Simplified Thermal Model and Optimal Control for Chilled Water Pipe in Air-Conditioning System

As an important subsystem in air-conditioning system, water system connects the chiller and terminal equipment. Therefore, it is necessary to develop an accurate thermal model for chilled water pipe to create the suitable indoor temperature and humidity environment. In this paper, the thermal model of the pipe was considered by utilizing the simplified thermal time-delay statespace model with the mass, energy balance, and heat consumption equations. Based on this improved model, the preview control as control strategy for water pipe temperature was proposed, and its robustness and stability were discussed. Subsequently, the performances of this model and control strategywere tested in a fan-coil system simulating withMATLAB and TRNSYS (Transient System Simulation Program). Explicitly, the results show that this model accurately predicts the thermal characteristic, and the average mean squared errors for water temperature were 11.14% and 12.82%, respectively. Meanwhile, the tracking effect of valve controller was better than the control strategy with no preview control.


Introduction
Currently, the main energy consumption in buildings is caused by air-conditioning system.As the basic subsystem, air conditioning water system (ACWS) creates a comfortable and healthy indoor environment for people through delivering the cold and heat energy, and the pipe network is the most important component.In particular, ACWS which increases the energy consumption and investment has become more complex.Therefore, the development of pipe thermal model is propitious to effectively optimize the control schemes and reduce the energy consumption.
In the past, many researches focused on the heat transfer and transition in ACWS [1].For instance, Schnedel et al. [2] formulated the flow path model of various components and pipes in the cooling water system similar to the actual electronic system.Ankur et al. [3] analyzed the hydraulic characteristics of the water pipe using the 3D numerical simulation and optimized the heat transfer power of the heat exchanger.Based on the simulation result, an improvement measure for the condenser was proposed.Bram et al. [4] developed a steady-state thermal behavior model of double thermal network pipes and analyzed the influence of temperature and pipe size on the heat loss.However, the thermodynamic state in the actual pipe is dynamic, so some scholars put the direction of pipe thermal model to the dynamic model.In the dynamic model, state-space model and its special types have been considered as the typical ones.State-space model can clearly represent the dynamic relationship between the input and output at system level, and it is easy to study the dynamic coupling and the essence of component models.Then, the accurate prediction model and pipeline structures optimization could be achieved by combining with various algorithms.For instance, Thosar et al. [5] established a nonlinear water system pipe dynamic model with state-space model and energy balance principle.Yao et al. [6,7] proposed a transient response model for constituent elements in the refrigeration cycle system with state-space model.Results showed that these models, with high prediction accuracy, were simple and easy to understand.As an important classification in statespace models, time-delay state-space model is more close to the real system and reflects the time delay.Therefore, this model provides a new direction for air conditioning system modeling.To sum up, time-delay state-space model is just widely used in mathematics, electronic information, and control engineering [8][9][10][11].Unfortunately, the applications of time-delay state-space in the air conditioning system are relatively less than other basic models.Based on the air conditioning model, the performance of controller can be improved using some intelligence algorithm, for instance, artificial neural network, fuzzy theory, model predictive control, particle swarm optimization, and genetic algorithm.In these control algorithms, preview control, which can be closely correlated to state space model, is one of the approaches available for producing a good performance by utilizing future information of the reference signal in the controller.Currently, preview control had been researched in servo-system control, robot control [12], spacecraft control [13], and generator set control [14].Meanwhile, some future state signals and interference signals in the air conditioning system can be predicted, even without model.So, preview control can be used in the air conditioning control.
The main purpose of this study is to present a simple thermal time-delay state-space model (T-SII model) for chilled water pipe in air-conditioning system, and a temperature control strategy for water pipe was proposed using preview control.Subsequently, the performances of T-SII model and preview controller were verified by an existing experimental platform.Results show that this model ensured high reliability and accuracy, and it will provide a guidance and reference for modeling in other air conditioning subsystems.Meanwhile, the delay time was compensated using preview control with the robustness and stability, and the tracking effect of valve controller was improved.

Model
. .Modeling Process.The modeling flow diagram is shown in Figure 1.
Step .For modeling of the water loop, the following simplified conditions are necessary: (a) The water temperature difference between the supply and the return is 5 ∘ C∼7 ∘ C, so the density of water can be regarded as a constant.
(b) The temperature and flow are only considered in the description of water state in the pipe.
(c) The average water temperature of the inlet and outlet pipe is regarded as a mixed value.
(d) The thermal resistance of the water pipe shell is converted into the insulating layer.
Then, the subspaces with different performance can be obtained [15].The complex air conditioning water system pipe network is established by connecting these components in turn, and the energy conservation in pipes includes heat transfer and convection.For the normal pipe, its thermal dynamic characteristics can be expressed as [16] Step .The basic variables (temperature and flow rate) can be considered as the fundamental and the lumped variables.The fundamental variables include the temperature and water flow rate.These variables can be considered as the sum of initial value and small increment.The lumped variables include mass specific heat (c) and coefficient of convection heat transfer (h), and these variables can be linearized using first-order Taylor formula.
Step .Through parameter linearization, the input variables, output variables, and state variables can be determined according to the import and export status.Generally, the entry parameter and exit parameter are considered as the input variable and output variable, respectively, and the selection about state variable can be the entrance or exit parameter.Then, all equilibrium equations are summarized into unified matrix representations.Finally, the T-SII model will be obtained with respect to the above process.
. .Pipe ermal Model.Pipe occupies the large proportion in the ACWS, and the thermal time delay in the pipe is the most obvious phenomenon.Water pipe network system can be divided into a certain number of straight pipes, which has analogous inlet and outlet parameters.The parameters of the pipe are shown in Figure 2.
The mass conservation equation for the water flowing through pipe can be expressed as  According to (1) and the assumed condition (c), mass and energy equation for water thermal state along the flow direction can be expressed using linear increment [17]: The mass and energy equation for water thermal state along the shell direction can be expressed as Two kinds of variables are considered in the equations: the fundamental and the lumped variables.The water temperature and water flow rate are included in fundamental variables, which can be considered as the summation of initial value and a small increment The lumped variables mainly include the coefficient (h) of heat, which can be linearized using the first-order Taylor formula: Through the linearized parameters, ( 3) and ( 4) can be transformed as follows: The mass and energy equation for water thermal state along the flow direction where The mass and energy equation for water thermal state along the shell direction where Combining ( 7) and ( 9), the basic thermal time-delay state-space model for the water pipe can be expressed as where the differential term is regarded as the state variable, inlet temperature and flow are regarded as input variables, and the outlet temperature and flow are regarded as output variables.
. .Valve.In pipe network, the relation between the valve opening and the water flow rate can be expressed as follows: The mass and energy equation can be expressed as Through parameter linearization, the state-space model of valve can be expressed as where

Control Strategy for the Temperature of Water Pipe
. .Design for Preview Controller.The control problem for pipes is to maintain temperature and flow at some specified set point using valve.For improving the performance of the air conditioning control system, controller is required to track the output signal and input signal closely, and the error signal between the state value and set value is often regarded as the target signal.In general, the feedback control of the traditional central air-conditioning needs the current system state and overcomes numerous external distractions.However, some future state signals and interference signals in the air conditioning system can be predicted, such as weather, planned schedule of people, pumps start or stop, and load requirement.To improve the control performance of the system, the method using future information is preview control known as linear quadratic optimal control, and Figure 3 shows the control principle.The input maintaining the pipe temperature state equals the output of the controller.Hence, based on the pipe thermal T-SII model, the control strategy is to search the optimal input signal.In order to simplify the controller designing algorithm and improve the calculation speed, T-SII model should be discretized using difference equation (Δx(k)=x(k)-x(k-1)).In this process, it is assumed that the system is sampled by an equal interval.When t=kT, the sample values will change.In (11),  p is a minimal variable; therefore it can be neglected.Then, the discretization is defined where k is sampling value (1,2,..,N).The parameter of temperature or flow changes periodically, and the preview length is  R .The error signal is where R(k) is the target signal.The first-order differential for x(k+1) is The error signal (e(k+1)) is Then, ( + 1) =  () + Δ ( + 1) (25) Thus, the error system is defined as follows: where The performance index function is defined as [18]  = 1 2 where J⩽  * (the maximum value of performance index function) Through a series of transformations, the optimal preview control is changed into the optimal control of error system with the performance index 'J' , and the control variable is Δu(k).The minimum performance index function of the optimal control for air conditioning system pipe is where . .Robustness Performance.Robust stability refers to the ability of remaining stable for control system [19].In other words, the preview controller should track the target signal R(k) with no steady-state error in spite of being affected by an uncertain disturbance, and the relationship between y(k) and r(k For any given invertible matrix X, the operator Δ : () → () enjoys the property ‖ XΔ X−1 ‖ ∞ ≤ 1, and the robustly asymptotically stable of system (26) with the constraint (J) can be proved according to the lemmas in [20].Meanwhile, the state feedback for system (20) will be defined Therefore, the robustness is determined by the gain matrix (K) which can be solved by LMI.It means that the optimization problem has an optimal solution with constraint, and the following condition can be derived according to the principles introduced using the result in [21]: where

Experimental Setup
In this work, data is registered in a fan-coil system, which is installed in Southwest Jiaotong University.Figure 4(a) shows the schematic diagram, illustrating that this system employs an air source heat pump with a rated power of 2200 W, and two fan-coil units (FCU) with an air flow rate of 1020 m According to (11) and ( 18), the coefficient matrix in system (20) In this work, the simulation system for models and preview control strategies is set up using TRNSYS (Transient System Simulation Program) and MATLAB.The models are validated against actual data gathered from actual experimental plant, and the performance analysis of preview control strategies is validated in the simulation system.Figure 5 shows the simulation diagram of this experiment.

Results and Discussions
. .Experimental Results.The mean squared error (MSE) is used to evaluate the model accuracy by comparing the calculated results with the experimental data during the transient response process, which is defined where y is the difference between the response parameters and the initial value, the subscript 'i' is the output value of the T-SII model, and n is sampling point total number, n=50.The superscript 'd' is the value acquiring from the actual data.The initial conditions, which are necessary for model simulation, are measured under the steady conditions before perturbations beginning to be applied in pipe system.The actual measured data are compared with the output of the T-SII model for the pipes temperature of room 1 and room 2 in summer and winter conditions.Figures 6, 7, 8, and 9 show the comparison between the measured value and predicted value of the simulation system in the inlet of the chilled water supply and return pipe for room 1 and room 2 separately.Results show that the T-SII model has     ( R ).Expediently, the LMI toolbox of MATLAB is used to determine the gain matrix of preview controller [22], and three situations ( R =0;  R =10;  R =20) are discussed to obtain the optimal coefficient for this experiment.In this simulation, the based coefficient matrix of T-SII model is shown in (37), and the reference signal is the regulating variable of temperature, and it can be taken as a step signal: where T is the regulating variable of temperature, Further, investigation will be taken to study the effect of three preview lengths of the reference signal on the closed-loop system.In this simulation, Figure 10 shows the output response of the closed-loop system when regulating variable is 1 ∘ C, and the tracking error is reduced and the settling time can be shortened.Moreover, by increasing the preview length of the reference signal, the output can track the reference signal faster.For analyzing the robustness and stability with preview control, the output response in different regulating variable and interference is discussed.Figure 11 shows the output response of pipe temperature in different regulating variable which represents various operating conditions including start and stable.It can be seen that the temperature response is asymptotically stable under any regulating variable and meets the control requirements.The parameters of the system may change during operation due to the interference.Figure 12 shows the output response of pipe temperature with different temperature interference, when the regulating variable is 1 ∘ C. It can be proved that system with preview control is robust stability.
According to the simulation result of TRNSYS and MATLAB, Figure 13 shows the response curve between valve opening target signal and tracking signal with the controller in which the preview feed forward compensation is not considered.It can be seen that the output signal cannot immediately trace the target signal with the variation of the valve-opening target signal.Although there is a time delay and deviation, it can trace the target signal later.Figure 14 shows the response curve when controller adopts the preview feed forward compensation.In this condition, the valve opening target signal can be promptly traced using the coefficients in (42).Meanwhile, the time delay does not appear and the error is minor in comparison with no preview control.Therefore, the tracking performance is improved, and the effect of time delay can be reduced.

Conclusions
The main goal of this study is to design the simplified thermal time-delay state-space model (T-SII model) and preview control for chilled water system in air-conditioning system.T-SII model adopted small increments as model state variables in order to fully reflect the thermal variation process, and preview control which behaves robustness and stability was applied based on the T-SII model.The results of experiment showed that T-SII model is capable of predicting the performance of the water thermal variation satisfactorily, and the delay time has been compensated by the preview control.Meanwhile, the results of simulation showed that the output of the preview control can track the reference signal faster by increasing the preview length of the reference signal, and the system with this control was robust stability.And also, the tracking performance has been improved in comparison with no preview control.

𝜏:
T i mede la y ,s c: M a s ss pe c i fi ch e a t ,J / k g ∘ C T: T em pera ture, ∘ C t: T i m e ,s L: Thepi pelen gth,m : Den si ty ,kg/m 3 F: Area,m 2 G: Mass flow rate of water, kg/s h: Coefficient of convection heat transfer, W/(m 2∘ C) M: Mass, kg R: Thermal resistance, ∘ C/W : Additional coefficient : Coefficient of convection, W/(m 2∘ C) : Thermal conductivity, W/m N: Turndown ratio of valve, % H: Relative opening degree of valve, % : Local resistance coefficient P: pressure, Pa E: U ni tma trix LMI: Linear Matrix Inequation CHW: Return water pipe CONW: Supply water pipe.

Subscripts env:
Environment pl: P u r el a g p and p(k): Pipe and pipe number w: W a t e r v: V a l v e iw: Inlet water temperature ow: Outlet water temperature ps: P i p es u r f a c e il: I n s u l a t i o nl a y e r ps: Pipe shell.

Figure 2 :
Figure 2: Schematic for the working process of pipe.

Figure 3 :
Figure 3: The preview control structure.

Figure 4 :Figure 5 :
Figure 4: Schematic diagram of the experiment house.(a) Schematic diagram of chilled water loop; (b) experimental equipment; (c) the layout of room 1.

Figure 6 :
Figure 6: Comparison of measured value and predicted value of the simulation system for CHW temperature in room 1.

Figure 7 :Figure 8 :Figure 9 :
Figure 7: Comparison of measured value and predicted value of the simulation system for CHW temperature in room 2.

Figure 10 :Figure 11 :
Figure 10: Comparison of tracking error for simulation system with different preview lengths.

Figure 12 :Figure 13 :Figure 14 :
Figure 12: Comparison of output response for pipe temperature with different temperature interferences.

Table 1 :
Structural information of equipment.

Table 2 :
Summarized initial parameter of the experimental system.