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State space model and robust control of plate heat exchanger for dynamic performance improvement
Accepted Manuscript State space model and robust control of plate heat exchanger for dynamic performance improvement Yaran Wang, Shijun You, Wandong Zheng, Huan Zhang, Xuejing Zheng, Qingwei Miao PII: DOI: Reference:
S1359-4311(16)31762-8 https://doi.org/10.1016/j.applthermaleng.2017.09.120 ATE 11177
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
19 September 2016 15 August 2017 23 September 2017
Please cite this article as: Y. Wang, S. You, W. Zheng, H. Zhang, X. Zheng, Q. Miao, State space model and robust control of plate heat exchanger for dynamic performance improvement, Applied Thermal Engineering (2017), doi: https://doi.org/10.1016/j.applthermaleng.2017.09.120
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State space model and robust control of plate heat exchanger for dynamic performance improvement Yaran Wang a, Shijun You a,b, Wandong Zheng a,b*, Huan Zhang a,b, Xuejing Zheng a,b, Qingwei Miao a a
School of Environmental Science and Engineering, Tianjin University, Yaguan Road 135#,
Jinnan District, Tianjin 300350, China b
National Engineering Laboratory for Digital Construction and Evaluation Technology of
Urban Rail Transit, Tianjin University, Yaguan Road 135#, Jinnan District, Tianjin 300350, China * Corresponding Author: Tel. / Fax: +86 22 2789 2626. E-Mail address: [email protected]
Highlights A state space model of plate heat exchanger was established. The proposed state space model of plate heat exchanger was verified with test data. A two-degrees-of-freedom
controller was designed for plate heat exchanger.
Control performances of the proposed
and PI controllers were compared.
Keywords plate heat exchanger, state space model, robust control, two-degrees-of-freedom control, dynamic performance improvement
Abstract Plate heat exchanger takes an important role in the modern industrial engineering. For large scale district heating system, plate heat exchanger is the key equipment in controlling the heat output from the heating substation. High performance control of 1
plate heat exchanger is essential for stable and efficient operation of the heating substation. The governing equation of plate heat exchanger is a pair of partial differential equations, which is not suitable for model based control. In this work, a linear state space model was developed to describe the dynamic characteristic of plate heat exchanger. Validation results show that the proposed model can capture the dynamics of plate heat exchanger with satisfied accuracy. Based on the state space model, the two-degrees-of-freedom
loop-shaping controller was designed for
plate heat exchanger to enhance the control performance. The reference tracking and disturbance
rejection performances of the proposed control algorithm were
studied and compared with the PI control. Results show that the presented two-degrees-of-freedom
loop-shaping control is superior to the PI control.
Nomenclature matrix of state space model width of the plate heat exchanger flow channel ( ) matrix of state space model matrix of state space model matrix of state space model matrix of state space model specific heat capacity of water distance between neighboring plates hydraulic diameter intermediate variable of two-degrees-of-freedom
loop-shaping control
scheme relative error of the simulated outlet temperature of high temperature side relative error of the simulated outlet temperature of low temperature side uncertain open-loop model transfer function from inlet temperature of the low temperature side to outlet temperature of low temperature side
2
Nomenclature transfer function from mass flow rate of the high temperature side to outlet temperature of high temperature side transfer function from mass flow rate of the low temperature side to outlet temperature of high temperature side transfer function from inlet temperature of the high temperature side to outlet temperature of high temperature side transfer function from inlet temperature of the low temperature side to outlet temperature of high temperature side overall heat transfer coefficient overall heat transfer coefficient of a specific steady state transfer function matrix of two-degrees-of-freedom ,
loop-shaping controller
transfer function matrix components of the two-degrees-of-freedom loop-shaping controller length of the flow channel number of flow channels in each side coprime factorization of the shaped model
,
number of the control volumes in a flow channel volume flow rate volume flow rate of a specific steady state maximum flow rate of high temperature side time temperature steady state temperature compensator transfer function specified to shape the open-loop model the reference transfer function model that describes the desired response of the closed-loop system coordinate along the flow channel intermediate variable of two-degrees-of-freedom
loop-shaping control
scheme heat conductivity coefficient (
)
density of the water small deviation & increment symbol of variable intermediate variable of two-degrees-of-freedom scheme scalar value specified by controller designer length of a control volume ,
coprime uncertainty 3
loop-shaping control
Nomenclature Subscripts high temperature side inlet low temperature side outlet plate
1. Introduction The plate heat exchangers are extensively used in energy industries, such as district heating systems [1], absorption chilling systems [2] and electricity production systems [3]. Efficient control of the plate heat exchanger is a key problem to enhance the dynamic performance of processes involved with plate heat exchanger. In large scale district heating systems, the plate heat exchangers are key components in controlling the heat outputs to secondary networks. Efficient control of plate heat exchanger improves the dynamic response and enhances the stability of the control system. Mathematical modeling is one of the most effective approaches to investigate the dynamic characteristics and control performances of plate heat exchangers. In the literatures, numerical modeling approaches were extensively studied and applied to heat exchanger control and operation analysis. Thermal dynamic model of a cross-flow heat exchanger was established and solved numerically to predict the transient response for multiple variable input operating analysis in Gao et al.’s work [4]. In Korzeń and Taler’s study, new equation set describing transient heat transfer process in tube and fin cross-flow heat tube exchanger was given and subsequently solved using the finite volume method, and good agreement between numerical and 4
experimental results was obtained with the method [5]. Using this technique, a model based control method was developed by Taler, and the experiment results show that the proposed control method is more stable compared with the PID controller [6]. Recently, studies on applying numerical methods to optimal design of heat exchangers have gained more attentions. Lyytikäinen et al. derived a practicable modeling method for plate heat exchanger geometry optimization by reducing the equations from 3D to 2D, which reduces the elapsed time of CFD simulations from hours to minutes [7]. Zhang et al. developed a distributed parameter model utilizing correlations of the heat transfer and flow friction to avoid solving partial differential equations for optimal design of the plate-fin heat exchanger, which largely reduces the computation burden in comparison with CFD method [8]. By dividing the plate heat exchanger into multiple slices in the direction of fluid flow and assuming the wall temperatures to be constant for the channels in each slice, Qiao developed a new model for the analysis of plate heat exchangers with multi-fluid, multi-stream and multi-pass configurations, which was solved without the need of knowing the fluid condition in the adjacent channels [ 9]. Despite the effectiveness of numerical calculation techniques for optimal design, it’s difficult to be applied to real-time control due to the computational pressure. Control-oriented modeling and feedback controller design methods for heat exchangers have also been paid wide attention, especially the Model Based Predictive Control. Model Based Predictive Control is one of the most effective solutions for industrial applications, due to its capabilities in dealing with nonlinearities, constraints,
5
uncertainties and multivariable coupling characteristics involved in the modern complex industrial processes. Fischer et al. developed a fuzzy model-based predictive controller for the temperature control of an industrial-scale cross-flow water/air heat exchanger, and a local recursive least-square algorithm was proposed for on-line adaptation of the fuzzy model [10]. In Akman and Uygun’s research, a nonlinear model predictive control scheme of a heat exchanger network was established and solved sequentially with an algebraic steady state optimization model, while satisfying the temporal and steady-state hard/soft constraints imposed on the target temperatures of the heat exchanger network [ 11 ]. Abdelghani-Idrissi et al. implemented a predictive functional control framework on a counter current heat exchanger, with a global model representing the response to inlet temperature variations [12]. González et al. discussed the online optimization and control of a heat-exchanger network through a two-level control structure, where the low level is a constrained model predictive control and the high level is a supervisory online optimizer [13]. Arbaoui et al. implemented a predictive functional control algorithm on a counter-current tubular heat exchanger to control the outlet temperature based on an approximated first order model, corresponding to the response of the heat exchanger to a step change of the flow rate [14]. A robust model predictive control algorithm for heat exchanger network was established and shows more satisfied performance and less cooling medium consumption with parameter uncertainties in comparison with the optimal linear quadratic control in Bakošová and Oravec’s research [15]. The combination of neural network and predictive control concepts has
6
also attracted some researches. Vasičkaninová and Bakošová proposed the combination of neural network predictive controller and the fuzzy controller in the complex control structure of heat exchanger with an auxiliary manipulated variable [16]. The main drawback of model based predictive control is its large computational burden due to solving online optimization problem. Other advanced control methods for heat exchangers such as: robust control, nonlinear control and fuzzy logic control have also been concerned. A nonlinear robust control approach for counter flow heat exchanger was developed based on the Galerkin method by Kanoh et al. [17]. An outlet temperature control algorithm for double-pipe heat exchangers, which takes into account and actually exploits the analytical and stability properties inherent to the open-loop dynamics was proposed to ensure positivity and boundedness of the input flow rate in Zavala-Río et al.’s study [18]. Based on nonlinear control theory, Maidi proposed a state-feedback control law that ensured a desired performance of a measured output defined as the spatial weighted average temperature of the internal fluid [19]. Dulău and Oltean designed a robust controller for heat exchanger with a second order time delay model by synthesis to ensure stable control performance with model uncertainties [20]. In Al-Dawery’s work, a first order model with time delay has been developed to suggest the transient responses of a plate heat exchanger, and a fuzzy logic controller of the plate heat exchanger was designed to achieve less settling time and oscillatory behavior [21]. Michel and Kugi developed a control strategy without knowledge of the heat transfer for plate heat exchanger based on controlling the total thermal energy
7
stored in the heat exchanger and a Kalman Filter to estimate the states [22]. Chentouf studied the Boundary feedback stabilization of a class of heat exchangers governed by hyperbolic non-autonomous evolution systems with non-smooth coefficients [23]. In this paper, a new control algorithm, the two-degrees-of-freedom
robust
controller was developed for the plate heat exchanger to enhance the dynamic performance. The two-degrees-of-freedom
robust control was a newly developed
control technique in recent decades to improve the control performance of dynamical systems [24]. The PID control is a standard and effective industrial control system solution, which has been applied to more than half of the industrial applications, while in most industrial applications, instead of PID, the PI control is usually used, due to the sensitive responses of the differential term to sensor noises [25]. Salam et al.’s work [21] showed that the dynamic performance of plate heat exchanger under PI control is superior to that of the plate heat exchanger under PID control. Hence, in order to illustrate the feasibility and validity of the proposed two-degrees-of-freedom controller for plate heat exchanger, the dynamic performances of the proposed controller are compared with the PI controller. The rest of this paper is organized as follows. The next section derives the governing equations of plate heat exchanger. After spatial discretization and linearization of the governing equations, the state space model was developed. Subsequently, the two-degrees-of-freedom
robust
control was introduced and applied to the control of plate heat exchanger. Next, the model precision of the state space model was validated with the test data from the literature. Then, the reference tracking and disturbance rejection performances of the
8
robust
control were evaluated through case studies in comparison with the PI
control. Finally, this paper closes with a summary of conclusions.
2. Modeling the plate heat exchanger The schematic of heat transfer process in plate heat exchanger is shown in Fig. 1, where the high temperature and low temperature fluids are separated by metal plates. The derivation of plate heat exchanger governing equation is introduced in subsection 2.1, and the linear state space model is established in subsection 2.2.
2.1. Governing equations of plate heat exchanger The governing equations of plate heat exchanger can be derived by simplifying the convective-diffusion equations of the fluid flow in the flow channels, and the following assumptions are considered [26]: 1. The dissipation effect of the fluid is negligible; 2. The thermo-physical properties of fluids are constants; 3. The fluids are assumed to be incompressible; 4. Heat losses from the heat exchanger to the surroundings are negligible; 5. Thermal dynamics of the metal plates are negligible; 6. The heat conduction of the fluid along the flow direction is negligible in comparison with the heat convection; 7. No phase changes occur inside the heat exchanger.
9
High temperature side
x L
Tl ql n
Tl dx x
Th
qh n
dx
dx
Th
Tl
Th dx x
Low temperature side
Fig. 1. Schematic of plate heat exchanger heat transfer process. The control volumes of the fluids and the coordinate system are shown in Fig. 1. For each high temperature side control volume, the heat balance equation is listed as follows: (1) For low temperature side, a similar heat balance equation is expressed as: (2) Thus the governing equations of the plate heat exchanger can be written as: (3) (4) This is a pair of partial differential equations. It is noted that system described by Eq. (3) and Eq. (4) is nonlinear distributed parameter system, which is also called the infinite dimensional system. For further study on the controller synthesis of the plate heat exchanger, the system model needs to be reduced to a finite dimensional form, which is to reduce the partial differential equations to ordinary differential equations. There are generally two kinds of effective approaches. One is the Galerkin method, 10
the other is the finite volume & difference method. In reference [17], the former technique was applied to simplify the distributed parameter model of counter flow heat exchanger into a nonlinear state space form. Reference [26] used the latter approach to obtain an accurate low order model for plate heat exchanger. In this paper, the finite volume integral method is adopted, and a linear state space model of the plate heat exchanger is derived for robust controller synthesis.
2.2. Model simplification method In this subsection, the finite volume integral method was adopted to simplify Eq. (3) and (4). Linearization approach by Taylor expansion was used to derive the linear state space model of plate heat exchanger. To apply the finite volume integral method, each flow channel of the high temperature and low temperature sides is divided into
control volumes, as is
shown in Fig. 2. The control volumes in each side are numbered consecutively from to
. Integration of the Eq. (3) and (4) within the th control volume along the
flow channels leads to the following equations: (5) (6)
11
Fig. 2. Control volumes in the flow channels. Assuming uniform temperature distributions within the th control volumes, Eq (5) and (6) can be approximated as: (7) (8) where
, and
. Then the following equations can be obtained: (9) (10)
Eq. (9) and (10) are finite dimensional model, which is nonlinear and difficult for controller design. For further simplification of the model, steady state solutions of Eq. (9) and (10) are solved. Steady state solutions of differential equations satisfy the steady state version of the differential equations, in which the time derivative terms are zero. The steady state forms of Eq. (9) and (10) are the following algebraic equations: (11)
12
(12) Let , ,
,
,
and
,
,
which are all
matrices, and
Eq. (11) and (12) can be written into matrix equation form: (13) where
is the steady state temperatures of control volumes and needs to be
calculated. The steady state solution of Eq. (9) and Eq. (10) are solved as:
.
To linearize Eq. (9) and Eq. (10), each variable is replaced by the sum of steady state value and deviation about the steady value. The variables are independent variables. Let
,
,
represents the independent variable,
13
and can be
written into the following form: (14) where
is a steady state value of
heat transfer coefficient
, and
is the deviation about
. The overall
is a dependent variable concerned with the mass flow rates
of the high temperature and low temperature sides. The overall heat transfer coefficient
can be calculated by the following formula [26]: (15)
where
is the heat transfer coefficient of the plate.
and
are
determined by the following formulas [26]: , , where
,
,
,
,
is the hydraulic diameter. The overall heat transfer coefficient
(16) (17) is a
dependent variable, which can be replaced by the following one-order Taylor expansion: (18) where the partial derivatives
and
can be calculated from Eq. (16)
and Eq. (17). Substituting Eq. (14) and Eq. (18) into Eq. (9) and Eq. (10), the high order terms are neglected. Therefore, the linearized equations are derived as follows:
(19)
14
(20) Do the following parameter replacement: , ,
, , , ,
(21)
and
,
,
(22) ,
,
and
are
matrices.
,
,
,
-dimensional vectors. The state equation of the state space model is written as follows: 15
and
are
(23) where
is the temperature vectors of the high temperature and low temperature side
control volumes.
is the inlet temperature of the high temperature side, and
is the inlet temperature of the low temperature side. According to the control volumes shown in Fig. 2, the outlet temperatures of the two sides are the state temperatures: , where
(24)
is the outlet temperature of the high temperature side;
is the outlet
temperature of the low temperature side. Therefore the outlet temperatures can be written in the following form: , where
(25) ,
. Combining Eq.
(23) and Eq. (25), the linear state space model of the plate heat exchanger is derived.
3. Robust
Control of plate heat exchanger
3.1 Input-output representation of plate heat exchanger The control scheme of plate heat exchangers in most industrial applications is to maintain the outlet temperature of one side around the desired values by regulating flow rate of the other side. Usually, different disturbances, such as the inlet temperature and flow rate variations may occur, which will have an impact on the control process. Hence, an efficient control system should have the ability to reject the disturbances and keep good tracking of the desired values. In order to study the relationship between the inputs and outputs, the transfer functions from the inputs to the outputs are defined and calculated as follows: 16
(26) (27) (28) (29) (30) (31) (32) (33) The input-output structure of plate heat exchanger system is illustrated by block diagram shown in Fig. 3. The system inputs can be divided into two categories: the manipulated inputs and disturbance inputs. For the district heating application of plate heat exchangers, the high temperature water flow rate input, and the other inputs outlet temperature
,
and
is taken as the manipulated are as disturbance inputs. The
is the controlled output and needs to track the desired setpoint.
Fig. 3. Input-output structure of the plate heat exchanger.
The input-output structure can also be written in the following form:
17
(34) (35) Since the controlled output is
, only subsystem in the blue dashed box is needed
for controller synthesis. This subsystem is described in Eq. (34).
3.1. Two-degrees-of-freedom
loop-shaping control of plate heat exchanger
The control objective is to keep the outlet temperature of the low temperature side ( side (
) tracking the desired value, by controlling the flow rate of high temperature ), under the variations of the inlet temperature of high temperature side (
),
the flow rate of low temperature side ( ) and the inlet temperature of low temperature side (
). Robust control of plate heat exchanger is to achieve acceptable control
performance, that is, to keep the controlled output
within specified bound from
its desired value, in spite of unknown but bounded disturbances (
,
and
and system parameter uncertainties (such as the overall heat transfer coefficient using constrained input (
) and available measurement (
), ),
). There are several
design approaches of robust controllers, such as the mixed sensitivity method, synthesis method,
loop-shaping method and LMI based time domain methods
etc. [24, 27]. These methods are all applicable and effective robust controller design tools. Among these methods, the two-degrees-of-freedom
control approach is
adopted for plate heat exchanger control in this paper, due to its convenient design procedure and satisfactory performances. The rest of this subsection explains the design procedure of the two-degrees-of-freedom
controller for plate heat
exchanger. Fig. 4 depicts the system structure of the two-degrees-of-freedom 18
loop-shaping control. Define
, where
function to shape the loop transfer function [24].
is a specified transfer
and achieve a stable and fast response
is the reference transfer function model that describes the desired
response of the closed-loop system. Transfer functions normalized coprime factorization of transfer function
and
are the
. To enhance the robustness
of the closed-loop plate heat exchanger system, the model uncertainty should be considered in the design stage and the uncertain model set
can be factorized as:
,
(36)
Fig. 4. Block diagram of the two-degrees-of-freedom where
and
control.
are unknown stable transfer function set, which describe the
system uncertainties of the specific model
. The uncertain model
can handle
varieties of model uncertainties [24], and the uncertainties of the seven assumptions in subsection 2.1 can also be incorporated in transfer function set.
.
is the bound of the unknown stable
is a specified reference model which represents the
desired response of the closed-loop plate heat exchanger system. And
is a scalar
parameter specified during the controller design. The two-degrees-of-freedom controller design problem can be formulated into an stabilizing controller
which minimizes the 19
control problem: find the norm of the transfer
matrix between
and
[24], see Appendix A for details.
The controller design procedure is as follows: Step 1, preliminarily specify the transfer functions . The technique for specifying
and
Step 2, formulate and solve the . The
and
, and the scalar
is given in [24]. control problem to obtain the controller
control problem solving methods are discussed more
detailedly in Ref. [24, 27], which can be conducted via MATLAB robust control toolbox. Step 3, simulate the closed-loop plate heat exchanger system to validate if satisfied reference tracking and disturbance rejection performances have been achieved. If not, go to step 1.
4. Results and discussion In this section, the state space model derived in this paper is validated with the test data from the literature. Then the two-degrees-of-freedom
controller is
designed and compared with the PI controller. Reference tracking and disturbance rejection performances of the two controllers were analyzed and discussed.
4.1. Model validation In order to validate the state space model of the plate heat exchanger proposed in this paper, the test data and system parameters in Ref. [26] are adopted. The state space matrices:
,
,
,
and
for plate heat exchanger provided in
Ref. [26] are calculated and listed in Appendix B. Dynamic simulation of the
20
proposed state space model with the test data of and the outlet temperatures temperatures
and
and
,
,
and
is conducted,
are calculated. The simulated outlet
are compared with the test data to validate the state space
model of plate heat exchanger Eq. (20). The comparison of and the relative error of
is shown in Fig. 5-a,
is shown in Fig. 5-b. The comparison of
is shown
in Fig. 6-a, and the relative error is shown in Fig. 6-b. The relative errors are calculated as: ,
(37)
, where
and
(38) are the calculated values at time instant ;
and
are the test values at time instant . 32 Model Test
30 28
Th,o ( )
26 24 22 20 18 16 0
100
200
300 Time (s)
400
500
Fig. 5-a. Comparison between the test data and model calculation of
21
600
.
10 8 6 4
Eh,o (%)
2 0 -2 -4 -6 -8 -10 0
100
200
300 Time (s)
Fig. 5-b. Relative error of
400
500
600
.
34 Model Test
32 30
Tl,o ( )
28 26 24 22 20 18 0
100
200
300 Time (s)
400
500
Fig. 6-a. Comparison between the test data and model calculation of
The relative errors of
and
are both varying within
600
.
, which indicates
that the proposed state space model of plate heat exchanger is effective, since the presented robust control technique can tackle and compensate the model uncertainty to ensure the control performance [24]..
22
10
8
El,o (%)
6
4
2
0
-2 0
100
200
300 Time (s)
400
Fig. 6-b. Relative error of
4.2. Comparison of robust
600
.
and PI control
In order to illustrate the two-degrees-of-freedom
500
feasibility and
validity of the proposed
loop-shaping control algorithm for the plate heat
exchanger, comparison between the
controller with PI controller was conducted
via a case study. As the effectiveness of the state space model of the plate heat exchanger has been validated in subsection 4.1, the plate heat exchanger parameters can be directly used in developing the state space model. The parameters of the plate heat exchanger studied in this subsection are from the manufacturer and listed in Table 1. In this subsection, the two-degrees-of-freedom
loop-shaping controller
and PI controller for plate heat exchanger are designed. Then the reference tracking and disturbance rejection performances of the two controllers are compared. Table 1. System parameters Description
Symbol
23
Value
Unit
channel width
0.8
channel length
1.36
water specific heat capacity
4220
empirical parameters
0.64
/
distance between neighboring plates
4.5
mm
plate thickness
0.5
number of flow channels in each side
137
/
empirical parameters
0.23
/
empirical parameters
0.75
/
water density
970
thermal conductivity of high temperature side water
0.68
thermal conductivity of low temperature side water
0.67
thermal conductivity of plate
15
dynamic viscosity of high temperature side water
0.00028
dynamic viscosity of low temperature side water
0.00041
steady state inlet temperature of high temperature side
80
o
steady state inlet temperature of low temperature side
50
o
steady state flow rate of high temperature side
0.03
steady state flow rate of low temperature side
0.03
Maximum flow rate of high temperature side
0.06
m
C C
The state space model of the plate heat exchanger studied in this subsection is derived and summarized in Appendix C. According to the design procedure of the two-degrees-of-freedom
loop-shaping controller illustrated in subsection 3.1, the
designing of the two-degrees-of-freedom specify the two transfer functions
and
frequency response of the open loop system 24
loop-shaping controller needs to .
is designed to shape the , and make the closed loop
system stable and have a faster response.
and
are designed by trial or error.
The transfer functions are selected as follows. (39) (40) According to the design procedures in subsection 3.1, the two-degrees-of-freedom loop-shaping controller is designed with MATLAB Robust Control Toolbox. Since the model order of the plate heat exchanger is order controller
, this leads to high
which is not applicable for implementation. In this
paper, in order to obtain a low order controller, the model reduction for the obtained high order controller is conducted. Details on model order reduction have been summarized in Appendix D. The three order controller is calculated and the transfer function form is as follows. (41) (42) The PI controller is designed with Ziegler-Nichols method, and tuned by trial and error to ensure the closed loop stability and fast response with the control input in an allowable range. The PI controller for the plate heat exchanger studied in this section was tuned as: (43) To study the closed loop performance of the proposed two-degrees-of-freedom loop-shaping controller, comparison between the controller were conducted in the following subsections.
25
controller and PI
4.2.1. Reference tracking performance Dynamic responses of PI and the proposed
in tracking a sequence of desired step variations with
controllers are shown in Fig. 7-a. As is shown in this figure,
the overshoots of
control and PI control are 2.5% and 6% respectively. The
settling times for
control and PI control are 250s and 500s respectively.
Therefore, the application of
controller ensures good tracking of the desired step
references, while using PI controller leads to larger overshoots and longer settling time. Fig. 7-b shows the variations of high temperature side flow rate in tracking the sequence of step references. Fig. 7-b indicates that the ensure the flow rate within the maximum value
control and PI control can .
70 68 66
Tl,o ( )
64 62 60 58 56 Reference Robust control PI control
54 52 0
500
1000
1500
Fig. 7-a. Responses of
2000 Time (s)
2500
3000
in tracking step reference of
26
3500
.
4000
0.06 Robust control PI control 0.05
qh (m3/s)
0.04
0.03
0.02
0.01 0
500
1000
1000
Fig. 7-b. Responses of
2000 Time (s)
2500
3000
in tracking step reference of
3500
4000
.
70 68 66 64
Tl,o ( )
62 60 58 56 54 Reference Robust control PI control
52 50 0
200
400
600
Fig. 8-a. Responses of
800
1000 Time (s)
1200
1400
1600
in tracking ramp reference of
27
1800
.
2000
0.06
0.05
qh (m3/s)
0.04
0.03
0.02
0.01 Robust control PI control 0 0
200
400
600
Fig. 8-b. Responses of
The dynamic responses of with PI and
800
1000 Time (s)
1200
1400
1600
in tracking ramp reference of
1800
2000
.
in tracking a sequence of desired ramp variations
control are shown in Fig. 8-a. This figure denotes that for both
control algorithms there are small steady-state errors when tracking the desired ramp references. However, PI control leads to larger steady-state error than The variations of high temperature side flow rate
control.
in tracking the step references is
shown in Fig. 8-b, which denotes that anti-saturation can be ensured with both controllers.
4.2.2. Disturbance rejection performance Responses of
in rejecting low temperature side flow rate variations (shown
in Fig. 10-a) with PI and the indicates that the oscillation of the
controllers are compared in Fig. 9-b. This figure controlled by PI controller is more intense than
controller. The maximum deviations of
are 0.9 oC and 3.7 oC respectively. Responses of
28
for
control and PI control
in Fig. 9-c means that the control
inputs for the two controllers are both within an allowable range. 0.034
0.032
ql (m3/s)
0.03
0.028
0.026
0.024
0.022 0
200
400
600
800 Time (s)
1000
1200
1400
1600
Fig. 9-a. Variations of 70 68 66 64
Tl,o ( )
62 60 58 56 54 Reference Robust control PI control
52 50 0
200
400
600
Fig. 9-b. Responses of
800 Time (s)
1000
1200
in rejecting variations of
29
1400
1600
0.06
0.05
qh (m3/s)
0.04
0.03
0.02
0.01 Robust control PI control 0 0
200
400
600
Fig. 9-c. Responses of
Fig. 10-a. shows a sequence of in rejecting the variations of
800 Time (s)
1000
1200
in rejecting variations of
1400
1600
.
variations. The dynamic responses of
for PI control and
10-b. This figure denotes that the settling time of control. And the maximum deviations of
control are shown in Fig. is longer for PI control than
for
control and PI control are
0.4 oC and 0.6 oC respectively. Fig. 10-c illustrates the dynamic responses of
,
which indicates that both controllers can ensure control inputs within an allowable range.
30
70 65 60
Tl,in ( )
55 50 45 40 35 30 0
200
400
600
800 Time (s)
Fig. 10-a. Variations of
1000
1200
1400
1600
.
68 66 64
Tl,o ( )
62 60 58 56 Reference Robust control PI control
54 52 0
200
400
600
Fig. 10-b. Responses of
800 Time (s)
1000
1200
in rejecting variations of
31
1400
.
1600
0.06
0.05
qh (m3/s)
0.04
0.03
0.02
0.01 Robust control PI control 0 0
200
400
600
Fig. 10-c. Responses of
The dynamic responses of for PI control and
1000
1200
in rejecting variations of
1400
1600
.
in rejecting the disturbances of
(Fig. 11-a)
control are shown in Fig. 11-b. This figure also indicates that
the settling time of deviations of
800 Time (s)
for
is longer for PI control than
control and PI control are 1.25 oC and 2 oC respectively.
The responses of high temperature side flow rate both the PI control and
control. The maximum
shown in Fig. 11-c indicate that
control can ensure control inputs within an allowable
range.
32
82
Th,in ( )
81
80
79
78 0
200
400
600
800 Time (s)
Fig. 11-a. Variations of
1000
1200
1400
1600
.
68 66 64
Tl,o ( )
62 60 58 56 Reference Robust control PI control
54 52 0
200
400
600
Fig. 11-b. Responses of
800 Time (s)
1000
1200
in rejecting variations of
33
1400
.
1600
0.06
0.05
qh (m3/s)
0.04
0.03
0.02
0.01 Robust control PI control 0 0
200
400
600
800 Time (s)
Fig. 11-c. Responses of
1000
1200
in rejecting variations of
1400
1600
.
4.2.3. Performances with parameter uncertainties It is impractical to obtain the precise thermal and physical parameters of the plate heat exchanger without accurate measurements, especially for the heat transfer coefficient. In this study, the heat transfer coefficient was calculated with the correlation of Nusselt numbers number
and
in Eq. (17). The correlation of Nusselt
is usually given by the manufacturers. However, the parameters in the
correlation might not be precise for a specific plate heat exchanger. To consider such factor, the parameter
in the Nusselt number correlation was assumed to decrease
30%, and the controllers were compared under such parameter uncertainty. The responses of disturbances of
(Fig. 9-a),
under the parameter uncertainty to the combined (Fig. 10-a) and
(Fig. 11-a) were shown in
Fig. 12-a. This figure indicates that,
controlled by the PI controller fluctuates
more intensely than the proposed
controller. In Fig. 12-a, the maximum
deviation of
controlled by PI controller is 3 oC. Nevertheless, the 34
controller
is able to keep indoor air temperature within
1 oC from its desired value and ensure
less steady-state error when tracking ramp reference, despite the disturbances of ,
and the parameter uncertainty, with the flow rate in an allowable range
(Fig. 12-b). Therefore, the proposed
controller is robust to disturbances and heat
transfer coefficient uncertainty. 70 68 66 64
Tl,o ( )
62 60 58 56 54 Reference Robust control PI control
52 50 0
200
400
Fig. 12-a. Responses of
600
800 Time (s)
1000
in rejecting variations of
1200
,
1400
and
1600
.
0.06
0.05
0.04
qh (m3/s)
,
0.03
0.02
0.01 Robust control PI control 0 0
200
400
Fig. 12-b. Responses of
600
800 Time (s)
1000
in rejecting variations of
35
1200
,
1400
and
1600
.
4. Conclusions In this paper, the state space model of the plate heat exchanger was established. Based on the state space model, the two-degrees-of-freedom
loop-shaping
controller was presented to improve the dynamic performance of the plate heat exchanger. Then the state space model was verified with test data. The
controller
and the PI controller were designed for a plate heat exchanger. Reference tracking and disturbance rejection responses were compared and analyzed to validate the dynamic performance of the proposed
controller. According to the above analysis and
discussions, the conclusions are drawn as follows: (1) Comparison of the test data and model simulation results in subsection 4.1 shows that the proposed state space model of plate heat exchanger can provide sufficient accuracy. The relative errors of the two outlet temperatures are both varying within
and
10%, which are in a satisfied range. Therefore, the state
space model of plate heat exchanger proposed in this paper can describe the thermal dynamics of plate heat exchanger precisely. (2) The reference tracking responses of
controlled by the PI controller in
tracking the step and ramp references all have larger overshoots and longer settling time. For step references, the overshoots of 6% respectively. The settling times for
control and PI control are 2.5% and control and PI control are 250s and 500s
respectively. There is larger steady state error of PI control than
control when
tracking desired ramp references. These indicate that the presented
controller is
superior to the PI controller in tracking desired values.
36
(3) The disturbance responses of the low temperature side flow rate,
controlled by ,
controller in rejecting
variations are more oscillatory than PI control and PI control are 0.9 oC
controller. The maximum deviations of
for
and 3.7 oC respectively, when tracking
variations. The maximum deviations of
control and PI control are respectively 0.4 oC and 0.6 oC, when tracking
for
variations. The maximum deviations of
for
also 1.25 oC and 2 oC respectively in tracking presented
control and PI control are variations. Therefore the
control algorithm can guarantee better disturbance rejection
performances when disturbances exist. (4) The proposed two-degrees-of-freedom
loop-shaping controller is able to
ensure the satisfied dynamic performance for both reference tracking and disturbance rejection under system capacity parameter uncertainties. When the key parameter in the Nusselt number correlation decreases 30%, which means that the heat transfer coefficient becomes smaller, the proposed deviations within
controller is still able to keep the
1 oC, while the maximum deviation of PI control is 3 oC. And the
steady state error of
controller is smaller than PI control. Besides the
controlled by the PI controller is more oscillatory than the proposed
controller.
These confirm the robustness of the presented two-degrees-of-freedom loop-shaping controller of the plate heat exchanger.
Appendix A. The
two-degrees-of-freedom
loop-shaping
37
control
design
problem
illustrated in Fig. 4 can be transformed to the following standard Find the stabilizing controller
control problem:
which minimizes the
transfer matrix between the signals
and
The transfer matrix from the signals
norm of the
. to
is
(A.1)
norm of
is the following maximum singular value: (A.2)
Then the design problem becomes minimizing the maximum singular value of over frequency range
. This problem can be solved by algebraic Riccati
equations or LMI methods with MATLAB Robust Control Toolbox [24, 27].
Appendix B. State space model for model validation The state space model matrices for plate heat exchanger parameters provided in Ref. [26] are calculated with the method proposed in this paper. The matrices are listed as follows.
1.892 1.647 1.892 1.647 1.892 A11
1.647 1.892 1.647 1.892 1010
38
1.243 0.998 1.243 0.998 1.243 A22 0.998
1.243 0.998 1.243 1010
And the state matrix ;
;
; ; ;
Appendix C. State space model for controller comparison The state space model matrices for the plate heat exchanger parameters listed in Table 1. are calculated and shown as follows.
0.678 0.447 0.678 0.447 0.678 A11
0.447 0.678 0.447 0.678 1010
39
0.678 0.447 0.678 0.447 0.678 A22 0.447
0.678 0.447 0.678 1010
And the state matrix ;
;
Appendix D. Controller order reduction The model order of the two-degree-of-freedom
loop-shaping controller
developed for plate heat exchanger is high, which is complicated for implementation. Hence, model order reduction of the controller is needed. Balanced truncation is one of the effective and applicable approaches for model order reduction. The general idea of balanced truncation is to neglect those parts of the original system that are less observable or/and less controllable [27]. Assume that the controller to be reduced has the following state space form (D.1) 40
(D.2) where
is called the balanced realization, if the solutions
and
to the following Lyapunov equations (D.3) (D.4) are such that and
, with
.
are called the controllability Gramian and observability Gramian, respectively.
When the system is balanced, both Gramians are diagonal and equal.
,
is the th Hankel singular value of the system. For a general system not in the balanced realization form, a state similarity transformation (balancing transformation) is required, for details of the transformation, see [27]. If the controller to be reduced has already been transformed to the balance realization, the matrix
of the controller can be written as
, and
and
, with . The matrices
,
be partitioned compatibly as ,
,
Then a reduced-order controller
(D.5)
can be defined by (D.6)
The reduced controller
is of th order and is called the balanced truncation of the
full order ( th) controller
. And the following error bound is satisfied (D.7)
where
denotes the trace of the matrix
41
, i.e.
, the
sum of the last
Hankel singular values. To reduce the original model into an
th-order system, there should be a large gap between
and
, i.e.
.
The preceding procedures of obtaining the reduced-order controller can be realized in Matlab [27].
Acknowledgement This work was supported by the State Oceanic Administration of China (Grant No. cxsf-43); and the National Natural Science Foundation of China (No. 51106110).
Reference [1] H. İ. Tol, S. Svendsen, A comparative study on substation types and network layouts in connection with low-energy district heating systems, Energy Convers. Manage. 64 (2012) 551-561 [2] M. D. Vega, J. A. Almendros-Ibañez, G. Ruiz, Performance of a LiBr–water absorption chiller operating with plate heat exchangers, Energy Convers. Manage. 47 (2006) 3393-3407. [3] D. Walraven, B. Laenen, W. D’Haeseleer, Comparison of shell-and-tube with plate heat exchangers for the use in low-temperature organic Rankine cycles, Energy Convers. Manage. 87 (2014) 227-237. [4] T. Gao, B. Sammakia, J. Geer, Dynamic response and control analysis of cross flow heat exchangers under variable temperature and flow rate conditions, Int. J. Heat Mass Tran. 81 (2015) 542-553. [5] A. Korzeń, D. Taler. Modeling of transient response of a plate fin and tube heat exchanger, Int. J. Therm. Sci. 92 (2015) 188-198.
42
[6] D. Taler, Mathematical modeling and control of plate fin and tube heat exchangers, Energy Convers. Manage. 96 (2015) 452-462. [7] M. Lyytikäinen, T. Hämäläinen, J. Hämäläinen, A fast modelling tool for plate heat exchangers based on depth-averaged equations, Int. J. Heat Mass Tran. 52 (2009) 1132-1139. [8] L. Zhang, C. Yang, J. Zhou. A distributed parameter model and its application in optimizing the plate-fin heat exchanger based on the minimum entropy generation, Int. J. Therm. Sci. 49 (2010) 1427-1436. [9] H. Qiao, V. Aute, H. Lee, K. Saleh, R. Radermacher, A new model for plate heat exchangers with generalized flow configurations and phase change, Int. J. Refrig. 36 (2013) 622-632. [10] M. Fischer, O. Nelles, R. Isermann. Adaptive predictive control of a heat exchanger based on a fuzzy model, Control Eng. Pract. 6 (1998) 259-269. [ 11 ] U. Akman, K. Uygun, Nonlinear model predictive control of retrofit heat-exchanger networks, Comput. Chem. Eng. 23 (1999) S231-S234. [12] M. A. Abdelghani-Idrissi, M. A. Arbaoui, L. Estel, J. Richalet, Predictive functional control of a counter current heat exchanger using convexity property, Chem. Eng. Process 40 (2001) 449-457. [13] A. H. González, D. Odloak, J. L. Marchetti, Predictive control applied to heat-exchanger networks, Chem. Eng. Process. 45 (2006) 661-671. [14] M. A. Arbaoui, L. Vernières-Hassimi, D. Seguin, M. A. Abdelghani-Idrissi, Counter-current tubular heat exchanger: Modeling and adaptive predictive functional control, Appl. Therm. Eng. 27 (2007) 2332-2340. [15] M. Bakošová, J. Oravec, Robust model predictive control for heat exchanger
43
network, Appl. Therm. Eng. 73 (2014) 924-930. [16] A. Vasičkaninová, M. Bakošová, Control of a heat exchanger using neural network predictive controller combined with auxiliary fuzzy controller, Appl. Therm. Eng. 89 (2015) 1046-1053. [17] H. Kanoh, T. Itoh, N. Abe, Nonlinear H∞ control for heat exchangers controlled by the manipulation of flow rate, Nonlinear Anal-Theor. 30 (1997) 2237-2248. [18] A. Zavala-Río, C. M. Astorga-Zaragoza, O. Hernández-González, Bounded positive control for double-pipe heat exchangers, Control Eng. Pract. 17 (2009) 136-145. [19] A. Maidi, M. Diaf, J. P. Corriou, Boundary control of a parallel-flow heat exchanger by input–output linearization, J. Process Contr. 20 (2010) 1161-1174. [20] M. Dulău, S. Oltean, A. Gligor, Conventional Control vs. Robust Control on Heat-exchangers, Procedia Technology 19 (2015) 534-540. [21] S. K. Al-Dawery, A. M. Alrahawi, K. M. Al-Zobai, Dynamic modeling and control of plate heat exchanger, Int. J. Heat Mass Tran. 55 (2012) 6873-6880. [ 22 ] A. Michel, A. Kugi, Model based control of compact heat exchangers independent of the heat transfer behavior, J. Process Contr. 24 (2014) 286-298. [ 23 ] B. Chentouf, On boundary stabilization of heat exchangers systems with non-smooth time-varying coefficients, Appl. Math. Comput. 233 (2014) 398-403. [24] S. Skogestad, I. Postlethwaite, Multivariable feedback control: analysis and design, 2nd ed., Wiley, New York, 2007. [25] G.F. Franklin, D.J. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems, 7th ed., Pearson, New York, 2015. [26] A. Michel, A. Kugi, Accurate low-order dynamic model of a compact plate heat
44
exchanger, Int. J. Heat. Mass. Tran. 61 (2013) 323-331. [27] D. W. Gu, P. H. Petkov, M. M. Konstantinov, Robust control design with MATLAB®, Springer, London, 2005.
Figure Captions
Fig. 1. Schematic of plate heat exchanger heat transfer process. Fig. 2. Control volumes in the flow channels. Fig. 3. Input-output structure of the plate heat exchanger. Fig. 4. Block diagram of the two-degrees-of-freedom
control.
Fig. 5-a. Comparison between the test data and model calculation of Fig. 5-b. Relative error of
.
45
.
Fig. 6-a. Comparison between the test data and model calculation of Fig. 6-b. Relative error of
.
Fig. 7-a. Responses of Fig. 7-b. Responses of
in tracking step reference of
.
in tracking step reference of
Fig. 8-a. Responses of Fig. 8-b. Responses of
.
.
in tracking ramp reference of
.
in tracking ramp reference of
.
Fig. 9-a. Variations of Fig. 9-b. Responses of Fig. 9-c. Responses of
in rejecting variations of in rejecting variations of
Fig. 10-a. Variations of
.
Fig. 10-b. Responses of
in rejecting variations of
Fig. 10-c. Responses of
in rejecting variations of
Fig. 11-a. Variations of
.
Fig. 11-b. Responses of Fig. 11-c. Responses of
.
. .
in rejecting variations of in rejecting variations of
Fig. 12-a. Responses of
in rejecting variations of
Fig. 12-b. Responses of
in rejecting variations of
Table Captions
Table 1. System parameters.
46
. . , ,
and and
. .
S1359-4311(16)31762-8 https://doi.org/10.1016/j.applthermaleng.2017.09.120 ATE 11177
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
19 September 2016 15 August 2017 23 September 2017
Please cite this article as: Y. Wang, S. You, W. Zheng, H. Zhang, X. Zheng, Q. Miao, State space model and robust control of plate heat exchanger for dynamic performance improvement, Applied Thermal Engineering (2017), doi: https://doi.org/10.1016/j.applthermaleng.2017.09.120
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
State space model and robust control of plate heat exchanger for dynamic performance improvement Yaran Wang a, Shijun You a,b, Wandong Zheng a,b*, Huan Zhang a,b, Xuejing Zheng a,b, Qingwei Miao a a
School of Environmental Science and Engineering, Tianjin University, Yaguan Road 135#,
Jinnan District, Tianjin 300350, China b
National Engineering Laboratory for Digital Construction and Evaluation Technology of
Urban Rail Transit, Tianjin University, Yaguan Road 135#, Jinnan District, Tianjin 300350, China * Corresponding Author: Tel. / Fax: +86 22 2789 2626. E-Mail address: [email protected]
Highlights A state space model of plate heat exchanger was established. The proposed state space model of plate heat exchanger was verified with test data. A two-degrees-of-freedom
controller was designed for plate heat exchanger.
Control performances of the proposed
and PI controllers were compared.
Keywords plate heat exchanger, state space model, robust control, two-degrees-of-freedom control, dynamic performance improvement
Abstract Plate heat exchanger takes an important role in the modern industrial engineering. For large scale district heating system, plate heat exchanger is the key equipment in controlling the heat output from the heating substation. High performance control of 1
plate heat exchanger is essential for stable and efficient operation of the heating substation. The governing equation of plate heat exchanger is a pair of partial differential equations, which is not suitable for model based control. In this work, a linear state space model was developed to describe the dynamic characteristic of plate heat exchanger. Validation results show that the proposed model can capture the dynamics of plate heat exchanger with satisfied accuracy. Based on the state space model, the two-degrees-of-freedom
loop-shaping controller was designed for
plate heat exchanger to enhance the control performance. The reference tracking and disturbance
rejection performances of the proposed control algorithm were
studied and compared with the PI control. Results show that the presented two-degrees-of-freedom
loop-shaping control is superior to the PI control.
Nomenclature matrix of state space model width of the plate heat exchanger flow channel ( ) matrix of state space model matrix of state space model matrix of state space model matrix of state space model specific heat capacity of water distance between neighboring plates hydraulic diameter intermediate variable of two-degrees-of-freedom
loop-shaping control
scheme relative error of the simulated outlet temperature of high temperature side relative error of the simulated outlet temperature of low temperature side uncertain open-loop model transfer function from inlet temperature of the low temperature side to outlet temperature of low temperature side
2
Nomenclature transfer function from mass flow rate of the high temperature side to outlet temperature of high temperature side transfer function from mass flow rate of the low temperature side to outlet temperature of high temperature side transfer function from inlet temperature of the high temperature side to outlet temperature of high temperature side transfer function from inlet temperature of the low temperature side to outlet temperature of high temperature side overall heat transfer coefficient overall heat transfer coefficient of a specific steady state transfer function matrix of two-degrees-of-freedom ,
loop-shaping controller
transfer function matrix components of the two-degrees-of-freedom loop-shaping controller length of the flow channel number of flow channels in each side coprime factorization of the shaped model
,
number of the control volumes in a flow channel volume flow rate volume flow rate of a specific steady state maximum flow rate of high temperature side time temperature steady state temperature compensator transfer function specified to shape the open-loop model the reference transfer function model that describes the desired response of the closed-loop system coordinate along the flow channel intermediate variable of two-degrees-of-freedom
loop-shaping control
scheme heat conductivity coefficient (
)
density of the water small deviation & increment symbol of variable intermediate variable of two-degrees-of-freedom scheme scalar value specified by controller designer length of a control volume ,
coprime uncertainty 3
loop-shaping control
Nomenclature Subscripts high temperature side inlet low temperature side outlet plate
1. Introduction The plate heat exchangers are extensively used in energy industries, such as district heating systems [1], absorption chilling systems [2] and electricity production systems [3]. Efficient control of the plate heat exchanger is a key problem to enhance the dynamic performance of processes involved with plate heat exchanger. In large scale district heating systems, the plate heat exchangers are key components in controlling the heat outputs to secondary networks. Efficient control of plate heat exchanger improves the dynamic response and enhances the stability of the control system. Mathematical modeling is one of the most effective approaches to investigate the dynamic characteristics and control performances of plate heat exchangers. In the literatures, numerical modeling approaches were extensively studied and applied to heat exchanger control and operation analysis. Thermal dynamic model of a cross-flow heat exchanger was established and solved numerically to predict the transient response for multiple variable input operating analysis in Gao et al.’s work [4]. In Korzeń and Taler’s study, new equation set describing transient heat transfer process in tube and fin cross-flow heat tube exchanger was given and subsequently solved using the finite volume method, and good agreement between numerical and 4
experimental results was obtained with the method [5]. Using this technique, a model based control method was developed by Taler, and the experiment results show that the proposed control method is more stable compared with the PID controller [6]. Recently, studies on applying numerical methods to optimal design of heat exchangers have gained more attentions. Lyytikäinen et al. derived a practicable modeling method for plate heat exchanger geometry optimization by reducing the equations from 3D to 2D, which reduces the elapsed time of CFD simulations from hours to minutes [7]. Zhang et al. developed a distributed parameter model utilizing correlations of the heat transfer and flow friction to avoid solving partial differential equations for optimal design of the plate-fin heat exchanger, which largely reduces the computation burden in comparison with CFD method [8]. By dividing the plate heat exchanger into multiple slices in the direction of fluid flow and assuming the wall temperatures to be constant for the channels in each slice, Qiao developed a new model for the analysis of plate heat exchangers with multi-fluid, multi-stream and multi-pass configurations, which was solved without the need of knowing the fluid condition in the adjacent channels [ 9]. Despite the effectiveness of numerical calculation techniques for optimal design, it’s difficult to be applied to real-time control due to the computational pressure. Control-oriented modeling and feedback controller design methods for heat exchangers have also been paid wide attention, especially the Model Based Predictive Control. Model Based Predictive Control is one of the most effective solutions for industrial applications, due to its capabilities in dealing with nonlinearities, constraints,
5
uncertainties and multivariable coupling characteristics involved in the modern complex industrial processes. Fischer et al. developed a fuzzy model-based predictive controller for the temperature control of an industrial-scale cross-flow water/air heat exchanger, and a local recursive least-square algorithm was proposed for on-line adaptation of the fuzzy model [10]. In Akman and Uygun’s research, a nonlinear model predictive control scheme of a heat exchanger network was established and solved sequentially with an algebraic steady state optimization model, while satisfying the temporal and steady-state hard/soft constraints imposed on the target temperatures of the heat exchanger network [ 11 ]. Abdelghani-Idrissi et al. implemented a predictive functional control framework on a counter current heat exchanger, with a global model representing the response to inlet temperature variations [12]. González et al. discussed the online optimization and control of a heat-exchanger network through a two-level control structure, where the low level is a constrained model predictive control and the high level is a supervisory online optimizer [13]. Arbaoui et al. implemented a predictive functional control algorithm on a counter-current tubular heat exchanger to control the outlet temperature based on an approximated first order model, corresponding to the response of the heat exchanger to a step change of the flow rate [14]. A robust model predictive control algorithm for heat exchanger network was established and shows more satisfied performance and less cooling medium consumption with parameter uncertainties in comparison with the optimal linear quadratic control in Bakošová and Oravec’s research [15]. The combination of neural network and predictive control concepts has
6
also attracted some researches. Vasičkaninová and Bakošová proposed the combination of neural network predictive controller and the fuzzy controller in the complex control structure of heat exchanger with an auxiliary manipulated variable [16]. The main drawback of model based predictive control is its large computational burden due to solving online optimization problem. Other advanced control methods for heat exchangers such as: robust control, nonlinear control and fuzzy logic control have also been concerned. A nonlinear robust control approach for counter flow heat exchanger was developed based on the Galerkin method by Kanoh et al. [17]. An outlet temperature control algorithm for double-pipe heat exchangers, which takes into account and actually exploits the analytical and stability properties inherent to the open-loop dynamics was proposed to ensure positivity and boundedness of the input flow rate in Zavala-Río et al.’s study [18]. Based on nonlinear control theory, Maidi proposed a state-feedback control law that ensured a desired performance of a measured output defined as the spatial weighted average temperature of the internal fluid [19]. Dulău and Oltean designed a robust controller for heat exchanger with a second order time delay model by synthesis to ensure stable control performance with model uncertainties [20]. In Al-Dawery’s work, a first order model with time delay has been developed to suggest the transient responses of a plate heat exchanger, and a fuzzy logic controller of the plate heat exchanger was designed to achieve less settling time and oscillatory behavior [21]. Michel and Kugi developed a control strategy without knowledge of the heat transfer for plate heat exchanger based on controlling the total thermal energy
7
stored in the heat exchanger and a Kalman Filter to estimate the states [22]. Chentouf studied the Boundary feedback stabilization of a class of heat exchangers governed by hyperbolic non-autonomous evolution systems with non-smooth coefficients [23]. In this paper, a new control algorithm, the two-degrees-of-freedom
robust
controller was developed for the plate heat exchanger to enhance the dynamic performance. The two-degrees-of-freedom
robust control was a newly developed
control technique in recent decades to improve the control performance of dynamical systems [24]. The PID control is a standard and effective industrial control system solution, which has been applied to more than half of the industrial applications, while in most industrial applications, instead of PID, the PI control is usually used, due to the sensitive responses of the differential term to sensor noises [25]. Salam et al.’s work [21] showed that the dynamic performance of plate heat exchanger under PI control is superior to that of the plate heat exchanger under PID control. Hence, in order to illustrate the feasibility and validity of the proposed two-degrees-of-freedom controller for plate heat exchanger, the dynamic performances of the proposed controller are compared with the PI controller. The rest of this paper is organized as follows. The next section derives the governing equations of plate heat exchanger. After spatial discretization and linearization of the governing equations, the state space model was developed. Subsequently, the two-degrees-of-freedom
robust
control was introduced and applied to the control of plate heat exchanger. Next, the model precision of the state space model was validated with the test data from the literature. Then, the reference tracking and disturbance rejection performances of the
8
robust
control were evaluated through case studies in comparison with the PI
control. Finally, this paper closes with a summary of conclusions.
2. Modeling the plate heat exchanger The schematic of heat transfer process in plate heat exchanger is shown in Fig. 1, where the high temperature and low temperature fluids are separated by metal plates. The derivation of plate heat exchanger governing equation is introduced in subsection 2.1, and the linear state space model is established in subsection 2.2.
2.1. Governing equations of plate heat exchanger The governing equations of plate heat exchanger can be derived by simplifying the convective-diffusion equations of the fluid flow in the flow channels, and the following assumptions are considered [26]: 1. The dissipation effect of the fluid is negligible; 2. The thermo-physical properties of fluids are constants; 3. The fluids are assumed to be incompressible; 4. Heat losses from the heat exchanger to the surroundings are negligible; 5. Thermal dynamics of the metal plates are negligible; 6. The heat conduction of the fluid along the flow direction is negligible in comparison with the heat convection; 7. No phase changes occur inside the heat exchanger.
9
High temperature side
x L
Tl ql n
Tl dx x
Th
qh n
dx
dx
Th
Tl
Th dx x
Low temperature side
Fig. 1. Schematic of plate heat exchanger heat transfer process. The control volumes of the fluids and the coordinate system are shown in Fig. 1. For each high temperature side control volume, the heat balance equation is listed as follows: (1) For low temperature side, a similar heat balance equation is expressed as: (2) Thus the governing equations of the plate heat exchanger can be written as: (3) (4) This is a pair of partial differential equations. It is noted that system described by Eq. (3) and Eq. (4) is nonlinear distributed parameter system, which is also called the infinite dimensional system. For further study on the controller synthesis of the plate heat exchanger, the system model needs to be reduced to a finite dimensional form, which is to reduce the partial differential equations to ordinary differential equations. There are generally two kinds of effective approaches. One is the Galerkin method, 10
the other is the finite volume & difference method. In reference [17], the former technique was applied to simplify the distributed parameter model of counter flow heat exchanger into a nonlinear state space form. Reference [26] used the latter approach to obtain an accurate low order model for plate heat exchanger. In this paper, the finite volume integral method is adopted, and a linear state space model of the plate heat exchanger is derived for robust controller synthesis.
2.2. Model simplification method In this subsection, the finite volume integral method was adopted to simplify Eq. (3) and (4). Linearization approach by Taylor expansion was used to derive the linear state space model of plate heat exchanger. To apply the finite volume integral method, each flow channel of the high temperature and low temperature sides is divided into
control volumes, as is
shown in Fig. 2. The control volumes in each side are numbered consecutively from to
. Integration of the Eq. (3) and (4) within the th control volume along the
flow channels leads to the following equations: (5) (6)
11
Fig. 2. Control volumes in the flow channels. Assuming uniform temperature distributions within the th control volumes, Eq (5) and (6) can be approximated as: (7) (8) where
, and
. Then the following equations can be obtained: (9) (10)
Eq. (9) and (10) are finite dimensional model, which is nonlinear and difficult for controller design. For further simplification of the model, steady state solutions of Eq. (9) and (10) are solved. Steady state solutions of differential equations satisfy the steady state version of the differential equations, in which the time derivative terms are zero. The steady state forms of Eq. (9) and (10) are the following algebraic equations: (11)
12
(12) Let , ,
,
,
and
,
,
which are all
matrices, and
Eq. (11) and (12) can be written into matrix equation form: (13) where
is the steady state temperatures of control volumes and needs to be
calculated. The steady state solution of Eq. (9) and Eq. (10) are solved as:
.
To linearize Eq. (9) and Eq. (10), each variable is replaced by the sum of steady state value and deviation about the steady value. The variables are independent variables. Let
,
,
represents the independent variable,
13
and can be
written into the following form: (14) where
is a steady state value of
heat transfer coefficient
, and
is the deviation about
. The overall
is a dependent variable concerned with the mass flow rates
of the high temperature and low temperature sides. The overall heat transfer coefficient
can be calculated by the following formula [26]: (15)
where
is the heat transfer coefficient of the plate.
and
are
determined by the following formulas [26]: , , where
,
,
,
,
is the hydraulic diameter. The overall heat transfer coefficient
(16) (17) is a
dependent variable, which can be replaced by the following one-order Taylor expansion: (18) where the partial derivatives
and
can be calculated from Eq. (16)
and Eq. (17). Substituting Eq. (14) and Eq. (18) into Eq. (9) and Eq. (10), the high order terms are neglected. Therefore, the linearized equations are derived as follows:
(19)
14
(20) Do the following parameter replacement: , ,
, , , ,
(21)
and
,
,
(22) ,
,
and
are
matrices.
,
,
,
-dimensional vectors. The state equation of the state space model is written as follows: 15
and
are
(23) where
is the temperature vectors of the high temperature and low temperature side
control volumes.
is the inlet temperature of the high temperature side, and
is the inlet temperature of the low temperature side. According to the control volumes shown in Fig. 2, the outlet temperatures of the two sides are the state temperatures: , where
(24)
is the outlet temperature of the high temperature side;
is the outlet
temperature of the low temperature side. Therefore the outlet temperatures can be written in the following form: , where
(25) ,
. Combining Eq.
(23) and Eq. (25), the linear state space model of the plate heat exchanger is derived.
3. Robust
Control of plate heat exchanger
3.1 Input-output representation of plate heat exchanger The control scheme of plate heat exchangers in most industrial applications is to maintain the outlet temperature of one side around the desired values by regulating flow rate of the other side. Usually, different disturbances, such as the inlet temperature and flow rate variations may occur, which will have an impact on the control process. Hence, an efficient control system should have the ability to reject the disturbances and keep good tracking of the desired values. In order to study the relationship between the inputs and outputs, the transfer functions from the inputs to the outputs are defined and calculated as follows: 16
(26) (27) (28) (29) (30) (31) (32) (33) The input-output structure of plate heat exchanger system is illustrated by block diagram shown in Fig. 3. The system inputs can be divided into two categories: the manipulated inputs and disturbance inputs. For the district heating application of plate heat exchangers, the high temperature water flow rate input, and the other inputs outlet temperature
,
and
is taken as the manipulated are as disturbance inputs. The
is the controlled output and needs to track the desired setpoint.
Fig. 3. Input-output structure of the plate heat exchanger.
The input-output structure can also be written in the following form:
17
(34) (35) Since the controlled output is
, only subsystem in the blue dashed box is needed
for controller synthesis. This subsystem is described in Eq. (34).
3.1. Two-degrees-of-freedom
loop-shaping control of plate heat exchanger
The control objective is to keep the outlet temperature of the low temperature side ( side (
) tracking the desired value, by controlling the flow rate of high temperature ), under the variations of the inlet temperature of high temperature side (
),
the flow rate of low temperature side ( ) and the inlet temperature of low temperature side (
). Robust control of plate heat exchanger is to achieve acceptable control
performance, that is, to keep the controlled output
within specified bound from
its desired value, in spite of unknown but bounded disturbances (
,
and
and system parameter uncertainties (such as the overall heat transfer coefficient using constrained input (
) and available measurement (
), ),
). There are several
design approaches of robust controllers, such as the mixed sensitivity method, synthesis method,
loop-shaping method and LMI based time domain methods
etc. [24, 27]. These methods are all applicable and effective robust controller design tools. Among these methods, the two-degrees-of-freedom
control approach is
adopted for plate heat exchanger control in this paper, due to its convenient design procedure and satisfactory performances. The rest of this subsection explains the design procedure of the two-degrees-of-freedom
controller for plate heat
exchanger. Fig. 4 depicts the system structure of the two-degrees-of-freedom 18
loop-shaping control. Define
, where
function to shape the loop transfer function [24].
is a specified transfer
and achieve a stable and fast response
is the reference transfer function model that describes the desired
response of the closed-loop system. Transfer functions normalized coprime factorization of transfer function
and
are the
. To enhance the robustness
of the closed-loop plate heat exchanger system, the model uncertainty should be considered in the design stage and the uncertain model set
can be factorized as:
,
(36)
Fig. 4. Block diagram of the two-degrees-of-freedom where
and
control.
are unknown stable transfer function set, which describe the
system uncertainties of the specific model
. The uncertain model
can handle
varieties of model uncertainties [24], and the uncertainties of the seven assumptions in subsection 2.1 can also be incorporated in transfer function set.
.
is the bound of the unknown stable
is a specified reference model which represents the
desired response of the closed-loop plate heat exchanger system. And
is a scalar
parameter specified during the controller design. The two-degrees-of-freedom controller design problem can be formulated into an stabilizing controller
which minimizes the 19
control problem: find the norm of the transfer
matrix between
and
[24], see Appendix A for details.
The controller design procedure is as follows: Step 1, preliminarily specify the transfer functions . The technique for specifying
and
Step 2, formulate and solve the . The
and
, and the scalar
is given in [24]. control problem to obtain the controller
control problem solving methods are discussed more
detailedly in Ref. [24, 27], which can be conducted via MATLAB robust control toolbox. Step 3, simulate the closed-loop plate heat exchanger system to validate if satisfied reference tracking and disturbance rejection performances have been achieved. If not, go to step 1.
4. Results and discussion In this section, the state space model derived in this paper is validated with the test data from the literature. Then the two-degrees-of-freedom
controller is
designed and compared with the PI controller. Reference tracking and disturbance rejection performances of the two controllers were analyzed and discussed.
4.1. Model validation In order to validate the state space model of the plate heat exchanger proposed in this paper, the test data and system parameters in Ref. [26] are adopted. The state space matrices:
,
,
,
and
for plate heat exchanger provided in
Ref. [26] are calculated and listed in Appendix B. Dynamic simulation of the
20
proposed state space model with the test data of and the outlet temperatures temperatures
and
and
,
,
and
is conducted,
are calculated. The simulated outlet
are compared with the test data to validate the state space
model of plate heat exchanger Eq. (20). The comparison of and the relative error of
is shown in Fig. 5-a,
is shown in Fig. 5-b. The comparison of
is shown
in Fig. 6-a, and the relative error is shown in Fig. 6-b. The relative errors are calculated as: ,
(37)
, where
and
(38) are the calculated values at time instant ;
and
are the test values at time instant . 32 Model Test
30 28
Th,o ( )
26 24 22 20 18 16 0
100
200
300 Time (s)
400
500
Fig. 5-a. Comparison between the test data and model calculation of
21
600
.
10 8 6 4
Eh,o (%)
2 0 -2 -4 -6 -8 -10 0
100
200
300 Time (s)
Fig. 5-b. Relative error of
400
500
600
.
34 Model Test
32 30
Tl,o ( )
28 26 24 22 20 18 0
100
200
300 Time (s)
400
500
Fig. 6-a. Comparison between the test data and model calculation of
The relative errors of
and
are both varying within
600
.
, which indicates
that the proposed state space model of plate heat exchanger is effective, since the presented robust control technique can tackle and compensate the model uncertainty to ensure the control performance [24]..
22
10
8
El,o (%)
6
4
2
0
-2 0
100
200
300 Time (s)
400
Fig. 6-b. Relative error of
4.2. Comparison of robust
600
.
and PI control
In order to illustrate the two-degrees-of-freedom
500
feasibility and
validity of the proposed
loop-shaping control algorithm for the plate heat
exchanger, comparison between the
controller with PI controller was conducted
via a case study. As the effectiveness of the state space model of the plate heat exchanger has been validated in subsection 4.1, the plate heat exchanger parameters can be directly used in developing the state space model. The parameters of the plate heat exchanger studied in this subsection are from the manufacturer and listed in Table 1. In this subsection, the two-degrees-of-freedom
loop-shaping controller
and PI controller for plate heat exchanger are designed. Then the reference tracking and disturbance rejection performances of the two controllers are compared. Table 1. System parameters Description
Symbol
23
Value
Unit
channel width
0.8
channel length
1.36
water specific heat capacity
4220
empirical parameters
0.64
/
distance between neighboring plates
4.5
mm
plate thickness
0.5
number of flow channels in each side
137
/
empirical parameters
0.23
/
empirical parameters
0.75
/
water density
970
thermal conductivity of high temperature side water
0.68
thermal conductivity of low temperature side water
0.67
thermal conductivity of plate
15
dynamic viscosity of high temperature side water
0.00028
dynamic viscosity of low temperature side water
0.00041
steady state inlet temperature of high temperature side
80
o
steady state inlet temperature of low temperature side
50
o
steady state flow rate of high temperature side
0.03
steady state flow rate of low temperature side
0.03
Maximum flow rate of high temperature side
0.06
m
C C
The state space model of the plate heat exchanger studied in this subsection is derived and summarized in Appendix C. According to the design procedure of the two-degrees-of-freedom
loop-shaping controller illustrated in subsection 3.1, the
designing of the two-degrees-of-freedom specify the two transfer functions
and
frequency response of the open loop system 24
loop-shaping controller needs to .
is designed to shape the , and make the closed loop
system stable and have a faster response.
and
are designed by trial or error.
The transfer functions are selected as follows. (39) (40) According to the design procedures in subsection 3.1, the two-degrees-of-freedom loop-shaping controller is designed with MATLAB Robust Control Toolbox. Since the model order of the plate heat exchanger is order controller
, this leads to high
which is not applicable for implementation. In this
paper, in order to obtain a low order controller, the model reduction for the obtained high order controller is conducted. Details on model order reduction have been summarized in Appendix D. The three order controller is calculated and the transfer function form is as follows. (41) (42) The PI controller is designed with Ziegler-Nichols method, and tuned by trial and error to ensure the closed loop stability and fast response with the control input in an allowable range. The PI controller for the plate heat exchanger studied in this section was tuned as: (43) To study the closed loop performance of the proposed two-degrees-of-freedom loop-shaping controller, comparison between the controller were conducted in the following subsections.
25
controller and PI
4.2.1. Reference tracking performance Dynamic responses of PI and the proposed
in tracking a sequence of desired step variations with
controllers are shown in Fig. 7-a. As is shown in this figure,
the overshoots of
control and PI control are 2.5% and 6% respectively. The
settling times for
control and PI control are 250s and 500s respectively.
Therefore, the application of
controller ensures good tracking of the desired step
references, while using PI controller leads to larger overshoots and longer settling time. Fig. 7-b shows the variations of high temperature side flow rate in tracking the sequence of step references. Fig. 7-b indicates that the ensure the flow rate within the maximum value
control and PI control can .
70 68 66
Tl,o ( )
64 62 60 58 56 Reference Robust control PI control
54 52 0
500
1000
1500
Fig. 7-a. Responses of
2000 Time (s)
2500
3000
in tracking step reference of
26
3500
.
4000
0.06 Robust control PI control 0.05
qh (m3/s)
0.04
0.03
0.02
0.01 0
500
1000
1000
Fig. 7-b. Responses of
2000 Time (s)
2500
3000
in tracking step reference of
3500
4000
.
70 68 66 64
Tl,o ( )
62 60 58 56 54 Reference Robust control PI control
52 50 0
200
400
600
Fig. 8-a. Responses of
800
1000 Time (s)
1200
1400
1600
in tracking ramp reference of
27
1800
.
2000
0.06
0.05
qh (m3/s)
0.04
0.03
0.02
0.01 Robust control PI control 0 0
200
400
600
Fig. 8-b. Responses of
The dynamic responses of with PI and
800
1000 Time (s)
1200
1400
1600
in tracking ramp reference of
1800
2000
.
in tracking a sequence of desired ramp variations
control are shown in Fig. 8-a. This figure denotes that for both
control algorithms there are small steady-state errors when tracking the desired ramp references. However, PI control leads to larger steady-state error than The variations of high temperature side flow rate
control.
in tracking the step references is
shown in Fig. 8-b, which denotes that anti-saturation can be ensured with both controllers.
4.2.2. Disturbance rejection performance Responses of
in rejecting low temperature side flow rate variations (shown
in Fig. 10-a) with PI and the indicates that the oscillation of the
controllers are compared in Fig. 9-b. This figure controlled by PI controller is more intense than
controller. The maximum deviations of
are 0.9 oC and 3.7 oC respectively. Responses of
28
for
control and PI control
in Fig. 9-c means that the control
inputs for the two controllers are both within an allowable range. 0.034
0.032
ql (m3/s)
0.03
0.028
0.026
0.024
0.022 0
200
400
600
800 Time (s)
1000
1200
1400
1600
Fig. 9-a. Variations of 70 68 66 64
Tl,o ( )
62 60 58 56 54 Reference Robust control PI control
52 50 0
200
400
600
Fig. 9-b. Responses of
800 Time (s)
1000
1200
in rejecting variations of
29
1400
1600
0.06
0.05
qh (m3/s)
0.04
0.03
0.02
0.01 Robust control PI control 0 0
200
400
600
Fig. 9-c. Responses of
Fig. 10-a. shows a sequence of in rejecting the variations of
800 Time (s)
1000
1200
in rejecting variations of
1400
1600
.
variations. The dynamic responses of
for PI control and
10-b. This figure denotes that the settling time of control. And the maximum deviations of
control are shown in Fig. is longer for PI control than
for
control and PI control are
0.4 oC and 0.6 oC respectively. Fig. 10-c illustrates the dynamic responses of
,
which indicates that both controllers can ensure control inputs within an allowable range.
30
70 65 60
Tl,in ( )
55 50 45 40 35 30 0
200
400
600
800 Time (s)
Fig. 10-a. Variations of
1000
1200
1400
1600
.
68 66 64
Tl,o ( )
62 60 58 56 Reference Robust control PI control
54 52 0
200
400
600
Fig. 10-b. Responses of
800 Time (s)
1000
1200
in rejecting variations of
31
1400
.
1600
0.06
0.05
qh (m3/s)
0.04
0.03
0.02
0.01 Robust control PI control 0 0
200
400
600
Fig. 10-c. Responses of
The dynamic responses of for PI control and
1000
1200
in rejecting variations of
1400
1600
.
in rejecting the disturbances of
(Fig. 11-a)
control are shown in Fig. 11-b. This figure also indicates that
the settling time of deviations of
800 Time (s)
for
is longer for PI control than
control and PI control are 1.25 oC and 2 oC respectively.
The responses of high temperature side flow rate both the PI control and
control. The maximum
shown in Fig. 11-c indicate that
control can ensure control inputs within an allowable
range.
32
82
Th,in ( )
81
80
79
78 0
200
400
600
800 Time (s)
Fig. 11-a. Variations of
1000
1200
1400
1600
.
68 66 64
Tl,o ( )
62 60 58 56 Reference Robust control PI control
54 52 0
200
400
600
Fig. 11-b. Responses of
800 Time (s)
1000
1200
in rejecting variations of
33
1400
.
1600
0.06
0.05
qh (m3/s)
0.04
0.03
0.02
0.01 Robust control PI control 0 0
200
400
600
800 Time (s)
Fig. 11-c. Responses of
1000
1200
in rejecting variations of
1400
1600
.
4.2.3. Performances with parameter uncertainties It is impractical to obtain the precise thermal and physical parameters of the plate heat exchanger without accurate measurements, especially for the heat transfer coefficient. In this study, the heat transfer coefficient was calculated with the correlation of Nusselt numbers number
and
in Eq. (17). The correlation of Nusselt
is usually given by the manufacturers. However, the parameters in the
correlation might not be precise for a specific plate heat exchanger. To consider such factor, the parameter
in the Nusselt number correlation was assumed to decrease
30%, and the controllers were compared under such parameter uncertainty. The responses of disturbances of
(Fig. 9-a),
under the parameter uncertainty to the combined (Fig. 10-a) and
(Fig. 11-a) were shown in
Fig. 12-a. This figure indicates that,
controlled by the PI controller fluctuates
more intensely than the proposed
controller. In Fig. 12-a, the maximum
deviation of
controlled by PI controller is 3 oC. Nevertheless, the 34
controller
is able to keep indoor air temperature within
1 oC from its desired value and ensure
less steady-state error when tracking ramp reference, despite the disturbances of ,
and the parameter uncertainty, with the flow rate in an allowable range
(Fig. 12-b). Therefore, the proposed
controller is robust to disturbances and heat
transfer coefficient uncertainty. 70 68 66 64
Tl,o ( )
62 60 58 56 54 Reference Robust control PI control
52 50 0
200
400
Fig. 12-a. Responses of
600
800 Time (s)
1000
in rejecting variations of
1200
,
1400
and
1600
.
0.06
0.05
0.04
qh (m3/s)
,
0.03
0.02
0.01 Robust control PI control 0 0
200
400
Fig. 12-b. Responses of
600
800 Time (s)
1000
in rejecting variations of
35
1200
,
1400
and
1600
.
4. Conclusions In this paper, the state space model of the plate heat exchanger was established. Based on the state space model, the two-degrees-of-freedom
loop-shaping
controller was presented to improve the dynamic performance of the plate heat exchanger. Then the state space model was verified with test data. The
controller
and the PI controller were designed for a plate heat exchanger. Reference tracking and disturbance rejection responses were compared and analyzed to validate the dynamic performance of the proposed
controller. According to the above analysis and
discussions, the conclusions are drawn as follows: (1) Comparison of the test data and model simulation results in subsection 4.1 shows that the proposed state space model of plate heat exchanger can provide sufficient accuracy. The relative errors of the two outlet temperatures are both varying within
and
10%, which are in a satisfied range. Therefore, the state
space model of plate heat exchanger proposed in this paper can describe the thermal dynamics of plate heat exchanger precisely. (2) The reference tracking responses of
controlled by the PI controller in
tracking the step and ramp references all have larger overshoots and longer settling time. For step references, the overshoots of 6% respectively. The settling times for
control and PI control are 2.5% and control and PI control are 250s and 500s
respectively. There is larger steady state error of PI control than
control when
tracking desired ramp references. These indicate that the presented
controller is
superior to the PI controller in tracking desired values.
36
(3) The disturbance responses of the low temperature side flow rate,
controlled by ,
controller in rejecting
variations are more oscillatory than PI control and PI control are 0.9 oC
controller. The maximum deviations of
for
and 3.7 oC respectively, when tracking
variations. The maximum deviations of
control and PI control are respectively 0.4 oC and 0.6 oC, when tracking
for
variations. The maximum deviations of
for
also 1.25 oC and 2 oC respectively in tracking presented
control and PI control are variations. Therefore the
control algorithm can guarantee better disturbance rejection
performances when disturbances exist. (4) The proposed two-degrees-of-freedom
loop-shaping controller is able to
ensure the satisfied dynamic performance for both reference tracking and disturbance rejection under system capacity parameter uncertainties. When the key parameter in the Nusselt number correlation decreases 30%, which means that the heat transfer coefficient becomes smaller, the proposed deviations within
controller is still able to keep the
1 oC, while the maximum deviation of PI control is 3 oC. And the
steady state error of
controller is smaller than PI control. Besides the
controlled by the PI controller is more oscillatory than the proposed
controller.
These confirm the robustness of the presented two-degrees-of-freedom loop-shaping controller of the plate heat exchanger.
Appendix A. The
two-degrees-of-freedom
loop-shaping
37
control
design
problem
illustrated in Fig. 4 can be transformed to the following standard Find the stabilizing controller
control problem:
which minimizes the
transfer matrix between the signals
and
The transfer matrix from the signals
norm of the
. to
is
(A.1)
norm of
is the following maximum singular value: (A.2)
Then the design problem becomes minimizing the maximum singular value of over frequency range
. This problem can be solved by algebraic Riccati
equations or LMI methods with MATLAB Robust Control Toolbox [24, 27].
Appendix B. State space model for model validation The state space model matrices for plate heat exchanger parameters provided in Ref. [26] are calculated with the method proposed in this paper. The matrices are listed as follows.
1.892 1.647 1.892 1.647 1.892 A11
1.647 1.892 1.647 1.892 1010
38
1.243 0.998 1.243 0.998 1.243 A22 0.998
1.243 0.998 1.243 1010
And the state matrix ;
;
; ; ;
Appendix C. State space model for controller comparison The state space model matrices for the plate heat exchanger parameters listed in Table 1. are calculated and shown as follows.
0.678 0.447 0.678 0.447 0.678 A11
0.447 0.678 0.447 0.678 1010
39
0.678 0.447 0.678 0.447 0.678 A22 0.447
0.678 0.447 0.678 1010
And the state matrix ;
;
Appendix D. Controller order reduction The model order of the two-degree-of-freedom
loop-shaping controller
developed for plate heat exchanger is high, which is complicated for implementation. Hence, model order reduction of the controller is needed. Balanced truncation is one of the effective and applicable approaches for model order reduction. The general idea of balanced truncation is to neglect those parts of the original system that are less observable or/and less controllable [27]. Assume that the controller to be reduced has the following state space form (D.1) 40
(D.2) where
is called the balanced realization, if the solutions
and
to the following Lyapunov equations (D.3) (D.4) are such that and
, with
.
are called the controllability Gramian and observability Gramian, respectively.
When the system is balanced, both Gramians are diagonal and equal.
,
is the th Hankel singular value of the system. For a general system not in the balanced realization form, a state similarity transformation (balancing transformation) is required, for details of the transformation, see [27]. If the controller to be reduced has already been transformed to the balance realization, the matrix
of the controller can be written as
, and
and
, with . The matrices
,
be partitioned compatibly as ,
,
Then a reduced-order controller
(D.5)
can be defined by (D.6)
The reduced controller
is of th order and is called the balanced truncation of the
full order ( th) controller
. And the following error bound is satisfied (D.7)
where
denotes the trace of the matrix
41
, i.e.
, the
sum of the last
Hankel singular values. To reduce the original model into an
th-order system, there should be a large gap between
and
, i.e.
.
The preceding procedures of obtaining the reduced-order controller can be realized in Matlab [27].
Acknowledgement This work was supported by the State Oceanic Administration of China (Grant No. cxsf-43); and the National Natural Science Foundation of China (No. 51106110).
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Figure Captions
Fig. 1. Schematic of plate heat exchanger heat transfer process. Fig. 2. Control volumes in the flow channels. Fig. 3. Input-output structure of the plate heat exchanger. Fig. 4. Block diagram of the two-degrees-of-freedom
control.
Fig. 5-a. Comparison between the test data and model calculation of Fig. 5-b. Relative error of
.
45
.
Fig. 6-a. Comparison between the test data and model calculation of Fig. 6-b. Relative error of
.
Fig. 7-a. Responses of Fig. 7-b. Responses of
in tracking step reference of
.
in tracking step reference of
Fig. 8-a. Responses of Fig. 8-b. Responses of
.
.
in tracking ramp reference of
.
in tracking ramp reference of
.
Fig. 9-a. Variations of Fig. 9-b. Responses of Fig. 9-c. Responses of
in rejecting variations of in rejecting variations of
Fig. 10-a. Variations of
.
Fig. 10-b. Responses of
in rejecting variations of
Fig. 10-c. Responses of
in rejecting variations of
Fig. 11-a. Variations of
.
Fig. 11-b. Responses of Fig. 11-c. Responses of
.
. .
in rejecting variations of in rejecting variations of
Fig. 12-a. Responses of
in rejecting variations of
Fig. 12-b. Responses of
in rejecting variations of
Table Captions
Table 1. System parameters.
46
. . , ,
and and
. .