Optimization of Dual-Design Operation Ventilation System Network Based on Improved Genetic Algorithm

Abstract

The COVID-19 pandemic has emphasized the crucial role of ventilation systems in mitigating cross-infections, especially in infectious-disease hospitals. This study introduces a dual-design operation ventilation system that can operate under two sets of ventilation conditions for normal and epidemic times. A challenge is optimizing duct diameters for required airflow while maintaining hydraulic balance. We designed an improved genetic algorithm with an adaptive penalty factor and velocity constraint, as well as the improved crossover probability and mutation probability. The improved genetic algorithm is suitable for ventilation system networks, which can find a better combination of air duct diameters to improve the hydraulic balance rate and reduce the usage of air valves, resulting in efficient hydraulic balancing commissioning. A supply air ventilation system of an actual hospital in China was selected as a case study, and the number of imbalanced air ducts was reduced from 14 to 4. Compared with the traditional genetic algorithm, it has a faster search speed and a better global search ability, which is effective for the optimal design of ventilation system networks.

1. Introduction

The coronavirus 2019 (COVID-19) is a highly infectious virus that has a significant effect on public health [1,2]. The World Health Organization (WHO) reported that at least 17,000 healthcare workers lost their lives due to COVID-19, primarily as a consequence of cross-infection [3]. As infectious-disease hospitals have the largest number of confirmed patients, they confront more difficult challenges than other places. Therefore, the air quality in the hospital deserves attention to minimize the risk of cross infection.

Research indicates that the proper use of ventilation systems has been a crucial factor in the spread of COVID-19 [4,5,6]. The rate of transmission is higher in buildings with lower fresh-air rates [7,8]. Inappropriate air circulation has been the cause of multiple viral transmission cases in indoor environments [9]. However, general hospitals and infectious disease hospitals serve different functions. When facing sudden epidemics, general hospitals cannot meet standards for ventilation systems in infectious-disease hospitals, leading to a risk of cross-infection when treating infectious patients. Meanwhile, infectious-disease hospitals are easily exposed to resource shortages, and the rapid construction of emergency hospitals on a large scale, such as the Thunder Mountain God Hospital [10], comes at a high cost [11].

Therefore, an effective way to address these issues is to adopt dual-design operation ventilation systems (DOV). For example, in patient wards, it means to operate under two sets of ventilation conditions: (1) During normal times, the fresh air rate is three air changes per hour (ACH) [12], emphasizing increased cleanliness to create a slightly positive-pressure environment. (2) During epidemic times, however, the fresh air rate is 6 ACH, and the exhaust air rate needs to be at least 150 m³/h more than the fresh-air rate [13], emphasizing the prevention of viral spread by creating a negative-pressure environment.

For DOV, one of the challenges is to ensure the required airflow rate during normal and epidemic times while achieving hydraulic balance. In the design of ventilation systems, selecting more appropriate duct diameters can enhance the system’s inherent hydraulic balance rate without air valves and reduce the usage of air valves, resulting in efficient hydraulic balancing commissioning during the transition from normal to epidemic times. This leads to faster responses during the transition as well as easier operation and maintenance. However, the data of duct diameter combinations in ventilation systems are discrete and extensive, making it difficult to adjust them only based on design experience. This real-world pipe network problem is a type of discrete combination problem [14,15] which can be solved by heuristic algorithms [16,17,18].

Previous studies have used genetic algorithms to optimize duct diameter combinations [19,20,21]. There are various improved genetic algorithms such as using adaptive penalty functions [22], improved crossover operator [23] and mutation operators [24,25], as well as in combination with hydraulic simulation software such as EPANET 2.2 [26]. However, the optimization objects of previous studies are typically urban water supply networks, while this study focuses on DOV networks. There are differences between water supply networks and DOV networks: (1) Optimization goal: urban water supply networks primarily aim to reduce network construction costs, which typically account for 50% to 70% of the total investment in water supply systems. However, the optimization goal for DOV networks is to increase the hydraulic balance rate and reduce the usage of air valves. (2) Constraint condition: the terminals of urban water supply networks are buildings with higher resistance and can easily use pumps for hydraulic balance. In contrast, the terminals of ventilation systems are air outlets with lower resistance and greater difficulty in achieving hydraulic balance.

Considering the above factors, it is significant to improve the hydraulic balance rate of DOV networks. To achieve this, we should design proper objective functions and improved genetic algorithms to optimize duct diameter combinations of DOV. This study selects a ventilation system of a real hospital in China as a case study.

This study aims to establish a method applicable for the design optimization of ventilation systems, which can be quickly solved to obtain a better combination of pipe diameters based on the original design; the hydraulic balance rate and the economy of the ventilation system network will be improved. The remainder of this paper is organized as follows. Section 2 describes the mathematical model of ventilation system network optimization. Section 3 presents the improved design and the optimization process of the genetic algorithm. Section 4 presents results and discussion based on a case study. Finally, Section 5 presents the conclusions drawn from the study.

2. Mathematical Model of Ventilation System Network Optimization

The optimization of a ventilation system network in this study is to find a better combination of air duct diameters to improve the hydraulic balance rate. The hydraulic balance rate is related to the position, direction, diameter and length of air ducts, as well as local resistance components [27,28].

2.1. Hydraulic Imbalance Rate and Imbalanced Air Ducts

To obtain the hydraulic balance rate, it is necessary to calculate the frictional resistance and local resistance of each air duct, expressed as follows:

$$\Delta P=\Delta P_f+\Delta P_l$$

(1)

$$\Delta P_f=\frac{\lambda }{d_e}·\frac{\rho V^2}{2}$$

(2)

$$\Delta P_l=\xi ·\frac{\rho V^2}{2}$$

(3)

where $\Delta P$ is the total resistance (Pa); $\Delta P_f$ is the frictional resistance (Pa); $\Delta P_l$ is the local resistance (Pa); $\lambda$ is the frictional resistance coefficient; $d_e$ is the equivalent diameter (m); $\frac{\rho V^2}{2}$ is the dynamic air pressure (Pa), $\rho$ is the density of air (kg/m³), $V$ is the air velocity (m/s); $\xi$ is the local resistance coefficient, and it can be obtained by interpolating the experimental data of a design manual [28].

For the frictional resistance coefficient ($\lambda$), it can be calculated by the Colebrook–White equation [29,30], expressed as follows:

$$\frac{1}{\sqrt{\lambda }}=-2\lg \left(\frac{K}{3.71de}+\frac{2.51}{Re\sqrt{\lambda }}\right)$$

(4)

$$Re=\frac{Vde}{\nu }$$

(5)

where$K$ is the absolute roughness of air duct (m); $Re$ is the Reynolds number, and $\nu$ is the kinematic viscosity (m²/s).

After calculating the total resistance of each air duct, the max downstream cumulative resistance (MR) of each air duct can be calculated, expressed as follows:

$$MR=max\left({{\displaystyle \sum }}_{i=1}^{downsteam air ducts of air duct i}\Delta P\right)$$

(6)

For example, the case of the supply air ventilation system network in this study is shown in Figure 1. As can be seen in Figure 1, the fan supplies air at air duct 1. Obviously, for the air duct 53, the supply air can flow through the air duct 53, 54 and 55 or 53 and 56 to reach the downstream terminal air duct 55 or 56. Then, the MR of the air duct 53, i.e., $M{R}_{53}$ can be calculated, expressed as follows:

$$M{R}_{53}=max\left(\Delta P_{53}+\Delta P_{54}+\Delta P_{55},\Delta P_{53}+\Delta P_{56}\right)$$

(7)

where$\Delta P_{53}$ is the total resistance of the air duct 53, which can be calculated by using (1), and the same for the rest of the items.

When an air duct has more downstream terminal air ducts, e.g., air duct 1, then $M{R}_1$ can be calculated similarly to the calculation of $M{R}_{53}$. That is, to get the maximum value among more sets of cumulative total resistance flowing from air duct 1 to different terminal air ducts. Actually, $M{R}_1$ is the most unfavorable resistance of this ventilation system, because the supply air can flow through air duct 1 to reach all the terminal air ducts. When an air duct has only one downstream terminal air duct, e.g., air duct 54, then $M{R}_{54}=\Delta P_{54}+\Delta P_{55}$. When an air duct is a terminal air duct, e.g., air duct 55, then $M{R}_{55}=\Delta P_{55}$.

The hydraulic imbalance rate is the deviation of the MR between two parallel air ducts. Hydraulic imbalance is defined as the hydraulic imbalance rate being greater than 15%. The number of imbalanced air ducts is expressed as follows:

$$N_{ib}={{\displaystyle \sum }}_{i=1}^n \left\{\begin{array}{c}0, imbalrat{e}_i\le 15\% \\ 1, imbalrat{e}_i>15\% \end{array}\right.$$

(8)

$$imbalrat{e}_i=\frac{\max \left(M{R}_i,M{R}_{i-parallel}\right)-M{R}_i}{M{R}_i}$$

(9)

where n is the number of air ducts; $\max \left(M{R}_i,M{R}_{i-parallel}\right)$ is the max MR of air duct i and its parallel ducts.

For example, air duct 53 and 57 are a pair of parallel air ducts, because the supply air can flow through air duct 52 to reach air duct 53 or 57. Then, the hydraulic imbalance rate of air duct 57 can be calculated, expressed as follows:

$$imbalrat{e}_{57}=\frac{\max \left(M{R}_{57},M{R}_{53}\right)-M{R}_{57}}{M{R}_{57}}$$

(10)

If $M{R}_{53}$ is significantly greater than $M{R}_{57}$, resulting in the $imbalrat{e}_{57}$ being greater than 15%, then air duct 57 is an imbalanced air duct. In the real ventilation system, it is required to install a valve to balance the resistance [28].

2.2. Velocity Constraint and Uneconomical Air Ducts

The economic velocity of the air duct in normal times is designed according to the recommended velocity with the requirements of noise reduction and vibration isolation [31], and the range is 3 m/s to 4 m/s for main ducts and 2~3 m/s for branch ducts and terminal ducts. In this way, when the ventilation system is transformed for epidemic times, the velocity doubles, and the range is 6 m/s to 8 m/s for main ducts and 4~6 m/s for branch ducts and terminal ducts, which is still within a reasonable range. The number of uneconomical air ducts is expressed as follows:

$$N_{ue}={{\displaystyle \sum }}_{i=1}^n \left\{\begin{array}{c}0, V_i^{min}<V_i<V_i^{max} \\ 1, V_i<V_i^{min} or V_i>V_i^{max}\end{array}\right.$$

(11)

where $V_i^{min}$ and $V_i^{max}$ are the minimum and maximum of economic velocity, which together constitute the velocity constraint [32].

2.3. Penalty Function

The air duct velocity will be unstable if the traditional genetic algorithm is applied without the velocity constraint. The hydraulic performance of the entire ventilation system will be impacted by some velocity that greatly deviates from the economic velocity. Consequently, to lower the fitness of air duct diameter combinations whose velocity deviates from the economic velocity, a penalty function limited by the velocity constraint should be constructed. The penalty function constructed by using (8), (11) and the penalty factor is as follows:

$$f=N_{ib}+k·N_{ue}$$

(12)

$$k={10}^{p\left(1-z\right)}$$

(13)

where k is the adaptive penalty factor [33]; $p$ is the adjustable parameter, $p>0$; z is the proportion of feasible solutions in the current population, which is defined as the $N_{ib}$ being smaller than when using the original design air duct diameter combination.

There are few feasible solutions in the population at the beginning of the evolution. At this point, k should take a larger value to make the search quickly enter the feasible region [34]. When the proportion of feasible solutions is rising, a smaller k should be taken to make the feasible solutions that are being searched for better and better. This is crucial for searching for the global optimal solution. Consequently, the proportion of feasible solutions influences the penalty factor [35,36].

3. Improved Design of Genetic Algorithm

3.1. Coding Method

Ventilation system network optimization is a discrete combination problem for which integer coding can be used to increase efficiency. The air duct diameters and the corresponding integer encoding rule are shown in

Table 1. Encoding rule.

Width (mm)	Height (mm)	Area (m2)	Integer Code
120	120	0.0144	1
160	120	0.0192	2
200	120	0.0240	3
… (Omitted here; see supplementary data for full table)
800	400	0.3200	30
800	500	0.4000	31
1000	400	0.4000	32

.

There are 32 types of rectangular air duct sizes that may be used in the case of this study, arranged according to the cross-sectional area from small to large, and encoded as 1 to 32. After encoding, an air duct diameter combination will be converted into a chromosome; its length is 57 (number of air ducts in the case of this study), and each integer on it represents a type of air duct size.

3.2. Fitness Function and Selection Operator

The fitness function is constructed by using the reciprocal of (12). The formula is as follows:

$$F^{\prime }=\frac{1}{f}$$

(14)

$$F=\frac{F^{\prime }}{\max \left(F^{\prime }\right)}$$

(15)

where $F^{\prime }$ is the reciprocal of the penalty function of each individual, and $F$ is the fitness of each individual.

Obviously, the best air duct diameter combination, i.e., the elite individual of the population, have a fitness of 1 and the rest of the individuals have a fitness less than 1. The greater $N_{ib}$ and $N_{ue}$, the greater the penalty function $f$ and the smaller the fitness $F$ of the individual, which is more probably to be eliminated during the evolution.

The selection operator is based on fitness; it will select individuals with greater fitness and eliminate individuals with smaller fitness by the proportional selection method, expressed as follows:

$$P_j=\frac{F_j}{{{\displaystyle \sum }}_{j=1}^N{F}_j}$$

(16)

where $P_j$ is the probability that the individual j in the population is selected; $N$ is the number of individuals, i.e., the population size; in this study, N = 50.

Therefore, individuals with greater fitness are more likely to be selected and passed on to the next generation, while individuals with smaller fitness are more likely to be eliminated.

3.3. Crossover Probability and Mutation Probability

The crossover probability and mutation probability should be set to adjust automatically according to the fitness during the evolution, which will improve the global search ability. This study achieves adjustment based on the expectation of fitness, expressed as follows:

$$E=\frac{{{\displaystyle \sum }}_{j=1}^N{F}_j}{N}$$

(17)

$$P_c=\frac{1}{1+e^{-h_1/E}}-0.1$$

(18)

$$P_m=\frac{1}{5\left(1+e^{h_2/E}\right)}$$

(19)

where $E$ is the expectation of fitness, i.e., the mean fitness of all the individuals in the population; $P_c$ is the crossover probability; $P_m$ is the mutation probability; $h_1$ and $h_2$ are adjustable parameters; and $h_1,h_2>0$.

As mentioned before, individuals with greater fitness are more likely to be selected and inherited by the next generation, causing $E$ to increase. Then, the crossover probability $P_c$ will decrease and the mutation probability $P_m$ will increase, which is more similar to the actual biological evolution process, so that the improvement of the crossover probability and mutation probability can be achieved.

3.4. Crossover Operator and Mutation Operator

The crossover probability will determine the number of parent individuals in each generation. After each pair of parent individuals are selected by the selection operator, the crossover operator will perform a one-point crossover, i.e., randomly select the crossover position from which they exchange with each other.

For example, assuming that a pair of selected parent individuals are:

$x_1^n$  30 18 21 12 … 5 (length is 57)

$x_2^n$  25 15 18 11 … 2

Supposing that the crossover operator starts from the third position, then the child individuals will be:

$x_1^{n+1}$  30 18 18 11 … 2

$x_2^{n+1}$  25 15 21 12 … 5

The mutation probability will determine the number of integers which will mutate on all the chromosomes in each generation; the mutation operator randomly selects the mutation position and executes the mutation. In addition, to prevent elite individuals from being destroyed, the elite individuals in each generation do not participate in mutation.

Considering the characteristics of the ventilation system network, this study restricts the interval of mutation operator to be ±1 around the original integer.

For example, assuming that a parent individual is:

$x_1^n$  30 18 21  12 … 5

Supposing that the mutation operator occurs at the third and last position, then the child individual will be:

$x_1^{n+1}$  30 18 20~22  12 … 4~6

Since the air duct diameters are arranged according to the cross-sectional area from small to large during the encoding, this kind of mutation operator can prevent the air velocity from excessive change, which would result in exceeding the economic velocity range. The improved mutation operator can not only ensure the diversity of the population, but also prevent individuals with high fitness from being destroyed.

3.5. Ventilation System Network Optimization Process

The flow chart of ventilation system network optimization process is shown in Figure 2.

The optimization process is: (1) Started: Read the original design air duct diameter combination of the ventilation system to obtain the initial individual. (2) Initialization: New individuals are generated randomly along with the initial individual to get the initial population. (3) Calculate fitness: The fitness of each individual is calculated by penalty function. (4) Selection: According to the fitness of each individual in each generation, parent individuals are selected by the proportional selection method. (5) Crossover and mutation: According to the improved crossover probability and mutation probability, crossover operator and mutation operator are performed to obtain child individuals. (6) Iteration: Iterate to the max generation, or the number of imbalanced air ducts ($N_{ib}$) has reduced to the optimization goal. (7) End: Output optimization results.

The bold part in Figure 2 illustrates the advantages of the improved genetic algorithm compared with the traditional genetic algorithm: (1) Adaptive penalty parameters are adjusted according to the proportion of feasible solutions during the evolution, which can improve the global search ability. In contrast, the penalty parameter and the velocity constraint are not used in the traditional genetic algorithm. (2) The improved crossover probability gradually decreases, and the improved mutation probability gradually increases with the increase in the expectation of fitness, which is more similar to the actual biological evolution process. In contrast, the crossover probability and the mutation probability in the traditional genetic algorithm are constants. (3) The improved mutation operator can ensure the diversity of the population and prevent individuals with high fitness from being destroyed. In contrast, the mutation operator in the traditional genetic algorithm will perform completely randomized mutations.

4. Result and Discussion

In this study, a supply air ventilation system of an actual hospital in China was selected as a case study, as shown in Figure 1. The ventilation system network is a branching network with 23 nodes and 57 air ducts. The total supply airflow is 5150 m³/h in normal times and 10,300 m³/h in epidemic times. The supply airflow and the length of each air duct are known; see Supplementary Data.

4.1. Comparison of the Improved Genetic Algorithms

A traditional genetic algorithm and an improved genetic algorithm are used to optimize the ventilation system network. The number of imbalanced air ducts ($N_{ib}$) during the evolution is shown in Figure 3 and Figure 4.

It can be seen that $N_{ib}$ decreases from 14 to 9 when the traditional genetic algorithm is iterated to the max generation (100), while $N_{ib}$ decreases to 4 when the improved genetic algorithm is iterated to the 57th generation, reaching the optimization goal (below 5) and quitting the iteration. It shows that the improved genetic algorithm has a faster search speed and a better global search ability.

The number of uneconomical air ducts ($N_{ue}$) during the evolution is shown in Figure 5 and Figure 6.

Obviously, $N_{ue}$ increases instead of decreases during the evolution of the traditional genetic algorithm, while the improved genetic algorithm adopts the velocity constraint, so it decreases from 26 to 18 during the evolution. As a result, the most unfavorable resistance during the evolution is shown in Figure 7 and Figure 8.

Because the uneconomical air ducts often have greater resistance, the most unfavorable resistance increases during the evolution of the traditional genetic algorithm. On the contrary, the most unfavorable resistance is slightly reduced during the evolution of the improved genetic algorithm. This point can also be seen from the best solution (the air duct diameter combination on the most unfavorable path of the elite individual) during the evolution. In the original design, the most unfavorable path of the ventilation system is, starting from the air duct 1, flowing through the air duct 2, 39, 51, and 52, and reaching the terminal air duct 57. The integer codes for the sizes of these six air ducts are 30, 30, 17, 8, 8, and 5, which are changed during the evolution, as shown in Figure 9 and Figure 10.

As mentioned before, the improved mutation operator can prevent the air velocity from excessive change. In contrast, the mutation operator of the traditional genetic algorithm performs random mutation, which would result in exceeding the economic velocity range. Combined with Figure 7 and Figure 8, it can be seen that the improved mutation operator prevents the increase in the most unfavorable resistance caused by the destruction of the more economical pipe diameter combinations, which is advantageous when searching for the optimal solution.

4.2. Validation of the Improved Genetic Algorithm

The proportion of feasible solutions, the crossover probability and the mutation probability during the evolution are shown in Figure 11, Figure 12 and Figure 13.

The proportion of feasible solutions gradually increases during the evolution, indicating that the searched solutions obtained are getting better and better. The crossover probability and mutation probability gradually decrease and increase during the evolution. As mentioned before, this indicates that the expectation of fitness is increasing, which is consistent with the expected effect of the improved crossover probability and mutation probability.

4.3. Limitation

This study also has some limitations: (1) Computation speed: In real larger-scale ventilation systems, the computation speed of the algorithm may encounter challenges. (2) Computation accuracy: The calculation of the frictional resistance coefficient ($\lambda$) in this study relies on the Colebrook–White equation [29,30], and local resistance coefficients ($\xi$) are based on experimental data from a design manual [28]. In real project applications, deviations may occur, indicating that more accurate resistance coefficients may need to be measured or simulation methods such as VentSim [37] may be used to obtain more accurate resistance coefficients. (3) Algorithmic design: Despite the adaptability of penalty factor, crossover probability, and mutation probability during the evolution, the choice of adjustable parameters in their calculation equations primarily depends on experience. Inappropriate selections may lead to computational instability. In addition, the algorithm exhibits dependence on the initial population. In real project applications, it is often necessary to begin with a reasonably designed pipe diameter combination as an initial individual, and new individuals are generated randomly along with the initial individual to get the initial population. This ensures a faster convergence speed. Consideration of incorporating other heuristic algorithms may enhance the performance of the algorithm.

5. Conclusions

The improved genetic algorithm introduced in this study can be applied to the design and optimization of the ventilation system. The optimization process can be summarized as: Initialize the population to obtain integer codes for multiple combinations of pipe diameters. Then, decode them into pipe diameter combinations and use (15) to calculate the fitness of all individuals. Based on the fitness ranking, the integer codes are substituted into the genetic algorithm to perform selection, crossover, and mutation operations and obtain the second-generation population. Repeat until the max generation or the $N_{ib}$ optimization goal is reached.

As a result, a better air duct diameter combination can be found during the evolution. In the case of this study, the number of imbalanced air ducts is reduced from 14 to 4, which will improve the hydraulic balance rate of the ventilation system network and reduce the usage of air valves. It will help to achieve efficient hydraulic balancing commissioning, especially for a dual-design operation ventilation system during the transition from normal to epidemic times. This leads to faster response during the transition as well as easier operation and maintenance.

Meanwhile, the number of uneconomical air ducts is reduced from 26 to 18, and the most unfavorable resistance after optimization is slightly reduced. This indicates that the reduction in imbalanced and uneconomical air ducts is not at the cost of increasing energy consumption.

The improved genetic algorithm adopted the adaptive penalty factor and the velocity constraint in the penalty function, and the improved crossover probability and the mutation probability will adjust automatically according to the fitness during the evolution. Compared with the traditional genetic algorithm, it has a faster search speed and a better global search ability. In conclusion, this method has a good application value in the optimal design of the ventilation system network.