RCGA based PID controller with feedforward control for a heat exchanger system

1. Introduction

A heat exchanger is widely used in chemical plants and/or ships because it can sustain a wide range of temperatures and pressures. There are different types of heat exchangers, but a shell and tube heat exchanger system is most widely used [1]. The main purpose of a heat exchanger system is to transfer heat from a hot fluid to a cooler fluid; thus, controlling the temperature of the outlet fluid is of prime importance. An accurate mathematical model is required to design a controller. However, it is difficult to obtain a suitable mathematical model because the dynamics of a heat exchanger depends on many factors such as temperature difference, heat transfer area, flow rate of fluids, and flow patterns. Various studies have proposed different temperature control methods for heat exchangers such as feedforward PID control [2], optimal linear fuzzy control [3], PI type fuzzy control [4], PID controller, and internal model control [5][6].

PID control is a classical method, and it has been studied for the last several decades and is widely used in the field. The inherent limitation of a PID controller, however, is that the controller acts after disturbance distorts the required control objective. If disturbances occur frequently, a PID controller would not be able to attain the desired steady state response. To overcome this drawback, feedforward control must be adopted with PID control.

This paper presents a linear PID controller with feedforward control that shows a better control performance and systematic tuning than conventional PID controllers. The temperature control of the heat exchanger may cause time delays and disturbances; thus a PID controller combined with a feedforward controller is designed to improve control performance.

The parameter tuning methods of the PID controller include the closed-loop and open-loop tuning methods such as the Z–N (Ziegler–Nichols) tuning method, IMC (Internal Model Control) tuning method, and RCGA (Real Coded Genetic Algorithm) tuning method [7]-[9]. Among them, the RCGA tuning method provides an optimal solution by using the evaluation function under given conditions. This study, therefore, uses the RCGA technique to tune the PID controller. In order to improve the tracking performance of the control system, the PID controller is tuned while the disturbance is fixed, and the set value is changed stepwise. Further, the PID controller tuned by RCGA and a feedforward controller are combined to examine disturbance rejection while the measured disturbance is applied to the heat exchanger system.

A set of simulations are carried out to compare set-point tracking and disturbance rejection performances with conventional PID controllers and show the feasibility of the proposed method.

2. Modeling of heat exchanger

Figure 1 shows the temperature control system of the shell and tube heat exchanger used in industrial fields. This system consists of a heat exchanger, a controller, an I/P converter, a control valve, and a temperature transmitter (TT). Overheated steam at a temperature of 180 [℃] supplied from the boiler flows into the tubes after the control valve adjusts the amount of steam. On the other hand, fluid at a temperature of 30 [℃], pumped by a centrifugal pump, flows through the shell. These conditions are assumed to be equilibrium states for simulations.

Figure 1:

Temperature control system of shell and tube heat exchanger

Different assumptions have been considered in this study. The first assumption is that the inflow and outflow rates of fluid are same, so that the level of fluid is constant in the heat exchanger. The second assumption is that the heat storage capacity of the insulating wall is negligible.

The inlet and outlet temperatures measured by a thermocouple are converted to current signals from 4 [mA] to 20 [mA] and are implemented in the feedback path of the control system.

The controller implements the control algorithm, compares the cold fluid output with the desired value, and then provides the necessary command to the final control element (valve) via the actuator unit. The actuator unit is a current-to-air pressure converter, and the final control unit is an air-to-open (fail-close) valve. The actuator unit converts the controller output within a range of 4-20 [mA] into a standardized air pressure signal within a range of 0.2-1.0 [bar]. The valve is actuated according to the controller decisions.

The experimental data available for the heat exchanger system is summarized in other studies [10][11]. A linearized mathematical model of heat exchanger is developed from the experimental data.

2.5 Disturbance

Assuming that the transfer function does not have a time delay, G_d(s) can be expressed as a first order system as follows.

$\[ G_d\left(s\right)=\frac{k_d}{1+{\tau }_{x}s} \]$

(5)

k_d and τ_x are the gain and time constants of disturbance, respectively. Further, it is assumed that τ_x is equal to the time constant of the heat exchanger.

Figure 2 shows the control system consisting of a saturator, actuator, and heat exchanger and the disturbance.

Figure 2:

Block diagram of controlled system

This can be expressed by the following equation.

$\[ Y\left(s\right)=\frac{1}{1+{\tau }_{x}s}\left[k_X{e}^{-Ls}{U}_a\left(s\right)+k_d{Y}_d\left(s\right)\right] \]$

(6)

Y(s) is the outlet temperature of the fluid, and Y_s(s) is the inlet temperature of the fluid under disturbance.

Table 1 summarizes the various parameters used in this study. The minimum and maximum values of the actuator are 4 [mA] and 20 [mA], respectively.

Table 1:

Parameters of the heat exchanger system

3. PID controller combined with feedforward controller

3.2 Feedforward controller

The inherent limitation of a feedback controller is that the controller acts after disturbance distort the required control objective. If disturbances occur frequently, the PID controller would not be able to attain the desired steady state response. To overcome this limitation, feedforward control must be adopted with PID control. Feedforward control limits the deviation caused by the disturbance while it works under the condition that the disturbance would be measured or estimated. Feedforward control works in combination with feedback control, as it cannot operate alone.

Figure 3 shows a control system incorporating the feedforward controller. In this system, the closed-loop transfer function of the output temperature under disturbance is as follows.

$\[ \frac{Y\left(s\right)}{D\left(s\right)}=\frac{G_f\left(s\right)G_p\left(s\right)+G_d\left(s\right)}{1+G_c\left(s\right)G_p\left(s\right)G_s\left(s\right)} \]$

(8)

Figure 3:

PID control system combined with feedforward controller

G_f(s) is the transfer function of the feedforward controller. G_p(s) is the transfer function of the actuator and heat exchanger.

In case the numerator of Equation (8) becomes 0, the disturbance can be removed.

$\[ G_f\left(s\right)G_p\left(s\right)+G_d\left(s\right)=0 \]$

(9)

Rewriting the above equation, the transfer function of the feedforward controller is as follows.

$\[ G_f\left(s\right)=-\frac{G_d\left(s\right)}{G_p\left(s\right)} \]$

(10)

Because there is a time delay in G_p(s) , the order of the numerator is larger than that of the denominator in G_f(s) . In order to replace the time delay, (1+λs)ⁿ is added in the denominator of G_f(s) , and then, Equation (10) is expressed as follows.

$\[ G_f\left(s\right)=-\frac{K_d}{K_a}\frac{\left(1+{\tau }_{a}s\right)\left(1+{\tau }_{x}s\right)}{\left(1+{\tau }_{x}s\right){\left(1+\lambda s\right)}^n},\left(\lambda >0\right) \]$

(11)

λ is the filter time constant, and its value varies between 0 and 1. In this paper, λ was set to 0.9 and n to 1 through trial and error.

4. Tuning of the PID controller

4.2 IMC tuning technique

In order to use the IMC technique, the transfer function of the plant have to be given by the FOPTD (First Order Plus Time Delay) system. Figure 4 shows the process of searching k, τ, and the L values of FOPTD obtained using RCGA, and we can obtain k = 4.976, τ = 28.557, L = 2.228.

Figure 4:

Search for, k, τ and L using RCGA

Figure 5 shows the step response for the SOPTD(Second Order Plus Time Delay) system and FOPTD system as the approximated model. Comparing the two responses shows that the step responses showed good agreement.

Figure 5:

Step responses of SOPTD and FOPTD system

4.3 RCGA-based tuning technique

First, in order to improve the tracking performance, the parameters of the PID controller are tuned while the disturbance is fixed and the set value is changed stepwise.

This paper proposes RCGA as a tuning method and the IAE(Integral Absolute Error) is used as an evaluation function.

$\[ J\left(\varphi \right)={\int }_0^{t_f}\left|e\left(t\right)\right|dt \]$

(13)

φ is [K_p, K_i , K_d]∈ R³ and e(t) is the error between the set value and the output, and the integration time t_f is a sufficiently large value enough to ignore integral values thereafter.

Because optimization techniques operate on a probabilistic basis in general, the solutions obtained vary slightly depending on the configuration of the initial group. Therefore, in order to ensure that the solution searched with RCGA is optimal, simulations are performed five time by using random seed values, and the average value of the results is determined as final solutions. Figure 6 shows the concept of searching each parameter of the PID controller by RCGA.

Figure 6:

Parameter optimization of PID controller using RCGA

In order to obtain ing the optimal parameter using RCGA, the number of generations, population, crossover, mutation were set to 50, 30, 0.9 and 0.05, respectively. Figure 7 shows the process of searching K_p, K_i , K_d of the PID controller using RCGA.

Figure 7:

RCGA-based tuning process for PID controller

Table 2 summarizes the parameters obtained from the proposed technique and the other techniques.

Table 2:

Parameters of the PID controllers

5. Simulation and review

In order to validate the proposed controller, simulations were performed with the sampling time of 0.01[s], and the proposed controller was compared with the Z–N and IMC technique.

5.1 Tracking performance of PID controller

Figure 8 shows the outputs and control inputs when the set value is changed from 50 [°C] to 60 [°C], under the assumption that disturbance does not exist.

Figure 8:

Tracking responses when yr is step-wisely increased from 50 [°C] to 60 [°C]

Table 3 summarizes the rising time, overshoot, setting time and absolute error integration.

Table 3:

Tracking performance

From Figure 8 and Table 3, it can be seen that rising time of the Z–N technique is the shortest, while the overshoot value and settling time are very large. The overshoot value of the IMC technique is slightly smaller than those of other methods, but the rising and settling times are very large. The overshoot value of the proposed RCGA technique is very similar to that of other techniques, while the rising and settling times are very small, corresponding to excellent results.

5.2 Disturbance rejection performance of PID controller without feedforward controller

In the above discussion, it was found that the RCGA-based PID controller has a relatively good performance in the tracking problem compared with other methods.

Next, simulation is performed to examine the disturbance rejection performance when a measured disturbance is applied to the heat exchanger system. The temperature of cold fluid is assumed to decrease stepwise from 30 [°C] to 20 [°C] after 50[s]. Figure 9 shows the control results, and Table 4 summarizes MV (maximum variation) and MV time, recovery time, and absolute error integration.

Figure 9:

Disturbance rejection responses without feedforward controller when d decreases stepwise from 30 [°C] to 20 [°C]

Table 4:

Disturbance rejection performance without feedforward controller

5.3 Disturbance rejection performance of PID controller combined with feedforward controller

Because the feedforward controller does not affect the tracking performance, the disturbance rejection performance with feedforward controller is shown intensively in Figure 10.

Figure 10:

Disturbance rejection responses with feedforward controller when d is step-wisely decreased from 30 [°C] to 20 [°C]

Table 5 summarizes the MV and MV time, recovery time, and absolute error integration.

Table 5:

Disturbance rejection performance with feedforward controller

According to Figure 9 and Figure 10, the PID controller combined with the feedforward controller shows better disturbance rejection performance. Especially, the RCGA-based PID controller shows excellent disturbance rejection performance when the feedforward controller is combined.

From these simulations, it is clear that a PID controller combined with a feedforward controller is a much better option for disturbance rejection control problem rather than conventional PID control.

6. Conclusion

This paper implemented a conventional PID controller and a PID controller with a feedforward controller to control the outlet temperature of a shell and tube heat exchanger system. A mathematical model of the heat exchanger was developed using experimental data of previous studies. The parameters of the PID controller were tuned using the RCGA technique. The performance of the controllers was evaluated using transient characteristics and error indices. From the simulation results, it was found that the performance of the RCGA-based PID controller combined with the feedforward control was superior under disturbance.

The classical PID controller showed a higher MV and MV time whereas the RCGA-based PID controller with feedforward controller reduced the MV, MV time and also obtained a good recovery time.

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