Transient protection is an important issue in pipeline design. As protective devices impose a huge cost on the project, it is better and more efficient to use optimization models for determination of their position and type with the aim of cost reduction. Except for the cost, the other important issue in obtaining the number and locating of protective devices is the consideration of important operational parameters during the utilization of the pipelines. This paper introduces a new objective function called ‘serviceability factor’ for achieving the best layout for protection devices by considering five main operational parameters. A double-objective model has been used to optimize the protective devices to obtain the minimum of cost and the most appropriate level of operational parameters. The presented model utilizes the non-dominated sorting genetic algorithm (NSGAII) simultaneously with transient analysis through the method of characteristics. A real pipeline has been optimized using this model and the results are presented in the form of Pareto optimal solutions.
INTRODUCTION
Water is one of the main elements in human communities and its transmission from resources to consumers is a very significant topic. A common and reliable method for this purpose is the use of pressurized pipelines. However, the huge implementation cost of such projects, on the one hand, and their technical sensitivities, on the other hand, intensify the necessity for optimal designing with minimum cost and maximum safety and efficiency.
Transient flow due to water hammer is one of the factors considerably affecting the cost and safety of pipelines. In pipelines with pumping stations, this mostly occurs due to sudden pump trip following a power failure.
Transients can introduce large pressure forces and rapid fluid accelerations into the system. These disturbances may result in pump and device failures, system fatigue, or pipe ruptures. Also, transients in pumping systems can lead to water column separation, which can lead to catastrophic pipeline failures. To prevent undesired effects of these pressures, the use of protective devices is necessary, including air-chambers and air-inlet valves. In pumping pipelines, the air-chamber can appropriately control positive and negative pressures (Stephenson 2002); however, its use imposes a considerable cost on the project. The air-inlet valve can effectively reduce negative pressure consequences (Zhuqing et al. 2011) and its price is less than an air-chamber. Thus, for designing a lower cost system, it is better to use air-inlet valves in the pipeline to reduce negative pressure and, consequently, decrease the number and volume of air-chambers; in this way, due to the lower price of air-inlet valves, the total cost of the system would also decrease.
In this circumstance, locating air-chambers and air-inlet valves in appropriate positions could have considerable effects in reduction of the number and sizes of these protective devices and decrease the system protection cost. On the other hand, apart from the cost, the other important issue in obtaining the number, types, and locating the protective devices, is the consideration of important parameters during the utilization of pipelines.
In the last three decades, numerous studies have been performed on the transients analysis methods in water transmission and distribution systems. Karney & McInnis (1990) analyzed the distribution systems under a wide range of flow conditions, with relatively few restrictions, using modern computer techniques. They presented examples of the dangers of oversimplifying either the physical system or the operating conditions. In another work, McInnis & Karney (1995) addressed the relatively unexplored area of transients in complex pipe networks. They presented a new formulation, permitting system demands to be represented as a distributed pipe flux. Their approach was compared with two conventional methods for modeling demands in pipe networks. Moreover, they compared the results of their model with a field test in a major transmission.
Boulos et al. (2005) provided a basic understanding of the physical phenomena and context of transient conditions and presented practical guidelines for their suppression and control; finally, they compared the formulation and computational performance of widely used hydraulic transient simulation schemes.
Some researchers have studied the effects of transient protection devices in pipelines and presented methods for the proper design of these devices. Lee & Leow (1999) studied the effect of air-inlet valve specifications on mitigation of pressurized waves in pipelines due to sudden pump trip and showed that locating air-inlet valves in the higher points of the pipeline could have a more significant effect on controlling negative pressures.
Wang et al. (2013) studied the effects of air-chambers in controlling water hammer in pumping systems at high pressures and concluded that in these systems, the key parameter is the volume of the air-chamber.
Stephenson (2002) presented good nomographs for estimation of required volume for the compressed air-chamber as a transient protection design and Zhuqing et al. (2011) researched the correct orifice diameter of air-inlet valves.
Many methods have been proposed for the optimization of pipelines and protective devices. Laine & Karney (1997) applied an optimization model for a simple pipeline connected to a pump and a reservoir. They used a complete enumeration scheme for steady state and transient flow analysis and system design.
Lingireddy et al. (2000) developed an optimization method to minimize the cost of compressed air-chambers simultaneously with respect to the pressure constraints. Their study was performed based on a bi-level genetic algorithm (GA) model.
Jung & Karney (2004) studied the effects of transient flow in a selection of optimum diameters in a water supply system with consideration of the design criteria of steady state. They used GA and particle swarm optimization (PSO) for problem-solving and perceived that the combination of GA and PSO in a transient analysis method could considerably improve the efficiency and cost of protection against the transient flow.
Jung & Karney (2006) combined GA and PSO methods for optimization of the size and location of water hammer protection devices in water distribution systems. They minimized maximum pressures, maximized minimum pressures, and minimized minimum and maximum pressure differences. Moreover, they presented several strategies based on surge tank application and pressure regulating valves. However, their study only included the hydraulic performance of protective strategies without consideration of its cost.
Jung et al. (2009) formulated the optimum design of a water supply system under transient conditions using a multi-objective optimization algorithm. Their optimization objectives were the minimization of pipeline cost and maximization of hydraulic reliability. In contrast to most optimization models where the demand is considered at the highest possible level, their method assumes demand during system designing as different.
Jung et al. (2011) considered a multilevel optimization method for determination of the pipeline sizes in a water supply system as influenced by water hammer. They introduced criteria, called wave damage potential factor, as the first objective function and construction cost of the system as the second objective function. Then, they utilized non-dominated sorting genetic algorithm (NSGA) for optimization of objective functions. In their study, no method or device was used for protection against the transient flow and optimization was only performed on the pipeline size.
Fathi-Moghaddam et al. (2013) used a GA method for the optimization of a water penstock tunnel and specification of surge tank for a hydroelectric power plant where the objective function was a benefit to cost ratio. Jung & Karney (2013) optimized the design of a water distribution system for the worst transient condition in two steps. In the first step, a PSO model was used to introduce the critical points with the highest effect on the transient flow. In the second step, a double-objective optimization based on NSGA was performed to determine the optimal sizes of the pipeline to simultaneously minimize the cost and the probability of occurrence of transient flow damage measured through wave damage potential factor.
In the context of the previous studies, the importance of transient effects and the necessity of pipeline protection against them and the application of optimization models for more economical use of transient protection devices is an approved issue. However, what has been neglected is the proper layout of protective devices concerning the utilization and serviceability issues. Devices located in various points of the pipeline might be different regarding serviceability and maintenance, for example, the possibility of quick serviceability. In this regard, since the pipelines pass through suburban areas, other aspects, such as environmental effects and general security of installing devices in each location also should be considered. Thus, in order to perform optimization, it is necessary to consider an objective function that takes into account the serviceability in addition to the cost of water hammer protection devices in pipelines and make a double-objective optimization.
Thus, in this paper, in the optimization of pipeline transient protection devices along with the cost, another objective called ‘serviceability factor’ is proposed and a double-objective optimization is presented. The cost objective is to reach the minimum cost via the proper combination of protective devices, such as different types of air-inlet valves besides the air-chambers. The serviceability factor objective is to achieve the best operation during the project utilization with a selection of the most appropriate layouts for protection devices by considering factors such as: difficulties in servicing during utilization, environmental effects of the use of each device in any location, and general security of installation locations.
The value of the cost objective can be determined by adding the price of devices and their installation cost. However, for calculation of serviceability, various criteria should be considered where some have quantitative, and some have qualitative aspects. Here, the use of multi-attribute decision-making (MADM) could be beneficial for adding together the various mentioned criteria in the form of an objective function, called serviceability factor. On the other hand, since the general aim of this modeling is to protect the pipeline against the pressures due to water hammer, determining the two mentioned objectives should be done with preservation of system pressures at a permissible level. Thus, the presented optimization model should be constrained by the allowable pressures.
To prepare this model, it is necessary to use the double-objective optimization algorithm simultaneously with a method of transient analysis of the pipeline. In this paper, the method of characteristics (MOC) has been utilized for the transient analysis while it is combined with NSGAII optimization algorithm.
In the following, after explaining the MOC transient analysis method, the approach of defining objective functions and the used optimization algorithm are presented, and then the combination of flow analysis and double-objective algorithm is described. Finally, the model is used for a super large-scale real pipeline and the results of optimization are presented in the form of Pareto optimal solutions.
GENERAL ANALYSIS OF WATER HAMMER
It is worth noting that the analysis of transient fluid was developed based on the following assumptions: (1) the flow is one-dimensional, (2) the fluid is slightly compressible, (3) the pipe walls are linear elastic, and (4) the conduit has expansion joints throughout its length.
The friction factor for unsteady flow is also determined using Brunone formula (Brunone et al. 1991).
PUMP POWER FAILURE, AIR-CHAMBER, AND AIR-INLET VALVE
Air-chamber in a pipeline.
Air-inlet valve in a pipeline.
Boundary conditions of the pump trip are explained in Wylie & Streeter (1993) and equations and details of air-chamber and combination air-valve conditions can be found in Chaudhry (2014). In these references, the boundary conditions are developed based on the following assumptions: (1) for air-chambers it is assumed that the air enclosed at the top of the chamber follows the polytropic relation for a perfect gas; (2) for air-combination valves it is assumed that the airflow into the pipeline is isentropic, the entrapped air remains at the valve location and is not carried away by the flowing liquid, and the expansion or contraction of the entrapped air is isothermal.
OBJECTIVE FUNCTIONS
The optimization model herein presented considers two objectives: the first objective evaluates the cost of the project and the second objective considers servicing and maintenance during utilization, which is defined as the project serviceability factor. Since the NSGAII method has been used and this algorithm acts based on the evolution of a population of solutions, it is necessary to calculate the value of objectives for each member of the population. Here, every member is a specific protection plan with some certain protective devices in the specified locations of the pipeline.
Cost
Serviceability factor
is the average amount of jth serviceability index in a protection plan.
where A1 to Am are pipeline protection plans (members of optimization population), each one with specified numbers, types, and locations of protective devices, and r is the value of each index in any protective plan.
Qualitative spectrum with bipolar scale.
Based on this scale, the best possible value can be measured for the index from 1 to 9. The value 5 is the breakpoint between desired and undesired condition.
and are, respectively, the maximum and minimum possible values for jth indexes.
); moreover:
CONSTRAINTS HANDLING
is the number of computational nodes of the pipeline, and are, respectively, the minimum and maximum pressures at node j, is the minimum allowable pressure at node j, i.e., a safe pressure value far enough from the water vapor pressure at that node, and is the maximum allowable pressure at node j, which is determined according to the wall thickness of the pipe. Equation (13) indicates that should be zero when all nodal pressures are in the allowable ranges and it equals a positive large number when they are violated. The penalty factor M is an arbitrary positive number determined by trial and error to achieve the best performance of GA in terms of both constraint handling and searching for the global optima in the feasible decision space. Using the penalty function causes that performance impact of protection devices to be tested in the model. Smaller values of indicate better capability of devices (air-chambers and air-valves).
OPTIMIZATION MODEL
Since the main issue of optimization is minimization, the functions must be such that their minimization is on the desired side. F1 is related to the cost, and its minimization is desired. Also the minimum amount of F2, that depends on serviceability indices, is appropriate (based on what was previously mentioned; the reduction of G1 to G5 is desired and improves the serviceability of the system). Thus, the reduction of serviceability factor and F2 is favorable.
Unlike one-objective optimization, multi-objective problems yield a collection of solutions rather than a certain solution such that each solution from this set could be an optimum. Multi-objective optimization methods can be divided into three general categories: evaluating methods, non-Pareto methods, and Pareto methods. Among these, Pareto methods are more flexible and stronger for engineering problems since they can obtain optimum multi-purpose solutions of Pareto in one implementation. In this paper, NSGAII (Deb et al. 2002) has been used, and is an evolutionary elitist algorithm through the Pareto method and based on GA.
Decision variables in objective functions are cost and serviceability indexes that are dependent on the location of air-chambers and their volume, the location of air-inlet valves and the diameter of their orifice. In this study, to define the permissible space of decision variables, first some locations are chosen to place each protection device and for the specification of each protection device, certain predetermined selections are considered according to a list. The location and type of the protection devices can vary in this way in the defined space. To determine objective functions, the price of each device is included in its specification list and for each candidate place, the distance from the service station, access road and power station, the environmental index, and the general security index are defined beforehand.
- 1.
The same as the GA method, an initial population of decision-making variables are randomly produced in the permissible space. Each member of this population includes a set of decision variables that are called chromosomes.
- 2.
With defined specifications for each chromosome, hydraulic analysis of water hammer is performed in the pipeline due to the predetermined scenario for pump power failure and the objective target values for each chromosome is obtained using its results. It is such, that concerning the protective devices in each member of the population, the total cost is computed and five indexes constituting serviceability factor also obtained. Then, using Equations (4)–(15), F1 and F2 values for each chromosome are calculated.
- 3.
The population is sorted based on non-dominated sorting according to the values of functions allocated to chromosomes, such that the members in the first front are a fully non-dominated set by all other members of the population. The members of the second front are dominated just by the members of the first front, and this trend continues in this way on other fronts so that all members in each front will get a rank based on the number of the category (according to Figure 4). This sorting is the basis for non-dominated sorting.
- 4.Crowding distance controlling parameter for any member of each front can be calculated through objective functions from the following equation:where k is the number of members,17
is crowding distance, k− 1 and k+ 1 are the number of members beside the intended member in the related front, Fi is ith objective function and
and
are, respectively, the minimum and maximum values of ith function in the intended front. Crowding distance indicates the closeness of the member to other members of the population in each front. Now, the population members are sorted first based on the front number and the second based on the crowding distance criterion such that the member with a lesser front number is in a higher rank and the member with higher dj value is in a higher rank if some of the members have the same fronts.
- 5.
A new generation of chromosomes is produced using the parent selection method through the binary tournament method (Haupt 2004) and application of crossover through BLX-a (Eshelman & Shaffer 1993), and they replace bad chromosomes in the population.
- 6.
A few genes except genes of the best chromosomes that are in the first front are randomly mutated.
- 7.
Convergence criterion is controlled. If GA chromosomes are sufficiently similar, optimization is stopped; otherwise, this algorithm returns to step 2 to iterate with a new population.
Non-dominated sorting.
Pareto optimal solutions.
Using the obtained solutions and based on the engineering judgments, it is possible to select the appropriate answer to preserve the values of objective functions in the desired level from the Pareto front.
CASE STUDY
The specification of pipes and pumping station of the Qadir project
| Pipeline properties | Pump station properties | ||
|---|---|---|---|
| Pipe diameter | 2,250 mm | Pump type | Horizontal centrifuge |
| Max allowable pressure | 37 bar | Number of pumps | 10 |
| Material | Steel | Total head | 160 m |
| Min allowable pressure (water vapor pressure) | 0.13 bar | Total discharge | 8.5 m3/s |
| Thickness | 19 mm | Rotational speed of each pump | 1,760 rpm |
| Wave speed | 1,000 m/s | Outlet diameter | 520 mm |
| Pipeline properties | Pump station properties | ||
|---|---|---|---|
| Pipe diameter | 2,250 mm | Pump type | Horizontal centrifuge |
| Max allowable pressure | 37 bar | Number of pumps | 10 |
| Material | Steel | Total head | 160 m |
| Min allowable pressure (water vapor pressure) | 0.13 bar | Total discharge | 8.5 m3/s |
| Thickness | 19 mm | Rotational speed of each pump | 1,760 rpm |
| Wave speed | 1,000 m/s | Outlet diameter | 520 mm |
The longitudinal profile and plan of the Qadir pipeline.
Length of pipes in system
| Pipe # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Length (m) | 150 | 2,000.09 | 2,000.08 | 2,000.01 | 2,808.93 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 | 2,850 | 2,000.01 |
| Pipe # | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| Length (m) | 2,000.01 | 2,000.02 | 1,213.67 | 2,000.01 | 2,000 | 2,000 | 2,000.05 | 2,000.05 | 1,696.86 | 2,000.07 | 1,999.87 | 2,000.21 | 2,000 | 2,000 | 2,000 |
| Pipe # | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 |
| Length (m) | 2,083.88 | 2,000.03 | 2,000.02 | 2,000.03 | 2,000.02 | 1,317.12 | 2,000.05 | 2,000 | 2,000 | 2,000.04 | 2,000 | 2,000.05 | 2,000 | 2,000 | 2,000 |
| Pipe # | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| Length (m) | 2,000 | 2,072.78 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 | 2,819.41 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 |
| Pipe # | 61 | 62 | 63 | 64 | |||||||||||
| Length (m) | 2,015.33 | 2,000 | 2,000 | 2,244.13 |
| Pipe # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Length (m) | 150 | 2,000.09 | 2,000.08 | 2,000.01 | 2,808.93 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 | 2,850 | 2,000.01 |
| Pipe # | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| Length (m) | 2,000.01 | 2,000.02 | 1,213.67 | 2,000.01 | 2,000 | 2,000 | 2,000.05 | 2,000.05 | 1,696.86 | 2,000.07 | 1,999.87 | 2,000.21 | 2,000 | 2,000 | 2,000 |
| Pipe # | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 |
| Length (m) | 2,083.88 | 2,000.03 | 2,000.02 | 2,000.03 | 2,000.02 | 1,317.12 | 2,000.05 | 2,000 | 2,000 | 2,000.04 | 2,000 | 2,000.05 | 2,000 | 2,000 | 2,000 |
| Pipe # | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| Length (m) | 2,000 | 2,072.78 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 | 2,819.41 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 | 2,000 |
| Pipe # | 61 | 62 | 63 | 64 | |||||||||||
| Length (m) | 2,015.33 | 2,000 | 2,000 | 2,244.13 |
The division of the pipeline and the status of serviceability indexes in the pipeline.
The division of the pipeline and the status of serviceability indexes in the pipeline.
According to the explained model, for optimization of transient protection devices against pump station power failure in this pipeline, all connection points of pipes have been considered as candidate points for locating protective devices. Seven types of air-chamber with different volumes and three types of air-inlet valves with different orifices were considered as possible selections for candidate points. Table 3 shows the list of protective device specifications that are usable in this optimization along with the cost of each one. The optimization model selects decision variables for each candidate point from these two lists. Although in the model there is the possibility of a lack of protective device for each candidate point, it is the state that nothing is selected from the options presented in a candidate point.
The specification of protective devices used in optimization model
| Air-inlet valve | Air-chamber | ||||
|---|---|---|---|---|---|
| Type | Orifice diameter Diameter (mm) | Cost ($) | Type | Volume (m3) | Cost ($) |
| A | 300 | 7,000 | A | 10 | 55,000 |
| B | 450 | 9,000 | B | 20 | 100,000 |
| C | 600 | 12,000 | C | 25 | 250,000 |
| D | 30 | 300,000 | |||
| E | 35 | 340,000 | |||
| F | 40 | 367,000 | |||
| G | 60 | 530,000 | |||
| Air-inlet valve | Air-chamber | ||||
|---|---|---|---|---|---|
| Type | Orifice diameter Diameter (mm) | Cost ($) | Type | Volume (m3) | Cost ($) |
| A | 300 | 7,000 | A | 10 | 55,000 |
| B | 450 | 9,000 | B | 20 | 100,000 |
| C | 600 | 12,000 | C | 25 | 250,000 |
| D | 30 | 300,000 | |||
| E | 35 | 340,000 | |||
| F | 40 | 367,000 | |||
| G | 60 | 530,000 | |||
To determine the weights of serviceability indexes, it is necessary to form a relative weight matrix with engineering judgment. This matrix is formed for this issue based on the regional condition and the facilities for servicing and maintenance according to Table 4.
The weight matrix of the Qadir project problem
| W = | |||||
| W = | |||||
The population is considered to be 200 and the mutation rate is assumed to be 0.02. A maximum number of permissible generations for implementation are also set as 1,000.
RESULTS
The optimum Pareto solutions for the Qadir project.
In Table 5, the specification and location of protection devices related to some Pareto solutions that are shown in Figure 8 are presented.
The specification of some of the protection plans present in Pareto solutions
| Solution# | Air-inlet valves (location and type) | Air-chambers (location and type) | F1 cost (1,000*$) | F2 (serviceability factor) |
|---|---|---|---|---|
| 1 | 4(A),8(A),10(A),15(A),33(A),47(A),51(A),56(A),61(A) | 27(A), 49(A) | 173 | 0.906 |
| 3 | 4(A),10(A),15(A),29(A),33(A),47(A),56(A),61(A) | 2(B), 27(A),49(B) | 311 | 0.642 |
| 4 | 4(A),14(A),33(A),50(A),56(A) | 2(B),8(A),27(B),49(A) | 345 | 0.582 |
| 7 | 4(A),10(A),14(A),56(A) | 2(B),27(B),32(A),49(B) | 383 | 0.265 |
| 8 | 4(A),14(A),56(A) | 2(B),27(B),32(B),49(B) | 421 | 0.196 |
| Solution# | Air-inlet valves (location and type) | Air-chambers (location and type) | F1 cost (1,000*$) | F2 (serviceability factor) |
|---|---|---|---|---|
| 1 | 4(A),8(A),10(A),15(A),33(A),47(A),51(A),56(A),61(A) | 27(A), 49(A) | 173 | 0.906 |
| 3 | 4(A),10(A),15(A),29(A),33(A),47(A),56(A),61(A) | 2(B), 27(A),49(B) | 311 | 0.642 |
| 4 | 4(A),14(A),33(A),50(A),56(A) | 2(B),8(A),27(B),49(A) | 345 | 0.582 |
| 7 | 4(A),10(A),14(A),56(A) | 2(B),27(B),32(A),49(B) | 383 | 0.265 |
| 8 | 4(A),14(A),56(A) | 2(B),27(B),32(B),49(B) | 421 | 0.196 |
Minimum pressure along the pipeline.
Concerning the obtained Pareto front in the optimization of this issue, there is the possibility of selecting a diverse range of protection plans with consideration of technical and economic criteria and considering utilization parameters.
CONCLUSION
In this study, an optimization model is presented to obtain the optimum specification of the pipeline protection plan against water hammer due to pump power failure. Here, the determination of the locations and specification of air-chambers and air-inlet valves in the pipeline have been considered. The presented model provides designers with a collection of Pareto optimal solutions for decision-making with consideration of two objectives. The first objective is cost and the second objective, that is presented for the first time in this study, is the serviceability factor which is obtained by a combination of five effective indexes on system servicing and maintenance using MADM. In the mentioned optimization model, NSGAII was used as a multi-objective optimization method combined with transient flow analysis through the MOC method. This model has been used in a real large project in the southwest of Iran. The results of this case study show that the use of serviceability objective along with cost can provide a broad range of solutions and help designers in the selection of the proper specification of protection systems against water hammer with simultaneous consideration of cost and utilization issues. The other advantage of the presented model is pipeline hydraulic constraint handling using the penalty functions in objectives. This approach means that all solutions are achieved with the certainty of pressure justifiablity.
However, given that the model is based on GA and MOC transient analysis, its main weakness is that it is very time-consuming. This issue creates significant restrictions for the optimization of long pipelines. Applying transient analysis in the frequency domain instead of time domain, and using faster optimization models, can be useful to improve the model convergence.
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