Unmeasured temperature control in the presence of unknown drifting parameters in a cruise ship application

Journal of Process Control 47 (2016) 22–34

Contents lists available at ScienceDirect

Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont

Unmeasured temperature control in the presence of unknown drifting parameters in a cruise ship application Philipp Nguyen ∗ , Robert Tenno School of Electrical Engineering, Aalto University, Otaniementie 17, 02150 Espoo, Finland

a r t i c l e

i n f o

Article history: Received 25 November 2014 Received in revised form 22 April 2016 Accepted 2 September 2016 Keywords: Fresh-Water-Cooling system on ships Waste heat recovery Modelling Estimation of drifting parameters Conditionally-Gaussian-Filtering Stabilizing feedback control

a b s t r a c t In cruise ship applications several waste heat recovery systems have been developed over the years in order to harvest the excess heat produced by the diesel engine and is stored in the coolant as thermal energy for instance. In this paper, a detailed model of a Fresh-Water-Cooling system is developed based on first principles and a given circuit scheme provided by the vessel manufacturer. The Fresh-WaterCooling system consists of multiple interconnected subsystems; in this paper the main focus lies on the heat exchanger, where the waste heat recovery process takes place. Due to the presence of unmeasured states and uncertain model parameters the heat exchanger is modelled as partially measured stochastic system. Further the process is proved to be conditionally Gaussian, which makes the use of a conditionally Gaussian filter possible. It produces optimal estimates for the states and the unknown drifting parameters in the mean square sense. The process state is locally stabilized by the estimated process values using stabilizing feedback control. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Efficient use of energy is a focal point in any branch of industry due to the finite nature of fossil fuel supplies among other reasons. In marine vessel applications, the diesel engine is the main source of power supply for the entire ship. Although fairly developed, its thermal efficiency is only 50% so that only half of the fuel energy is converted into mechanical work. The other half appears as thermal energy in the form of exhaust gas, coolant and radiation. This is referred as waste heat and considerable of work has been deployed in developing different kinds of waste heat recovery systems. Here one particular system is considered known as the Fresh-WaterCooling system (FWC). In this paper the main focus lies on the heat exchanger, where the waste heat recovery takes place. In brief, the heat exchanger is divided into two channels referred as the primary and secondary side. The cold seawater feed on the secondary side is heated up by the hot coolant on the primary side, which contains the excess thermal energy produced by the diesel generator. However, the heat exchanger is subject to several kinds of uncertainties; e.g. its geometrical configuration is not known accurately as it changes from ship to ship. Moreover the heat

∗ Corresponding author. E-mail address: philipp.nguyen@aalto.fi (P. Nguyen). http://dx.doi.org/10.1016/j.jprocont.2016.09.002 0959-1524/© 2016 Elsevier Ltd. All rights reserved.

transfer process strongly depends on the unknown flow conditions inside the heat exchanger, which might cause the associated heat transfer coefficient to fluctuate over a wide range during operation. Many approaches exist across different fields in engineering dealing with the determination of the heat transfer coefficient. A brief outline of those approaches is provided next.

1.1. Thermodynamics/fluid mechanics Within the thermodynamics and fluid mechanics community, the main objective is the accurate modelling of the governing physical phenomena. That often leads to complex and coupled field problems, where analytical solutions are difficult to obtain; e.g. the Navier–Stokes equations. Empirical and numerical methods have been developed as a consequence and the results have been well reported in literature. For instance, in [1,2], empirical correlations are available to estimate the heat transfer coefficient for numerous geometries in the case of forced convection. The obtained correlations are expressed in terms of dimensionless numbers: the Nusselt (Nu), Prandtl Pr and Reynolds number (Re). However, the analogies are only valid for very specific conditions under which the experiments have been carried out. With increasing computing power, Computational Fluid Dynamics (CFD) provides an option to solve the Navier–Stokes equations numerically. In [3,4], the heat transfer coefficient is

P. Nguyen, R. Tenno / Journal of Process Control 47 (2016) 22–34

Nomenclature Abbreviations CFD Computational Fluid Dynamics CGF Conditionally-Gaussian Filter CV control valve EKF extended Kalman Filter Fresh-Water-Cooling system FWC HT high temperature LT low temperature mixing valve MV SV splitting valve Conditionally-Gaussian system, estimation and control ε¯ innovation process  conditional covariance matrix  unconditional mean of unknown parameter vector  dynamics of drift  covariance of noise   standard deviation  unknown drifting parameter ε1 , ε2 white noise indirect control variable; flow rate through the ϕ shell-side of the heat exchanger a0 (t, yt ), A0 (yt ) time-varying vector functions dependent on measurements yt a1 (t, yt ), A1 (t, yt ) time-varying matrix functions dependent on measurements yt a2 (t) time-varying vector function covariance matrices of process and measurement b, B noise d stability resource gain of innovation G h sampling interval K upper bound for TE,out Kp proportional gain of controller conditional mean m Td desired constant temperature at the heat exchanger outlet u direct control variable; set-point temperature in CV independent Wiener processes V, W y measurements z vector of unmeasured parameters and state FWC system characteristic function for control and splitting

1 , 2 valves Qj flow rate through j Tj temperatures at j volume at j Vj WHeat transferred heat from engine to coolant Physical and geometrical quantities  density of water and brine heat transfer area on the interior surface of the tube Ai bank Ao heat transfer area on the exterior surface of the tube bank heat capacity of water and brine cp hi convective heat transfer coefficient on the tube-side ho convective heat transfer coefficient on the shell-side thermal conductivity of walls k L length of the tube bank

23

Abbreviations mW mass of the tube bank ri interior radius of on tube ro exterior radius of on tube Subscripts and superscripts initial 0 CL closed-loop OL open-loop after mixing valve 1 a MV1 BP by-pass E engine fix fixed set-point from the low-temperature circuit fr LT Hex heat exchanger in at inlet LT low temperature circuit out at outlet shell S Sea sea water T tube t instant in time t towards the low-temperature circuit to LT to MV1 towards mixing valve 1 interior surface of tube bank W, i W, o exterior surface of tube bank waste heat recovery WHR

determined in the boundary layer of a wall for laminar and turbulent air flow and in atmospheric flow conditions with low and high Reynolds numbers. 1.2. Control engineering For several industrial processes, estimation of the heat transfer coefficient is a crucial task; it is evident from the different applications such as chemical reactors, biochemical and food industries or in a turbo-charged diesel engine [5–8]. The well-known extended Kalman Filter (EKF) has been applied in all of those cases, where the posterior filtering densities of the states is assumed to be a Gaussian distribution. These are formed by utilizing Taylor series approximations of the nonlinear state equations. Application of the EKF requires deterministic model coefficients (Jacobians). However, in our case study the model coefficients depend on noisy measurements introducing stochastics to the system. Theoretically, this circumstance is not covered by the EKF framework leading to non-proper solutions as the mean and variance are incorrect biased estimates of the state and unknown parameters. They are biased since the estimation problem is infinite dimensional and the statistical moments do not form a closed system for the calculation of the mean and variance. One must find the probability density first before calculating the mean and variance. The probability density can be found as the solution of the Zakai equation or Kushner equation developed for nonlinear systems. For instance Zakai-filtering has been applied to a nonlinear parameter estimation problem in deposition process control [9]. In less severe cases, when the coefficients are only nonlinear dependent on the measurements, (and not nonlinear regarding the state) the mean and variance can still be calculated in closed form with the conditionally Gaussian filter. We follow this conventional approach in the manuscript. This paper is organized in eight sections. In Section 2 the control objectives are stated. Based on a given circuit scheme and

24

P. Nguyen, R. Tenno / Journal of Process Control 47 (2016) 22–34 Evaporator Control:

Evaporator

LT circuit

LT circuit

Output:

Sea water Disturbance:

CV

State: Heat exchanger

Sea water CV

Output:

SV

Engine

Heat exchanger Engine HT circuit

MV1

Fig. 1. Control system.

MV2 first principles a detailed model of the FWC system is derived in Section 3. Section 4 presents the FWC system in computationallyclosed form. In Section 5 the heat exchanger is presented as a nonlinear, partially measured system along with a general drift model representing the unknown states and parameters. Section 6 specifies the filtering and control problem including all the necessary conditions of a conditionally Gaussian process; those assumptions are proved to hold for the system on hand. In that case, a Conditionally-Gaussian Filter (CGF) is applied to estimate the unknown states and parameters yielding optimal results in the mean-square sense. Based on the estimation results, stabilizing control is used to drive the estimated state to a desired constant reference value at the outlet of the heat exchanger. The simulation environment is specified and the numerical results are presented and discussed in Section 7 and conclusions are drawn in Section 8. 2. Control objectives In Fig. 1, the control system is depicted along with the main components and relevant variables.

HT circuit

Fig. 2. Scheme of the FWC system.

At the outlet of the engine, the flow QE is divided by the control valve (CV). One part is recirculated through a by-pass QBP back to the inlet of the engine. The remaining part QHex is directed into shellside of the heat exchanger denoted as the waste-heat recovery section. Cold seawater (brine) enters from the tube-side and is heated up due to the existing temperature gradient between hot engine coolant and brine. This process is referred as waste-heat recovery. The FWC system consists of two cooling circuits, a HighTemperature (HT) and Low-Temperature (LT) circuit, which are connected through two valves: SV and MV1, see Fig. 2. An exchange of heat flows takes place in order to maintain a design temperature at the mixing valve (MV1), which is specified by the ship operator. Here the process variables of the LT-circuit are only modelled as an exterior input given by QLT with temperature TLT for simplicity reasons. 3.2. Diesel engine and cooling jacket

• The engine load WHeat is regarded as an uncontrolled disturbance of the system, which directly depends on the known engine load program during a cruise and is provided by the ship operator. • The thermostat set-point is the control variable denoted with u, which is used to manipulate the position of the control valve (CV). Its model is presented later (2) • Temperature measurements are taken from the heat exchanger at the tube-side outlet y1 = TSea,out and on the exterior surface of the tube bank y2 = TW,o . • Local stabilization of the unmeasured temperature z4 = THex,out to a constant reference value Td is performed by using feedback controls. The main challenge is that the control law depends on the unmeasured states und unknown model parameters; therefore an estimator must be established prior to the control design. The aim of the stabilizing feedback control is to maintain a steady recovery of waste heat in the heat exchanger and providing steady heat flow conditions towards other interfacing subsystems such as an evaporator or the LT circuit. 3. FWC system: first principle modelling 3.1. Working principle A scheme of the FWC system on hand is shown in Fig. 2. During operation, the diesel engine must be cooled sufficiently in order to avoid any thermal damage. A pump provides a constant flow rate feed QE through the cooling jacket of the engine. The generated excess heat from the diesel engine is transferred to the coolant and stored as thermal energy.

A relatively simple heat transfer model of the engine is presented. The engine itself is modelled as a heat source, which is proportional to the current loading profile during operation. The transferred heat to the coolant can be deduced as a consequence. The heat balance is taken for the cooling jacket.

 WHeat dT E,out QE  = TE,in − TE,out + , VE  cp VE dt

(1)

where TE,out is the temperature at the engine outlet, TE,in is the temperature at the engine inlet, QE is the constant net flow rate through the cooling jacket, VE is the volume of the cooling jacket, WHeat is the transferred heat,  is the density and cp is the specific heat capacity of water. 3.3. Control valve This is a thermostat-controlled splitting valve, that determines the water distribution towards the by-pass and waste-heat recovery section. Its characteristic curve is described by the following nonlinear function:



1 =

1 1 + tanh 2



TE,out − u(t) 2



.

(2)

The thermostat set-point is denoted with u(t) and is our direct control variable. The mass balance for the control valve is: QBP = (1 − 1 ) QE QHex = 1 QE ≡ ϕ(u),

(3)

P. Nguyen, R. Tenno / Journal of Process Control 47 (2016) 22–34

25

Control valve 1

0.8 u1 = 90 °C u2 = 92 °C

κ1 [−]

0.6

u3 = 88 °C 0.4

0.2

0 82

84

86

88

90 92 temperature [°C]

94

96

98

Fig. 4. Shell-and-Tube heat exchanger. Fig. 3. Operating curve of control valve for three different thermostat set-points: u1 = 90 ◦ C, u2 = 92 ◦ C and u3 = 88 ◦ C.

where the flow rate towards the heat exchanger QHex is replaced with the indirect control variable ϕ(u). This change in notation is utilized later during the control design, see Section 6.4. An illustrative example of the CV operating curve is displayed in Fig. 3, for three different operating points, when TE,out = ui for i = 1, 2, 3. In those cases the flow QE is equally split into the by-pass and heat exchanger due to 1 = 0.5. The thermostat set-point u(t) of the CV shifts the operating curve horizontally while maintaining its original nonlinear characteristic. The control task is the stabilization of the unmeasured temperature to a design temperature Td at the heat exchanger outlet THex,out , which is achieved by manipulating the adaptive CV through u(t). The flow is distributed accordingly to the heat exchanger branch in order to fulfil the control task. Further, it is important to note that the control – yet to be developed – is subject to the following impediments: • The CV is subject to fluctuation (disturbance) in the outlet temperature of the engine TE,out , which is a result of a varying engine load program during operation. • Further, the control depends on the estimated state and drifting parameters associated with the heat exchanger; this is outlined in Section 5. Before designing the stabilizing control a reliable estimator must be found first. For small deviations T = TE,out − u(t), the division of the flows in the CV is linearly dependent on the control u(t)



1 =

TE,out − u(t) 1 1+ 2 2



(4)

and for large deviations, it equals 0 or 1. In other terms, for small T , the CV operates in the linear region around the set-point u(t), whereas for big variations it operates in the saturation region, see Fig. 3. 3.4. Heat exchanger One of the most commonly used heat exchanger is the Shelland-Tube heat exchanger, depicted in Fig. 4. A bank of tubes is embedded into the shell of the heat exchanger. The cold brine (secondary fluid) enters from the tube inlet and gets heated up along the entire tube length by the hot engine water (primary fluid) flowing on the shell-side. Baffles are installed in order to create a higher degree of turbulence improving the heat transfer as a consequence. In this paper, the heat exchanger is modelled as a lumped parameter system based on the following assumptions: • Only the type of heat exchanger is known, whereas the precise geometry dimensions remain uncertain.

• Flow conditions on the shell- and tube-side are assumed to be turbulent; however the exact Reynolds number and the degree of turbulence is not known. • The outer shell of the heat exchanger is perfectly insulated. • For the calculation of the heat transfer areas – only one single tube is considered, which has the equivalent combined heat transfer area as the entire tube bank. This approach has been adapted by [10]. • Dependencies along the spatial coordinate of the heat exchanger are neglected. Energy balances are taken at the shell-side and tube-side outlet and on the interior and exterior surfaces of the tube walls. This leads to a system of four coupled differential equations of stochastic nature due to the outlined uncertainties described in the list above. The model uncertainties are accounted by means of multiplicative and additive uncertainties. The multiplicative uncertainty is represented by an unknown model parameter  and additive uncertainty is introduced by a Wiener process W(t). The system of Wiener processes in the models is independent from each other. The inclusion of the Wiener processes W(t) make (5)–(7) stochastic differential equations, that gain their exact meaning through Ito’s integrals. 3.4.1. Shell-side

 dT Hex,out QHex  = TE,out − THex,out − Ao ho   VS dt



THex,out − TW,o



 cp VS

1 (t)

˙ (t). + W

(5)

In (5), QHex is the flow rate at the shell inlet, VS is the volume of the shell-side, TE,out is the temperature of the primary fluid at the shell inlet, THex,out is the temperature of the primary fluid at the shell outlet. The temperature on the exterior surfaces of the tube wall is TW,o , the exterior heat transfer area of the tube bank is Ao and ho is the convective heat transfer coefficient on the shell-side. The parameters Ao and ho are both unknown and lumped into one single drifting parameter  1 , that is estimated later. 3.4.2. Tube-side

 dT Sea,out QSea  = TSea,in − TSea,out − Ai hi VT dt  



TSea,out − TW,i



 cp VT

2 (t)

˙ (t). +W

(6)

26

P. Nguyen, R. Tenno / Journal of Process Control 47 (2016) 22–34

In (6), QSea is the constant flow rate of seawater to the tube inlet, VT is the volume of the tube-side, TSea,in is the inlet temperature and TSea,out is the temperature of seawater at the tube outlet, TW,i is the temperature on the interior surface of the tube wall, Ai is the interior heat transfer area of the tube bank and hi is the convective heat transfer coefficient on the tube-side. The parameters Ai and hi are both unknown and lumped into one single unknown drifting parameter  2 to be estimated.

3.6. Mixing valve 1

3.4.3. Outer and inner walls of tubes

TMV =

The flow Qto MV 1 towards MV1 is mixed with Qfr LT from the LTcircuit of temperature TLT . Since the net flow in the HT-circuit is constant, it follows: Qa MV 1 = QHex . The mass balance for MV1 is: Qa MV 1 = Qfr LT + Qto MV 1 . The heat balance for MV1 is: fix

1

dT W,o L =− dt mW ln(ro /ri )







 2 k  TW,o − TW,i cp

− Ao ho

 

TW,o − THex,out

˙ (t). +W

(7)

In (7), L is the length of the heat exchanger, mW is the mass of the tube bank, ro and ri is the exterior and interior radius of one single tube and k is the thermal conductivity of the tube. In realcase scenarios the geometry is rather uncertain, albeit on a small scale. Therefore, a weak drift model is used for parameter  3 , in which the unknown geometrical parameters are lumped.





 2 k  TW,i − TW,o cp

3 (t)



− Ai hi

TW,i − TSea,out

 

 cp VT

 .

(8)

2 (t)

3.5. Splitting valve The splitting valve (SV) controls the amount of water and thus the heat flow exchange between the HT- and LT-circuit. At the outlet of the heat exchanger, QHex is divided into the mixer Qto MV 1 and the LT-circuit Qto LT . The net water flow in the HT-circuit shall remain constant, what leads to: Qto LT = Qfr LT . The mass balance for the splitting valve (SV) is: (9)

2 QHex .

2 = with



QHex

THex,out .

(12)

(13)

TE,in =

QBP QHex fix TE,out + T . QE QE MV 1

(14)

4. Computationally closed form of the FWC system The numerous models from Section 3 represent the FWC system; in order to provide a more transparent structure of the computationally-closed system, it is partitioned and summarized in three parts: • (A) Mass balance for flows Q, (16). • (B) Heat balance for temperatures T, (18). • (C) Partially measured heat exchanger, Section 5. Regarding the mass and heat balances average values for a noisefree system are computed and together with the partially measured heat exchanger it forms a computational-closed form of the FWC system. In this section only the system of flows (A) and temperatures (B) are presented as the whole next section is dedicated for the presentation of the partially measured heat exchanger (C). 4.1. System of flows Based on the mass balances in (3), (9), (11), (13), a vector of flow rates Q is defined as Q = QBP

QHex

Qto LT

Qto MV 1

Qfr LT

Qa MV 1

T

.

(15)

The flows satisfy the following system:

In order to provide simultaneous cooling performance for the engine and efficient waste heat recovery, the desired reference fix temperature after MV1 is set to TMV . The according control law 1 / TLT : for 2 is obtained by inserting (9) in (12) for THex,out =



Qto MV 1

QE = QBP + QHex .



Qto LT = (1 − 2 ) QHex Qto MV 1 =

TLT +

The heat balance for MV2 is:

1 (t)



QHex

3.7. Mixing valve 2



 cp VS

dT W,i L =− dt mW ln(ro /ri )

Qfr LT

At MV2 the mass balance is:

3 (t)



(11)

fix

TMV − TLT

1

THex,out − TLT fix

0 ≤ 2 ≤ 1 ⇒ TMV

1



(10)

≤ THex,out .

Eq. (10) regulates the heat flow exchange between the HT- and fix LT-circuit in order to stabilize TMV during the whole operation. It 1 presents a trade-off solution between efficient waste heat recovery fix and maintaining thermal safety of the engine and the set-point TMV 1 is provided by the ship operator.

⎡

1 0

0

1

0

0

1

0

0

0

1

1

0

⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎣0 

0

⎡ ⎤ QBP ⎡ (1 − ) Q ⎤ E 1 ⎤ ⎢Q ⎥ 0 0 0 ⎢ Hex ⎥ ⎢ 1 QE ⎥ ⎥ ⎢ ⎥ 0 0 0 ⎥⎢ ⎥ ⎥⎢ Qto LT ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ 0 0 0 ⎥ = ⎢ 1 (1 − 2 ) QE ⎥ ⎥⎢ ⎢ ⎥ ⎢ ⎥ ⎥ 1 0 0 ⎥⎢ Qto MV 1 ⎥ ⎢ ⎥

1 2 QE ⎢ ⎥ ⎢ ⎥ ⎦ 0 −1 0 ⎢ Qfr LT ⎥ ⎣ 0 ⎦ ⎣ ⎦ 0 0 −1

 Qa MV 1 0

 AQ    B Q

(16)

Q

For the known constant net flow rate QE and given valve characteristics 1 and 2 , the system (16) can be solved by: Q = A−1 · BQ . Q

(17)

P. Nguyen, R. Tenno / Journal of Process Control 47 (2016) 22–34

4.2. System of temperatures Here heat balances are presented for (1), (8), (14), which forms a deterministic system for the computation of temperatures

⎡ QE ⎡˙ ⎤ − TE,out ⎢ VE ⎢˙ ⎥ ⎢ ⎣ TW,i ⎦ = ⎢ 0 ⎣ 0

  T˙

−

1 − 1



⎤ ⎡ ⎤ T ⎥ E,out ⎥⎢ ⎥ ⎣ TW,i ⎦ 0 ⎥ ⎦ QE VE

0 2 k 3 2 − cp  cp VT

0

−1



TE,in

   T

AT ( 1 ,2 ,3 )

⎡W

⎤

Heat

⎢ cp VE ⎢ 2 y1 2 k 3 y2 +⎢ ⎢ c V + cp ⎣ p T

⎥ ⎥ ⎥. ⎥ ⎦

(18)

fix



1 TMV

1





BT (2 ,3 )

The first and second coordinate are dynamic models, whereas the third coordinate is a static model representing the mixing process at MV2. In shorter notation the deterministic system is summarized as: T˙ = AT · T + BT .

(19)

The system of flow rates and temperatures is solved based on the assumption, that the drifting parameters  and control u(t) are known, which is not the case initially. Altogether, the system of flows and temperatures is simultaneously solved for the FWC system. All the variables from (17) to (19) are assumed to be available for further analysis – in particular for the estimator and control design of the partially measured heat exchanger, which is presented in the next section. 5. Partially measured heat exchanger The partially measured heat exchanger is the main focus of our analysis including its stochastic characterization, which will be discussed in this section. 5.1. Unknown drifting parameters The first principle model of the heat exchanger from (5) to (8) is subject to uncertainties specified by the vector of lumped model parameters :



 = 1

2

3

T

(20)

They are listed in Table 1 along with their original form and units. A general drifting model is introduced for the parameters  as a linear stochastic drift:





d =  (t) −  dt +  dW. Here



matrix

T



determines

(21) the

drift

dynamics,

=

is the unconditional mean of , the covari1 2 3 ance matrix is  and the standard Wiener process is W. The latter

two determine the randomness of the drift. It is a flexible representation, which can be tailored to different specific processes by selecting the tuning parameters ,  and  accordingly. This general uncertainty model includes two particular cases: the constant drift d = 0 and the random walk drift d = dW. A stochastic approach is reasonable since parameters  1 and  2 contain the convective heat transfer coefficients ho and hi , which are subject to the unknown degree of turbulence inside the pipe. The effect of turbulence significantly alters the heat transfer process. The models of parameters  1 and  2 are considered to have the degree of freedom to change according to uncertain flow conditions. Parameter  3 describes the uncertainty regarding the physical configuration and geometry of the heat exchanger. The general type of heat exchanger is known; but no information is available about specific components such as pipe diameters, tube length etc. Here, a model for a weak drift is chosen to represent the geometrical uncertainty that is initially unknown. The unknown drifting parameters  and unmeasured state THex,out are collected in the augmented vector of unmeasured states z and the measurements are y.



z = 1

2



3

THex,out

y = TSea,out

TW,o

T

T

,

(22)

.

5.2. List of given parameters and time-dependent variables The following list contains all the constants and known parameters/variables of the FWC system, which are relevant for the filter and control design. The list serves as on overview and reference point for the reader when it comes to the presentation of the partially measured system (23), its coefficients (26) and the filter and control design in Section 6. The characteristics of the parameters/variables are in the left column of Table 2; ‘c’ indicates constants, whereas ‘t’ denotes time-dependent variables. Additional reasoning is provided on the nature of the variables/parameters, which gives further insight on the processes of the relatively large FWC system. 5.2.1. Parameters – constant All the model parameters 1 , , cp , VS , VT , and VE are constant and have been described earlier in Section 3. 5.2.2. Flow rates – constant A supply pump provides a constant feed of seawater QSea into the tube-side of the heat exchanger. Concerning the HT-circuit, the net flow is given to be constant, which means that the exchanged flow rates between the HT- and LT-circuits are equal; it is regulated by the splitting valve (10). 5.2.3. Temperatures – constant The temperature Td denotes the constant reference value that shall be achieved by feedback control of the unmeasured state z4 . The incoming seawater temperature is assumed to be constant since the investigated time period is only 14 h. The simulation has been carried out for summer conditions in the Baltic Sea with Table 2 Table of known parameters and variables.

Table 1 Unknown parameters of the heat exchanger

Symbols

Lumped parameters

Original parameters

Units

1 2 3

Ao ho Ai hi

[W/K] [W/K] m/kg

L mW ln(ro /ri )

27

Parameters – ‘c’ Flow rates – ‘c’ Temperatures – ‘c’

1 , , cp , VS , VT , VE QSea , QE fix Td , TSea,in , TLT , TMV

Temperatures – ‘t’ Disturbance – ‘t’

TE,out , TE,in , TW,i WHeat

1

28

P. Nguyen, R. Tenno / Journal of Process Control 47 (2016) 22–34

TSea,in = 15 ◦ C. If a longer period is considered – an entire year for instance – then the seasonal variation of the seawater temperature shall be considered; but this is not the case here. The coolant from the LT-circuit has constant temperature TLT = 35 ◦ C by specification. The control law (10) constantly stabilizes the temperature at fix MV1 to its design value TMV = 80 ◦ C. The operating point is pro1 vided by the ship’s operator and presents a trade-off solution between a safe operation of the diesel engine and waste heat recovery inside the heat exchanger.

of the operating points is specified based on the control task in Section 6.4. The occurring bilinear terms are

5.2.4. Temperatures – time dependent The temperatures TE,in , TE,out and TW,i are processes computed through the deterministic system of temperatures (18).

After linearization and time discretization a weak-sense analogy of the heat exchanger model is obtained as a stochastic modification of (23):

5.2.5. External disturbance – time dependent The injected heat WHeat is directly dependent on the current engine load during a cruise; a validated model for the heat transfer has been provided by the ship operator.

zt+h =

5.2.6. Measurements – time dependent Temperatures measurements relevant for the filter design are



summarized in the measurement vector y = TSea,out

TW,o

T

(22).

5.2.7. Unmeasured states – time dependent The drifting parameters  and unmeasured temperature THex,out are summarized in the augmented unmeasured state vector z =



T

(22). 1 2 3 THex,out If not mentioned otherwise, any other variable apart from y, z, TE,in , TE,out , TW,i and WHeat is assumed to be constant for the remainder of the paper.

z4 · z1 ≈ −Td 1 + 1 z4 + Td z1 ,

with Td being the constant reference temperature at the heat exchanger outlet, 1 is the unconditional mean of z1 , and ϕ0,t is the indirect reference control defined later, see (29). 5.5. Discrete time and linear approximation

dy = (

⎡

y(0) = y0 , where z ∈ R4 is the vector of unmeasured states, y ∈ R2 is the vector of measurements, ϕ (u) ∈ R1 is the indirect control variable, u ∈ R1 is the direct control variable, W ∈ R4 and V ∈ R2 are independent Wiener processes, representing the process and measurement noise respectively. The coefficients of the system are a1 ∈ R4×4 , a2 ∈ R4 and A0 ∈ R2 , A1 ∈ R2×4 , b ∈ R4×4 and B ∈ R2×2 . The initial conditions for the unknown parameters and state z0 is specified by a Gaussian distribution: z0 ∼N (m0 , 0 ), where m0 ∈ R4 is the initial mean and 0 ∈ R4×4 is the initial covariance matrix. These guestimates represent the initial uncertainty. 5.4. Linearization The system (23) is nonlinear due to the occurrence of bilinear terms with respect to the unmeasured states z and the indirect control variable ϕ(u). It is caused by the coefficients a1 (t, yt , z1 ), a2 (t, z4 ) and A1 (t, yt , z4 ), which all contain the unmeasured state z. Those expressions are required to be linearized by a first order Taylor expansion in order to meet the conditions of a conditionally Gaussian system. A conditionally Gaussian framework is desired for the process since it inherits numerous beneficial properties that are presented later in Section 6.3. The linearization and determination

⎡

0

0

⎢ ⎢ 1 0 0 ⎢0 a1 (t, yt ) = ⎢ 0 1 0 ⎢0 ⎣ y −T 2 d cp VS 0 0

13 0

⎢ ⎢0 b=⎢ ⎢ ⎣0

1.5 0

0

0

⎤

⎥ ⎢ ⎥ ⎢ ⎥ ⎢0 ⎥ , a2 (t) = ⎢ ⎥ ⎢0 ⎦ ⎣T

1 d (ϕ0,t + ) VS⎡ cp 1 0

⎡ (23)

⎤

0

⎢ ⎢ ⎢0 a0 (t, yt ) = ⎢ ⎢0 ⎣T

a1 (t, yt , z1 ) z(t) + a2 (t, z4 ) ϕ (u) )dt + b dW

A0 (yt ) + A1 (t, yt , z4 ) z(t))dt + B dV

(25)

where h is the sampling interval, ε1 and ε2 are independent white noises representing the process and measurement uncertainties. Note how the coefficients a1 (t, yt ), a2 (t) and A1 (t, yt ) are no longer dependent on the unmeasured states z. Due to the linearization procedure a new coefficient a0 (t, yt ) has been introduced. The coefficients of the linearized, time-discrete system are:

Based on the provided models, the heat exchanger is represented as a continuous, partially measured, stochastic system:

z(0) = z0 ,

(a0 (t, yt ) + a1 (t, yt )zt + a2 (t)ϕ(ut )) h √ +b h ε1 (t + h), √ yt + (A0 (yt ) + A1 (t, yt )zt ) h + B h ε2 (t + h),

yt+h =

5.3. Nonlinear, partially measured stochastic system

dz = (

(24)

z4 · ϕ(u) ≈ −Td ϕ0 + ϕ0 z4 + Td ϕ(u)

0

0

0

0

10

0

0

−4

0

1−

⎤

⎥ ⎥ ⎥ ⎥, ⎥ ⎦ E,out − Td VS ⎤

ϕ0,t 1 − VS  cp VS

⎥ ⎥ ⎥ ⎥, ⎥ ⎦

(26)

⎡ −3 ⎤ ⎥ 10 0 ⎥ ⎥, B = ⎣ ⎦. ⎥ ⎦ 10−4 0

1.5 · 10−2

The noise intensities corrupting the process and measurements are specified through the coefficients b and B. Further sensitivity analysis is carried out regarding the estimation accuracy for increased noise intensities for both process and measurements. Next, an expression for the time-dependent, indirect reference control ϕ0,t is derived. For that purpose the vector  is defined as



 = 1

2

3

Td

T

(27)

,

which consists of the unconditional mean  of the unknown parameters z and the desired temperature set-point Td . The indirect reference control ϕ0,t is only apparent in the fourth coordinate z4 , which is evident from coefficient a2 (t) (26). Assuming that the reference  represents a physically reachable target, ϕ0,t is obtained from the noise-free SISO system, which is extracted from (25). z4 (t + h) − z4 (t) 0 ⇒ ϕ0

= a0 + a1 z4 (t) − z4 (t) + a2 ϕ0 = a0 + a1  −  + a2 ϕ0 =−

1 a2,4

(a0,4 + a1,4  − )

(28)

P. Nguyen, R. Tenno / Journal of Process Control 47 (2016) 22–34

Here, a2,4 and a0,4 are the fourth coordinates of the coefficients a2 (t) and a0 (t, yt ), respectively. The fourth row of a1 (t, yt ) is denoted with a1,4 and  is the reference vector as defined in (27). The resulting indirect reference control is: ϕ0,t =

Td − y2 TE,out − Td

1  cp

(29)

6. Estimation and control The outline of this section is as follows: • (A): It is proved, that the linearized, discrete time system governed by (25) is a conditionally Gaussian process. The required assumptions are presented and verified against the system on hand. • (B): The general method of Conditionally Gaussian Filtering (CGF) developed for nonlinear systems can be applied – [11]. The recursive, discrete time algorithm is presented for state and parameter estimation. • (C) After proving the conditionally Gaussian properties of our system and introducing the CGF algorithm, its main benefits and advantageous properties are explained. • (D): The estimated temperature at the heat exchanger outlet is stabilized to a constant reference value by applying feedback control. The derivation of the control law is presented in this section. • (E): As a result of the local temperature stabilization the waste heat recovery inside the heat exchanger is discussed. Moreover, the heat flow towards interconnected energy subsystems such as evaporator and LT-circuit is considered, as well. 6.1. Conditionally-Gaussian process According to the Theorem 13.3 by Lipster and Shiryaev (1977), a stochastic system is conditionally Gaussian, if the following assumptions are satisfied:

29

cold mediums – as specified in the second law of thermodynamics. Implication of the aforementioned statement is that none of the temperatures inside the heat exchanger can exceed the one of the primary fluid TE,out . As a consequence, it is sufficient to find an upper bound for TE,out in order to guarantee the boundedness of all occurring coefficients. For that purpose, the cooling jacket model (1) is reformulated by inserting the heat balance of MV2 (14). dT E,out WHeat QE fix

1 (ut ) (TMV − TE,out ) + = 1 VE VE  cp dt

(30)

The lower and upper bound of the control valve are fixed by the saturation characteristics of the CV: 0 ≤ 1 (TE,out , ut ) ≤ 1, see Fig. 3. The injected heat WHeat is directly proportional to the known engine load program. That fact is exploited to find the corresponding lower and upper bound. The upper bounds are defined for full max , which are load operation and are designated as 1max = 1 and WHeat inserted in (30). It yields a linear ODE with constant coefficients, the solution of which is known to be: TE,out (t) = TE,out (0) e−at − with

b (1 − e−at ), a

fix MV

+W max ( cp )−1 )

fix MV

QE T

(31)

(QE T

Heat

1 a= V 1 and b = . VE E An upper bound is found for TE,out (t), which is simultaneously the global upper bound of the FWC system designated with K:

| TE,out (t) | ≤ max (TE,out (0),

b ) ≡ K. a

(32)

Therefore, all the coefficients are guaranteed to be bounded by the upper threshold K. Assumptions (ii) and (iii) are satisfied: E | a0 (t, yt )|2 < ∞, | a2 (t) |≤ K,

E | A0 (yt )|2 < ∞, | a1 (t, yt ) |≤ K,

| A1 (t, yt ) |≤ K

(33)

From assumptions (ii)–(iv) – it is concluded – that the process of (25) does not disperse at any instant in time

i Each coefficient a0 (t, yt ), a1 (t, yt ), a2 (t), A0 (yt ) and A1 (t, yt ) is a non-anticipative (causal) function. ii Finiteness: E | a0 (t, yt )|2 < ∞, E | A0 (yt )|2 < ∞, iii Boundedness: |a1 (t, yt ) | ≤ K, |a2 (t) | ≤ K, |A1 (t, yt ) | ≤ K iv Initial state: z0 ∼N (m0 , 0 ),  0 > 0

E( xt 2 + yt 2 ) < ∞

The aforementioned assumptions (i)–(iv) are elaborated for the partially measured heat exchanger (25). Assumptions (i) and (iv) are obvious: None of the coefficients depend on future measurements, which is necessary for causality. Due to the presence of uncertain parameter values, an initial Gaussian distribution with non-zero initial covariance is a natural choice. Assumptions (ii) and (iii) are most critical for the system and are elaborated next. Each element of the coefficients from (26) is required to be finite or bounded in the conditionally Gaussian framework. The upcoming proof of the aforementioned conditions solely relies on physical mechanism in heat transfers; therefore the more suitable, original physical notation is used for the variables involved. Clearly, the highest temperature occurs at the outlet of the cooling jacket TE,out due to the injected heat from the engine (1). In the coefficients (26), all temperatures associated with the heat exchanger TSea,out , TW,o and TW,i , (6)–(8) are lower than TE,out at any instant in time as the heat flux is always assured to go from hot to

6.2. Conditionally-Gaussian-Filter

and is conditionally Gaussian in addition (Theorem 13.3, [11]). Furthermore, the initial conditional probability density p (z0 | y0 ) = N(m0 , 0 ) exists and is Gaussian due to initial uncertainties  0 > 0, assumption (iv). N(m0 , 0 ) =

0

1 √ e−(1/2)((z0 −m0 )/0 ) 2

(34)

Filtering theory addresses the problem of estimating the unmeasured states zt of a dynamical system, which is measured through the entire history of noisy measurements (ys : 0 ≤ s ≤ t) up to the current instant in time. A recursive, discrete-time CGF can be applied for state estimation, because all the aforementioned conditions (i)–(iv) of a conditionally Gaussian process are satisfied (Theorem 13.4, [11]): mt+h = a0 + a1 mt + a2 ϕ(ut ) + G ε¯ t+h , t+h =



a1 t aT1



T

+ bb − GG ,

 T

G = a1 t A1

T

 T −1/2

BBT + A1 t A1

ε¯ t+h = BBT + A1 t AT1

−1/2

(35) (36)

,

(yt+h − A0 − A1 mt ) ,

(37) (38)

where m is the conditional mean,  is the conditional covariance matrix. The gain of innovation is G and ε¯ t ∈ R2 is the innovation process.

30

P. Nguyen, R. Tenno / Journal of Process Control 47 (2016) 22–34

6.3. Guaranteed properties and benefits of a conditionally Gaussian system By conducting a successful proof of a conditionally Gaussian properties in Section 6.1, one benefits from the numerous advantageous properties when the CGF is applied in state or parameter estimation. The following results have been all proved in theory [11]. The conditional probability density p (zt | y0 . . . yt ) can be computed by a relatively simple recursive algorithm for the mean mt = E (zt | y0 . . . yt ) and covariance t = E (t | y0 . . .yt ) conditioned on the entire history of measurements, which is Gaussian at any instant in time. The statistics can be represented in closed-form meaning that all the higher statistical moments can be computed from the first two statistical moments only: mt and  t . The filter is said to be finite dimensional. The CGF is optimal in the mean-square error sense and there is no way for better prediction of the state than by this algorithm, y y [11]: mt = E(zt | Ft ) and t = cov(zt | Ft ). The innovation process has guaranteed white noise properties: ε¯ t ∼IN(0, I). This fact holds for continuous and time-discrete systems (Theorem 13.5, [11]). The amount of information in the innovation process is exactly the same as in the measured process. If ε¯ t ∼IN(0, I) is not met; possible reasons are: • • • •

systematic error (nonzero mean) correlation in time bigger or smaller variance than unity time-dependent variance

In that case, the filter should be reconsidered, because there are still predictable components – which can be recovered by the filter. All of the aforementioned properties are conserved by the CGF even in the presence of stochastic coefficients due to the dependence on noisy measurements in linear or nonlinear fashion. Further, the inclusion of a nonlinear feedback control does not deteriorate the filter performance. 6.4. Control

ϕ(u, t) = −KP (m4 − Td ).

The purpose of the local temperature stabilization at the shell outlet is to have steady waste heat recovery and heat flows towards connected subsystems such as an evaporator and the LT-circuit; the heat flows are given by WHex





=  cp QHex TE,out − THex,out ,

WEv

=  cp QSea TSea,out ,

WLT

=  cp QtoLT THex,out ,

(42)

where WHex is the recovered waste heat in the heat exchanger, WEv is the heat flow to the downstream evaporator and WLT is the heat flow towards the LT-circuit, see Fig. 1. 7. Process simulation 7.1. Simulation settings The filter and controller are validated against the first principle models (1)–(14) and the linear stochastic drift model for the unknown parameters (21); the following specifications are made for the simulation environment: Seawater and coolant • heat capacity cp = 4184 J/(kg K) • density  = 1000 kg/m3 Heat exchanger • conductivity of copper pipes k = 400 W/(m K) • volume of shell side VS = 8.60 m3 • volume of tube side VT = 3.6 m3



TE,out − u(t) 2



QE .

(40)

Ultimately, it yields the nonlinear control law that is implemented in the CV, (2): u(t) = TE,out + 2 atanh

• total volume of cooling jacket VE = 700 m3 Constant flow rates and temperatures

(39)

It is a proportional feedback control for the indirect control variable ϕ(u, t) given the regulation error defined as the difference between m4 and Td . The estimated temperature at the outlet of the heat exchanger is computed by the CGF is designated with m4 . Since the last coordinate (a2 (t))4 is uniquely positive, then the closed loop system satisfies: | (a1 (t, y))4,4 − Kp (a2 (t))4 |< 1 leading to closed-loop stability. For implementation, the indirect control variable ϕ(u, t) – given by the relatively simple proportional control – needs to be converted into the direct control variable u(t) given by the following conversion formula (3):



6.5. Heat flows

Engine

A linear feedback of the estimated state is applied for stabilizing the process state z4 and its estimate m4 to a desired constant reference Td .

1 ϕ(u) = 1 + tanh 2

The conditionally Gaussian system (25), the direct control, u(t) (41), the estimated state and drifting parameters and the mass and heat balances by (17) and (19) form a computationally-closed system.

2K

P

(m4 − Td ) +1 QE

(41)

• flow rate through engine QE = 360 m3 /h • flow rate through secondary side of heat exchanger QSea = 3.6 m3 /h • temperature of sea water TSea,in = 15 ◦ C • temperature of LT-circuit water TLT = 35 ◦ C • set-point after mixer 1 T fix = 80 ◦ C MV 1

Control • reference temperature Td = 84 ◦ C • control gain KP = 0.15 Measurement noise intensity • see coefficient B in (26) Process noise intensity • see coefficient b in (26)

P. Nguyen, R. Tenno / Journal of Process Control 47 (2016) 22–34

Unknown parameters and states

31

Estimation and tracking of z2

4

x 10 3.56

• drifting parameters z1 . . . W/K, z2 . . . W/K, and z3 . . . m/kg • outlet temperature of heat exchanger z4 . . . ◦ C

process estimation

3.54 3.52 3.5

[W/K]

Computing • time step h = 1 s • simulation time T = 14 h

4

x 10

3.58

3.48

3.56

3.46

3.54

3.44

3.52 3.5

3.42

3.48 3.4

3.46

3.38

7.2. Numerical results

0

50

100

150

time [s]

3.36 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

time [h] Fig. 7. Estimation and tracking of parameter z2 .

Estimation and tracking of z3 0.8 process estimation

0.7 0.6

[m/kg]

During the entirety of the 14 h cruise, different steering maneuvers of low and high loading operation are performed. The corresponding heat WHeat , which is transferred from the engine to the coolant is depicted in Fig. 5. The system is continuously perturbed by WHeat , which was defined as our system input in Section 2. Under this condition, the unknown drifting parameters and unknown states z are reasonably estimated by the CGF and both – the process state z4 and its estimate m4 are locally stabilized to the desired reference value Td . In Fig. 6, the parameter estimate z1 converges within 3 h to its true value can be tracked fairly well over the duration of the simulation. In Fig. 7, parameter z2 is estimated rapidly as it converges within 2 min. Due to the rapid convergence, a composite graph with appropriate scaling is presented in order to highlight the transition phase when the initial uncertainty is removed.

0.5 0.4 0.3 0.2 0.1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

time [h] Injected heat: WHeat

Fig. 8. Estimation and tracking of parameter z3 .

2.5

Estimation and control of z4

2

88

[MW]

88 87.5

1.5

86

86.5

1

85

[°C]

86

0.5 0

84

85.5

83

85

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

time [h]

process estimation reference

87

87

0

1000

2000 time [s]

3000

84.5 84

Fig. 5. Injected heat WHeat from the diesel engine to the coolant during 14 h cruise.

83.5 83

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

time [h]

Estimation and tracking of z1

4

5.5

x 10

Fig. 9. The unmeasured state z4 (blue) and estimated state m4 (red) are controlled to the vicinity of the desired reference value Td (green). Greater accuracy has been obtained for the estimated state due to the observation performance of the filter. (For interpretation of the color reference in this figure legend, the reader is referred to the web version of the article.)

process estimation 5

[W/K]

4.5

4

3.5

3

0

1

2

3

4

5

6

7

8

9

10

11

time [h] Fig. 6. Estimation and tracking of parameter z1 .

12

13

14

In Fig. 8, parameter z3 converges in about 2 h and is tracked well for the remainder of the simulation. A model for weak drift has been used to represent the uncertainties associated with the geometry of the heat exchanger model. In Fig. 9, the unmeasured state z4 (blue) is stabilized to the vicinity of the reference value Td ; albeit with visible fluctuation around it. Its deviation is due to the process noise intensity b in the process model – see first line of (23) – and is irreducible as long as the noise intensity is maintained at this level. On the other hand, its estimate m4 (red) is stabilized with greater accuracy in about than 1 h. This is

32

P. Nguyen, R. Tenno / Journal of Process Control 47 (2016) 22–34

Temperature residual

Control variable u 0

temperature [°C]

100 98

°C

96 94 92 90 88

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

−1 −2 −3 −4 −5

time [h] Indirect control variable

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

time [h] Operating regions of CV

0.04

1 0.8

0.02

κ1 [−]

[m3/s]

0.03

0.01 0

operating curve steady−state region transition region

0.6 0.4 0.2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

time [h]

0 82

84

86

Table 3 Reduction of estimation accuracy of all estimates when both process and measurement noise intensity is increased by 20%.

Reduction of estimation accuracy

m1

m2

m3

m4

−11%

−12%

−19%

−2%

an expected result due to the presence of the estimator, albeit there is an undershoot in the transition phase. Subsequently the estimate approaches the reference value relatively slowly in an asymptotic manner. The settling time can be reduced by increasing the control gain KP , which would cause higher agitation in the initial phase. The corresponding control u(t) and the indirect control variable ϕ(u) are depicted in Fig. 10. The depicted estimation results in Figs. 6–9 are based on the specified process and measurements noise intensities (26), which serve as the reference case for the following noise sensitivity analysis. It is of interest how the estimation accuracy degenerates when both process and measurement noise intensities are increased. The filter performance can be quantified through the standard deviation of a residual defined as the difference between the process and its estimate. =

 



E (z − m)2 ,

(43)

where E denotes the mathematical expectation operator. It has been found in simulations that the estimation accuracy of the unknown states decreases by 11% on average when both the process and measurement noise are increased by 20% with respect to the reference case. The details are summarized in Table 3. Next, it is investigated in which region the CV operates regarding its nonlinear characteristics dependent on the perturbed outlet temperature TE,out . It is interpreted as an external disturbance due to the uncontrolled heat injection from the engine load program. Its key property is the equal flow distribution to both branches – bypass and heat exchanger in case of TE,out = u(t). For that purpose the temperature residual is defined as T = TE,out − u(t), which indicates the deviation from the linear operating region of the CV. It is observed that most of the time the temperature residual is within the range of T =−2.3 . . . −1.8 ◦ C. The main operation of the CV is in the nonlinear region (red) for most of the time. Only during the transition period it operates closer to the linear region (green). Those observations are displayed in Fig. 11.

88

90

92

94

96

98

temperature [°C]

Fig. 10. Top: Direct control variable u(t). Bottom: Indirect control variable ϕ(u).

Fig. 11. Top: The temperature residual T indicates the difference between TE,out and the thermostat set-point u(t). Bottom: Shift of the operating region of the CV during the simulation.

Fig. 12. Innovation process between 1 h and 1.1 h.

Table 4 Computed statistics of the innovation process over T = 14 h. Mean

Standard deviation

0.00 0.00

1.00 1.01

The innovation process ε¯ t is shown for the time period between 1 h and 1.1 h in Fig. 12 for clearer visualization. However, its characteristics are computed for the entire simulation time and are confirmed to have white noise properties as the applied CGF is optimal in the mean-square error sense. The statistics of the computed innovation process are listed in Table 4. A simple stability test is conducted for the system (25), where a random walk model is applied to the first three coordinates, which makes the corresponding subsystem neutral; this is evident from the drift model (26). The fourth coordinate of (25) – which represents the heat exchanger temperature dynamics – is considered in open-loop configuration first. It is stable, if the last element of the matrix a1 (t, yt )4,4 is less than one in terms of absolute values. The stability resource d is defined as: dOL = ||sOL | − 1| with

sOL = a1 (t, yt )4,4

(44)

P. Nguyen, R. Tenno / Journal of Process Control 47 (2016) 22–34

Open Loop 1 0.998 0.996 stability threshold open loop

0.994 0.992 0.99

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

time [h] Closed Loop 1

33

the downstream (green) is achieved within 2 h of transition time. On the other hand, the heat flow between the interface of the HTand LT circuit is more sensitive to the external disturbance of a rapid loading change in the diesel engine. For instance, at roughly t = 9 h the engine load is changing as a step, which is detected from the corresponding heat injection in Fig. 5. This causes a transitional period between 9 and 12 h when the heat flow slowly converges back to relative steady conditions. After stabilizing the unmeasured state z4 to the reference Td , the transferred heat towards the LT-circuit is only dependent on the control law (10) through QLT , which regulates the exchanged flows between HT- and LT-circuit in order to maintain the fixed set-point.

0.9

8. Conclusion

0.8 stability threshold closed loop

0.7 0.6

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

time [h] Fig. 13. Top: Open loop stability. Bottom: Closed loop stability with linear feedback control Kp = 0.15.

Transferred heat 6 WHex

5

WEv

[MW]

4

WLT

3 2 1 0 −1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

time [h] Fig. 14. Recovered waste heat WHex and heat flows towards the evaporator WEv and LT-circuit WLT .

or in other words, it must lie inside the unit circle of the complex plane in general case. This is the stability test for the open-loop system. A simple stability test is conducted also for closed-loop system as the control law (39) is inserted. dCL = ||sCL | − 1| with sCL = a1 (t, yt )4,4 − KP a2 (t)4

In this paper, a dynamical model of a FWC system including its subsystems was developed based on a provided scheme and on first principles. The main focus has been on the development of the heat exchanger model including its numerous uncertainties. It has been modelled as a partially measured stochastic system due to insufficient knowledge about its geometry, flow conditions and heat transfer processes. It was proved, that the linearized heat exchanger is conditionally Gaussian, which makes the use of a conditionally Gaussian filter possible for parameter and state estimation and benefiting from its numerous advantages. In particular the estimation quality of the CGF is optimal in the mean square sense as the innovation process has shown to possess white noise properties. All higher statistics can be represented in closed form and are calculated from the mean and variance only. The conditional distribution of state and parameters for given measurements is Gaussian. Moreover, the CGF maintains all the aforementioned properties in the presence of stochastic coefficients, which depend on noisy measurements in a nonlinear fashion. This is the most beneficial upside compared to the EKF or UKF algorithm since it does not tolerate stochastic coefficients, which would result in flawed estimates through bias, efficiency and statistics. Furthermore, the inclusion of a nonlinear feedback control does not harm the filter performance. These results have been verified in simulations by using the derived first principle models and estimation algorithm along with a feedback control. One state and the three drifting parameters of the heat exchanger could be successfully estimated by the CGF with satisfying results regarding accuracy, convergence time and tracking capability while the unmeasured state could be stabilized to its reference. As a result the waste heat recovery process and the heat flows towards interconnected subsystems are fairly steady processes as demanded in the specifications.

References (45)

In Fig. 13, sOL and sCL (red) are shown along with the stability threshold 1 (blue) for the open-loop and closed-loop configuration, respectively. It is observed, that the stability resource of the open loop system is small (dOL < 0.005); the open loop system is close to neutral. Applying the feedback control with Kp = 0.15 makes the system more stable as the stability resource increases to (dCL ≈ 0.1 . . . 0.2). In Fig. 14, the recovered waste heat is shown in closed-loop configuration along with the heat flows towards the evaporator and LT-circuit. After a 3 h transition period, the waste heat recovery in the heat exchanger (blue) is more or less a steady process, which rejects the disturbance heat injection WHeat well from the diesel engine side. The effect of the process and measurement noises are evident. A steady heat flow towards the connected evaporator in

[1] F.P. Incropera, S. DeWitt, T. Bergman, A. Lavine, Fundamentals of Heat and Mass Transfer, 6th ed., John Wiley & Sons, Inc., 2007. [2] J.H. Lienhard IV, J.H.V. Lienhard, A Heat Transfer Textbook, 4th Edition (Dover Civil and Mechanical Engineering), Dover Publications, 2011. [3] A. Neale, D. Derome, B. Blocken, J. Carmeliet, Determination of surface convective heat transfer coefficients by CFD, in: 11th Canadian Conference on Building Science and Technology, Banff, Alberta, Canada, 2007. [4] T. Defraeye, B. Blocken, J. Carmeliet, {CFD} analysis of convective heat transfer at the surfaces of a cube immersed in a turbulent boundary layer, Int. J. Heat Mass Transf. 53 (1-3) (2010) 297–308. [5] A. Benkouider, J. Buvat, J. Cosmao, A. Saboni, Fault detection in semi-batch reactor using the {EKF} and statistical method, J. Loss Prevent. Process Ind. 22 (2) (2009) 153–161 [Online]. Available at: http://www.sciencedirect.com/ science/article/pii/S0950423008001423. [6] Y. Chetouani, Design of a multi-model observer-based estimator for fault detection and isolation (FDI) strategy: application to a chemical reactor, Br. J. Chem. Eng. 25 (4) (2008) 777–788. [7] K. Graichen, V. Hagenmeyer, M. Zeitz, Adaptive feedforward control with parameter estimation for the chylla-haase polymerization reactor, in: 44th

34

P. Nguyen, R. Tenno / Journal of Process Control 47 (2016) 22–34

IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC’05, IEEE, 2005, pp. 3049–3054. [8] M. Nyberg, T. Stutte, Model based diagnosis of the air path of an automotive diesel engine, Control Eng. Pract. (2004) 513–525. [9] R. Tenno, Nonlinear parameters estimation via zakai filtering in deposition process control, IFAC-PapersOnLine 48 (28) (2015) 1124–1129.

[10] A. Hajdu, T. Mach, P. Medina, E. Yu, P. Kügler, E. Lindner,;1; Team 7 Control of a Shell and Tube Heat Exchanger. [11] R.S. Lipster, A.N. Shiryaev, Statistics of Random Processes II (Applications), 2nd ed., Springer, 1977.