat recovery Organic Rankine Cycles. Applied Energy,88:2183–2190, 2011.
0 500 1000 15002
2.5
3
3.5
Mheat
0 500 1000 1500
140
160
time [s]
Theat
0 500 1000 1500
0
10
20
time [s]
500 1000 150080
100
120
140
0
Tsh [°C]
Tsat [°C] Ref
Figure 5: failing experiment with constraint MPC control using RTO
300 400 500 600 700 800 900 1000100
110
120
130
140
150
time [s]
Theat [°C]
300 400 500 600 700 800 900 1000
6
8
10
Pev [bar] Ref
300 400 500 600 700 800 900 1000
40
60
time [s]
Tsh [°C]
Figure 6: experiment with non-linear constraint NMPC control using RTO
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Contents
Preface ii
Summary iii
List of Symbols iv
Extended Abstract vi
1 Introduction 1
1.1 The Organic Rankine Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Specific properties ORC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 The working fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Variable heat source . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 ORCnext: collaboration and test set-up . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 The IWT project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.3 Simulation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Optimal Operation 8
2.1 Performance of the cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Direct performance: efficiency and generated power . . . . . . . . . . 8
2.1.2 Indirect performance: superheating & saturation temperatures . . . . 9
2.2 Input sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Uncontrollable inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Controllable inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Optimisation saturation temperature . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Derivation optimiser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
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3 Identification 20
3.1 Objective & method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Identification nominal design . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Input-output model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.2 Disturbance to output . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.3 Unmodelled dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 System Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.2 Input-Output coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Control Strategies 29
4.1 Objective & control strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 PI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4 Real-time optimised control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Model Predictive Control 35
5.1 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1.1 MPC algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.1.3 Constrained control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Limitation of linear control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6 Low Order Non-linear Model 42
6.1 Objective and techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.2 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.2.1 Heat exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.2.2 Pump & expander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.3 Interconnection subsystems & simulation . . . . . . . . . . . . . . . . . . . . . 52
6.3.1 Initial conditions & simulation parameters . . . . . . . . . . . . . . . . 53
6.3.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.4 State estimation: Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . 55
6.4.1 Linearised dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.4.2 EKF equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.4.3 Performance filter with artificial noise . . . . . . . . . . . . . . . . . . 60
x
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7 Nonlinear Model Predictive Control 62
7.1 NEPSAC algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.2.1 Experiment with artificial noise . . . . . . . . . . . . . . . . . . . . . . 65
7.2.2 Decreasing computation time . . . . . . . . . . . . . . . . . . . . . . . 68
7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8 Conclusions 71
8.1 Discussion results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
8.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Appendix A: Simulation Model 73
Appendix B: EPSAC Algorithm 76
Appendix C: Moving boundary model and the CoolProp library 80
Bibliography 85
xi
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Chapter 1
Introduction
For a good understanding of this work, the working principle of on organic rankine cycle (short:
ORC) is explained, the typical terminology and applications are given. For more detailed
thermodynamic information about cycle architecture, fluid selection, component design, etc...
other sources must be addressed.
1.1 The Organic Rankine Cycle
The organic rankine cycle is as the name states, based on the well-known rankine cycle, see
figure 1.1. This steam cycle is the core mechanism in most power plant and especially nuclear
plants. The main difference is that no steam or water is used as working fluid in an ORC,
but in stead an organic fluid is used. The choice of the fluid is discussed later in section
1.2.1. The ORC is a cycle that is capable of producing electricity out of waste heat or other
usually not used heat sources (e.g. geothermal energy, biomass, ...) that have a relative low
temperature. The cycle produces electricity with an acceptable efficiency, compared to other
technologies.
The working principle is as follows. Low grade waste heat, meaning relatively low temper-
ature (typically below 200◦C), is lead through an evaporator where heat is exchanged with
the organic working fluid in the cycle. This working fluid is, before it enters the evaporator,
pumped to a higher pressure level. The liquid phase fluid is then evaporated by the waste
heat just above its saturation temperature. This hot and pressurised gas is led through an
expander that acts as a turbine for an electrical generator. As the gas expands the pressure
and temperature of the gas decreases. The fluid is led through the condenser and collected
in the liquid receiver. It can now be pumped up again and the cycle is closed.
The ORC cycle has been introduced for various purposes and its architecture can vary with it.
This work focusses on medium range applications with varying heat source and therefore a set-
up with recuperator is used, because this is advantageous if the temperature after expansion
is still considerably higher than the pump outlet temperature, and will improve the thermal
1
Page 16
Expander
Pump
Receiver
Condenser
Evaporator
Figure 1.1: The (organic) rankine cycle
Expander
Recuperator
Pump
Receiver
Condenser
Evaporator
Figure 1.2: Schematic of an ORC with recuperator
2
Page 17
efficiency of the cycle, but it does not affect the output power. Therefore is must be said
that including a recuperator is not of interest in waste heat applications where extracting the
maximum power is the only objective. A schematic of this improved cycle is shown in figure
1.2. This set-up is available in Howest, Kortrijk, where experimental research is done on the
cycle, in a collaboration project named ORCnext. In section 1.3.1 this project is presented.
1.2 Specific properties ORC
For control purpose it is necessary to introduce some properties that are typical for ORCs.
These are important for insight in the dynamic analysis in further chapters.
1.2.1 The working fluid
The choice of the organic fluid used in the cycle is important, thus far many studies have
been made to obtain the best results for WHR. For detailed information on fluid selection
references are made to, Dai et al. [2009], Quoilin et al. [2011b]. Independently of this choice
all these organic fluids have some similar characteristics, but we will focus on R245fa as this
is the fluid being used in the ORCnext test set-up in Kortrijk.
In figure 1.3 the (T,s) diagram is shown of the working fluid. The main reason for using
organic fluids is because their critical temperature (The temperature at the maximum of
the curve in the (T,s) diagram, where saturated vapor and saturated liquid coincide) is low
compared to steam (145◦ C for R245fa compared to 374◦C for steam). The evaporation
temperature of water becomes very high when the pressure is increases, this is not the case
for the organic fluids used. The ORC was introduced because of its ability to handle low
temperature waste heat, and the specific fluid makes this possible, Dai et al. [2009], Liu et al.
[2004]
Notice the vapor saturation line is bending over, this is another special characteristic that can
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20
20
40
60
80
100
120
140
160
s [kJ/kg]
T [°C
]
1
2s
3
4
5s6
2 5
hf
cf
waste heat
cooling water
organic fluid
Figure 1.3: (T,s) diagram and ORC cycle for R245fa
3
Page 18
be used. Such fluids are called dry (bending over) or isentropic (vertical line). Liquid droplets
must be avoided in the expander in any case. This would cause the component to degrade
much more quickly and severe damage can be the result. Because we are working with dry
fluids, these accidents are less likely to occur, and no superheating (see later) is needed to
avoid this.
1.2.2 Variable heat source
Waste heat used in ORC has the typical property that amount of heat available is not al-
ways constant due to changes in environment, processes that are further upstream or other.
Producing the maximum power possible at a time, requires the cycle to adapt its operating
point to the heat available. This requires an advanced control system. Another possibility to
physically filter the high variability in the waste heat is the use of an intermediate oil loop,
see figure 1.4. This extra loop makes the control easier, but decreases the thermal efficiency
of the whole cycle. This extra loop is not considered in this work.
Expander
Pump
Receiver
Condenser
Evaporator
Intermediate oil loop
Waste heat
Figure 1.4: Schematic of an ORC with intermediate oil loop
1.2.3 Applications
The ORC was introduced for its ability to effectively use low temperature heat and the concept
has been adopted since the 1970’s. First applications were geothermal energy (< 170◦C)
use where heat from down in the earth crust was used to heat the fluid. Nowadays most
application come from biomass plants. The third biggest application is industrial waste heat
(< 300◦C) but others exist as well, exhaust gas from ICEs or gas turbines or solar heat
competitive with solar panels, just to name a few.
All of these application typically have low temperature heat and the ORC is able to produce
4
Page 19
electricity from these wastes in a more efficient way than other technologies.
In Maraver et al. [2014] an optimisation is performed for different types of waste heat to
compare different fluids. It is shown that depending on the temperature either a subcritical
or a transcritical operation is optimal. As we focus on low grade waste heat here (< 200◦C)
only subcritical operation is considered.
1.3 ORCnext: collaboration and test set-up
1.3.1 The IWT project
The current interest in ORC technology is mainly due to the efforts towards a more efficient,
durable and green technology. The Flemish, Belgian institute IWT, standing for innovation
through science and technology, has granted several funds because of the believe in the field
of ORC. A SBO, strategical basic investigation, lead to a collaboration of several Belgian
partners, named ORCnext. University of Gent, University of Liege, University of Antwerp,
Hogeschool West-Vlaanderen are all elaborated in the project together with some institutes
and companies, Atlas Copco, Power Link and SET.
The aim of the project is to improve the ORC technology and make it more suitable for
industrial use. The improvement is focussing on efficiency of the cycle, the operability etc....
The research includes:
• cycle architecture
• design expanders
• dynamic modelling and prototyping
• dynamic control and optimization
• economical analysis
1.3.2 Test setup
A test rig is provided at the college of West-Vlaanderen in Kortrijk, where experimental stud-
ies are performed. This setup has been updated lately (2013) but was originally developed by
BEP Europe called E-RATIONAL, and extended especially for this project with extra control
parameters and sensors. It has a nominal power Pn of 11 kW. The different components of
the cycle are briefly analysed in this section, read Melotte and Quoilin [2012] for more detailed
information.
heat exchangers
In total three heat exchangers are used within the cycle, the evaporator, the condenser and the
recuperator. All three are in brazed plate exchangers. The evaporator is the most important
5
Page 20
component to be controlled, for safety and efficiency reasons.
In figure 1.5 it is shown how the temperature of the working fluid changes when it flows
through the evaporator, indicating important points: The evaporator outlet temperature Tev,
the saturation temperature Tsat and the difference between these two temperature is the
superheating ∆Tsh.
Subcooled Two phase Superheated
Tin
Tsat
Tout
T
ΔTsh
Figure 1.5: Temperature profile of the fluid through the evaporator
Expander
The expander is especially adapted for ORC technology because turbines are not available
for low power ratings, typical for ORC. Therefore a volumetric expander is used, this type
of expander is also able to withstand a certain amount of droplets without suffering severe
damage. The expander speed will be one of the main control inputs as it is able to control
the volume flow rate in the cycle after the evaporator and can thereby regulate the expander
inlet pressure to an optimal value.
Turbopump
To reach a higher pressure level before heating the fluid and to overcome the pressure drops
in the cycle, a turbo pump is installed before the recuperator.
Sensors
Pressure sensors are installed before and after every component. their accuracy is 0.25%
over the whole pressure range (0 to 16 bar with the biggest error 0,04 bar). The pressure
measures are not very useful for control as the simulation models are not accurate enough to
6
Page 21
predict them. But averaged pressures can be used for superheating calculation or the average
evaporator and/or condenser pressure.
Temperature measurements are available at the same place in the cycle, with accuracy of
±0,05 K. These measurements are more useful for control purposes as the temperature time
constants are slower and the simulation models are able to predict them.
Also a coriolis mass flow meter is installed, such that the working fluid mass flow can be
measured.
1.3.3 Simulation model
All experiments in this work are done through computer simulation and not on
the real set-up of the cycle. For the simulations a dynamic model that has been developed
in modelica language is used and integrated in a dymolar environment. The model and the
modelica library (www.thermocycle.net) are developed by Sylvain Quoilin at the University
of Liege for the ORCnext project. In figure 1.6 the model of the ORC cycle in the dymola
interface is shown. Check Appendix A for information on the simulation model. For control
design the dymola environment is not well suited and therefore the dynamic model is exported
to Matlabr. using FMI toolbox.
Some remarks about the simulations and the model:
• a sampling time of 5s is used in the simulations (time-scales of the cycle are minutes)
• the simulations are only valid in certain regions: e.g. superheating must be non-zero.
source_Cdot
MT
dp_hp
expander
N_rot
duration=0
generatorNext
dp_lp
heat_sinkMT
pump
f_pp
duration=0
tank
Figure 1.6: The dynamic model used for simulation in Dymola
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Page 22
Chapter 2
Optimal Operation
In this chapter open loop experiments are performed in order to investigate the input output
behaviour of the cycle when one variable is changed. Especially the influence on the output
power is of interest, so that the cycle can be optimally controlled.
2.1 Performance of the cycle
2.1.1 Direct performance: efficiency and generated power
The ORC cycle’s performance can be analysed from different points of view. This performance
must of course be based on the intended purpose and have a good correlation with it. In this
context two major performance parameters have been used in ORC. One is the thermal
efficiency of the cycle, which is the ratio of net power that is produced in the cycle over the
total heat taken from the waste heat stream, coming from the hot fluid. The net power is the
generated power diminished with the power subtracted by the pump.
ηth =PgenQhf
=Pexp − Ppp
Qhf(2.1)
The efficiency has the important feature that it can say how well the cycle uses the heat that
it subtracts from the waste heat that is provided. This can be important if the hot fluid, that
delivers the waste heat, will be used after leaving the ORC evaporator and has been used in
literature frequently to validate results obtained from simulation or experiments.
A second performance parameter for the cycle that has been widely used is the net generated
power of the cycle. Here we will neglect the energy transfer efficiency from turbine to grid,
because it has no meaning for control of the ORC. We thus assume that power generated in
the turbine equals the generated electrical power. This simplification neglects the mechanical,
electromechanical efficiency of the gear box and respectively generator as well as the inverters
efficiency to convert the electricity to the grid frequency. Using the power as a performance
indicator is justified if the waste heat has no more functioning after leaving the evaporator of
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Page 23
the ORC. Thus it is better to get as much energy out of the waste stream, such that as much
power as possible can be generated.
Important to remark is that the power generated can only say something about the cycle
under review. This power is inherently determined by the mass flow rate of the waste heat
stream and its temperature at the input, and is also constraint to be lower than the rated
power of the cycle. A third performance parameter can be introduced. Consider the exergy
flow of the waste heat, defined as the maximum work that can be extracted from the waste
stream if it is brought to the equilibrium state of the environment. The exergy Ex or exergetic
power Pexe can be calculated by:
Pexe = m(hhf,in − hhf,amb) (2.2)
With this exergetic power we can define an exergetic efficiency or overall efficiency of how
well the cycle generates its power, compared to the maximal output power available. This
efficiency takes into account the waste heat temperature and mass flow rate. Using this
efficiency it is also possible to compare different ORC cycle architectures or operations with
different fluids.
ηexe =PnetPexe
(2.3)
Conclusion:
There are several ways to determine the performance of an ORC cycle, here we discussed
the two most straightforward indicators: thermal efficiency and generated power. The third
indicator overall efficiency is less used and has a more theoretical practise. In this work we
have a fixed fluid and cycle architecture, therefore it is allowed to use the net generated power
as performance indicator.
2.1.2 Indirect performance: superheating & saturation temperatures
The net generated power is a complex function of all the parameters in the cycle. For control
we want easy measurable variables. It is not an easy task to directly control the output power
as this has the disadvantage that suboptimal operation is likely to happen. It has been shown
three or four important temperatures determine the optimal operation point of the ORC cycle
Quoilin et al. [2011b].
the superheating ∆Tsh is a parameter that has to be taken into consideration. Decreasing
the superheating leads to higher efficiency of the cycle. This can be explained: From the
thermodynamic properties of the (organic) fluid the heat transfer in the evaporator differs if
the fluid is in liquid form, two-phase or gaseous form. It is shown by Yamamoto et al. [2001]
or Dai et al. [2009] that for organic fluids used for ORC purposes, the two-phase region is the
most beneficial for heat transfer because of the low latent heat value, Yamamoto et al. [2001].
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Page 24
We thus want our cycle to operate as close as possible to the zero superheating point. Care
has to be taken, if the superheating falls below zero. Liquid droplets can enter the expander
which causes severe damage to this component. Thus when controlling the superheating of
the cycle, safety margins have to be taken into account. A minimum stable superheating is
being used by Fallahsoho et al. [2010] in refrigeration systems.
Having an accurate control for the superheating, the evaporator saturation temperature
is the most important parameter as it determines the expanders inlet pressure. In Quoilin
et al. [2011b] an optimal value of the evaporation temperature is calculated for maximum
power generation depending on the waste heat condition and the ambient conditions. In
section 2.3 a simple method based on theoretical foundations is used in order to calculate the
optimal value.
Tsat,ev = f(mheat, Theat, Tcool) (2.4)
The third parameter is the condenser saturation temperature. This parameter deter-
mines the expanders outlet pressure and can increase the generated power. Care has to be
taken here as well, if the outlet temperature is higher than the saturation temperature, mean-
ing null subcooling, cavitation can occur in the pipes and turbo pump, causing damage as
well. In most ORC applications null subcooling is obtained by adding a liquid receiver.
Conclusion:
• superheating ∆Tsh must be as low as possible
• saturation temperature in the evaporator Tsat,ev has an optimal value
• saturation temperature in the condenser Tsat,cd should be as low as possible
• subcooling in the condenser above zero.
2.2 Input sensitivity
Now that we discussed all important parameters in the cycle, it is important to know how all
input variables of the cycle influence these parameters. A stepwise increase in disturbances
and actuators is simulated and the response is analysed. All variations are around the nominal
design point:
10
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Pnet 11 kW
Theat 145◦C
mheat 3 kg/s
mcool 4 kg/s
Npp 1800 rpm
Nexp 3000 rpm
Tsat 129◦ C
Tev,out 139◦ C
∆Tsh 10◦ C
Table 2.1: Conditions under nominal operation
2.2.1 Uncontrollable inputs
At first the response to uncontrollable inputs is considered, these could be seen as disturbances
of the environment from control point of view, but they also determine the optimal operating
point of the cycle so the effect on saturation temperature is analysed. The main disturbances
are the mass flow rate of the waste heat and its input temperature. Off less concern is the
condenser inlet temperature, as will come clear.
Input temperature waste heat
The temperature of the waste heat entering the evaporator, Theat is varied stepwise from below
the nominal operating point 140◦ C to 170◦ C, while all other inputs are kept constant at
their nominal value (see table above). An increase of ±2 kW in the output power is observed.
The saturation temperature Tsat increases with 15 degrees. This result already indicates the
importance adjusting the optimal saturation temperature in the evaporator.
Also notice the non-linearity in the system: while Theat is increased with a constant step
the considered dependant variables have a not constant stepwise increase. This non-linearity
makes that controlling the cycle is more difficult. The effect on output power and saturation
temperature is clearly under damped, and the overshoot becomes stronger with increasing
Theat.
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Page 26
0 1000 2000 3000140
150
160
time[s]
Theat [
°C]
0 1000 2000 3000
130
135
140
time[s]
Tsat,ev [
°C]
0 500 1000 1500 2000 2500 3000 3500
7
8
9
time[s]
Pgen [
kW
]
Figure 2.1: effect of variable waste heat temperature on generated electrical power and saturation
temperature Tsat,ev
Mass flow rate waste heat
The mass flow rate of the waste heat is changed from 1,5 kg/s to 6 kg/s. The increase in
output power is only 500 W. It seems strange that the amount of waste is not the most
important parameter for power generation. This can be explained: The ORC is designed for
certain mass flow rate of waste heat. Below this design value the power fluctuates strongly
reacting on the available waste heat. When it is higher than the design value, the ORC cannot
efficiently extract the heat, the evaporator is too small to extract all heat.
Also the non linearity is even stronger here and the overshoot decreases as the mass flow
increases.
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Page 27
0 1000 2000 3000
2
3
4
5
time[s]
Mheat [
kg
/s]
0 1000 2000 3000
128
129
130
time[s]
Tsat,ev [
°C]
0 500 1000 1500 2000 2500 3000 3500
6.8
7
7.2
time[s]
Pgen [
kW
]
Figure 2.2: effect of variable waste heat mass flow rate on generated electrical power and saturation
temperature Tsat,ev
Condenser inlet temperature
During the year and during one day the ambient temperature changes typically in a range
of maximum 40◦C. In figure 2.3 the temperature is changed over 7.5 degrees. The change
in output power amounts to 700 W. The generated power decreases because the condenser
pressure varies proportional with the saturation temperature.
0 500 1000 1500 2000
18
20
22
24
time[s]
Tcool [
°C]
0 500 1000 1500 2000
32
34
36
time[s]
Tsat,cd [
°C]
0 200 400 600 800 1000 1200 1400 1600 1800 2000
6.8
7
7.2
time[s]
Pgen [
kW
]
Figure 2.3: effect of variable condenser inlet temperature on generated electrical power and satura-
tion temperature Tsat,cd
13
Page 28
2.2.2 Controllable inputs
In previous section it is shown that the and saturation temperature must be carefully con-
trolled to an optimal value. In this section all controllable inputs are varied one-by-one in
order to investigate their impact on the generated power, saturation temperature and super-
heating.
Pump speed
The pump speed increases stepwise from 1500 rpm to 1950 rpm. In figure 2.4 the effect on
saturation and evaporator outlet temperature, superheating, expander inlet and condensing
pressure, output power and mass flow rate of the working fluid is shown.The pump speed has
a big effect on the superheating and saturation temperature.
Increasing the pump speed while keeping expander speed constant means increasing the tur-
bine inlet pressure, the pressure ratio over the expander is changed and this way the output
power is increased. Also here the non-linearity is present.
0 1000 2000 3000
1700
1800
1900
time[s]
Npp [
rpm
]
0 1000 2000 3000
6
6.5
7
time[s]
Pgen [
kW
]
0 1000 2000 3000
5
10
15
20
time[s]
Tsh [
°C]
0 1000 2000 3000
124
126
128
time[s]
Tsat,ev [
°C]
Figure 2.4: Effect pump speed on cycle temperatures, pressures and other variables
14
Page 29
Expander speed
0 1000 2000 3000
2600
2800
3000
3200
time[s]
Nexp [
rpm
]
0 1000 2000 3000
6.5
7
7.5
time[s]
Pgen [
kW
]
0 1000 2000 30000
5
10
15
time[s]
Tsh [
°C]
0 1000 2000 3000
127
128
129
130
131
time[s]
Tsat,ev [
°C]
Figure 2.5: Effect expander speed on cycle temperatures, pressures and other variables
The expander speed is increased from 2460 rpm to 3360 rpm.
The mass flow rate is kept constant, because the pump speed, moving the liquid state, more
or less fixes this value due to the not compressible liquid phase. Because of the volumetric
expander, the volume flow rate changes linearly with its speed and therefore the inlet density,
and also inlet pressure, changes inversely proportional. Notice the expanders effect on the
superheating that is a lot faster than the pump speed. Also a fast response on the saturation
temperature is observed, because the expander speed affects the inlet pressure.
For controlling we notice some difficulties. The response on the superheating response shows
non-minimum phase behaviour. Both variables first change in the opposite direction before
reaching their steady state value.
Mass flow rate condenser
At last it is shown that controlling the condenser mass flow rate has only a minor effect on
the cycle. It is thus a less important actuator to use, although it can easily be controlled
by controlling the speed of the pump in the cooling water circuit. In figure 2.6 the cooling
water mass flow rate mcool is varied from 1 kg/s to 5 kg/s and the effect on generated electrical
power is plotted. We can conclude that there is only a very small effect (increase of less than
300 W). It is also shown that Mcool strongly affects the condenser saturation temperature
and thus the expander outlet pressure.
We can conclude that controlling the condensing is not necessary and this has the economical
benefit that only two main actuators need active control.
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Page 30
0 500 1000 1500 2000
2.53
3.54
4.5
time[s]
Mcool [
kg
/s]
0 500 1000 1500 2000
33
34
35
time[s]
Tsat,cd [
°C]
0 200 400 600 800 1000 1200 1400 1600 1800 2000
7
7.1
7.2
time[s]
Pgen [
kW
]
Figure 2.6: effect of variable cooling water mass flow rate on generated electrical power
2.3 Optimisation saturation temperature
In this section an optimiser is build that calculates the optimal evaporator saturation tem-
perature. It is based on the work of He et al. [2012]. It is based on theoretical thermodynam-
ics. Another approach is to build a realistic steady state model, this however requires good
knowledge of the cycles physics. An example of this second approach is given in Quoilin et al.
[2011a].
2.3.1 Derivation optimiser
The aim is to produce the maximum net power:
Wnet = Wexp −Wpp = f(mwf , pev, pcd) (2.5)
Both the work of the pump and expander are determined by the high and low pressures in
the cycle and the mass flow rate. These pressures should thus be controlled to an optimal
value, being equivalent to controlling optimal saturation temperatures. (one-one relationship
temperature-pressure in the two-phase region).
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Page 31
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20
20
40
60
80
100
120
140
160
s [kJ/kg]
T [
°C]
1
2s
3
4
5s
6
2 5
hf
cf
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20
20
40
60
80
100
120
140
160
s [kJ/kg]
T [
°C]
1
2s
3
4
5s6
2 5
hf
cf
∆1
Figure 2.7: Ts-diagram ORC cycle with and without superheating
The net work can be written (approximately) as a function of the saturation temperature.
To do so consider the Ts-diagram in figure 2.7. If it is assumed that the superheating is
neglectible small than the cycle can be approximated by the right Ts-diagram. Now the net
power extracted by the cycle equals net heat extracted:
Wnet
mwf= Qcycle = Qin −Qout (2.6)
= Area(1− 2− 3− 4− 5− 6− 1) (2.7)
If the Area(1-2-3-1) is neglected because it is so small the expression becomes:
Wnet ≈ mwf ×Area(1− 3− 4− 5− 6) (2.8)
≈ mwf × (Tsat − T1)[(s4 − s3) +
1
2(s3 − s1)
]ηs (2.9)
Knowing that:
∆hvap = Tsat(s4 − s3) (2.10)
s3 − s1 ≈ lnTsatT1
(2.11)
mwf∆hvap = mheatcp,heat(Theat − Tsat −∆T1) (2.12)
Where equation 2.11 is obtained assuming a constant heat capacity: cpdT = dh = Tds. The
net work finally becomes:
Wnet = ηsmheatcp,heat(Theat −∆T1 − Tsat)(Tsat − T1)
Tsat
(1 +
Cp,wfTsat2∆hvap
ln
(TsatT1
))(2.13)
Now the net output power is function of the waste heat, the latent heat, the saturation
temperature, the condenser outlet and the pinch point in the evaporator. Here the latent
heat is strongly dependant on the saturation temperature and Chao suggests the following
correlation for the latent heat:
∆hvap = RgTc
[7.08
(1− Tsat
Tc
)0.354
+ 10.95ω
(1− Tsat
Tc
)0.456]
(2.14)
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Page 32
Where ω is the acentric factor, which is a chemical property of a fluid and thus a constant.
Now equation 2.13 and 2.14 determine the work together with the known parameters. To
optimise the power the derivative of the net work is set to zero:
d
dTsatWnet(Tsat,∆hvap) = 0 (2.15)
m (2.16)
dWnet
dTsat+∂Wnet
∂Tsat
d∆hvapdTsat
= 0 (2.17)
Solving this equation iteratively using the Newton-Raphon method yields the optimal value:
Tsat = f(∆T1, T1, Theat) (2.18)
2.3.2 Mapping
If this procedure is carried out for several values of the dependant variables, a mapping can be
made yielding tables of optimal working points. This is visualised in figure 2.8 where a pinch
point of 5 degrees is assumed and figure 2.9 where the waste is 145 degrees and condenser
outlet 25 degrees.
100
150
200
0
5050
100
150
Theat
[°C]T
cd,out [°C]
Tsat [
°C]
60
70
70
70
80
80
80
90
90
90
100
100
100
110
110
Theat
[°C]
Tcd,o
ut [
°C]
100 120 140 160 18010
15
20
25
30
35
40
45
50
Figure 2.8: Optimal Tsat in function of condenser outlet and heat source temperature
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Page 33
0 5 10 15104
106
108
110
112
114
116
118
Pinch point ∆ T1 [°C]
Tsat [
°C]
Optimal saturation temperature
Figure 2.9: Optimal Tsat in function of the evaporator pinch point
In general the optimal saturation temperature increases with the waste heat inlet temperature
and the condenser outlet. The variation goes from 40◦C to 140◦C, so a variation of 100 degrees
seems possible. However this optimal value decreases when the pinch point in the evaporator
is increased. However it must be noted that the pinch point is a design parameter and almost
not depending on the operation.
Important remark here is that transcritical cycles are not considered here. When the waste
heat temperature becomes larger than 200◦C, the cycle operation should be operated as a
trancritical one, meaning that the fluid never enters the two-phase region when it is heated.
For this operation the optimiser presented here is not valid because a saturation temperature
has no meaning for transcritical cycles.
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Page 34
Chapter 3
Identification
3.1 Objective & method
In the previous chapter an open loop analysis was performed. The input-output and disturbance-
output behaviour was investigated. It was shown that their are 4 important parameters that
influence the output power, namely: the waste heat temperature Theat, the waste heat mass
flow rate mheat, the speed of the pump Npp and the speed of the expander Nexp. The first
two are disturbances that influence the output power strongly, and both are measurable. The
last two are the main actuators that determine the operating point of the cycle.
Other parameters that influence the cycle are the condensing stream mass flow rate Mcool
and temperature Tcool. The condensing stream temperature is determined by the ambient
temperature and is therefore a disturbance, but its variations have two very long time-scales
compared to the cycle dynamics: daily trend and seasonal trend. mcool has only a minor
effect on the output power, as was shown through simulation in section 2.2.2.
An expansion valve between expander and evaporator could be installed in order to have a
possibly faster control on the expander inlet pressure, but this is not the case here. This
choice has been adapted when a non-volumetric expander is used in the cycle. (Zhang et al.
[2014], Padula et al. [2012])
In order to design a controller, a model of the cycle is needed. The restriction is that a low
order model is required in order to design a fast controller. Two possibilities for the modelling
are: or creating a physical realistic model or using identification techniques.
The advantage of the first method, physical modelling from first principles is that a physi-
cal interpretation of the results is straightforward. This technique has been used in the past:
Quoilin uses a discretised model for the heat exchangers Quoilin et al. [2011a] whereas Wei
et al. [2008] and Zhang et al. [2012] use moving boundary (MB) models, both lead to high
order models (MB lower order than discretised) that are not very usefull for real-time opti-
misation due to the highly non-linear and complex models that are obtained, partly caused
20
Page 35
by the thermodynamic states (density, heat transfer coefficients, heat capacities...) that vary
with pressure and temperature.
The second method, using identification, has the advantage that good results can be ob-
tained already with low order models. the physical interpretation then becomes more difficult,
but can still be obtained from good knowledge of the thermodynamic cycle. This identifica-
tion is based on experimental data obtained from open loop experiments on a real setup or
as done in this work using the higher order simulation models obtained from first principles.
(see section 1.3.3)
First an identification is performed in the design point and in the second part the non-linearity
present in part load operation is addressed. In chapter 4 and 5 the obtained models are used
for controller design. In order to have a full model of the system the identification is done
for the full 3 inputs (Npp, Nexp, mcool) 3 outputs (∆Tsh, Tsat,ev, Tsat,cd) system and a system
analysis is done. Taking all this in consideration we obtain the following model for the ORC:
Disturbances
Inputs
ṁheat
Theat
Nexp
Npp
OutputTsat,ev
ṁcool
ΔTsh
Tsat,cd
Figure 3.1: ORC model showing main disturbances and inputs
3.2 Identification nominal design
For operation at design conditions a model that is valid in this point is needed. Therefore a
linearised MIMO model around the operating point is identified. The proposed model accord-
ing to the choices made before is shown on the block diagram in figure 3.2. G(s) and Gd(s)
are two matrices of transfer functions and u, y and d are respectively the inputs, outputs
and disturbances in concatenated form.
The identification is done using the system identification toolbox in matlab. The pem (pre-
21
Page 36
diction error method, Ljung L) was used with a multi sine excitation signal in a frequency
range of [0,0001Hz 0,08Hz] the lower bound taken in order to identify the gain correctly and
the higher bound chosen to identify the dynamics around the bandwidth of the system. This
type of signal is used in order to avoid numerical issues caused by the solver of the simulator.
The excitation signal has an amplitude of 100rpm for expander and pump speed and 0,5kg/s
for the cooling water. A sampling time of 5s, a identification period of 5000s and a validation
period of 5000s was used.
G(s)
Gd(s)
d
u+
y+
Figure 3.2: Block diagram system model
u =
Npp
Nexp
mcool
y =
∆Tsh
Tsat,ev
Tsat,cd
d =
[Theat
mheat
](3.1)
3.2.1 Input-output model
Using the identification method described above the following model is obtained. The transfer
functions were obtained by finding best fitting process models (86% fitting by validation, not
considering the zero-relations).
G(s) =
−4.34e−32.6s
92.8s+ 10.76−17.9s+ 1
76.7s+ 10
0, 64e−37.9s
83.2s+ 1−0.47
1
45.1s+ 10
0.79e−45.8s
93.8s+ 10.01
1653.0s+ 1
874.8s2 + 78.7s+ 1−0.397
1
18.8s+ 1
(3.2)
Consider first the left column i.e. the dynamics from pump speed to output. The response can
be approximated by a first order model with delay. This model has been adopted in the past
and gives satisfying results. The delays in the response are caused by transport delays: The
pump speed directly alters the mass flow rate of the working fluid, this fluid needs a finite
time to flow through the pipes and evaporator before it is able to influence the saturation
temperature and superheating and the same goes up for influencing the condenser.
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500 1000 1500 2000 2500 3000 3500 4000 4500 5000
−4
−3
−2
−1
0
1
2
Time [s]
Tsh [°C
]
Data
Model: fit 79%
Figure 3.3: Validation model pump speed Npp to superheating ∆Tsh
The response of the expander speed (middle column) shows non-minimum phase behaviour:
a RHP zero is present from expander speed to superheating: Changing the expander speed
directly alters the volume flow rate going through the expander. Because of the inertia
of the liquid phase in the evaporator the pressure stays unchanged (thus also saturation
temperature). Thus all the dynamics in the first fraction of the response are in the gas that is
expanded. The gas density drops down causing the outlet temperature of the evaporator to
drop down, and since the saturation temperature is unchanged the superheating drops down.
Then after a while the pressure starts decreasing as well and this causes the superheating to
go up again. Because the expander can directly alter the gas properties no delays are present.
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Time [s]
Tsat,ev [
°C]
Data
Model fit: 94%
Figure 3.4: Validation model expander speed Nexp to evaporator saturation temperature Tsat,ev
23
Page 38
The model from cooling water to condenser saturation temperature (third column) is a first
order transfer function. The identification from cooling water to evaporator working condi-
tions was not possible, a physical interpretation a model would not easy as the condenser is
separated by from the hot pressure line by the pump and expander.
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
−0.1
−0.05
0
0.05
0.1
0.15
Time [s]
Tsat,cd [°C
]
Data
Model fit: 93%
Figure 3.5: Validation model cooling water mass flow rate Mcool to condenser saturation Tsat,cd
3.2.2 Disturbance to output
In this section the influence of the changing conditions in the waste heat stream are analysed.
A model is identified that tries to capture the dynamics of these disturbances to the outputs.
The same excitation signal was used with an amplitude of 5 degrees for Mheat and 0,5kg/s
for Mheat.
Gd(s) =
1, 41686, 9s+ 1
240, 6s+ 11, 548
−99, 8s+ 1
76, 6s+ 1e−26,5s
0, 679−8.3 + 1
12.5s+ 10.753
76.9s+ 1
57.7s+ 1e−23.7s
0, 027620.5s+ 1
107.5s+ 1e−63.1s 0.292
1
22.4s+ 1
(3.3)
A physical interpretation is not given, but we remark that the time scales are in the same
range as those of G(s), non-minimum phase behaviour is present and delays as well. The
obtained model was validated with a 77,6% fitting.
3.2.3 Unmodelled dynamics
In section 3.1 it was mentioned that the system is highly non-linear and has rather complex
dynamics. The identification above yields an linear MIMO system augmented with transport
delays. It is therefore important to know the consequences of approximating the cycle by a
24
Page 39
linear model.
The obtained linear model will only be valid around the operating point where the identifi-
cation is performed, and it can thus be expected that this model is a bad approximation in
other operating points. This problem will be tackled in chapter 6.
A second phenomena that is neglected are the so called secundary effects or cycle effects.
Therefore we look at figure 3.6. A change in the pump speed yields a discontinuity in the
response of Tsat,ev: Changing the pump speed will affect the operation of the evaporator and
changes the saturation temperature (Primary effect). This will then change the operation of
the condenser as well, affecting the pump again and the cycle closes again at the inlet of the
evaporator. This response could be modelled by:
Tsat(s) =(g1(s) + g2(s)e
−Tcs)Npp(s) (3.4)
Where g2(s) models the secondary effects and Tc is the time for the fluid to go through the
whole cycle and this value depends on the mass flow rate Mwf and thus on the pump speed.
These secondary effects are also present in the expander dynamics and on other outputs.
0 100 200 300 400 500 600 700−1
0
1
2
3
4
5
6
time [s]
Tsat
Figure 3.6: step response from Npp to Tsat,ev showing the secondary effect
3.3 System Analysis
3.3.1 Controllability
Scaling
In order to use systematic techniques to analyse to identified system the plant G(s) and Gd(s)
needs to be scaled such that:
y(s) = G(s)u(s) + Gd(s)d(s) (3.5)
|y| < 1 for every |u| < 1, |d| < 1 (3.6)
The inputs and disturbances are scaled by there maximum allowed deviations:
25
Page 40
• pump speed: 1800± 1200rpm
• expander speed: 3000± 900rpm
• cooling water 3± 2kg/s
• waste heat temperature: 150± 50◦C
• waste heat mass flow rate: 3± 2kg/s
the outputs by their maximum expected (deviating) reference values;
• superheating: 10± 10◦C
• saturation temperature: 90± 50◦C
• condenser outlet temperature: 30± 20◦C
Putting these values in diagonal matrices the new model is obtained:
G(s) = D−1y G(s)Du (3.7)
Gd(s) = D−1y G(s)Dd (3.8)
Out of simplicity the accents on G(s) and Gd(s) are omitted in the following.
Poles & transmission zeros
In figure 3.7 the poles and transmission zeros of G(s) are plotted.
−0.08 −0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0−2
−1.5
−1
−0.5
0
0.5
1
1.5
2x 10
−3
Pole−Zero Map
Real Axis (seconds−1
)
Imagin
ary
Axis
(seconds
−1)
Figure 3.7: pole-zero map of G(s)
26
Page 41
Limitations on control
• in figure 3.8 the singular values and the condition number of the plant G(s) are plotted.
It can be seen that the minimum singular value is very low at low frequencies. This
indicates that the system will be difficult to control in some output directions. Also the
condition number µ =σmaxσmin
goes to infinity for frequencies above 0, 5rad/s, indicating
that the system is hard to control especially at frequencies above this value.
• the closed loop bandwidth must be chosen 10 times slower than the sampling frequency:
ωbw < 0.1× 2π
5s= 0, 126
• if decentralised control is used, the delay from Npp to ∆Tsh limits the bandwidth:
ωbw <1
Td= 0, 0307rad/s
10−4
10−3
10−2
10−1
100
101
10−2
100
frequency [rad/s]
σmax
σmin
3dB line
10−4
10−3
10−2
10−1
100
101
101
102
103
104
Condition number µ
frequency [rad/s]
Figure 3.8: The singular values and condition number of G(s)
27
Page 42
3.3.2 Input-Output coupling
For decentralised control the influence of each input to every output must be known, so that
the pairing between the three inputs and three outputs is chosen correctly. A very handful
tool for this is the Relative Gain Array (RGA):
Λ(G) = G×(G−1
)T(3.9)
The diagonal elements should be the response of the chosen pairings, here the pairing is
chosen as: Npp −∆Tsh, Nexp − Tsat,ev and Mcool − Tsat,cd. If the RGA matrix Λ(G) is close
to identity, then a decentralised control is possible, if not there is a lot of coupling present in
the system and a multi-variable controller is advised. This yields an RGA matrix evaluated
at steady state and around the closed loop bandwidth (taken here as 0.02rad/s):
|Λ(j0)| =
1.31 0.31 0
0.31 1.31 0
0 0 1
(3.10)
|Λ(jωbw)| =
1, 06 0, 25 0
0, 25 1, 06 0
0 0 1
(3.11)
The pump and expander speed have a strong interaction on superheating and saturation
temperature of the evaporator. The coupling becomes even stronger at frequencies where
control is important: around ωbw. This indicates that decentralised control is discouraged.
The coupling between Mcool and Tsat,cd and decoupling from other outputs is very good. If one
chooses to control the the condensing stream, this control can be designed independently from
the rest of the system. From now on the condenser dynamics are neglected, meaning
we only consider inputs Npp, Nexp and outputs Tsh, Tsat,ev.
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Page 43
Chapter 4
Control Strategies
4.1 Objective & control strategy
In this chapter classic control strategies are designed. As mentioned in chapter 2 the following
requirements were concluded for optimal operation:
• superheating ∆Tsh must be as low as possible
• saturation temperature in the evaporator Tsat,ev has an optimal value
• saturation temperature in the condenser Tsat,cd should be as low as possible
Controller Systemu y
sensors
ym
optimiser
d
r
Figure 4.1: Control loop with reference values given by a steady state optimiser
The first two requirements are the most important and need a good control. It was already
mentioned that the optimal value of Tsat,ev is dependant on the ambient conditions and the
waste heat. To obtain this value typically a steady state optimiser is used that calculates this
29
Page 44
optimal value (see figure 4.1). This optimiser was build in section 2.3.
To compare the performance of the different controllers a standard experiment is executed
that represents a realistic working condition: The mass flow rate of the waste heat mheat is
varying around 3kg/s. And the waste heat temperature Theat, which is the most important
disturbance is first varying around 145◦C, and then its average value slowly decreases with
10 degrees to 135◦C (see figure 4.2). The optimiser keeps the superheating reference constant
at 8◦C and decreases the saturation temperature reference from 125◦C to 120◦C some time
after Theat has decreased its value.
0 500 1000 1500 2000 2500 3000 3500
2.9
3
3.1
3.2
Mheat
0 500 1000 1500 2000 2500 3000 3500
135
140
145
time [s]
Theat
Figure 4.2: The standard experiment
First a decentralised PI controller is designed in the design point and it is shown that this
leads to unsatisfying results. Secondly the system is dynamically decoupled, it is shown that
decoupling the system is beneficial and improves the performance.
4.2 PI
In this section two decentralised controllers are designed in order to control the superheating
∆Tsh and the saturation temperature Tsat,ev. In the past other centralised controller designs
have been used e.g. Elliott and Rasmussen [2010] uses a cascaded design for superheat control
in a refrigeration cycle in order to reduce the non-linearity.
30
Page 45
C(s) G(s)
Disturbances
r
+
e u y
sensors
−ym
Figure 4.3: Control loop for decentralised PI
4.2.1 Design
The decentralised controller C(s) has the following form:
C(s) =
[Csh(s) 0
0 Csat(s)
](4.1)
Using the matlab PID auto-tuner we obtain the following PIs:
Csh(s) = −0.286
(1− 1
0.00343s
)(4.2)
Csat(s) = −1.637
(1− 1
0.157s
)(4.3)
The specifications of the superheating control is 70% PM and 175s rise time (time until the
output crosses 1 for a step response). The controller is tuned such that no overshoot occurs
when a step is applied.
The specifications of the saturation temperature control is 60% PM and 50s rise time. This
faster response is required to narrowly follow its reference values.
4.2.2 Validation
To check the performance of the decentralised PI controller, the standard experiment is ex-
ecuted and plotted in figure 4.4. At the moment Theat starts decreasing the superheating
control fails to keep its reference value and drops strongly. If the heat source temperature
drop even more the expander is likely to be damaged when superheating drops below zero.
Notice that decreasing the saturation temperature reference with Theat enlarges the power
generation, at the time the heat source temperature drops, the net power generated drops
below zero, this means that the pump consumes more power than the expander produces.
Here we can already see the need for an optimiser to track optimal working conditions.
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0 1000 2000 30000
10
20
time [s]
Tsh [
°C]
0 1000 2000 3000
118
120
122
124
126
time [s]
Tsat [
°C]
0 500 1000 1500 2000 2500 3000 3500
−2
0
2
4
6
8
time [s]
Pnet [
kW
]
Figure 4.4: standard experiment with PI control:
tracking performance (upper) and net electrical power (below)
4.3 Decoupling
The system is strongly coupled as was shown by the RGA analysis in section 3.3.2. Because
of this coupling and the bad results obtained by decentralised control, decoupling the system
might improve the control.
The system is dynamical decoupled using the following feed-forward scheme:
R1
R2
C1
C2
G11
G12
G21
G22
D1
D2
−
−
Y1
Y2
Figure 4.5: decoupling scheme
Here D1 and D2 are chosen such that the outputs Y1 and Y2 are solely dependant on respec-
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Page 47
tively U1 and U2, the outcome (or actuator inputs) of both controllers:
Y1 = G11U1 +G12U2 (4.4)
= G11(UC1 − UD1) +G12U2 (4.5)
(4.6)
Same expression is obtained for Y2 and to decouple the system:
UD1 = D1 × U2 = (G11)−1G12 × U2 (4.7)
UD2 = D2 × U1 = (G22)−1G21 × U1 (4.8)
Now the decentralised controller is retuned on the decoupled system with the same specifica-
tions on PM and rise time. The response on the standard experiment is plotted below:
0 1000 2000 30000
5
10
15
time [s]
Tsh [
°C]
0 1000 2000 3000
118
120
122
124
126
time [s]
Tsat [
°C]
0 500 1000 1500 2000 2500 3000 3500
−2
0
2
4
6
8
time [s]
Pnet [
kW
]
Figure 4.6: standard experiment using dynamic decoupling and PI control
Decoupling the system clearly improves the control. The variations in superheating are smaller
and the superheating never drops to zero. The saturation temperature is even better decou-
pled and shows less overshoot.
4.4 Real-time optimised control
The mapping of optimal evaporator saturation temperatures that was build in section 2.3 has
not been tested yet. This is done here with the decoupled PI controllers. The waste heat
profile from the standard experiment is subjected to the system and the controller follows
references given by the optimiser. The results is plotted in figure 4.7
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Page 48
0 1000 2000 30000
5
10
15
time [s]
Tsh [
°C]
0 1000 2000 3000
90
100
110
120
time [s]
Tsat [
°C]
0 500 1000 1500 2000 2500 3000 35000
2
4
6
8
time [s]
Pnet [
kW
]
Figure 4.7: standard experiment with RTO control
The controller does not manage reach the optimal evaporator saturation temperature (red
line = optimised reference). The expanders speed saturates when lowering the saturation
temperature (or pressure). This phenomena was investigated and apparently the current
cycle is not designed for optimal operation. The heat exchangers are too big for the waste
heat stream and because of that the actuators are under-designed to get the evaporator in its
optimal working point.
Although the control does not manage to reach the reference, the net output power is a little
higher than without optimiser, the optimiser does its job but the system is badly designed.
It can be concluded that unless the cycle is designed for its nominal working conditions, the
cycles architecture must be taken into account during the optimisation process.
4.5 Conclusion
A control was designed based on the identified models from chapter 3. Controlled variables
were chosen to be superheating and saturation temperature as these determine the optimal
operation point (chapter 2). It was shown that both variables can be controlled using con-
ventional PI controllers under weak varying conditions of waste heat. It was shown that the
coupling in the system can be diminished by the use of decoupling to improve the perfor-
mance.
The controller was tested together with a steady state optimiser. The control could not reach
optimal conditions due to the fact that the current cycle is not designed for its waste heat
stream. The approach of steady state optimisation did not take into account the component
architecture, but could increase the net generated power.
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Chapter 5
Model Predictive Control
5.1 The method
5.1.1 MPC algorithm
In order to improve the performance of the control a Model Predictive Control strategy is
carried out. Or shortly written as MPC. Since the 80’s several MPC algorithms have been
introduced, one of the most well known is the General Predictive Control, known as GPC
control (see Clarke et al. [1987]). Another methodology was presented by De Keyser [2003],
known as the EPSAC algorithm, which stands for Extended Prediction Self-Adaptive Control.
In this work the EPSAC algorithm is used, just for the sake of completeness it is noted that
also state space MPC algorithms have been introduced over the years.
The user is free to choose any of these different algorithms, depending on their easy under-
standing or out of model considerations. But all algorithms have similar foundations, but
might have some special characteristics or properties that are usefull in a certain control ap-
plication.
The EPSAC algorithm is explained in detail in Appendix B. But we will introduce the most
important properties here. The basic idea behind an MPC controller is that a trustworthy
dynamic model of our system is available and that we know or can guess what the future set
points will be. The model is then used to find the optimal input to reach the set point in a
finite time.
The optimal inputs are obtained by minimising a cost function without violating constraints
on inputs, outputs and slew rates, for a 2 input 2 output system this problem can be sum-
35
Page 50
marised to:
min J =N2∑
k=N1
[r1(t+ k|t)− y1(t+ k|t)]2 +N2∑
k=N1
[r2(t+ k|t)− y2(t+ k|t)]2 (5.1)
subjected to:
Y 1 = Y1,base +G1Uopt
Y 2 = Y2,base +G2Uopt (5.2)
Umin < Uopt < Umax
|Uk − Uk−1| < 4UmaxYmin < Y < Ymax (5.3)
Here 7.1 is the solidary cost function, 5.2 are the EPSAC prediction equations using the
system model and 5.3 are the constraints that are imposed on the system.
At every sample this optimisation problem is solved yielding an optimal input vector Uopt,
this input is then imposed to the system during one sampling time.
Two important MPC parameters are introduced here: N1 and N2 which are the horizon
parameters. N1 defines from when the effect of the input is added to the cost and is usually
chosen equal to the minimum delay that the system needs to have an impact of one of the
different inputs. N2 is called the prediction horizon and defines until when the response of
the inputs is considered in the cost function and thus also how far the prediction is carried
out. This value can be based on physical grounds e.g. the rise time or settling time or any
other time constant that characterises this output. But in practise this parameter is seen as
a tuning parameter to obtain good control performance.
5.1.2 The model
The EPSAC algorithm uses a discrete step input model. So for each input-output pairing
the step response must be known for the interval [N1N2]. In chapter 3 a MIMO transfer
function model was identified. Taking the step response for each I/O pairing and sampling
the response every 5s gives the needed model:
G(s) =
−4.34e−32.6s
92.8s+ 10.76−17.9s+ 1
76.7s+ 1
0, 64e−37.9s
83.2s+ 1−0.47
1
45.1s+ 1
sampling−−−−−−→ G(z) =
z−7−0.12− 0.11z−1
1− 0.95z−1−0.18 + 0.23z−1
1− 0.94z−1
z−80.016 + 0.022z−1
1− 0.94z−1−0.05
1− 0.89z−1
(5.4)
Now from equation 5.4 the G matrices can be obtained by taking a step response.
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5.1.3 Constrained control
One of the advantages of MPC over conventional control is the possibility to include con-
straints in the control system. This is especially of importance for the organic rankine cycle
due to a lot of limitations that have to be ensured during operation. For the set-up of ORC-
next in Kortrijk the following limitations are imposed:
Actuator limits
The pump and expander are being used as main actuators in the system by controlling their
speed. This speed is limited because of structural design limitations of the pump and pipes
and electrical overheating of electro-motor or generator. Also acceleration limits are present
not to overload the actuators due to the fluid inertia.
• pump speed: up to 3000rpm
• pump acceleration: up to 100rpm/s
• expander speed: 2000 to 4000 rpm
• expander acceleration: up to 100rpm/s
Pressure limits
The cycles components must be able to withstand the high pressures in the cycle. This
thus concerns the high pressure line pipes, evaporator, pump and expander. They should be
designed for optimal operation:
• pressure: up to 12 bar
This limitation is equal to a maximum saturation temperature 100◦C for R245fa.
Waste heat temperature
The ORC was mainly designed for low grade waste heat, but its value can vary in time but is
should always be ensured to be lower than a certain temperature at the inlet of the evaporator
to not damage this component (due to thermal stresses e.o.) or degrade the organic fluid.
Therefore a maximum evaporator inlet temperature of 140◦C is needed.
Superheating and subcooling
As already mentioned in chapter 2, superheating and subcooling must be non-zero because this
can damage the expander (liquid droplets) and the pump (cavitation) therefore a minimum
safety value is imposed to the control:
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• superheating: > ∆Tsh,min
• subcooling: > ∆Tsub,min
5.2 Simulation
Standard experiment
The obtained MPC algorithm with constraints on inputs, outputs and slew rates was sim-
ulated for the standard experiment and the results is plotted in figure 5.1. The constraints
onsubcooling are not yet implemented. The MPC parameters were tuned to:
N1 = 1 sample (5.5)
N2 = 60 samples (= 300s) (5.6)
Nu = 1 sample (5.7)
α = 0.8 (5.8)
The control is able to track the reference and to reject disturbances.
0 1000 2000 30000
5
10
15
time [s]
Tsh [
°C]
0 1000 2000 3000
118
120
122
124
126
time [s]
Tsat [
°C]
0 500 1000 1500 2000 2500 3000 35000
2
4
6
8
time [s]
Pnet [
kW
]
Figure 5.1: standard experiment using solidary MPC control
RTO control
As was mentioned in chapter 4 the evaporator of the system is over-designed, giving rise to
inaccessible output pares (∆Tsh, Tsat) because of the limitations in pump and expander speed.
As the aim is optimal control of the ORC, controlling the optimal saturation temperature is
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of more concern than the superheating (if its value stays above zero). Therefore we introduce
the following optimisation problem:
min J =N2∑
k=N1
[Tsat,opt(t+ k|t)− Tsat(t+ k|t)]2 (5.9)
subjected to:
Y 1 = Y1,base +G1Uopt
Y 2 = Y2,base +G2Uopt (5.10)
Umin < Uopt < Umax
|Uk − Uk−1| < 4Umax∆Tmin < ∆Tsh < ∆Tmax (5.11)
Only the optimal saturation temperature is tracked but superheating is assured to be bounded.
The minimum and maximum values are 5◦C and 20◦C respectively where the upper limit was
chosen for efficiency reasons (arbitrary value). The standard experiment is executed with the
optimiser, the result is plotted in figure 5.2.
0 1000 2000 30000
10
20
time [s]
Tsh [
°C]
0 1000 2000 300080
100
120
time [s]
Tsat [
°C]
0 500 1000 1500 2000 2500 3000 35000
2
4
6
8
time [s]
Pnet [
kW
]
Figure 5.2: Constrained MPC with optimised saturation temperature tested with the standard ex-
periment
The controller is able to control the saturation temperature and obeys the constraints imposed
to the controller. Again it must be noted that the optimal saturation temperature is not
reached as a result of the over-sizing of the evaporator (or under-designing of pump and
expander).
If we compare the optimised and not optimised experiments the following can be concluded:
In the first half of the experiment, the (not-optimal) tracking yields a little higher net power
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than the optimised one. This is the over-sizing effect (working in suboptimal conditions),
whereas the tracking reference value is (approximately) the optimal value. However in the
second part of the experiment the optimiser performs better than the tracking control. Overall
seen we see that the RTO MPC strategy gives the best response.
5.3 Limitation of linear control
The ORC system is now subjected to very strong variations in the waste heat stream. The
waste heat profile over time is plotted in figure 5.3 together with the optimised inputs. This
could simulate an ORC system implemented in an ICE where the exhaust gases vary very
heavily over time. The ORC system will need to work out of nominal conditions and is chal-
lenged by the bandwidth of the waste heat profile.
It is already clear that the inputs do not satisfy the given constraints (figure 5.3): the ex-
pander speed exhibits sudden peaks because no allowable optimised input signal could be
found.
The response of the controlled variables is depicted in figure 5.4. Here it is clear that the
control is not able to obey the constraints subjected to the control. The superheating crosses
the upper and lower limits frequently so that no zero-superheating crossing cannot be guar-
anteed any more if the variations become even more aggressive. Also the net output power
shows peaks whenever the expander speed exhibits peaks, this is certainly not optimal.
0 500 1000 15002
2.5
3
3.5
Mheat
0 500 1000 1500
140
160
time [s]
Theat
0 200 400 600 800 1000 1200 1400 16001000
2000
3000
4000
5000
time [s]
Npp
[rpm]
Nexp
[rpm]
Figure 5.3: waste heat stream and inputs applied
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0 500 1000 15000
10
20
time [s]
Tsh [
°C]
0 500 1000 150080
100
120
140
time [s]
Tsat [
°C]
0 200 400 600 800 1000 1200 1400 16000
2
4
6
8
time [s]
Pnet [
kW
]
Figure 5.4: optimal tracking response for experiment with strongly varying waste heat source
5.4 Conclusions
Because of the constraints in the system optimal control is not guaranteed with conventional
control strategies (PI, Decoupling,...). Therefore an MPC algorithm (EPSAC approach) was
implemented that takes into account all constraints on inputs, outputs and slew rates. In the
nominal operating point the MPC controller shows good control performance, better than
the conventional control.
When strong variations in the waste heat stream are expected (e.g. exhaust of an ICE) a
linear model will not be sufficient for controller design or MPC. This non-linearity will be
modelled in the next chapter in order to use for non-linear control.
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Chapter 6
Low Order Non-linear Model
6.1 Objective and techniques
In this chapter the non-linearity and complexity of the dynamics are tackled. The aim is to
find a non-linear model that can be used in predictive control for real-time optimisation. In
chapter 3 a first attempt was made to capture the major dynamics through identification af
a 2 input 2 output MIMO system. The drawback of this technique is that it yields a linear
model and totally neglects the non-linear behaviour of the cycle that is inherently present
due to mass and energy balances of the cycle but also due to the changing fluid properties.
The goal is a low order model that is computationally efficient to be used on-line in an
optimisation process that drives the cycle for start-up, regulation and shut-down. It needs to
be used in real-time, this means that a procedure of state estimation, (iterative) prediction
and optimisation, and control must be executed within this small sampling period. This
period is between 5s to 10s.
In thermodynamic cycles each component (e.g. evaporator, pump,...) has its own dynamic
behaviour that determines the input-output response. This output on itself can be an input
of another component. This way a connection between two components is created and con-
sidering all subsystems and their connections an overall model is obtained.
The ORC consists of four or five main components: evaporator, pump, expander, condenser
(and recuperator). Taking the dynamics of all components into account would yield a rather
complex system and since we want of low order model we can select the main dynamics in the
system and ignore the fastest modes by replacing them with static equations. All dynamics
of the system are summarised in table 6.1.
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evaporator
liquid zone slow (mass inertia)
two-phase zone medium (mass and thermal interia)
gas zone (superheating) fast→static model
condenser
liquid zone slow
two-phase zone medium
gas zone (downcooling) fast→static model
pump fast→static model
expander fast→static model
Table 6.1: comparison of the timescales of the ORC components
6.2 Components
6.2.1 Heat exchangers
The most important component in the system is the evaporator as it substracts the waste
heat. The dynamics of the evaporator are caused by the thermal inertia of the metal compo-
nents and the fluid, as well as the mass inertia of the liquid phase.
Two choices for modelling are being used: finite element discretisation (FE) and the moving
boundary method (MB). The advantage of the latter is that it is a very intuitive and compu-
tational efficient, but is less robust and accurate as the FE method (Wei et al. [2008]).
43
Page 58
Subcooled Two phase Superheated
L0 L1 L2
Tw1
pev
Tw0
moutmin
hin hout
Figure 6.1: moving boundary model heat exchanger
For the MB technique the heat exchanger is divided in three zones: a liquid zone, two-phase
and a superheated zone. The first two determine the mean dynamics and for the superheated
region a static model can be used. For each zone a mass and energy balance is developed
which results in a system of differential equations assuming averaged temperatures and den-
sities. The work of Jensen [2003] is taken here as a reference for MB models. (See Appendix C)
The parameters of the evaporator [Mw, Vtot, Atot] (respectively mass of the steel heat exchange
plates, heat exchanger internal volume for the working fluid side, total heat exchange area)
can be obtained from the data-sheet. The other [Ltot, Oi, Oo, Aw, A] are implicitly determined
by the MB equations because a cylindrical geometry is assumed, the inner and outer diameters
and circumferences are equal:
44
Page 59
Di
A
Ltot
original geometry: MB geometry:
Atot Atot = LtotπDi
Vtot Vtot =LtotπD
2i
4= LtotA
Oi = πDi
Mw Aw =Mw
Ltotρw
ρw
Table 6.2: Mapping of evaporator geometry to MB geometry
The MB equations can be put in another form that is more suitable for simulation. The
obtained set of equations is therefore transformed in a non-linear state space system:
M(x)dx
dt= f(x, u) (6.1)
M(x) is the mass matrix depending on the states, f(x, u) is a non-linear vectorfunction.
Evaporator
The evaporator state space model consists of 5 states [L0, L1, p, Tw0, Tw1] and 5 inputs
[mpp, mexp, hin,dhindt
, Thf ] If the MB equations from Appendix C are rearranged to suit the
45
Page 60
state space model of equation 6.6 the following model is obtained:
m11 0 m13 0 0
m21 m22 m23 0 0
m31 m32 m33 0 0
0 0 0 m44 0
0 0 0 0 m55
L0
L1
p
Tw0
Tw1
=
f1
f2
f3
f4
f5
(6.2)
The coefficients mii and fi are given in table 6.4. And the model parameters are summarised
in table 6.3:
U heat transfer coefficient for the liquid region
(subscript denotes the region: l = liquid, tp = two-phase, g = gas)
Ush overall heat transfer coefficient for the superheated region
D diameter of MB tube cross section
Ltot total length MB tube
O circumference MB tube cross-section
A MB tube cross section area
cw specific heat capacity metal wall between waste heat and working fluid
mw mass of the metal wall
Aw cross section area metal wall
ρw density metal wall
γ mean void fraction (calculation see Appendix C)
Table 6.3: model parameters evaporator
46
Page 61
f1 mpp(hin − h′) +1
2AL0
dhindt
((h0 − hin)
∂ρ0∂h
∣∣∣∣p
− ρ0)
+OiUlL0(Tw0 − T0)
f2 mpp −1
2AL0
∂ρ0dh
∣∣∣∣p
∂hindt− mexp
f3 mpph′ − mexph
′′ − 1
2AL0
∂ρ0∂h
∣∣∣∣p
∂hindt
h′ +OiUtpL1(Tw1 − T1)
f4 OoUo(Thf − Tw0)−OiUl(Tw0 − T0)
f5 OoUo(Thf − Tw1)−OiUtp(Tw1 − T1)
m111
2A(ρ0(hin + h′)− 2ρ′h′)−A(ρ0 − ρ′)h′
m131
2AL0
(ρ0∂h′
∂p+ (hin + h′)
(∂ρ0∂p
∣∣∣∣h
+1
2
∂ρ0∂h
∣∣∣∣p
∂h′
∂p
)− 2
)−AL0
(∂ρ0∂p
∣∣∣∣h
+1
2
∂ρ0∂h
∣∣∣∣p
∂h′
∂p
)h′
m21 A(ρ′ − ρ′′) +A(ρ0 − ρ′)
m22 A(1− γ)(ρ′ − ρ′′)
m23 AL1
(γ∂ρ′′
∂p+ (1− γ)
∂ρ′
∂p
)+AL0
(∂ρ0∂p
∣∣∣∣h
+1
2
∂ρ0∂h
∣∣∣∣p
∂h′
∂p
)
m31 A(h′ρ′ − h′′ρ′′) +A(ρ0 − ρ′)h′
m32 A(1− γ)(h′ρ′ − h′′ρ′′)
m33 AL1
(γ
(ρ′′∂h′′
∂p+ h′′
∂ρ′′
∂p
)+ (1− γ)
(ρ′∂h′
∂p+ h′
∂ρ′
∂p
)− 1
)+AL0
(∂ρ0∂p
∣∣∣∣h
+1
2
∂ρ0∂h
∣∣∣∣p
∂h′
∂p
)h′
m44 cwAwρw
m55 cwAwρw
Table 6.4: The coefficients mii and fi of the evaporator state space system
The thermodynamic states [h0, h′, h′′, T0, T1, ρ0, ρ
′, ρ′′] and the derivatives of thermodynamic
states
[dρ0∂h
∣∣∣∣p
,∂ρ0∂p
∣∣∣∣h
,∂ρ′
∂p,∂ρ′′
∂p,∂ρ′′
∂h,∂ρ′′
∂h
]are calculated using the CoolProp library (see Ap-
pendix C).
47
Page 62
condenser
The same MB model can be used for the condenser, care has to be taken not to confuse the in-
puts and outputs of the system. Because the superheated region is a static model L2 is known
but the condenser output enthalpy hout is unknown, the state vector of the condenser thus be-
comes: [hout, L1, pcd, Tw0, Tw1]. The inputs vector of the condenser [mpp, mexp, L2,dL2
dt, Tcf ].
The coefficients of the mass matrix and vector function are given in table 6.5.
48
Page 63
f1 mout(h′ − hout) +
1
2A(ρ0(hout + h′)− 2ρ′h′)
dL2
dt−A(ρ0 − ρ′)h′
dL2
dt−OiUlL0(T0 − Tw0)
f2 min − mout +A(ρ′ − ρ′′)dL2
dt+A(ρ0 − ρ′)
dL2
dt
f3 minh′′ − mouth
′ +A(h′ρ′ − h′′ρ′′)dL2
dt+A(ρ0 − ρ′)h′
dL2
dt−OiUtpL1(T1 − Tw1)
f4 OiUl(T0 − Tw0)−OoUo(Tw0 − Tcf )
f5 OiUtp(T1 − Tw1)−OoUo(Tw1 − Tcf )
m11 −1
2AL0
((h0 − hin)
∂ρ0∂h
∣∣∣∣p
− ρ0)
m12 −1
2A(ρ0(hout + h′)− 2ρ′h′) +A(ρ0 − ρ′)h′
m131
2AL0
(ρ0∂h′
∂p+ (hout + h′)
(∂ρ0∂p
∣∣∣∣h
+1
2
∂ρ0∂h
∣∣∣∣p
∂h′
∂p
)− 2
)−AL0
(∂ρ0∂p
∣∣∣∣h
+1
2
∂ρ0∂h
∣∣∣∣p
∂h′
∂p
)h′
m211
2AL0
∂ρ0∂h
∣∣∣∣p
m22 A(1− γ)(ρ′ − ρ′′)−A(ρ′ − ρ′′)−A(ρ0 − ρ′)
m23 AL1
(γ∂ρ′′
∂p+ (1− γ)
∂ρ′
∂p
)+AL0
(∂ρ0∂p
∣∣∣∣h
+1
2
∂ρ0∂h
∣∣∣∣p
∂h′
∂p
)
m311
2AL0
∂ρ0∂h
∣∣∣∣p
h′
m32 A(1− γ)(h′ρ′ − h′′ρ′′)−A(h′ρ′ − h′′ρ′′)−A(ρ0 − ρ′)h′
m33 AL1
(γ
(ρ′′∂h′′
∂p+ h′′
∂ρ′′
∂p
)+ (1− γ)
(ρ′∂h′
∂p+ h′
∂ρ′
∂p
)− 1
)+AL0
(∂ρ0∂p
∣∣∣∣h
+1
2
∂ρ0∂h
∣∣∣∣p
∂h′
∂p
)h′
m44 cwAwρw
m55 cwAwρw
Table 6.5: The coefficients mii and fi of the condenser state space system
49
Page 64
6.2.2 Pump & expander
For the pump a static model is used. The pump speed directly imposes the mass flow rate,
given the nominal conditions:mpp
mnom=
Npp
Nnom,pp(6.3)
The output enthalpy is calculated as:
hpp,out = hcd,out +pev − pcdρcd,outεpp
(6.4)
The isentropic efficiency εpp is an important parameter for a pump that is depending on
the working conditions. In most cases an empirical black box model is used. Here εs was
determined as a function of the pressure ratio rp and the speed of the pump Npp, yielding the
mapping depicted in figure 6.2. A maximum of εs = 0.22 is encoutered for Npp = 2600rpm
and rp = 6.75. The efficiency shows a strong relation with both pump speed and pressure
ratio.
1000
2000
3000
246810120
0.05
0.1
0.15
0.2
0.25
Nexp
[rpm]rp
εs
0.0
3
0.03
0.03
0.030
.06
0.06
0.06
0.06
0.09
0.0
9
0.09
0.09
0.12
0.1
2
0.12
0.15
0.1
5
0.15
0.18
0.1
8
0.21
0.21
Npp
r p
1000 2000 3000
2
4
6
8
10
12
Figure 6.2: isentropic efficiency pump as a function of pressure ratio and pump speed
Also for the expander a static model is used. But because of the importance of this component
on the power generation a more detailed model is used. This model is taken from Lemort and
Quoilin [2009] and is the same used in the computer models from Modelica (see Appendix
A).
Also here a the model parameters cannot be used assumed to be constant. A mapping of
the isentropic efficiency is plotted in figure 6.3. The efficiency is strongly related to the
pressure ratio, but only very little related to the expander speed. A maximum of εs = 0.55 is
encountered for rp = 7.2 and Nexp = 3500rpm.
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Page 65
24
68
1012
20003000
4000
0
0.2
0.4
0.6
0.8
rp
Nexp
[rpm]
εs
0.0
80.0
80.0
80.1
60.1
60.1
60.2
40.2
40.2
4
0.3
2
0.3
20.3
20.4
0.4
0.4
0.4
0.4
8
0.4
8
0.48
0.4
8
0.4
8
rp
rpm
2 4 6 8 10 122000
2500
3000
3500
4000
4500
Figure 6.3: isentropic efficiency expander as a function of pressure ratio and expander speed
The mass flow rate of the expander is not only related to its speed (as was assumed for the
pump) because of the compressability of the gas. A mapping of the predicted mass flow rate
as function of the supply pressure and expander speed is given in figure 6.4. (Here a condenser
pressure of 2bar is assumed).
510
1520
25
2000
3000
4000
−0.5
0
0.5
1
1.5
psu
[bar]Nexp
[rpm]
mpre
d
0
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.9
0.9
1
psu
[bar]
rpm
5 10 15 20 252000
2500
3000
3500
4000
4500
Figure 6.4: predicted mass flow rate expander as a function of supply pressure and expander speed
51
Page 66
6.3 Interconnection subsystems & simulation
The four models obtained above must be interconnected to obtain the whole cycle. To improve
calculation time the dynamic models of evaporator and condenser are combined:
Mev 0
0 Mcd
dxevdt
dxcddt
=
fev
fcd
(6.5)
This way a state space system is obtained of the following structure:
M(x, TS)dx
dt= f(x, TS, u) (6.6)
Here TS denotes the thermodynamic states that are calculated with CoolProp. They are
assumed constant during one iteration and then updated before a new iteration is executed.
The simulation scheme is presented below:
sample
xev(t) xcd(t)
inputs
Npp Nexp
pump expander
ambient
Thf Tcf
CoolProp
TS(t)
M(x, u)dx
dt= f(x, u)
calculate outputs
xev(t+ Ts) xcd(t+ Ts)
∆Tsh ∆Tsub
store data
Figure 6.5: Simulation scheme
52
Page 67
6.3.1 Initial conditions & simulation parameters
When simulating the cycle care has to be taken with initial conditions of the states and
the thermodynamic states. Important is that the cycle operating conditions depend on the
amount of organic fluid in the cycle. The densities of the fluid have to be taken accordingly
when initialising a simulation:
Mtot =∑
i
(ρiVi) (6.7)
In practise the initialisation of a moving boundary model comes to choosing good guesses of
the initial lengths of the regions and the pressures in the system.
The simulations were performed using a stepsize of 0.1s so that the fluid properties are
updated each sample. This was needed for the simulation to have numerical convergence. The
simulation took about 400s in order to simulate 1000s (in open loop). The model is therefore
too slow to be used in real time unless this can be improved significantly. (A significant
improvement was made by including the CoolProp library inside the dynamic system: more
information in next chapter)
6.3.2 Simulation
The model was fitted to the real set-up of the ORCnext project. The following parameters
were used:
pump expander
mnom 0.22 rν 4
Npp,nom 1800
53
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evaporator condenser
Ul 2000 Ul 1000
Utp 2000 Utp 3000
Ug 2000 Ug 1000
Uo 850 Uo 1000
Vtot 0.0378 Vtot 0.0378
Atot 16.18 Atot 16.18
cw 500 cw 500
Mw 69 mw 69
Table 6.6: simulation model parameters
Using these parameters a step response in the nominal working point is carried out for both
pump and expander speed. This was compared to the last validated simulation models from
Universite de Liege (ULg). The result is plotted in figure 6.6.
54
Page 69
0 10 20 30 40 50 60 70 80 90 100−5
−4
−3
−2
−1
0
time [s]
model ULg
low order model
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
time [s]
model ULg
low order model
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
time [s]
model ULg
low order model
0 10 20 30 40 50 60 70 80 90 100−0.8
−0.6
−0.4
−0.2
0
0.2
time [s]
model ULg
low order model
Figure 6.6: Comparison step response low order model and simulation model
(left: pump step - right: expander step - upper: superheating - below: saturation tem-
perature)
For the pump the dynamic response corresponds very good to that of the simulation models.
Both superheating and saturation temperature fit well. It must be noted that the steady
state point at t=0 has been substracted from the stepresponse because of the non-zero steady
state error. However the error can be removed by adjusting the model parameters.
The response of the expander speed shows a big error in gain. The model is not valid for
changing expander speed and must be improved.
6.4 State estimation: Extended Kalman Filter
The obtained low order nonlinear model is a state space system with 10 states [L0, L1, pev, Tw0, Tw1|hout, L1, pcd, Tw0, Tw1]. With the first 5 the evaporator states and the last 5 the condenser
states. In a real set-up not all of these states are measurable, usually there are pressure and
temperature measurements at the inlet and outlet of each component.
For an ORC it is important to control pressures, superheating and subcooling. These 4 values
can be determined from the measurements. Based on those 4 values, the other states have to
be estimated in real time. Because of the non-linearity a discrete time extended kalman filter
(EFK) is advised:
55
Page 70
6.4.1 Linearised dynamics
The EFK principle is that the system is linearised around the current operating point:
M(xss + ∆x)d(xss + ∆x)
dt= f(xss + ∆x, uss + ∆u) (6.8)
m
M(xss)d∆x
dt=∂f
∂x
∣∣∣∣xss,uss
∆x+∂f
∂u
∣∣∣∣xss,uss
∆u (6.9)
Also for the outputs:
yk + ∆yk = h(xss + ∆x, uss + ∆u) (6.10)
m
∆yk =∂h
∂x
∣∣∣∣xss,uss
∆x+∂h
∂u
∣∣∣∣xss,uss
∆u (6.11)
Doing this for the evaporator we obtain:
∂fev∂x
=
f11 0 f13 f14 0
0 f22 0 0 0
0 f32 f33 0 f35
0 0 0 f44 0
0 0 f53 0 f55
(6.12)
56
Page 71
f111
2Adhindt
(h′ − hin
2
∂ρ0∂h
∣∣∣∣p
− ρ0)
+OiUl(Tw0 − T0)
f13 −mpp∂h′
∂p+
1
4Adhindt
(∂h′
∂p
∂ρ0∂h
∣∣∣∣p
)
f14 OiUlL0
f221
2A∂ρ0∂h
∣∣∣∣p
dhindt
f311
2A∂ρ0∂h
∣∣∣∣p
dhindt
h′
f32 UtpOi(Tw1 − T1)
f33 mpp∂h′
∂p− mexp
∂h′′
∂p− 1
2A∂ρ0∂h
∣∣∣∣p
dhindt
∂h′
∂p−OiUtpL1
∂T1∂p
f35 OiUtpL1
f44 −OoUo −OiUl
f53 OiUtp∂T1∂p
f55 −OoUo −OiUtp
Table 6.7: The coefficients fi of the linearised evaporator system
And for the condenser:
∂fcd∂x
=
f11 f12 f13 f14 0
0 0 f23 0 0
0 f32 f33 0 f55
0 0 0 f44 0
0 0 f53 0 f55
(6.13)
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Page 72
f11 −mpp +1
2Aρ0
dL2
dt
f12 OiUl(T0 − Tw0)
f13 mpp∂h′
∂p+
1
2AdL2
dt
(ρ0∂h′
∂p− 2
∂ρ′h′
∂p
)+A
∂ρ′h′
∂p
dL2
dt
f14 OiUlL0
f23 AdL2
dt
(∂ρ′
∂p− ∂ρ′′
∂p
)−AdL2
dt
∂ρ′
∂p
f32 −OiUtp(T1 − Tw1)
f33 mexp∂h′′
∂p− mpp
∂h′
∂p+A
dL2
dt
(∂ρ′h′
∂p− ∂ρ′′h′′
∂p
)−AdL2
dt
∂ρ′h′
∂p+A
dL2
dtρ0∂h′
∂p−OiUtpL1
∂T1∂p
f35 OiUtpL1
f44 −OiUl −OoUo
f53 OiUtp∂T1∂p
f55 −OiUtp −OoUo
Table 6.8: The coefficients fi of the linearised condenser system
Now only the output function yk = h(xk) needs to be linearised:
pev
∆Tsh
pcd
∆Tsub
= yk =
pev
(Thf − Tsat,ev)[1− exp
(−OiUsh(Ltot − L0 − L1
cpmexp
)]
pcd
(Tsat,cd − Tcf )
[1− exp
(−OiUsub(Ltot − L1 − L2
cpmpp
)]
(6.14)
So that Hk is obtained:
Hk =
0 0 1 0 0 0 0 0 0 0
h1 h1 h2 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0
0 0 1 0 0 0 h3 h4 0 0
(6.15)
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Page 73
h1 −(Thf − Tsat,ev)exp(−OiUsh(Ltot − L0 − L1
cpmexp
)(OiUshcpmexp
)
h2 −[1− exp
(−OiUsh(Ltot − L0 − L1
cpmexp
)]∂Tsat,ev∂p
h3 −(Tsat,cd − Tcf )exp
(−OiUsub(Ltot − L1 − L2
cpmpp
)(OiUsubcpmpp
)
h4
[1− exp
(−OiUsub(Ltot − L1 − L2
cpmpp
)]∂Tsat,cd∂p
Table 6.9: The coefficients hi of the linearised outputfunction
Combining the linear dynamics of evaporator and condenser:
F = M(xk|k−1)−1
∂fev∂x
0
0∂fcd∂x
(6.16)
Hk =∂h
∂x(6.17)
6.4.2 EKF equations
EKF
u(t−Ts)
ym(t) x(t)
Figure 6.7: EKF state estimation
Then the EFK filter equations are split in two steps, first when at tk a measurement is taken,
a prediction step is performed using the non-linear model:
˙x = f(x(t), u(t)) (6.18)
P (t) = F (t)P (t) + P (t)F (t)T +Q(t) (6.19)
Solving this system of differential equations starting from xk−1|k−1 and Pk−1|k−1 gives yields
xk|k−1 and Pk|k−1. So based on the previous (estimated) states and the model a prediction of
the new state and prediction of the error covariance matrix P is obtained. The update step
59
Page 74
then takes into account the measured outputs:
Kk = Pk|k−1HTk (HkPk|k−1H
Tk +Rk)
−1 (6.20)
xk|k = xk|k−1 +Kk(yk − h(xk|k−1)) (6.21)
Pk|k = (I−KkHk)Pk|k−1 (6.22)
This then yields the suboptimal state estimation (EFK does not guarantee optimal estima-
tion, unlike the linear kalman filter).
Obtaining the covariance matrices Q of the process noise and R of the measurement noise and
an initial choice P0|0 is a tricky thing and will affect the performance of the filter. However
a systematic procedure has been suggested in Schneider and Georgakis [2013]. The measure-
ment noise is taken from the device data-sheet. As mentioned in chapter 1 the used pressure
sensors have a mean error of 0.04bar and temperature sensors are accurate to 0.01 degrees.
This then yields the R matrix:
R = k
[40002 0.42
0.42 40002
](6.23)
Where k is a tuning parameter that determines how hard the estimate will rely on the mea-
surement. The covariance for subcooling and superheating of 0.4 is dominated by the pressure
measurement that calculates the saturation temperature. The suggestion for the error covari-
ance matrix uses the expected bounds on the states:
P0|0 = diag
[1
2(x0 − x0)T (xu − x0)
](6.24)
≈ 1
2
(xu − xl)T2
(xu − xl)2
(6.25)
=1
8diag(102, 102, 10000002, 1002, 1002, 1000002, 102, 2000002, 252, 252) (6.26)
In order to determine Q the process noise needs to be known. How much does the dynamic
model depends on its parameters to represent the real system?. A suggestion is given by the
authors that uses real data. A real plant is not considered here and thus Q is here defined
intuitively:
Q = diag(.01, .01, 100002, 10−4, 10−4, 10002, .01, 10002, 10−5, 10−5) (6.27)
6.4.3 Performance filter with artificial noise
To test the EKF estimator, the system is subjected to artificial noise that could in a real
set-up originate from sensor noise. As mentioned before pressure measurements are accurate
to 0.04bar and superheating can be calculated accurate to 0.4 degrees.
The covariance matrices derived above are used where a k factor of 5 is being used for R.
The result of this experiment where a pump step is applied is depicted in figure 6.8.
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Page 75
100 120 140 160 180 2007.4
7.6
7.8
8
8.2
8.4
8.6x 10
5
Plant pressure
EFK pressure estimation
measured pressure
105 110 115 120 125
7.95
8
8.05
8.1
8.15
8.2
x 105
Plant pressure
EFK pressure estimation
measured pressure
Figure 6.8: filter performance during a step response with artificial noisy measurement (k=5)
The performance of the filter does still depend strongly on the measurements, this can di-
minished by tuning the parameter k. By enlarging this parameter to 10 the following result
is obtained:
100 120 140 160 180 2007.4
7.6
7.8
8
8.2
8.4
8.6x 10
5
Plant pressure
EFK pressure estimation
measured pressure
115 120 125 130 135
8
8.05
8.1
8.15
8.2
8.25
x 105
Plant pressure
EFK pressure estimation
measured pressure
Figure 6.9: filter performance for k=10
It is clear that the filter behaves better with the enlarged k. The measurements are less
determining for the estimated states.
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Chapter 7
Nonlinear Model Predictive Control
The low order non-linear model build in the previous chapter is intended to be used in an
NMPC scheme on a real plant. Since the model is not yet able to fully predict the plant
outputs (see section 6.3.2) an implementation on a real plant is not yet possible. In this
chapter a NEPSAC controller is build for the low order model. The model is here used both
as model and plant where artificial noise has been added. This will then show the performance
of the EKF together with an NMPC controller.
7.1 NEPSAC algorithm
The working principle of the EPSAC algorithm has already been explained in chapter 5 and
Appendix C. The NEPSAC algorithm is just an extension of this, where the non-linear model
is used in stead of a linear one to obtain the G-matrix for prediction.
Because the superposition principle doesn’t hold for a non-linear model, the obtained optimal
input after optimisation cannot be just superposed on the base input. But the NEPSAC
algorithm does assume this is a good guess and applies this suboptimal input. The optimisa-
tion is then performed again with the new base input Ubase(k) = Ubase(k − 1) + Uopt and an
iteration procedure is done until a certain threshold. (maximum number of iterations reached,
or Uopt → 0). More information can be found in appendix C.
The NEPSAC parameters are summarised in table 7.1, and are the same for both pump and
expander and for both outputs superheating and evaporator pressure.
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N1 1
Nu 1
N2 10 (=50s)
itermax 1
Uopt,min 10rpm
Ustep 200rpm (G-matrix)
Table 7.1: NEPSAC parameters
Because of the slow simulation time no iterations are done (itermax = 1). This approach has
been used successfully in refrigeration systems by Leducq et al. [2006] in order to decrease
the computation time.
7.2 Simulation
Because of the slow simulation time an attempt to increase the computation speed was ob-
tained by including the calculation of thermodynamic states in the dynamic model. During
one sample the thermodynamic states are updated, whereas in previous simulations they were
assumed to be constant during one sample. This way the sampling time could be increased
to 5s. The following simulation scheme then results:
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sample
xev(t) xcd(t)
inputs
Npp Nexp
pump expander
ambient
Thf Tcf
M(x, u)dx
dt= f(x, u)
CoolProp
TS
calculate outputs
xev(t+ Ts) xcd(t+ Ts)
∆Tsh ∆Tsub
store data
Figure 7.1: Simulation scheme with CoolProp inside dynamic system
The overall control scheme includes the extended kalman filter, the NEPSAC controller and
the optimiser:
NEPSAC Systemu y
sensorsEKF ym
x
optimiser
d
r
Figure 7.2: Full NMPC control scheme
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7.2.1 Experiment with artificial noise
The plant is the exact copy of the model and is controlled with the NEPSAC controller, track-
ing optimised saturation temperatures. The plant is subjected to noise on the measurements
ym. The measured outputs are [pev ∆Tsh pcd ∆Tsub].
The EKF is designed with the earlier used Q, R and P0|0 and a k-factor of 1 is being used here.
The low value of k is due to the improvement made with including the CoolProp calculations
inside the dynamic model. The following optimisation problem results:
min J =
N2∑
k=N1
[Tsat,opt(t+ k|t)− Tsat(t+ k|t)]2 (7.1)
subjected to:
Y 1 = Y1,base +G1Uopt (7.2)
M(x, u)dx
dt= f(x, u) (7.3)
Umin < Uopt < Umax
|Uk − Uk−1| < 4Umax
∆Tmin < ∆Tsh < ∆Tmax (7.4)
A maximum (60 degrees) and minimum value (35 degrees) for superheating is included as
a constraint. Because the model was not yet able to predict superheating correctly, mainly
caused by the modelling choice of a not decreasing waste heat temperature through the
evaporator, no tracking of superheating is implemented. Also the over-sizing of the heat
exchangers would cause a superheat and pressure controller to fail. This effect was already
encountered in chapter 4.
The slew rate constraints are 500rpm, the prediction horizon N2 is 10 samples which means
that we take the average rise time (=50s) of the controlled variables to be equal to the
prediction horizon. Intuitively we can expect that the control should be able to work with
this horizon.
The waste heat experiment is plotted below:
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300 400 500 600 700 800 900 1000105
110
115
120
125
130
135
140
145
time [s]
Thf
[°C]
Figure 7.3: Strongly varying waste heat
The result of the control is depicted in figures 7.4 and 7.5.
300 400 500 600 700 800 900 10001000
1500
2000
2500
3000
3500
4000
4500
time [s]
N
pp [rpm] N
exp [rpm]
Figure 7.4: pump speed and expander speed
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300 400 500 600 700 800 900 1000
6
8
10
Pev [bar] Ref
300 400 500 600 700 800 900 100030
40
50
60
time [s]
Tsh [°C]
Figure 7.5: Tracking response saturation temperate and superheating
The NEPSAC controller is able to track the optimal evaporator pressure (or equivalent:
saturation temperature) taking into account the very strong variations over time (the waste
heat has about 3 peak values in every 1.5 minutes). The controller satisfies the constraints of
pump and expander speed.
Now that the full control loop has been tested it is interesting to know the calculation time
needed. The whole simulation of 1000s took about 1400s. This is still too slow. The needed
calculation time per sample is shown in picture 7.6.
60 80 100 120 140 160 1807
7.5
8
8.5
9
9.5
sample
co
mp
uta
tio
n t
ime
[s]
Figure 7.6: calculation time per sample in seconds for the overall control system
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It is clear that one sample on average takes about 8s where the maximum should be 5. To
improve control and allow for an iterative NEPSAC control system this should be decreased
even more to the order of 1s.
7.2.2 Decreasing computation time
For the controller to work on the real system the computation time per sample must be below
5s. Therefore the NMPC parameters can be retuned, especially the prediction horizon N2
determines the needed computation time per sample. N2 is lowered to 5 samples. The waste
heat stream is plotted below:
300 400 500 600 700 800 900 1000105
110
115
120
125
130
135
140
145
time [s]
Thf
[°C]
Figure 7.7: Strongly varying waste heat
The result of the control is depicted in figures 7.8 and 7.9.
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300 400 500 600 700 800 900 10001000
1500
2000
2500
3000
3500
4000
4500
time [s]
Nexp
[rpm] Npp
[rpm]
Figure 7.8: pump speed and expander speed
300 400 500 600 700 800 900 10004
6
8
10
12
Pev [bar] Ref
300 400 500 600 700 800 900 100030
40
50
60
time [s]
Tsh [°C]
Figure 7.9: Tracking response saturation temperate and superheating
The NEPSAC controller is still able to track the optimal evaporator pressure, but has become
more aggressive. The whole simulation of 1000s took about 850s. The needed calculation
time per sample is shown in picture 7.6.
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60 80 100 120 140 160 1804.2
4.4
4.6
4.8
5
5.2
sample
co
mp
uta
tio
n t
ime
[s]
Figure 7.10: calculation time per sample in seconds for the overall control system
We see that, neglecting some outliers, in general the computation time is below 5s. This was
the final goal and it is shown that it is possible.
7.3 Conclusion
A NEPSAC controller is designed and tested on the model itself. The real-time steady state
optimiser could be combined with a non-linear MPC controller giving satisfactory results that
were able to track very quick variations in time of the waste heat. A superheat control has
not been implemented yet due to the poor results on superheat modelling.
It was shown that a non-linear MPC can be used to control the cycle using moving boundary
models of the heat exchangers. A further decrease of the computation time could be able to
improve the control even more, such that an iterative NEPSAC can be used.
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Chapter 8
Conclusions
8.1 Discussion results
Based on previous work regarding the dynamic modelling or ORC cycles, the problem of
optimal control was addressed as this has not been investigated deeply. The method of
RTO was used by designing a steady state optimiser that calculates the optimal saturation
temperature based on the conditions of the waste heat.
Two control strategies have been implemented and compared. A conventional decentralised PI
and decoupled PI were designed based on identified transfer functions and their performance
was tested. For working in nominal conditions these controllers performed sufficient and
decoupling improves the control.
Since the ORC system has to be regulated obeying input and output constraints a linear
MPC strategy (EPSAC) was implemented successfully together with the RTO. Especially for
optimal control the MPC approach was preferable as it could satisfy even output constraints
(e.g. superheating limits).
For strongly varying waste heat systems such as an ICE exhaust, linear control is likely to fail
as it does guarantee to obey the safety constraints on superheating. In the end of this work
a first attempt is made for using NMPC (NEPSAC). A non-linear model and a non-linear
state estimator were build successfully. It was shown that it could be used to control the
evaporator pressure.
8.2 Future work
The aim of this thesis was to develop a deeper understanding in the needed optimisation
and control of organic rankine cycles. However all tests were performed using ideal computer
models and without considering a real set-up. Up to now only very few research has been
done on the control of ORCs in a real set-up.
In particular the model based controllers should be tested because their performance is
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strongly related to the validity of the model, which cannot be trustworthy until tested in
real experiments.
In this work a first attempt to use NMPC was carried out. Although the road been paved by
the development of a non-linear model and extended kalman filter for state estimation still
a lot of work needs to be done. Their is a need for model improvement as the model was
not fully able to guess steady state values but its dynamic behaviour did approach the real
dynamics. The model can be extended for start-up and shut-down procedures by the use of
switched MB models. A good reference can be found in Li and Alleyne [2009].
Also the EKF will need retuning for a real set-up. The process noise should be calibrated
to obtain correct values for the matrix Q. As mentioned a procedure has been suggested by
Schneider and Georgakis [2013].
A suggestion that has not been tested yet is to use adaptive linearisation. The non-linear
model developed can be difficult to implement for NMPC. However linearised dynamics have
been developed so that they can be adapted to the estimated state (EKF) at every instant.
This approach has been suggested in refrigeration systems and can be promising. This opens
new doors to use the advantages of linear control.
The optimiser developed in chapter 2 seemed inadequate for badly designed ORC set-ups
as was the case in the work. This is due to the fact that it does not take into account the
components geometry and model parameters. To have more accurate results that can be used
for real-time optimisation this need to be included in the steady state model.
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Appendix A: Simulation Model
The simulation model being used in this work will be explained in this section. The model
is developed by Sylvain Quoilin at the university of Liege and for detailed reports on the
dynamic modelling of the ORC cycle, the reader is referred to Quoilin et al. [2011a] and other
technical reports of the author. The aim here is to give the reader enough information and
insight in the physics of the ORC cycle and to adress the limitations of this model.
The models are created in modelica and fluid properties are computed with CoolProp.
Heat exchangers
The evaporator and possibly recuperator are modelled as discretised in (N-1) cells.
Ai =A
N − 1Vi =
V
N − 1Mi =
M
N − 1(8.1)
For each cell a mass and energy balance equation is used. Momentum balances are neglected,
which leads to constant pressure in the whole evaporator:
dMi
dt= Mi − Mi+1 (8.2)
Viρidhidt
+ ¯Mi(hi+1 − hi) = ¯Qi +Aidp
dt(8.3)
(8.4)
The heat exchange in the metal wall between hot fluid, working fluid and the metal wall in
between:
cwMwdTw,idt
= A(qwf,i + qhf,i) (8.5)
qwf,i = Uf,i(Twf,i − Tw,i) (8.6)
qhf,i = Uhf,i(Twf,i − Tw,i) (8.7)
The condenser model is restricted to constant cooling water mass flow rate and temperature.
A dynamic model for the condenser is not used. In stead a constant pinch point and subcooling
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are determined and with these the condensing temperature is calculated.
It should be noted that the accuracy of the model is depending a lot on the number of nodes
N .
Expander
A steady state model is assumed because of the fast dynamics. The expansion is divided in
two steps: isentropic expansion and constant volume expansion.
isentropic expansion (supply to internal condition):
w1 = hsu − hin (8.8)
constant volume expansion (internal condition to expander outlet):
w2 = νin(pin − pout) (8.9)
then the work in the expander and the exit enthalpy is computed as:
Wexp = m(w1 + w2)ηmech (8.10)
hex = hsu −Wexp
M(8.11)
The mass flow rate is computed as:
m =FFρsuVsNexp
60(8.12)
Pump model
As with the expander a steady state model is assumed for the pump.
The internal isentropic efficiency is defined by:
εs,pp =νsu,pp(pout,pp − psu,pp)
hout,pp − hsu,pp(8.13)
and calculated using the empirical law:
εs,pp = a0 + a1log(Xpp) + a2log(Xpp)2 + a3log(Xpp)
3 (8.14)
Receiver
The liquid level in the receiver is calculated as:
dφ
dt=
1
ρ′Vtank(mout,cd − msu,pp) (8.15)
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Cycle model
The net generated power, cycle efficiency and overall efficiency is calculated by:
Wnet =
∫ t2
t1
Wexp − Wppdt = ηcycleQev (8.16)
Qev =
∫ t2
t1
Qevdt (8.17)
= ηhr
∫ t2
t1
mhf (hsu,hf − hhf,amb)dt (8.18)
Then the overall efficiency is defined as:
ηoverall = ηcycleηhr (8.19)
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Appendix B: EPSAC Algorithm
This appendix explains the idea and implementation behind the EPSAC and NEPSAC control
algorithm for MIMO systems as described in De Keyser [2003].
EPSAC
prediction In EPSAC the following discrete time linear model is used:
y(t) = x(t) + n(t) (8.20)
x(t) = f(x(t− 1), x(t− 2), ..., u(t− 1), u(t− 2), ...) (8.21)
n(t) =C(q−1)
D(q−1)e(t) (8.22)
Here y represent the measured output vector, x the model output vector, u the input vector
and n the output noise vector. The model output is thus assumed to be perturbed by noise
that originates from zero mean white noise e. The operator q−1 represents the backward shift
operator also known as z−1
The key idea behind MPC is the prediction and optimisation process. We are therefore
interested in the outcome of y(t+ k). A prediction is thus carried out at time t:
y(t+ k|t) = x(t+ k|t) + n(t+ k|t) (8.23)
x(t+ k|t) = f(x(t− 1), x(t− 2), ..., u(t+ k − 1), u(t+ k − 2), ...u(t− 1), u(t− 2), ...) (8.24)
The prediction of the output is a function of the future noise and input signals. The future
inputs are calculated in the optimisation process (discussed later). The future noise is cal-
culated using the disturbance model 8.22. If no such model is available a default filter is:
C(q−1)
D(q−1)=
1
1− q−1 (8.25)
This filter takes the last available noise measurement and adds the white noise e at time t.
For the prediction of n(t + k|t), the future white noise e(t + k|t) is needed. Because of it’s
zero mean, the best prediction is zero. The predicted output noise is then calculated using
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stored values of the past output noise and white noise:
n(t+ k|t) = f(n(t− 1), n(t− 2), ..., e(t+ k − 1), e(t+ k − 2), ..., e(t− 1), e(t− 2), ...) (8.26)
e(t+ i) = 0 i = 0...k − 1
For the default filter the prediction is just the last measured noise value:
n(t+ k|t) = n(t|t) (8.27)
optimisation a 2 inputs 2 outputs system is assumed here. The predicted output is seen
as the superposition of two effects, a base response and an optimised response:
y1(t+ k|t) = y1,base(t+ k|t) + y1,opt(t+ k|t) (8.28)
y2(t+ k|t) = y2,base(t+ k|t) + y2,opt(t+ k|t) (8.29)
for k = N1...N2
The MPC parameters N1, N2 define the prediction interval, N2 is the prediction horizon. The
base response is the effect of past control inputs and predicted output noise and calculated
as described above. The optimised response is the effect of the optimising control inputs
u1,opt(t+ k|t) and u2,opt(t+ k|t). In concatenated form the prediction vector is calculated as:
Y 1 = Y1,base + Y1,opt = Y1,base +G11u1,opt +G12u2,opt
Y 2 = Y2,base + Y2,opt = Y2,base +G21u1,opt +G22u2,opt (8.30)
Writing out fully for output 1 this becomes:
y1,base(t+N1|t)
...
y1,base(t+N2|t)
=
y1,opt(t+N1|t)
...
y1,opt(t+N2|t)
+
g11,N1−Nu+1
...
g11,N2−Nu+1
u1,opt(t|t) +
g12,N1−Nu+1
...
g12,N2−Nu+1
u2,opt(t|t)
It is here assumed that every sample uopt is kept constant over the prediction interval. This
can be extended to non-constant optimising input over a so-called control horizon Nu but is
not done here. The vectors Gij are the step responses from input i to output j
The optimisation is done by minimisation of a cost function:
min J =
N2∑
k=N1
[r1(t+ k|t)− y1(t+ k|t)]2 +
N2∑
k=N1
[r2(t+ k|t)− y2(t+ k|t)]2 (8.31)
The next step is to determine the optimising inputs in order to minimise the cost function
without violating the given constraints. Using G1 = [G11 G12], G2 = [G21 G22] and
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Uopt = [u1,opt u2,opt] the problem can be summarised to:
min J = (R1 − Y1)T (R1 − Y1) + (R2 − Y2)T (R2 − Y2) (8.32)
subjected to: (8.33)
Y 1 = Y1,base +G1Uopt (8.34)
Y 2 = Y2,base +G2Uopt (8.35)
Umin < Uopt < Umax
|Uk − Uk−1| < 4UmaxYmin < Y < Ymax
Hard input and output constraints are introduced as well as a slew rate limitation. The
last step is to transform the problem such that it is compatible with constraint optimisation
software packages. In this report the matlab function quadprog is used. This package asks
for input arguments in following form:
min J = UToptHUopt + 2fTUopt (8.36)
st : AUopt = b
CUopt < d
By substitution of the equality constraints into the cost function we find:
H = GT1G1 +GT2G2
f = −GT1 (R1 − Y1,base)−GT2 (R2 − Y2,base)
Secondly we transform the 3 inequality constraints to be a function of only Uopt:
input constraints [1
−1
]Uopt <
[Umax − UbaseUbase − Umin
](8.37)
slew rate constraint
−4Umax < Ubase + Uopt − U(t− 1) < 4Umax (8.38)
This can be rewritten as:[
1
−1
]Uopt <
[4Umax + U(t− 1)− Ubase4Umax − U(t− 1) + Ubase
](8.39)
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output constraint Transforming the output constraint by substituting the equality con-
straint yields: [G
−G
]Uopt <
[Ymax − YbaseYbase − Ymin
](8.40)
The matrix C and vector d are now obtained from (13), (15) and (16):
C =
C1
C2
C3
d =
d1
d2
d3
(8.41)
It must be noted that using cost function 8.31 is known as solidary control because both
inputs together try to minimise the cost J .
NEPSAC
The EPSAC method described above can be extended to non-linear systems as well. Assum-
ing a non-linear model is available, the step response of the non linear system is calculated by
simulating the system for u = Ubase and u = Ubase + Ustep. The time response of the system
is found by subtracting both responses and dividing by the step amplitude. This gives the
desired step response: gk =xk,step − xk,base
Ustep
NEPSAC is able to handle non linear models, where relation 8.30 does not hold true due to
the non-linearity. This problem is solved by an iteration scheme:
1. calculate Ubase and Ybase = Xbase +N from past control and noise prediction
2. calculate the optimising input U (as described in section 3)
3. now update Ubase = Ubase + U
4. if U < Ustop stop the iterations
The solutions will eventually converge towards the non linear optimal solution. Important
remark is that the step response G has to be calculated every iteration again to have a correct
linearised approximation.
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Appendix C: Moving boundary
model and the CoolProp library
Moving boundary model
In this section the mass and energy balance equations of a moving boundary heat exchanger
model is presented. The model is based on the work of Jensen [2003]. A heat exchanger is
divided in three zones: liquid zone (0), two-phase (1) and superheated (2). Because we can
neglect the fast gas dynamics, the superheated region is just a static model. In each zone an
average temperature, density and enthalpy are assumed.
Liquid zone
The mass balance becomes:
A
[(ρ0 − ρ′)
dL0
dt+ L0
(∂ρ0∂p
∣∣∣∣h
+1
2
∂ρ0∂h
∣∣∣∣p
h′dp
)dp
dt+
1
2L0∂ρ0∂h
∣∣∣∣p
dhindt
]= min − mout (8.42)
The liquid energy balance becomes:
1
2A(ρ0(hin − h′
)− 2ρ′h′
)dL0
dt
+1
2AL0
(ρ0 +
1
2
(hin + h′
) ∂ρ0∂h
∣∣∣∣p
)hindt
+1
2AL0
(ρ0dh′
dp+(hin + h′
)(∂ρ0dp
∣∣∣∣h
+1
2
∂ρ0∂h
∣∣∣∣p
dh′
dt− 2
))dp
dt
= minhin − mouth′ +OiUlL0 (Tw0 − T0)
(8.43)
The wall energy balance:
Awρwcp,wdTw0dt
= OoUo(Thf − Tw0)−OiUl(Tw0 − T0) (8.44)
Two-phase zone
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The mass balance becomes:
A
[(ρ′ − ρ′′
) dL0
dt+ (−γ)
(ρ′ − ρ′′
) dL1
dt+ L1
(γdρ′′
dp+ (1− γ)
dρ′
dp
)dp
dt
]= min−mout (8.45)
The two-phase energy balance becomes:
A(ρ′h′ − ρ′′h′′
)dL0
dt(8.46)
+A (1− γ)(ρ′h′ − ρ′′h′′
)dL1
dt(8.47)
+AL1
(γd (ρ′′h′′)
dp+ (1− γ)
d (ρ′h′)
dp− 1
)dp
dt(8.48)
= minh′ − mouth
′′ +OiUlL1 (Tw1 − T1) (8.49)
The wall energy balance:
Awρwcp,wdTw1dt
= OoUo(Thf − Tw1)−OiUl(Tw1 − T1) (8.50)
Static superheated zone (evaporator):
The superheating is calculated by integration the following energy ballance equation of an
infinitesimal length along the superheating region:
mexpcpdT = OiUsh(Thf − T )dl (8.51)
m (8.52)
∆Tsh = (Thf − T1)[1− exp
(−OiUsh(L− L0 − L1)
cpmout
)](8.53)
(8.54)
Static downcooling zone (condenser):
The length of the downcooling zone in the condenser is calculated using the same approach
as for the superheating:
L2 = −ln(Tcf − T1Tcf − Tin
)cpmexp
UshOi(8.55)
The above equations are valid for an evaporator where the inlet is at the cold liquid side. The
equations for the condenser can be derived similarly by adjusting the right hand side of the
balance equations. This has been done in section 6.2.1.
Calculating thermodynamic states using the CoolProp library
The thermodynamic states used in the model are dependant on the organic working fluid
being used:
[h0, h′, h′′, T0, T1, ρ0, ρ
′, ρ′′, γ] = f(p, hin, f luid) (8.56)
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The CoolProp library Bell et al. [2014] is a free and computational efficient choice for the
calculation of fluid properties. The code for calculation of these values in Matlab is given
below:
1 function ts = TS(X,Hcold)
2 global fluid
3
4 % extract pressure (kPa -> Pa)
5 P = X(3)/1000;
6
7 % calculate thermodynamic states using CoolProp
8 T1 = Props('T','P',P,'Q',0,fluid)-273.15; % working fluid saturation temperature
9
10 D l = Props('D','P',P,'Q',0,fluid);
11 H l = 1000*Props('H','P',P,'Q',0,fluid); % enthpaly (kJ/kg -> J/kg)
12 dHdP l = dHdP q(P,0,fluid);
13 drhodP l = dDdP q(P,0,fluid);
14
15 D g = Props('D','P',P,'Q',1,fluid);
16 H g = 1000*Props('H','P',P,'Q',1,fluid);
17 dHdP g = dHdP q(P,1,fluid);
18 drhodP g = dDdP q(P,1,fluid);
19
20 H0 = (Hcold+H l)/2;
21 D0 = Props('D','H',H0/1000,'P',P,fluid);
22 T0 = Props('T','H',H0/1000,'P',P,fluid)-273.15;
23 drhodH p = dDdH p(P,H0/1000,fluid);
24 drhodP h = dDdP h(P,H0/1000,fluid);
25
26 % calculate the average void fraction
27 mu = D g/D l;
28 gam = (1-muˆ(2/3)*(1+log(muˆ(-2/3))))/(1-muˆ(2/3))ˆ2;
29
30 ts = [T0,T1,D0,D l,D g,H0,H l,H g,...
31 dHdP l,dHdP g,drhodP h,drhodH p,drhodP l,drhodP g,gam];
32 end
Figure 8.1: Matlab code for calculating fluid properties with CoolProp
The mean void fraction γ is defined as the cross section ratio of saturated gas over the totalA′′
A.
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It can be calculated by Jensen [2003]:
µ =ρ′′
ρ′(8.57)
γ =1− µ 2
3
(1 + ln
(µ−
23
))
(1− µ 2
3
)2 (8.58)
The derivatives of thermodynamic states:
[dρ0dh
∣∣∣∣p
,dρ0dp
∣∣∣∣h
,dρ′
dp,dρ′′
dp,dρ′′
dh,dρ′′
dh
](8.59)
are not directly available in CoolProp but can be obtained. In the code above functions
drhodH p, drhodP h, dhdp q, dDdp q calculate these values, the code is given here:
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1 function y = dDdH p(P,H,fluid)
2 deltaH = 0.001;
3
4 D1 = Props('D','P',P,'H',H,fluid);
5 D2 = Props('D','P',P,'H',H+deltaH,fluid);
6
7 y = (D2-D1)/(1000*deltaH);
8 end
1 function y = dDdP h(P,H,fluid)
2 deltaP = 0.001;
3
4 D1 = Props('D','H',H,'P',P,fluid);
5 D2 = Props('D','H',H,'P',P+deltaP,fluid);
6
7 y = (D2-D1)/(1000*deltaP);
8 end
1 function y = dDdP q(P,Q,fluid)
2 deltaP = 0.001;
3
4 D1 = Props('D','Q',Q,'P',P,fluid);
5 D2 = Props('D','Q',Q,'P',P+deltaP,fluid);
6
7 y = (D2-D1)/(1000*deltaP);
8 end
1 function y = dHdP q(P,Q,fluid)
2 deltaP = 0.001; % 0.01 bar
3
4 H1 = Props('H','P',P,'Q',Q,fluid);
5 H2 = Props('H','Q',Q,'P',P+deltaP,fluid);
6
7 y = (H2-H1)/(deltaP);
8 end
1 function y = dTsatdP(P,fluid)
2 dP = 0.001;
3
4 Tsat1 = Props('T','P',P,'Q',1,fluid);
5 Tsat2 = Props('T','P',P+dP,'Q',1,fluid);
6
7 y = (Tsat2-Tsat1)/(1000*dP);
8 end
Figure 8.2: Matlab code for calculating the derivatives of fluid properties
84
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Bibliography
Ian H. Bell, Jorrit Wronski, Sylvain Quoilin, and Vincent Lemort. Pure and pseudo-pure fluid
thermophysical property evaluation and the open-source thermophysical property library
coolprop. Industrial & Engineering Chemistry Research, 53(6):2498–2508, 2014.
D W Clarke, C Mohtadi, and P.S Tuffs. Generalized Predictive Control - Part I. The Basic
Algorithm. Automatica, 1987.
Yiping Dai, Jiangfeng Wang, and Lin Gao. Parametric optimization and comparative study
of organic Rankine cycle (ORC) for low grade waste heat recovery. Energy Conversion and
Management, 50(may):576–582, 2009.
R De Keyser. Model based predictive control for linear systems. In UNESCO Encyclopaedia
of Life Support Systems http://www.eolss.net, volume 35 pages. Eolss Publishers Co Ltd,
Oxford, 2003. (available online at: http://www.eolss.net/sample-chapters/c18/e6-43-16-
01.pdf).
Matthew S. Elliott and Bryan P Rasmussen. On reducing evaporator superheat nonlinearity
with control architecture. International journal of refrigeration, 33, 2010.
H. Fallahsoho, C. Changenet, S. Place, C. Ligeret, and X. Lin-Shi. Predicive Functional
Control of an expansion valve for minimizing the superheat of an evaporator. International
journal of refrigeration, 33:409–418, 2010.
Chao He, Chao Liu, Hong Gao, Hui Xie, Yourong Li, Shuangying Wu, and Jinliang Xu. The
optimal evaporation temperature and working fluids for subcritical organic Rankine cycle.
Energy, 38:136–143, 2012.
Jakob Munch Jensen. Dynamic modelling of thermo-fluid systems. PhD thesis, Technical
University of Denmark, 2003.
Denis Leducq, Jacques Guilpart, and Gilles Trystram. Non-linear predictive control of a
vapour compression cycle. International Journal of Refrigeration, (29):761–772, 2006.
85
Page 100
Vincent Lemort and Sylvain Quoilin. Designing scroll expanders for use in heat recovery
Rankine cycles. In Proceedings of The International Conference on Compressors and their
Systems, London, September 2009.
Bin Li and Andrew G Alleyne. A Full Dynamic Model of a HVAC Vapor Compression Cycle
Interacting with a Dynamic Environment. In American Control Conference, ThC13.2, 2009.
Bo-Tau Liu, Kuo-Hsiang Chien, and Chi-Chuan Wang. Effect of working fluids on organic
Rankine cycle for waste heat recovery. Energy, 29:1207–1217, 2004.
Daniel Maraver, Javier Royo, Vincent Lemort, and Sylvain Quoilin. Systematic optimization
of subcritical and transcritical organic Rankine cycles (ORCs) constrained by technical
parameters in multiple applications. Applied Energy, (117), 2014.
N Melotte and S Quoilin. Experimental Study and Dynamic Modelling of a Waste Heat
Recovery Organic Rankine Cycle. Master’s thesis, University of Liege, 2012.
F. Padula, R. Sandrini, and G. Cominardi. Adaptive PI Control of an Organic Rankine Cycle
Power Plant. In IFAC Conference on Advances in PID Control, March 2012.
Sylvain Quoilin, Richard Aumann, Andreas Grill, Andreas Schuster, Vincent Lemort, and
Hartmut Spliethoff. Dynamic modeling and optimal control strategy of waste heat recovery
Organic Rankine Cycles. Applied Energy, 88:2183–2190, 2011a.
Sylvain Quoilin, Sebastien Declaye, Bertrand F. Tchanche, and Vincent Lemort. Thermo-
economic optimization of waste heat recovery Organic Rankine Cycles. Applied Thermal
Engineering, 31:2885–2893, 2011b.
Renee Schneider and Christos Georgakis. How To NOT Make the Extended Kalman Filter
Fail. Industrial & Engineering Chemistry Research, 2013.
D Wei, X Lu, Z Lu, and J Gu. Dynamic modeling and simulation of an Organic Rankine
Cycle (ORC) system for waste heat recovery. Applied Thermal Engineering, 28, 2008.
Takahisa Yamamoto, Tomohiko Furuhata, Norio Arai b, and Koichi Mori. Design and testing
of the Organic Rankine Cycle. Energy, 26:239–251, 2001.
J Zhang, W Zhang, G Hou, and F Fang. Dynamic modeling and multivariable control of
organic Rankine cycles in waste heat utilizing processes. Computers and Mathematics with
Applications, 64, 2012.
J Zhang, Y Zhou, R Wang, J Xu, and F Fang. Modeling and constrained multivariable
predictive control for ORC (Organic Rankine Cycle) based waste heat energy conversion
systems. Energy, 2014.
86
