Pengfei Li

Chunhui Feng

Hongchuan Liu

Yong Zhao

Zhen Li
email: zheng.li@ifsttar.fr

Haocheng Xiong
email: xiong.haocheng@gmail.com

Development and Assessment of a Water Pressure Reduction System for Lining Invert of Tunnels in Saturated Grounds

In this study, the feasibility of a novel bottom-up drainage and water pressure reduction system for reducing the secondary lining external water pressure on tunnels has been validated by conducting laboratory evaluations. The tunnels environment including surrounding rock, lining, bottom drainage system and other supplementary components have been simulated to investigate the working pattern and efficiency of this drainage system. The drainage system has been further optimized by analyzing the measured

Introduction

Water penetration and dripping are always critical problems for tunnels in the water-saturated ground. The groundwater in the tunnels are typically treated in two ways: waterproofed or drained [START_REF] Yoo | Hydraulic deterioration of geosynthetic filter drainage system in tunnels -its impact on structural performance of tunnel linings[END_REF]. The waterproofed tunnel doesn't allow the groundwater to penetrate the secondary lining as the waterproof boards are set along the circumferential direction to form a complete. This issue very common in urban tunnel constructions due to the low groundwater discharging rate. A drained tunnel usually allows groundwater into the tunnel, which is finally discharged along with the various measures in such tunnel. Usually, the waterproof and drainage system of a drained tunnel consists of initial support, secondary lining, waterproof boards, blind tubes, and central drainage ditches. Existing waterproof and drainage system for tunnels is mainly designed to deal with the groundwater pressure applied on the upper structure of the tunnel [START_REF] Yuan | Tunnel waterproofing practices in China[END_REF][START_REF] Shin | Effective control of pore water pressures on tunnel linings using pin-hole drain method[END_REF]. The water pressure applied is supposed to be bearded by the tunnel invert. Unfortunately, it's been proved that the current design is not able to solve the problems due to the potential damages on the ballast bed, including uplift, mud pumping, cracking and so on [START_REF] Butscher | Swelling potential of clay-sulfate rocks in tunneling in complex geological settings and impact of hydraulic measures assessed by 3D groundwater modeling[END_REF].

Among the four causes [START_REF] Gamisch | Future trends in construction and maintenance management of drainage systems in traffic tunnels[END_REF] of these damages, the groundwater plays a much more important role than the other factors, such as construction defection, longterm train vibration load, and poor surrounding rock.

Many researchers have investigated how to control the external water pressure on the tunnel lining. [START_REF] Arjnoi | Effect of drainage conditions on porewater pressure distributions and lining stresses in drained tunnels[END_REF] conducted theoretical analysis and numerical simulation to study the effect of drainage conditions on pore water pressure distributions and lining stresses in a drained tunnel with two different boundary conditions. [START_REF] Wang | Theoretical and experimental study of external water pressure on tunnel lining in controlled drainage under high water level[END_REF] presented a theoretical model to predict the distribution of water pressure on tunnel lining by laboratory test and field evaluation. [START_REF] Fang | The effect of external water pressure on the liner behavior of large cross-section tunnels[END_REF] developed an apparatus to apply the appropriate external water pressure by evacuating the inner space of the tunnel and applying external air pressure to the liner to act as the water pressure. [START_REF] Stripple | Evaluation of two different drainage systems for rock tunnels[END_REF] presented a new design of drainage system for rock tunnels. [START_REF] Yee | Substrate Modification and Magnetic Water Treatment on the Maintenance of Tunnel Drainage Systems. II: Field Tests[END_REF] and [START_REF] Yee | Substrate Modification and Magnetic Water Treatment on the Maintenance of Tunnel Drainage Systems. II: Field Tests[END_REF] analyzed the influence between magnetic fields and calcium carbonate deposition through laboratory tests and field evaluations to prevent clogging. Besides, wave vibrations [START_REF] Xin | Reduction of Adherent Forces of Sedimentous Contaminants in Tunnel Drainage using Vibrations from Flexible and Transparent Organic Films[END_REF], Quantum Stick [START_REF] Jung | Evaluation of advanced drainage treatment for old tunnel drainage system in Korea[END_REF] and geosynthetic [START_REF] Jang | Evaluation of discharge capacity of geosynthetic drains for potential use in tunnels[END_REF], etc. were also investigated. [START_REF] Choi | An optimal lightweight foamed mortar mix suitable for tunnel drainage carried out using the composite lining method[END_REF] proposed an optimal lightweight-foamed mortar mix suitable for composite lining method to facilitate tunnel drainage. The lightweightfoamed mortar replaces the traditional initial support, and the drainage system works more efficiently due to its porous structure. The current studies mainly focus on the porewater pressure distributions on the whole tunnel lining or the water pressure that applies to the upper structure of the tunnel. There is still much research needs on the solution of the problems occurred on tunnel invert caused by high water pressure.

To prevent the damage of the tunnel invert caused by water pressure, [START_REF] Li | A bottom-to-up drainage and water pressure reduction system for railway tunnels[END_REF] developed a novel bottom-to-up drainage and water pressure reduction system (Fig. 1) to reduce the water pressure that applies on the secondary lining. Such system includes a transverse catchment system, a longitudinal water conducting system, and a bottomto-up water drainage system. Series of numerical analyses has been conducted to evaluate its drainage performance. Better waterproof and drainage function can be achieved on the entire tunnel by combining the new and existing technologies. This paper presents an experiment-based study on the bottom-to-up drainage and water pressure reduction system. The performance of the proposed system will be validated by experiments. The drainage efficiency of different types of drainage pipe and the influence to seepage field caused by the pipe diameter will also be analyzed.

The overall idea of this article is shown in flowchart 2.

Experimental setup

The equipment setup is shown in Fig. 3 including the structure of tunnel invert, the bottom-to-up drainage, water pressure reduction system and surrounding rock with groundwater, which simulates the structure part highlighted in Fig. 1. According to functions of the components, they can be divided into two systems: water supply system

Water supply system

The water supply system (Fig. 4) consists of 6 parts, providing power to make water flow and receiving water from the drainage system. Water tank(I) is a steel cylinder, which is 1.2m high with an inner diameter of 0.6m and a 2mm-thick wall. The Water pump(II) is the main power supply of the experiment to make water flow, which is 1.1m-long with a flow of 3 m 3 /h. Intake pipe(III) connects the water supply system and the simulating drainage system. There are two tees in the intake pipe(III), the first tee divides the flow path into two: one connects inlet valve(VI) to make water flow into the simulating drainage system, the other connects return valve(V) and return pipe(IV) to make water flow back into water tank(I); the second tee divides the other flow path into two to connect testing chamber as there are two intakes on both sides of the testing chamber.

The following two flow paths were designed in this experiment:

1. Path 1 is 1-water tank →2-water pump →3-intake pipe →the first tee →5return valve →4-return pipe →1-water tank; 2. Path 2 is 1-water tank →2-water pump →3-intake pipe →the first tee →the second tee →6-inlet valve →the simulating drainage system →1-water tank.

The volume of water flowing into the testing chamber can be controlled by regulating 5-return valve and 6-inlet valve, the water pressure in the testing chamber can also be controlled if superadding regulating 22-ball valve.

Drainage system

The drainage system makes up the testing chamber and the components in it. The testing chamber is a cuboid steel box with a size of 3m×1m×0.65m (length × width × height), which is used to simulate the tunnel invert structure and groundwater shows in Fig. 5. There is a buffer box(VIII) in each side of the testing chamber, and the size of the buffer box(VIII) is 0.2m×1m×0.65m (length × width × height). The surplus space of the testing chamber is the main container of simulating tunnel invert. The size of the testing chamber is 2.6m×1m×0.65m (length × width × height). There is a pressure gauge(VII-a) connected to the top of each buffer box(VIII). The Cover plate(XVII) is installed on the top of the testing chamber by screws. The Cover plate(XVII) is manufactured with two parts to reduce the weight for easy transport. On each plate, there is a hole corresponding to the erect drainage pipe and oblique drainage pipe.

The components in the testing chamber are shown in Fig. 6. From bottom to top, there are gravel filling(X), precast concrete board(XI), convex hull drainboard(XII), cast-in-place concrete filling(XIII), half-round tube(XIV), waterproof board(XV) and gravel filling Ⅱ(XVI), corresponding to the simulation of surrounding rock, initial support, convex hull drainboard, secondary lining, half-round tube and waterproof board on the edge in the bottom-to-up drainage and water pressure reduction system respectively. The permeability of initial support is relatively high in the tunnel because it is usually fabricated by shotcrete. It is very difficult to reproduce the shotcrete support in the laboratory with the same permeability performance. Therefore a gap about 3cm wide is put between two 11-precast concrete boards for water flowing through to reach a permeability similar to the real condition. Gravel filling Ⅱ(XVI) is used to fill the space between the cast-in-place concrete filling(XIII) and cover plate(XVII) and transfers force if necessary.

There are two types of bottom-to-up drainage pipes: oblique drainage pipe(XIX) and erect drainage pipe(XX). The performance of the drainage pipe can be analyzed through the data of water discharge and water pressure reduction, which helps to improve the design of bottom-to-up drainage and water pressure reduction system in tunnels.

The diameter of the half-round tube(XIV) is 200mm, and that of oblique drainage pipe(XIX) and erect drainage pipe(XX) is 100mm. There are convex parts on both sides of convex hull drainboards(XII). When convex hull drainboard(XII) is set on precast concrete boards(XI), there are plenty of flowing paths under the convex hull drainboard, and the depression area will be filled by cast-in-place concrete to meet the force-bearing demand. The half-round tube is connected to drainage pipes with a tee and waterproof glue. The edges of the waterproof board, half-round tube, and testing chamber will be connected to each other by waterproof glue. The whole structure of the waterproof board, half-round tube, and drainage pipe are shown in Fig. 7.

Monitoring and data acquisition system

In this experiment, the pore water pressure and flow charge of drainage pipes are investigated.

Pore water pressure monitoring

There are two types of pore pressure sensors: HC-25 micropore pressure sensor (0.2% in precision) and HCYB-25 micropore pressure sensor (0.5% in precision).

Water discharge monitoring

Water discharge monitoring system consists of a water container, an electronic scale, and a stopwatch. The weight of the water container was measured before the experiment. When adding water into the container, the corresponding elapsed time was also monitored. The water discharge of each drainage pipe was obtained by calculating the mass of the water in the container and the elapsed time. For better accuracy, 3 was used as an average value of water discharge in each experimental condition.

4 Experimental programme

The process of installation of experimental equipment

The process of installing the bottom-to-up drainage and water pressure reduction system for tunnels include filling gravel material, arranging and setting pore pressure sensors, installing drainage pipes, pouring concrete for simulating tunnel invert secondary lining, installation of inlet/outlet pipes and valves, testing of data acquisition equipment and so on. The detailed process is as follows:

(1) Transporting and fix the testing chamber and water tank etc. in the prepared testing field;

(2) Laying some non-woven fabric to cover the inlet holes to prevent any gravel getting into the buffer box;

(3) Filling gravel in the bottom of testing chamber with a thickness of 18cm, tamping to dense the gravel;

(4) Placing the Bottom Tier Sensors according to the prepared arrangement;

(5) Placing two precast concrete boards into the testing chamber and reserve a 3cm gap between them, seal the edge of the boards with waterproof glue;

(6) Placing the Medium Tier Sensors and gather all the wires of the sensors with a tarp;

(7) Installing half-round tube, convex hull drainboard, waterproof board and so on, connect and seal with waterproof glue;

(8) Installing the erect drainage pipe and oblique drainage pipe, set the Top Tier Sensors;

(9) Pouring self-compacting concrete with a thickness of 40cm, tamping gently and complete maintenance;

(10) Filling the upper gravel with a thickness of 10cm and tamping;

(11) Installing the cover plate and connecting it to the testing chamber with screws;

(12) Installing one-way valve, ball valves and other pipes;

(13) Installing intake pipe, valves, water pump and so on, connect the water supply system and the testing chamber;

(14) Installing the pressure gauges on both sides of the testing chamber, connecting sensors and the data acquisition equipment, debugging facilities, and prepare to carry out the experiment.

Simulations

The following experiments will be discussed in this section.

(

1) Steady seepage test

The distribution of water pressure in steady seepage will be analyzed in this study.

When performing the experiment, all intake pipes and 4-return pipe will be opened except the drainage pipes, which are controlled by the ball valves(XXII). It includes three conditions:

Only the ball valve of erect drainage pipe is opened, and the draining is conducted using erect drainage pipe (called "DEDP" for short);

Only the ball valve of oblique drainage pipe is opened, and the draining is conducted using oblique drainage pipe (called "DODP" for short);

Both ball valves are opened, and the draining is conducted using drainage pipes (called "DBDP" for short).

(2) Influence of the drainage pipes diameter on the effect of water pressure reduction

The effect of the water pressure reduction system will be evaluated, and the influence of the diameter of the drainage pipe will also be investigated. Due to the limitation of the laboratory instrumentation, the drainage pipe was not able to be replaced directly to complete the experiment of the diameter of the drainage pipe.

Different opening angles of the ball valve(XXII) result in different drainage discharge of the drainage pipe, which is similar to draining with different diameters of drainage pipes. In the experiment, return valve(V) was opened, and the inlet valve(VI) and the ball valve(XXII) was closed. The water pump(II) and the inlet calve(VI) opened gradually to increase the water pressure of the pressure gauge(VII) until 20kPa. The ball valve(XXII) of erect drainage pipe was then opened to about 30°, and the water pressure was measured during the process of the water pressure reduction. This process was repeated, and the water pressure was recorded with ball valve(XXII) of erect drainage pipe opened to 15° and 10°, and ball valve(XXII) of oblique drainage pipe opened to 30°, 15°, and 10°.

It is found that the inner diameter of the drainage pipe is 10cm when the ball valve(XXII) is opened to about 30°, 15°, and 10°. The equivalent diameters of the drainage pipe are 47mm, 27mm, and 18mm, respectively.

(3) Influence of inlet water pressure on the seepage field

In this section, the water pressure and water discharge will be discussed. The crosssection area of the two types of drainage pipe is as identical as possible to make the inlet water pressure the only variable. In the experiment, the return valve(V) was opened first, then the inlet valve(VI) was closed, and the ball valves(XXII) was then opened to about 10°. The water pump(II) was then started, and the inlet valve(VI) was opened gradually to make the water pressure to reach 5kPa, 10kPa, 15kPa, and 20kPa, which is monitored by pressure gauge(VII). Then the pore water pressure and water discharge of each pressure is obtained. Under the same condition, the average pore pressure measured from the bottom tier sensors is the highest. The one from the medium tier is lower than the bottom tier, and the one from the top tier is the lowest. The difference of the data measured by the bottom tier and the medium tier, the medium tier, and the top tier is 0.961kPa, 0.896kPa, respectively. The difference is almost the same as the hydrostatic pressure of the sensors in the different height of, which indirectly proves the correctness of these water pressure data.

Influence of the diameter of the drainage pipe

The water pressure of the secondary lining of the tunnel is primarily affected by the water pressure reduction system. The water pressure measured by No.10~No.22 pore pressure sensors is more accurate than No.1~No.9. Therefore, the water pressure data measured by No.10~No.22 sensors will be mainly discussed in this part. The measured water pressure is shown in Table 4. Due to the error of the data acquisition system, the water pressure measured by the 7-pressure gauge varies slightly around 20kPa, which should maintain stable at 20kPa accurately without system error.

Table 4 shows that when the equivalent diameters are 47mm, 27mm, and 27mm, the water pressure decrease for about 60%, 45%, and 30%, respectively, which indicates that the water pressure decreases with the increase of the equivalent diameter of drainage pipe. This result proves that if the drainage capacity of the drainage and water pressure reduction system is enough for a tunnel, the water pressure on the secondary lining will not increase too much, which agrees with the conclusion of [START_REF] Li | A bottom-to-up drainage and water pressure reduction system for railway tunnels[END_REF]. If the drainage capacity of the drainage pipe for a tunnel is not enough for the water inflow of the construction region, the effect of the bottom-to-up drainage and water pressure reduction system will decrease, and it may even result in the damage of the tunnel due to the additional drainage measures in the bottom of the tunnel.

The reduction of water pressure due to the influence of drainage pipe diameter is shown in Fig.

. When 22-ball valve is opened to about 45° (the equivalent diameter is 63mm), the water flow is in a critical state between whole pipe flow and partly-filled pipe-flow, which indicates that the water pressure state approaches to that of opening ball valve about 90°. So the water pressure data of fully opened ball valve will be regarded as that of 45°. And the angle of 0° means the ball valves fully closed, i.e., water pressure will not decrease.

In Fig.

, with the increase of the equivalent diameter of the drainage pipe, the water pressure reduction ratio gradually increases and tends to be stabilized. For the case that the initial pressure about 20kPa, the water pressure reduction ratio is approximately 60%. If the equivalent diameter is larger than 63mm, the whole pipe flow in each pipe will switch to partly-filled pipe-flow. It indicates the water pressure will not decrease any more even if increasing the diameter of the drainage pipe if the initial pressure is 20kPa. Therefore, the effect of reducing water pressure on tunnel secondary lining will be better with the increase of drainage pipe diameter as long as it is whole pipe flow.

For a certain tunnel region, the whole pipe flow will become partly-filled pipe-flow when the diameter of drainage pipe increases to a certain extent, which means the effect of pressure reduction will not get better any longer. Therefore, for a safe tunnel structure, in the design of the bottom drainage system for the tunnel, the drainage pipe should be large enough for the water inflow of the tunnel construction region.

The data of Table 4 and Fig.

shows that both the erect and oblique drainage pipe has good performance in water pressure reduction. However, under the same test condition, the water pressure reduction ratio of the erect drainage pipe is more than the oblique drainage pipe for about 4%. It indicates that the drainage effect of erect drainage pipe is better than the oblique drainage pipe to some extent.

The data of No.15 pore pressure sensor monitoring water pressure decreasing process is shown in Fig. . The water pressure decreasing process of both erect and oblique drainage pipe is great, which lasting about 2~3s, when the equivalent diameters are 47mm and 27mm. The water pressure of the pipe with an equivalent diameter of 18mm decreases very slowly, after about 50s the water pressure still shows some trend to decrease. This experimental phenomenon indicates that a large enough cross-section area of the drainage pipe contributes to a more rapid water pressure reduction process, which is helpful to deal with the sudden increase of water pressure like a sudden rainfall.

Contrarily, if the diameter of the drainage pipe is too small, the water pressure decreases rapidly at first and then very slow, which leads to a long drainage process with little effect. Thus, it is critical to have a large enough drainage pipe to satisfy the requirement of the tunnel structure. It is not necessary to use a partly-filled pipe-flow while a whole pipe flow can reduce water pressure on the secondary lining of a tunnel efficiently. The changing curve of water pressure with a time of 27mm-equivalent-diameter erect drainage pipe and 27mm-equivalent-diameter oblique drainage pipe is shown in Fig. .    With a comprehensive consideration of security, cost-effectiveness, etc., it is recommended that the drainage capacity should be designed for a partly-filled pipeflow for the dry season and a whole pipe flow for the rainy season.

Influence of inlet water pressure

The relationship between water pressure and intake water pressure is shown in flow rate increases with the increase in intake pressure. The flow rate of the erect drainage pipe is two times higher than the oblique drainage pipe with the same intake pressure. Therefore, it can be concluded that with the same cross-section area and whole pipe flow, the flow rate of the erect drainage pipe is about two times greater than the oblique drainage pipe. Furthermore, the slope of the flow rate curve of the erect drainage pipe is much larger than the oblique drainage pipe. It indicates that the water discharge of the erect drainage pipe would increase rapidly with the increase of the intake pressure. In other words, the erect drainage pipe can manage the sudden change of the water inflow more efficiently. Therefore, the erect type of drainage pipe is more effective than the oblique type in the water flow rate analysis.

Summary and Conclusion

This paper presents a study on assessing the feasibility of bottom-up drainage and water pressure reduction system for railway tunnels by simulation and analyzing the pressure reduction effect. The experiment equipment consists of two main parts: the water supply system and the drainage system. The experiment was set up imitating the bottom structure of the tunnel with the bottom drainage system under three test conditions: water pressure distribution of steady seepage in a regular working state; the influence of the change of the drainage pipes diameter on the effect of water pressure reduction; and inlet water pressure on the seepage field. The results of the experiment can be summarized as follows:

(1) The bottom-up drainage system can effectively drain the water in the bottom of the tunnel by the difference of natural hydraulic pressure.

(2) It is discovered that the erect drainage pipe is better than the oblique one in the drainage efficiency, flow rate, and percentage reduction of water pressure.

(3) The bottom-up drainage and water pressure reduction system can effectively reduce the secondary lining external water pressure, and there will be a well water pressure distribution at the bottom of the tunnel.

(4) If the water inflow is stable with whole pipe drainage, the reduction effect of water pressure is remarkable by increasing the diameter of the drainage pipe. In other words, if the drainage pipes are incapable of draining the water at the bottom of tunnels, the reduction effect of water pressure on the secondary lining is unremarkable, and the reduction process is very long.

(5) In the design of drainage pipes, the safety of the tunnel lining structure needs to be taken into consideration and guarantee as well as the water inflow difference of rainy season and dry season. Therefore, the design of partly-pipe flow in the dry season and whole pipe flow in the rainy season is recommended.

This study proved the bottom-to-up drainage and water pressure system is feasible for use on tunnels. Due to the difference between the construction site and laboratory test, more field test data is necessary to validate and optimize the performance of the proposed drainage system.

Fig. 2. The overall idea flowchart 476

  and drainage system. (Water tank(I), Water pump(II), Intake pipe(III), Return pipe(IV), Return valve(V), Inlet valve(VI), Pressure valve(VII), Buffer box(VIII), Inlet holes(IX), Gravel filling(X), Precast concrete board(XI), Convex hull drainboard(XII), Cast-inplace concrete filling(XIII), Half-round tube(XIV), Waterproof board(XV), Gravel filling Ⅱ (XVI), Cover plate(XVII), One-way valve(XVIII), Oblique drainage pipe(XIX), Erect drainage pipe(XX), Studdle(XXI), Ball valve(XXII). )

  HCSC-16 data acquisition equipment and YBY-2001 strain acquisition instrument were used for data acquisition. There are 3 tiers of pore pressure sensors installed at different heights (seeFig. ). The Bottom Tier Sensors is placed in the gravel filling(X) to monitor the pore pressure representing surrounding rock. The Medium Tier Sensors is placed between precast concrete boards(XI) and waterproof board to monitor the pore pressure that representing simulating secondary lining. The Top Tier Sensors is placed in each drainage pipe which is 15cm higher than the medium tier. The horizontal arrangement of each tier of pore pressure sensors is shown in Fig. . No.1~No.9 sensors are HCYB-25 micro pore pressure sensors, and No.10~No.22 sensors are HC-25 micro pore pressure sensors.

Fig

  Fig. shows the entire experimental system.

  pressure distribution of steady seepage In Fig. (a) and Fig. (b), the water discharge of the erect drainage pipe approximately equals to that of the oblique drainage pipe. The exact water discharge values measured from an erect drainage pipe and oblique drainage pipe are 1384.64g/s and 1248.97g/s respectively. The water discharge of erect drainage pipe is slightly greater than that of the oblique pipe. Fig. (c) shows the water discharge of both drainage pipes during draining. The measured water discharges of erect and oblique drainage pipe are 695.82g/s and 387.24g/s respectively. It is found that, under the same test condition of intake water pressure and opening angles of ball valves, the drainage efficiency of the erect drainage pipe is better than the oblique pipe. It may be due to the difference of the drainage paths since the drainage path of the erect drainage pipe is much shorter than the oblique drainage pipe. The water pressure values measured by each tier of sensors are shown in Error! Reference source not found., 2, and 3. Comparing to the value monitored by each sensor under different test conditions, the water pressure of most DEDP is less than that of DODP. It means that the water pressure reduction effect of the erect drainage pipe is greater than the oblique drainage pipe, which agrees well with the result proved by water discharge analysis. The water pressure values of most DODP are less than that of the DEDP since the active drainage area of DODP is larger than that of DEDP or DODP.It shows that in a certain range, multiple drain outlets are more conducive to reduce water pressure than a single drain outlet. It indicates that the distance between the two drain outlets needs rational design.From each tier of sensors, the water pressure measured has a similar trend. It indicates that the water pressure is well distributed and changed synchronously, which proves that the bottom-to-up drainage and water pressure reduction system is effective to reduce water pressure at the bottom of the tunnel. Since the average values of water pressure under each test condition are different, the standard deviation is unable to validate the data. Therefore, the coefficient of variation ("CV," the ratio of the standard deviation to average value) is chosen to validate the collected data. Only the CV values of the Bottom Tier Sensors in each test condition are more than 10%, which means that the water pressure at the bottom of the testing chamber flutters obviously. The maximum CV value of the Medium and Top tier is 6.10%, which means that the water pressure values in the position of each tier are almost not fluctuant, proving the good distribution of water pressure in another way. The reasons of such fluctuation can be concluded as: 1) The water flow paths are disordered and the flow rate is not as uniform as the bottom tier of pore pressure sensors placed in gravel filling, in which there are plenty disordered interspace; 2) The data measured by the bottom and the top tier of sensors have much better accuracy than the bottom tier.

Fig. .

 . Fig.. The water pressure increases linearly with the increase of intake pressure, which

Fig. 3 .Fig. 11 .FigFig. 14 .Fig

 31114 Fig. 3. Sketch of the experimental drainage system

Table 1 Water pressure measured by the first tier of pore pressure sensors

 1

												Standard
		CH1	CH2	CH3	CH4	CH5	CH6	CH7	CH8	CH9	Average		Coefficient
	Item											deviation
		/kPa	/kPa	/kPa	/kPa	/kPa	/kPa	/kPa	/kPa	/kPa	/kPa		of variation /%
												/kPa
	DBDP	10.694	10.161	/ *	9.985	11.925	8.490	10.153	10.206	12.044	10.457	1.138	10.88
	DEDP	11.450	10.094	/	9.989	10.749	8.204	10.153	10.305	12.047	10.374	1.139	10.98
	DODP	13.383	9.835	/	10.212	11.934	9.327	10.389	10.653	10.695	10.804	1.287	11.92

* :"/"means no available data because of sensor damage.

Table 2 Water pressure measured by the second tier of pore pressure sensors

 2

														Standard
		CH10	CH11	CH12	CH13	CH14	CH15	CH16	CH17	CH18	CH19	CH20	Average		Coefficient
	Item													deviation
		/kPa	/kPa	/kPa	/kPa	/kPa	/kPa	/kPa	/kPa	/kPa	/kPa	/kPa	/kPa		of variation /%
														/kPa
	DBDP	9.748	9.664	9.653	9.323	9.666	9.361	9.363	9.638	8.575	9.681	/ *	9.467	0.351	3.70
	DEDP	9.889	9.809	9.798	9.407	9.817	9.588	9.590	9.791	8.197	9.823	/	9.571	0.505	5.28
	DODP	10.072	9.987	9.961	9.619	10.000	9.733	9.716	9.962	8.084	10.005	/	9.714	0.592	6.10

* :"/"means no available data because of sensor damage.

Table 3 Water pressure measured by the third tier of pore pressure sensors

 3

					Standard
		CH21	CH22	Average		Coefficient
	Item				deviation
		/kPa	/kPa	/kPa		of variation /%
					/kPa
	DBDP	8.629	8.392	8.510	0.167	1.96
	DEDP	8.751	8.657	8.704	0.067	0.77
	DODP	8.949	8.753	8.851	0.139	1.57

Table 4 Water pressure value and pressure decrease percentage in different diameter of drainage pipe Equivalent diameter /mm Drain pipe

 4

			Item	CH10	CH11	CH12	CH13	CH14	CH15	CH16	CH17	CH18	CH19	CH20	CH21	CH22 Average
		Erect	Before	30.766	30.455	30.814	24.736	35.486	29.202	29.083	29.781	28.998	31.228	/ *	31.579	27.526
		drainage	After	10.055	9.910	9.901	9.356	10.012	10.276	11.023	10.019	6.776	9.945	/	8.827	8.334
		pipe	Percentage 67.32% 67.46% 67.87% 62.18% 71.79% 64.81% 62.10% 66.36% 76.63% 68.15%	/	72.05% 69.72% 68.04%
	47
		Oblique	Before	28.324	27.997	28.257	23.572	31.171	26.903	26.869	27.641	25.386	28.522	/	28.340	25.036
		drainage	After	10.596	12.050	12.052	11.266	12.169	12.021	12.749	12.137	8.916	12.084	/	10.984	10.046
		pipe	Percentage 62.59% 56.96% 57.35% 52.20% 60.96% 55.32% 52.55% 56.09% 64.88% 57.63%	/	61.24% 59.87% 58.14%
	27	Erect	Before	28.564	27.582	27.716	24.605	28.639	26.314	26.560	27.724	/	27.803	/	26.928	24.522

* :"/"means no available data because of sensor damage.

Acknowledgments

The support of project of research and development of science and technology from CHINA RAILWAY (CR) with Grant No. 2016G002-O and Grant No. 2014G004-C is highly appreciated.

Appendix A. Figures

Fig. 1. "bottom-to-up" drainage and pressure reduction system at the bottom of railway tunnels [START_REF] Li | A bottom-to-up drainage and water pressure reduction system for railway tunnels[END_REF]


Elissa El Rassy

Aurélie Delaroque

Patrick Sambou

Harish Kumar Chakravarty

Alexis Matynia
email: alexis.matynia@sorbonne-universite.fr

On the potential of Particle Swarm algorithm for the optimization of detailed kinetic mechanisms. Comparison with Genetic Algorithm

This work investigates the potential of the particle swarm algorithm for the optimization of detailed kinetic mechanisms. To that end, empirical analysis has been conducted to evaluate the efficiency of this algorithm in comparison with the genetic algorithm. Both algorithms are built on evolutionary processes according to which a randomly defined population will evolve, over the iterations, towards an optimal solution. The genetic algorithm is driven by crossover and mutation operators and by a selection method. The PSO approach is based on the experience of each individual and on the group experience to control the direction of its evolution. The success of the application of an algorithm can be sensitive to the choice of operators and the relative importance attributed to them. Therefore, to make the comparison as rigorous as possible, about a dozen strategies were proposed for each algorithm and the performances were evaluated. A degraded version of the GRI-Mech 3.0 mechanism (i.e. containing some of the kinetic constants randomly modified) was generated and then optimized by the two evolutionary algorithms to recover the predictive character of the original mechanism. The results show that, for the majority of the proposed strategies, PSO is more efficient than the GA, whereas the latter is generally much more used for the optimization of detailed kinetic mechanisms.

Introduction

The chemical processes that take place during combustion can be described using detailed kinetic mechanisms. Depending on the size of the fuel involved and the degree of precision desired, these mechanisms can be composed of a few very simplified chemical steps or, on the contrary, of several thousand elementary reactions. [START_REF] Lu | Toward accommodating realistic fuel chemistry in large-scale computations[END_REF] The reaction rates can be determined experimentally, 2 by means of theoretical approaches [START_REF] Carpenter | Ab Initio Computation of Combustion Kinetics. 1. Vinyl Radical + O2[END_REF] or by analogy with reactions of the same type (H-abstration / beta-scission, etc.) [START_REF] Sarathy | Alcohol combustion chemistry[END_REF] Nevertheless, the complexity of the phenomenology of a chemical reaction implies that the uncertainties on the rate constants generally remain relatively large. [START_REF] Wang | Combustion kinetic model uncertainty quantification, propagation and minimization[END_REF] Similarly, the development of models by combining sets of reactions, even with a high degree of confidence in the kinetic constants, does not always guarantee good predictivity. [START_REF] Frenklach | Optimization and analysis of large chemical kinetic mechanisms using the solution mapping method-combustion of methane[END_REF] The optimization of a kinetic mechanism implies the improvement of its predictability by the variation of the reaction rate parameters within their uncertainty limits. A distinctive feature inherent to the optimization of detailed models is the very large number of variables to be processed. For each reaction modeled by the extended Arrhenius law, three variables are subject to modification. Thus, a methane combustion mechanism such as the GRI-Mech, containing 325 reactions, potentially involves the modification of a thousand variables.

Establishing the optimal values of the unknown parameters represents a very complex inverse problem.

Different numerical approaches have been developed to address this type of problem.

Gradient-based optimization has been used to optimize low dimensional kinetic mechanisms containing a few tens of reactions. [START_REF] Bock | Modelling of Chemical Reaction Systems[END_REF][START_REF] Rabitz | Chemical sensitivity analysis theory with applications to molecular dynamics and kinetics[END_REF] Such methods can be effective in solving small scale systems. Nevertheless, gradient based methods are tailored to the search for the nearest optimum and the objective functions of the problems in chemical kinetics modelling are generally complex, having multiple ridges and valleys, especially if the dimension of the kinetic models is large. Therefore, the success of the method is very dependent on the initial guess since the algorithm can easily get trapped in a local optimum.

On the other hand, evolutionary algorithms are particularly well suited to finding the optimum for chemical reaction systems having a large number of variables with a highly structured response surface. [START_REF] Gallagher | Genetic algorithms: A powerful tool for large-scale nonlinear optimization problems[END_REF] These optimization tool of metaheuristic nature are also called "evolutionary population based method". These algorithms are stochastic and can be used as a local or global search procedure, tackling optimization problems of non-linear and potentially noisy multimodal functions. [START_REF] Michalewicz | Genetic Algorithms + Data Structures = Evolution Programs[END_REF][START_REF] Bäck | Handbook of evolutionary computation[END_REF] The Genetic Algorithm (GA) is an optimization method inspired by the theory of evolution. It is regularly used for optimizing kinetic models and represents a very efficient method for optimizing kinetic models. [START_REF] Balland | A genetic algorithm with decimal coding for the estimation of kinetic and energetic parameters[END_REF][START_REF] Rein | Application of genetic algorithms and thermogravimetry to determine the kinetics of polyurethane foam in smoldering combustion[END_REF][START_REF] Kadiva | Estimation of Kinetic Parameters for Hydrogenation Reactions Using a Genetic Algorithm[END_REF] In their review, Elliot et al. [START_REF] Elliott | Genetic algorithms for optimisation of chemical kinetics reaction mechanisms[END_REF] presented the efficiency of a genetic algorithm for the optimization of hydrogen, methane and kerosene mechanisms, based on experimental datasets. In addition to a reduction process, this algorithm can also be exploited in order to compensate for the errors resulting from the loss of information [START_REF] Elliott | Reaction mechanism reduction and optimisation for modelling aviation fuel oxidation using standard and hybrid genetic algorithms[END_REF][START_REF] Perini | Development of reduced and optimized reaction mechanisms based on genetic algorithms and element flux analysis[END_REF][START_REF] Jaouen | Automatic reduction and optimisation of chemistry for turbulent combustion modelling: Impact of the canonical problem[END_REF][START_REF] Cailler | Building-up virtual optimized mechanism for flame modeling[END_REF] .

Particle Swarm Optimization (PSO) is another evolution algorithm. It consists of exploiting the swarm intelligence that relies on the independent evolution of particles and their interactions in a biological type system. Although this algorithm is also well adapted to the optimization of large problems, it has been rarely used for the optimization of kinetic models. [START_REF] Guo | Estimation of kinetic parameters from adiabatic calorimetric data by a hybrid Particle Swarm Optimization method[END_REF][START_REF] Ding | Kinetic parameters estimation of pinus sylvestris pyrolysis by Kissinger-Kai method coupled with Particle Swarm Optimization and global sensitivity analysis[END_REF][START_REF] Ding | The accuracy and efficiency of GA and PSO optimization schemes on estimating reaction kinetic parameters of biomass pyrolysis[END_REF][START_REF] Sun | Optimization of a Fluid Catalytic Cracking Kinetic Model by Improved Particle Swarm Optimization[END_REF] Ding et al. [START_REF] Ding | The accuracy and efficiency of GA and PSO optimization schemes on estimating reaction kinetic parameters of biomass pyrolysis[END_REF] proved the better optimization performances of PSO compared to GA when applied for three-component parallel reaction mechanism of biomass pyrolysis. However, they used only one set of parameters (further referred as optimization "strategy"), which limits the comparison. Guo et al. [START_REF] Guo | Estimation of kinetic parameters from adiabatic calorimetric data by a hybrid Particle Swarm Optimization method[END_REF] compared different PSO strategies to the genetic algorithm (for which only one strategy was used). They showed that, although the PSO approach is prone to fall into a local optimal prematurely, the adaptation of the PSO coefficients could allow better optimization performance than the genetic algorithm. However, and based on the authors' knowledge, this tool has not been used for the optimization of large detailed kinetic mechanisms (i.e. with a number of reactions much higher than ten), so far. The efficiency of this algorithm has been proved in many other different fields [START_REF] Eberhart | Particle swarm optimization: developments, applications and resources[END_REF][START_REF] Gaing | A particle swarm optimization approach for optimum design of PID controller in AVR system[END_REF][START_REF] Jin | Particle Swarm Optimization for Antenna Designs in Engineering Electromagnetics[END_REF][START_REF] Valle | Particle Swarm Optimization: Basic Concepts, Variants and Applications in Power Systems[END_REF][START_REF] Alrashidi | A Survey of Particle Swarm Optimization Applications in Electric Power Systems[END_REF][START_REF] Mohammadi-Ivatloo | Combined heat and power economic dispatch problem solution using particle swarm optimization with time varying acceleration coefficients[END_REF][START_REF] Khare | A review of particle swarm optimization and its applications in Solar Photovoltaic system[END_REF][START_REF] El Rassy | Simultaneous and direct identification of thermophysical properties for orthotropic materials[END_REF] .

The main objective of our current study is to evaluate the efficiency of the Particle Swarm algorithm for the optimization of detailed kinetic mechanisms. The performances will be assessed and compared to those of the genetic algorithm. The optimization is performed using an open-source tool based on Python (Brookesia 32 ) and the well-known methane kinetic model GRI-Mech 3.0. [START_REF] Smith | [END_REF]

Method

Computing

Simulation conditions and degraded mechanism settings

The conditions evaluated in the present study are composed of six cases of reactor at atmospheric pressure (constant enthalpy) for CH 4 /air mixtures at the equivalence ratio Φ = 0.5, 1, 1.5 of initial temperature 1050 K and 1500 K. Simulations are performed with GRI-Mech 3.0 33 and the Cantera solver [START_REF] Goodwin | An Objectoriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes[END_REF] . Subsequent evaluation of important reactions and optimizations are performed with Brookesia. [START_REF] Matynia | An open-source tool for reduction and optimization of kinetic mechanisms[END_REF] In order to evaluate the ability of optimization algorithms to improve the predictivity of a detailed kinetic model, the GRI-Mech3.0. mechanism has been "degraded". To do this, the kinetic constants of the 49 th "most important" reactions were randomly modified. The importance of the reactions is evaluated on the basis of the direct interaction coefficients between a species A and a reaction r i , that are calculated according to the formulation of Pepiot-Desjardin and Pitch 35 as:

r Ar i = |ν i,A ω i | max (P A , C A ) (1)
with:

P A = n reactions i=1 max(0, ν i,A ω i ) C A = n reactions i=1 max(0, -ν i,A ω i ) (2)
ν i,A represents the stoichiometric coefficient of species A in reaction i and ω i the net reaction rates of the ith reaction.

The optimization targets are CH 4 , CO, CO 2 , C 2 H 4 , C 2 H 6 concentration profiles, the temperature and the ignition delay time. To calculate the interaction coefficients for the temperature and ignition delay time targets, CO 2 and CH 3 species were considered, respectively. Seven reactions with the highest interaction coefficients have been selected for each species, bringing the total number of modified reactions to 49.

By only modifying the important reactions for the targets, it is possible not to affect reactions having no, or very little, impact on the simulation results and whose constants could be strongly modified without control. The variations applied to the terms B, n, and C of the extended Arrhenius law, expressed in eq. 3 are 15, 5, and 5 %, respectively.

k = B • T n exp( -C RT ) (3)
Figure 1 shows an example of a reactor simulation carried out using the initial mechanism and then using the degraded mechanism. The differences between the results of the two simulations, and thus the loss of predictivity of the degraded mechanism are clearly observed.

Evaluation of the optimized mechanisms quality

For each optimized mechanism and at each iteration, the relative errors of the targets are calculated and averaged. The relative ignition time error is calculated as the difference between the ignition time obtained by the reference and the optimized mechanism divided by the time obtained by the reference. Concerning the variables evolving over time (temperature, species), errors are calculated as the relative differences between the areas under the curves obtained with the reference simulations and the simulations resulting from the optimized mechanisms. Details on the calculations are provided in the documentation of Brookesia. Initially, the averaged error calculated for the degraded mechanism is 40.4 %.

The fitness value for the optimization algorithms is calculated as the inverse of the averaged error. In doing so, the best individuals have the highest fitness.

Considerations for the comparison of stochastic optimization methods

The global search algorithms addressed in this work are stochastic in nature and therefore the quality of the optimization depends on many random parameters such as the quality of the first generation. In a stochastic approach, the output possesses some inherent randomness.

The same set of input parameters values and initial conditions can lead to a set of different outputs. Thus, to best evaluate the differences between the optimization schemes, each strategy was evaluated 20 times and the curves were averaged.

It is important to note that the set of GA and PSO strategies has been developed so that at each iteration, an equal number of simulations is performed. This prerequisite ensures that the algorithms can be compared for an equal numerical cost.

Genetic algorithm

The genetic algorithm, developed by Holland in the 1960s, [START_REF] Holland | Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control and Artificial Intelligence[END_REF] is a stochastic optimization approach inspired by Darwin's principle of natural selection. It can be used in the application of inverse methods, where the solution to be reached is known but the input parameters are not, or not accurately. The algorithm will then consist of performing many simulations with different sets of input parameters. At the end of this series of simulations, the sets of parameters giving the results closest to the optimal solution are retained (Selection).

Operators will have the function of interchanging these parameters or of crossing them by means of operations such as averages, weighted averages, etc. (Cross-Over), or of randomly modifying one or more variables (Mutation) to extend the search as far as possible in the space of the solutions and to avoid as far as possible the trapping of the optimum search in a local minimum. A more complete description of different operators can be found in the reference. [START_REF] Goldberg | Genetic Algorithms in search optimization and machine learning[END_REF] The GA algorithm coded in the Brookesia software integrates 4 selection operators (Roulette, Rank, Geometric Norm, and Elitism), 4 cross-over operators (Simple, Multiple, Arithmetic, Heuristic), and 4 mutation methods (Uniform, Non-uniform, Boundary). Details on the application of each of these operators are provided in the Brookesia documentation. [START_REF] Matynia | An open-source tool for reduction and optimization of kinetic mechanisms[END_REF] To evaluate the potential of the GA algorithm, the impact of each module was evaluated separately. The table 1 summarizes the choice of operators for the 11 evaluated strategies.

Strategies 1 to 4 each uses a different selection method. Strategies 5 to 8 / 9 to 11 assess the efficiency of the different crossing/mutation methods.

PSO

Principle

This algorithm was proposed by Russel Eberhart (electrical engineer) and James Kennedy (socio-psychologist) in 1995. [START_REF] Eberhart | A new optimizer using particle swarm theory[END_REF] Particle Swarm Optimization is a population-based, selfadaptive research method inspired by the social behavior of groups of birds or fish schools.

The PSO algorithm explores the search space, thanks to a population of particles interacting with each other and adapting their position and speed during the optimization process. Research of the optimum is guided by a local research method, based on personal experience, and complemented by a global research method, based on the experience of the group, trying to balance exploration and exploitation.

The dynamic of a particle i is thus governed by an "inertial" component, relative to its speed at the previous iteration t, a "cognitive" component based on the best position P i found by the individual during the optimization process and a "social" component based on the best location of the population found at the iteration t, P g . In a hyperspace of dimension d, the velocity and position of a particle i are, respectively, governed by the following equations:

v i,d (t + 1) = W • v i,d (t) inertial component + C 1 • a 1 • (P i,d (t) -x i,d (t)) cognitive component + C 2 • a 2 • (P g,d (t) -x i,d (t)) social component (4)
x i,d (t + 1) = x i,d (t) + v i,d (t + 1) (5)
where W represents the inertia weight, C 1 , C 2 represent the acceleration constants, and a 1 , a 2 are values, ranging from 0 to 1, randomly set at each PSO iteration t and for each particle i.

Constriction factor

The optimization of parameters by PSO algorithm was initially of an empirical nature. In 1999, the work of Clerc 39 demonstrated the necessity to use a constriction factor to ensure the convergence of the particle swarm algorithm. A new law, incorporating the constriction term, governs the velocity of a particle by the following equation:

40 v i,d (t + 1) = χ • [v i,d (t) + c 1 • a 1 • (P i,d (t) -x i,d (t)) + c 2 • a 2 • (P g,d (t) -x i,d (t))] ( 6
)
where χ is the constriction factor, such as:

χ = 2κ 2 -φ -φ 2 -4φ (7)
with

φ = c 1 + c 2 , φ 4, κ ∈ [0, 1]. 10
In this equation, the variable κ controls the speed of convergence: fast convergence to a stable point shall be obtained when κ equals 0, while a κ equal to 1 results in slower convergence.

Evolution of different components weights during the iterative optimization process

The success of a population-based algorithm depends on the subtle balance between local and global searching. This equilibrium must be managed globally for the entire optimization and can also be refined as the process progresses. Thus, it is possible to vary the values of the parameters w, c 1 , and c 2 to favor a global search at the beginning of the optimization and then the local search at the end. These approaches are discussed in section 2.3.6.

Score calculation

Other approaches have been developed to improve the efficiency of the algorithm in more specific contexts. For example, Cai et al. [START_REF] Cai | Particle Swarm Optimization[END_REF] introduced a score variable S to rank particles according to their fitness, following this equation:

S i (t) = f i (t) -f worst (t) f best (t) -f worst (t) (8)
This variable can be used to weight the inertial component: by affecting a high inertia to a particle having a low fitness level, the global research is promoted. Conversely, assigning a low inertia to a particle with a high fitness level favors the local search for the particles closest to an optimum (which is a maximum in the current study). The weighted inertia of the particle i ranges between a maximum limit value W max and a minimum limit value W min :

W i (t) = W min (t) + (W max (t) -W min (t)) • (1 -S i (t)) (9)

Convergence detector

The PSO algorithm implemented in Brookesia integrates a convergence detector. If the standard deviation between the fitness levels of the whole population is less than 10 % of the maximum fitness value, the direction of the inertial component is momentarily multiplied by a factor of -5 or +5:

W i (t) = W i (t) •      1 if σ(f) 0.1 • f max 5 • (-1) a 3 otherwise ( 10
)
with a 3 , a binary random variable on {0; 1}. In this way, the diversity of the population is guaranteed during the optimization process. When 90% of the iterations have been performed, this detector is deactivated in order to enable a local search, more relevant at the end of the optimization process.

Many other tools can be integrated into the PSO code for the improvement of the identification quality, depending on the application. 20,23,29,31,42

Panorama of PSO strategies

Like any evolutionary algorithm approach, the PSO method is prone to the problems of low convergence, convergence to a local minimum, and so forth. To design the best PSO configuration for detailed mechanism optimization, 14 strategies have been evaluated and compared.

In order to visualize more precisely the optimization schemes, criteria for the relative importance of the inertial, cognitive and social components were defined:

∼ W = W W + (1/2) • (C 1 + C 2 ) (11)
∼

C 1 = (1/2) • C 1 W + (1/2) • (C 1 + C 2 ) ( 12
) ∼ C 2 = (1/2) • C 2 W + (1/2) • (C 1 + C 2 ) (13)
Note that the absolute values of the terms C 1 and C 2 have been divided by two to take into account the influence of the random factors a 1 and a 2 .

The evolution of these criteria during the optimization process is presented, for each of the strategies described hereafter, in figure 2.

• Strategies with constriction factor (1 to 4)

The first strategy is a reference. It is based on the work of Clerc 39 who demonstrated the necessity to use a constriction factor to ensure the convergence of the particle swarm algorithm (see equation 6). The parameters of the equation are often set as

c 1 = c 2 = 2.
05 and κ = 1. [START_REF] Eberhart | Comparing inertia weights and constriction factors in particle swarm optimization[END_REF][START_REF] Hatanaka | Particle Swarm Optim[END_REF][START_REF] Tuppadung | Comparing nonlinear inertia weights and constriction factors in particle swarm optimization[END_REF] In such a case, the optimization scheme is similar to the equations 4 and 5 with the parameters W = 0.730 and

C 1 = C 2 = 1.496.
The second strategy proposes to keep the same values for the terms c 1 and c 2 . On the other hand, the value of κ is divided by 2 (κ = 0.5) to evaluate the effect of an a priori faster convergence.

Results & Discussion

Convergence curves comparison

Genetic algorithm

The convergence curves obtained using the genetic algorithm strategies are presented in graph a) of figure 3. The evolution of the mean error found for the estimation of the optimization targets is presented as a function of the number of iterations. It is important to remember here that each curve presented in the figure is the average result of 20 independent optimizations (see section 2.1.3). This approach intends to take into account the stochastic nature of the optimization algorithms and to ensure rigorous comparisons.

The assessment of the different strategies for the genetic algorithm shows an important sensitivity to the selection method. The algorithm efficiency with rank-based selection is much worse than that obtained with other methods. The selection by geometric norm presents slightly inferior results compared to the selection by roulette and elitism. Never-theless, considering the standard deviations between the optimizations (discussed in section 3.1.3), the performances can be considered as quite similar.

The GA_5 to GA_8 strategies differ in the importance devoted to the different crossover operators. The convergence curves presented in figure 3 a) show that the quality of optimization is very slightly affected by these choices. The choice of the mutation method, on the other hand, has a greater impact. The results indicate that the quality of optimization is higher when non-uniform or boundary mutation operators are predominantly used.

The GA_9 to GA_11 strategies have different importances in the use of mutation operators. Mutation consists of randomly varying the variables to be optimized. In the Brookesia code, about 30 % of the variables are modified. Note that the non-uniform mutation takes into account the progress of the optimization process. Thus, the more the number of iterations evolves, the less the modifications on the variables are important (more details on these methods are provided in the documentation of Brookesia 32 ). This allows to maximize the exploration of the search space at the beginning of the optimization and to focus on a local search at the end. The comparison shows that, in the context of the optimization of detailed kinetic models, the latter method should be preferred to uniform mutation. It should also be noted that an important use of the boundary mutation operator also gives good results. Nevertheless, the use of this method leads to a large increase in the standard deviation between optimizations (see figure 4) indicating that the quality of the optimization can vary significantly from one run to another.

This work highlighted the sensitivities in the choice of genetic algorithm operators for the optimization of detailed kinetic mechanisms. The main observations are that the algorithm is not very sensitive to the choice of selection operators (except the rank selection) and crossing operators and is more sensitive to the choice of mutation, for which the non-uniform operator should be preferred. Moreover, the convergence curves established for a large number of GA strategies provide a consistent basis for comparison to evaluate the efficiency of the PSO method, which has not, to the authors' knowledge, been used for the optimization of detailed kinetic mechanisms, so far.

Particle Swarm Optimization

The convergence curves for the PSO strategies are presented in figure 3 b). Strategies 1 to 4 integrate the implementation of a constriction factor to ensure theoretical convergence.

Strategies 1 to 4 integrate the implementation of a constriction factor to ensure theoretical convergence. As illustrated in figure 2, the relative influences of the inertial, cognitive and social components are constant for strategies 1 and 2 and are evolutive for strategies 3 and 4. Strategies 2 and 4 are affected by a constriction factor equal to 0.5, instead of 1 used for strategies 1 and 3, to accelerate the speed of convergence. It is noticeable that the distinction of the optimization quality between these four strategies is made almost exclusively on the choice of the convergence speed. Thus, strategies 2 and 4 present significantly better results than strategies 1 and 3. Given the very large number of variables to be optimized in a detailed mechanism optimization problem, it appears more appropriate to favor rapid convergence.

The set of strategies 5 to 8 gives, surprisingly, very different results and integrates both the strategy with the highest performance (6) and the least effective strategy (8). A more global analysis of the strategies seems to show that, for this type of problem, the higher relative importance of the cognitive component has a positive influence on the optimization, whereas the high importance of the inertial component is in contrast quite unfavorable.

Thus, strategies 7 and 8, which have a strong inertial weight at the beginning of the optimization, give poorer results. We also note, when comparing these two strategies, that the score parameter integrated into the strategy 8, and which was intended to promote global research for individuals with low fitness levels, has an unfavorable effect on the quality of the optimization. Interestingly, this difference, although relatively small, is found for all pairs of strategies with/without score (7/8, 9/10, 11/12, and 13/14).

The comparison between the pairs of strategies 9/10 and 11/12 shows that the inversion of the weight evolution for the cognitive component during the optimization process while keeping W and C2 coefficients equal, has little effect on the quality of the optimization.

Besides, the pair of strategies 13/14, with coefficients C1 and C2 comparable to those of pair 11/12 and whose influence of the inertial component is modified (less important initially but higher at the end) provides more efficient optimizations.

Comparison between GA and PSO strategies

A global comparison of the set of strategies shows that, overall, the PSO algorithm provides better optimization than the GA. Thus, among the 5 best strategies, 4 are obtained by the PSO algorithm while among the 5 least efficient, 3 were obtained when the GA algorithm was applied. Nevertheless, the efficiency of optimization is more variable between PSO strategies.

Thus, except for strategy 2, which stands out from the other trends, the average error at the end of optimization by GA is between 3 and 4.5 %. It is between 2 and 5 % for the PSO algorithm. This indicates a higher sensitivity to the choice of optimization parameters when applying the PSO algorithm.

The convergence rate of the PSO algorithm seems significantly faster than that of the genetic algorithm. Thus, the residual error for the 6 best PSO strategies drops to about 4 % or less after the first 20 generations, whereas equivalent performance is obtained after 40 generations for GA strategies 10 and 11.

Figure 4 shows the standard deviations calculated between the 20 optimizations performed for each strategy. Overall, the standard deviations of the PSO algorithm are higher than those of the GA. Excluding the value of GA_2, the mean standard deviation is 16 % lower for GA than for PSO ( This observation underlines the reliability of the optimization approach using evolutionary algorithms. It can also be noted that the difference magnitude between the kinetic constants of the optimized mechanism and the GRI-Mech 3.0 mechanism reflects, overall, the optimization quality. Thus, for the most efficient optimizations identified in the previous section, the difference amplitudes are among the smallest and can be reduced by up to a factor of 3 (PSO strategies 5, 6, 13, and 14, for example) compared to the degraded mechanism. PSO was compared to the genetic algorithm (GA), which is nowadays commonly used for the optimization of kinetic models. The results show that the best strategies of the algorithm offer higher performance than those obtained for the genetic algorithm. The speed of convergence observed for a large number of PSO approaches is also significantly faster. Besides, the evolution of the optimized kinetic constants converges, for all the strategies tested in this work, towards the initial values of the GRI-Mech 3.0 mechanism, which underlines the relevance of the approaches used. The statistical analysis of the results showed that the quality of the optimizations can vary significantly from one optimization to another. It appears that the dispersion of the results is generally higher with PSO than with GA.

In conclusion, this work attests to the good performance of the PSO algorithm compared to the equivalent GA method, which is much more widely used for the optimization of kinetic mechanisms. Given the simplicity of the algorithm, this work demonstrates the interest in considering Particle Swarm Optimization as a good alternative to the Genetic Algorithm, for such applications.

32

 32

Figure 1 :

 1 Figure 1: C 2 H 6 , CO, and CO 2 molar fractions evolution computed for CH 4 /air mixtures Φ = 0.5 and 1500 K. Simulations performed with the GRI-Mech 3.0 mechanism, the "degraded" version, then "optimized" versions via the GA and PSO algorithms.

c 1 and c 2 •

 2 parameters evolve from 2.5 to 3.5 and 1.5 to 3.5, respectively, in strategies 3 and 4 to favor a local search at the end of the optimization process. The value of κ is 1 in strategy 3 and is reduced to 0.5 in strategy 4. Strategies without constriction factor, with decreasing inertia, constant cognitive component, and increasing social component(5 to 8) To favor a global search at the beginning of the optimization then the local search at the end, strategies 5 to 8 and the corresponding parameters w, c 1 and c 2 have been defined such that the relative influence of the inertial component (favorable to the global search) decreases over iterations, the cognitive component remains constant, and the social component (favorable to the local search) increases. On this basis, several patterns are possible according to the initial importance given to each of the components at the beginning of the process. It is to be noted that strategy 8 is composed of the same parameters as strategy 7 but the inertia term W is weighted by the individual's score (see equation 8).• Strategies without constriction factor, with decreasing inertia, decreasing cognitive component, and increasing social component (9 to12) In these strategies, the relative weights of inertia and cognitive components decrease over the iterations while that of social component increases which favors local search as long as the optimization progresses. The parameters of strategies 10 and 12 are similar to those of strategies 9 and 11, respectively, but the inertia term W is weighted by the individual's score.• Strategies without constriction factor, with constant inertia, decreasing cognitive component, and increasing social component(13 to 14) In these strategies, the relative importance of inertia remains constant, which maintains a non-negligible share of randomness throughout the optimization. To tend towards a local search, the weight of the cognitive and social components decreases and increases, respectively. Strategy 14 integrates the individual's score in the calculation of the inertial weight.

Figure 2 :

 2 Figure 2: Relative importance of the different components in the strategies adopted for optimization by the PSO method

Figure 3 :Figure 4 :

 34 Figure 3: Mean error evolution as a function of the iteration number for the different optimization strategies tested

Figure 5

 5 Figure5shows the mean relative deviation of the B, n, and C coefficients of the modified reactions between the GRI-Mech 3.0 mechanism, the degraded version, and then the optimized versions. For all the strategies tested in this work, the optimization reduces the differences between the kinetic constants of the degraded version and the reference mechanism.

Figure 5 :

 5 Figure 5: Standard deviation between optimization processes

Table 1 :

 1 The choice of GA operators for the 11 evaluated strategies

	Strategies	Selection	Crossover (*) Mutation (*)
		Roulette	Rank	Geometric norm	Elitism	Simple	Multiple	Arithmetic	Heuristic	Uniform	Non-uniform	Boundary
	1	x				2 2 4 4 3 3	2
	2		x			2 2 4 4 3 3	2
	3			x		2 2 4 4 3 3	2
	4				x 2 2 4 4 3 3	2
	5			x		6 2 2 2 3 3	2
	6			x		2 6 2 2 3 3	2
	7			x		2 2 6 2 3 3	2
	8			x		2 2 2 6 3 3	2
	9			x		2 2 4 4 6 1	1
	10			x		2 2 4 4 1 6	1
	11			x		2 2 4 4 1 1	6

Table 2 :

 2 Selected parameters for the development of PSO strategies

	Strat	κ		c1 init final init final c2
	1	1		2.05 2.05 2.05 2.05
	2	0.5		2.05 2.05 2.05 2.05
	3	1		2.5	3.5	1.5	3.5
	4	0.5		2.5	3.5	1.5	3.5
	Strat	W init final init final init final Wmin C1 C2 Score
	5	0.6	0.2	1.1	1.2	1	2
	6	0.8	0.3	1.6	2	0.5	2
	7	1.1	0.4 1.75 1.4	1.3	2
	8	1.1	0.4 1.75 1.4	1.3	2	0.4
	9	1	0.2	1	2	1	2
	10	1	0.2	1	2	1	2	0.2
	11	1	0.2	2	1	1	2
	12	1	0.2	2	1	1	2	0.2
	13	0.7	0.7	2	0.8	0.8	2
	14	0.7	0.7	2	0.8	0.8	2	0.2

Acknowledgements

The authors gratefully acknowledge financial support from ED SMAER doctoral school of Sorbonne University as well as the ONERA grant No. F/12893DA-GGUE.

Author Contributions