esa-sceva/satcom-chunk-collection huggingface dataset:
# Heat Transfer Optimization of an Electronic Control Unit Immersed in Forced Liquid Coolant
## 1 Introduction
Depending on the fluid's viscosity, it may also act as a damper to help stop vibrations and shocks [47; 48]. The characteristics of typical dielectric liquids are low density, low boiling point, non-reactivity, non-corrosivity, and strong chemical and thermal stability. Although there is a large amount of literature on thermal immersion cooling and cost savings, there is little literature on the impact of immersion cooling on the reliability of information technology (IT) equipment such as PCBs and electronic packages. [PERSON] et al. [49], focusing on the change of thermo-mechanical properties of PCBs when immersed in dielectric fluid and the effect of the change on the reliability of electronic packages, showed that [PERSON]'s modulus is decreasing for PCBs after immersion in dielectric coolant, which is likely to increase the reliability of electronics packages. Moreover, research by [PERSON] [50] using a variety of dielectric liquids has demonstrated that the dielectric properties are satisfactorily stable and undergo minimal change over time, even when exposed to high temperatures. Additionally, electrical component oxidation and corrosion will be prevented, contamination from dust, debris, and particles will be decreased [34], and the performance of electronic components will increase [34; 51; 52]. Immersion cooling has not yet been used in the automotive industry to cool an ECU; instead, it is primarily used to cool data centers and occasionally to cool the batteries in electrical vehicles. The current paper proposes a cooling concept for future centralized platforms of ECUs in conventional vehicles and EVs that involves grouping multiple electronic devices into a single system and cooling them with recirculated dielectric coolant. According to this active cooling solution implemented to increase the heat transfer of the entire system, the current paper attempts to establish an optimal configuration between fan airspeed, pump airflow rate, and housing coolant volume. The system will be exposed to an ambient temperature of 85 \({}^{\circ}\)C. From international standards used in automotive ISO 16750-1 and ISO 16750-4 for environmental conditions and testing for electrical and electronic equipment of road vehicles, this temperature was selected to cover the thermal requirements for ECUs mounted on passenger, luggage, and load compartments or ECUs mounted on the exterior or in cavities. The choice of coolant fluid is obviously critical in this case. As a new cooling technology proposed for automotive, there is still a lack of research addressing the lifetime, fluid stability, material compatibility, and understanding of sustainability. To evaluate the cooling benefits, temperature measurements on the hot spots of these solutions were compared to a reference electronic module cooled only by thermal paste.
## 3 Control Strategy of Wind Storage System to Suppress Low-Frequency Oscillation
### Analysis of the Influence of Different Installation Positions of POD on Low Frequency Oscillation
When load disturbance \(\Delta P_{\rm L}\) occurs, the system energy is surplus, and the power absorbed by the wind storage device is \(\Delta P_{\rm SYS}\), so the equations of motion of the synchronous generator rotor are:  
\[T_{\rm J}\frac{d\omega}{dt}=P_{\rm m}-P_{\rm e}+\Delta P_{\rm L}-\Delta P_{\rm SYS }=\Delta P_{\rm L}-\Delta P_{\rm SYS} \tag{37}\]  
When the wind storage system is in the maximum power modulation state, \(\Delta P_{\rm SYS}=P_{\rm max}\) is obtained from Equation (32):  
\[\frac{d\omega}{dt}=\frac{\Delta P_{\rm L}-P_{\rm max}}{T_{\rm J}} \tag{38}\]  
At the initial moment of oscillation, the wind storage system power has not changed, \(\Delta P_{\rm SYS}=0\), and the maximum speed change rate at this time is:  
\[\frac{d\omega}{dt}|_{\rm max}=\frac{d\omega}{dt}|_{t=0}=\frac{\Delta P_{\rm L }}{T_{\rm J}} \tag{39}\]  
Taking inertia control as an example, the output of ESS under inertia control does not exceed the capacity limit \(P_{\rm max}\), namely:  
\[\Delta P_{\rm SYS}=K_{\rm d}\frac{d\omega}{dt}\leq P_{\rm max} \tag{40}\]So the gain \(K_{\rm d}\) of the D controller can be obtained from Equations (34) and (35):  
\[K_{\rm d}=\frac{T_{\rm J}P_{\rm max}}{\Delta P_{\rm L}} \tag{41}\]  
The Equations of motion of the rotor in the ESS inertia control mode are obtained by bringing Equation (36) into (32) and (35):  
\[T_{\rm J}\frac{d\omega}{dt}=\Delta P_{\rm L}-\frac{T_{\rm J}P_{\rm max}}{ \Delta P_{L}}\frac{d\omega}{dt} \tag{42}\]  
Obtain the synchronous machine speed change rate under inertia control as  
\[\frac{d\omega}{dt}=\frac{\Delta P_{L}^{2}}{T_{\rm J}(\Delta P_{\rm L}+P_{\rm max })} \tag{43}\]  
By subtracting from (33) and (38):  
\[\frac{\Delta P_{\rm L}-P_{\rm max}}{T_{\rm J}}-\frac{\Delta P_{L}^{2}}{T_{\rm J }(\Delta P_{L}+P_{\rm max})}<0 \tag{44}\]  
So the wind storage system is at maximum power output on \(\frac{d\omega}{dt}\) under maximum power modulation is better than that of inertia control.
## 2 Control Equations for Euler-Bernoulli Rotor
### Free Vibration Equation of the Beam Element
 \tag{18}\]  
Ultimately, the free vibration control equation for the beam element is derived as follows:  
\[EI\frac{\partial^{4}v}{\partial x^{4}}+m\ddot{v}-J_{p}\Omega\frac{\partial^{2} \ddot{w}}{\partial x^{2}}-J_{d}\frac{\partial^{2}\ddot{v}}{\partial x^{2}}=0 \tag{19}\]  
\[EI\frac{\partial^{4}w}{\partial x^{4}}+m\ddot{w}+J_{p}\Omega\frac{\partial^{2} \dot{v}}{\partial x^{2}}-J_{d}\frac{\partial^{2}\ddot{w}}{\partial x^{2}}=0 \tag{20}\]  
The weak form equation of the beam element's control equation is given by  
\[\int_{x_{i}}^{x_{j}}\Bigg{(}EI\frac{\partial^{4}v}{\partial x^{4}}+m\ddot{v}-J _{p}\Omega\frac{\partial^{2}\dot{w}}{\partial x^{2}}-J_{d}\frac{\partial^{2} \ddot{w}}{\partial x^{2}}\Bigg{)}\delta vdx+\int_{x_{i}}^{x_{j}}\Bigg{(}EI \frac{\partial^{4}w}{\partial x^{4}}+m\ddot{w}+J_{p}\Omega\frac{\partial^{2} \dot{v}}{\partial x^{2}}-J_{d}\frac{\partial^{2}\ddot{w}}{\partial x^{2}} \Bigg{)}\delta wdx=0 \tag{21}\]  
where \(\delta v\) and \(\delta w\) represent the virtual displacements. Integrating the weak form equation by parts yields  
\[ \begin{split}&\int_{x_{i}}^{x_{j}}\Bigg{(}EI\frac{\partial^{4}v}{ \partial x^{4}}\bigg{)}\delta vdx=EI\frac{\partial^{3}v}{\partial x^{3}}( \delta v)x_{x_{i}}^{x_{j}}-EI\frac{\partial^{2}v}{\partial x^{2}}\delta\bigg{(} \frac{\partial v}{\partial x}\bigg{)}_{x_{i}}^{x_{j}}+\int_{x_{i}}^{x_{j}}EI \frac{\partial^{2}v}{\partial x^{2}}\delta\Bigg{(}\frac{\partial^{2}v}{ \partial x^{2}}\bigg{)}dx\\ &\int_{x_{i}}^{x_{j}}J_{p}\Omega\frac{\partial^{2}\dot{w}}{ \partial x^{2}}\delta vdx=J_{p}\Omega\frac{\partial\dot{w}}{\partial x}( \delta v)x_{i}^{x_{j}}-\int_{x_{i}}^{x_{j}}J_{p}\Omega\frac{\partial\dot{w}}{ \partial x}\delta\bigg{(}\frac{\partial v}{\partial x}\bigg{)}dx\\ &\int_{x_{i}}^{x_{j}}J_{d}\frac{\partial^{2}\dot{v}}{\partial x^{2}} \delta vdx=J_{d}\frac{\partial\ddot{v}}{\partial x}(\delta v)x_{i}^{x_{j}}-x_{ i}^{x_{j}}J_{d}\frac{\partial\dot{v}}{\partial x}\delta\bigg{(}\frac{\partial v}{ \partial x}\bigg{)}dx\end{split}  \tag{22}\]  
The first integral is  
\[ \begin{split}&\int_{x_{i}}^{x_{j}}\Bigg{(}EI\frac{\partial^{4}v}{ \partial x^{4}}+m\ddot{v}-J_{p}\Omega\frac{\partial^{2}\ddot{w}}{\partial x^{2}}-J _{d}\frac{\partial^{2}\ddot{v}}{\partial x^{2}}\Bigg{)}\delta vdx\\ &=\int_{x_{i}}^{x_{j}}\Bigg{[}EI\frac{\partial^{2}v}{\partial x^{2}} \delta\bigg{(}\frac{\partial^{2}v}{\partial x^{2}}\bigg{)}+m\ddot{v}\delta v+J_ {p}\Omega\frac{\partial\dot{w}}{\partial x}\delta\bigg{(}\frac{\partial v}{ \partial x}\bigg{)}+J_{d}\frac{\partial\ddot{v}}{\partial x}\delta\bigg{(}\frac{ \partial v}{\partial x}\bigg{)}dx\Bigg{]}dx\end{split}  \tag{23}\]The intermediate step involves  
\[
# Numerical Modeling of Single Pile Behaviors Due to Groundwater Level Rising
## 4 Numerical Analysis
### Procedure of FE Simulation  
The simulation was divided into five stages (Figure 8). Times are days from t = 0 in 1957 for 29,200 days to 2037.  
1. Initial (In 1957, t = 0): This stage was the generation of initial distribution of vertical and horizontal stresses, the soil unit weight, and the coefficient of soil pressure was used to calculate the initial soil stress state. The initial pore water pressure was assigned assuming hydrostatic pressure only.
2. Groundwater levels dropped (t = 0 to 14,600 days): The fully coupled flow and deformation analysis was used for calculation. The groundwater head function was defined from \(-\)0.0 to \(-\)30.0 m, decreasing at a rate of \(-\)0.75 m/year. At the end of this step, the groundwater level was the lowest.
3. Pile installation (In 1997, t = 14,600 days): The piles were installed and loaded axially with the working load. The model assumed that no soil displacement occurred while the piles were installed. The pore water pressure was set as in the previous step. The piles were installed at a minimum groundwater level before rising. One sub-step (iii-substage) was included in this step: the piles were tested when the groundwater level was lowest, for comparing with the load settlement curve after the groundwater stopped rising.
4. Groundwater levels rose (t = 14,600 to 29,200 days): The fully coupled flows and deformations were calculated in this step. The groundwater level rose from \(-\)30.0 to \(-\)1.0 m at +0.725 m/year. Note that the groundwater level rise was constrained to 1.0 m to prevent groundwater seeping onto the surface. The working load of each pile remained constant. The pile head and soil surface movement, stress distribution along with the pile and the occurrence of plastic points in each time step, was registered.
5. End of groundwater rise (In 2037, t = 29,200 days): The groundwater level was highest and close to the hydrostatic pressure. The ultimate load of piles was determined from the load settlement curve, using the pile diameter displacement (10%D), which was defined as the ultimate load [35]. For (v-substage), the pile load settlement curve was compared with the piles at the lowest groundwater level in (iii-substage). Pile load reduction in each pile settlement was computed.  
Figure 7: Ultimate Loads vs. settlement.
## 3 Results and Discussion
### Ketones  
Ketones are products of fatty acids metabolism [42; 48]. In the present study seven aliphatic and one aromatic ketones were recorded in _P. erungii_ mushrooms but only four (3-octanone, 1-octen-3-one, 2,3-octanedione and 2-undecanone) in _P. ostreatus_ strains; however, the latter contained a higher amount of ketones (266.9 \(\pm\) 101.9 \(\mu\)g/g f.w. in _P. ostreatus_ vs. 68.59 \(\pm\) 20.60 \(\mu\)g/g f.w. in _P. erungii_) (Table 2). Nonetheless, no clear and consistent distribution pattern is observed when the cultivation substrates are taken into account (Figure 1). Among the ketones detected, 3-octanone is predominant in both species at concentrations of 51.01 \(\pm\) 15.44 and 249.9 \(\pm\) 98.8 \(\mu\)g/g f.w. in _P. erungii_ and _P. ostreatus_, followed by 1-octen-3-one, 2-undecanone and 2,3-octanedione in lower concentrations, while acetophenone, 2-octanone, 6-methyl-5-hepten-2-one, and 3-octen-2-one were detected only in _P. erungii_ strains (Tables S1 and S2). Regarding their aromas, 3-octanone is described having a sweet, fruity, musty and lavender aroma [21], while 1-octen-3-one represents the dried-mushroom flavor imparting wet ground smell [40; 49]. They both possess low odor detection thresholds, and are considered important contributors to the aroma of mushrooms [21; 40]. The ketones detected in the present study have been reported among volatiles of various edible mushroom species [2; 18; 20; 29; 40; 50; 51].
# Summertime Aerosol Radiative Effects and Their Dependence on Temperature over the Southeastern USA
## 2 Methodology
### Spaceborne Observations
#### 2.1.3 Modis  
The Moderate-Resolution Imaging Spectroradiometers (MODIS, [50]) are passive satellite instruments aboard two satellites: Terra and Aqua. Having a swath width of 2300 km, these instruments cover the Earth's surface every 1 to 2 days. The instruments have 36 spectral bands ranging in wavelength from 0.4 \(\upmu\)m to 14.4 \(\upmu\)m. Two of the bands are imaged at a nominal resolution of 250 m at nadir, five bands at 500 m and the remaining 29 bands at 1 km. The Terra satellite was launched in 1999 and Aqua in 2002.  
In order to analyse how the biogenic emissions depend on the underlying vegetation types we used the Land Cover Type Climate Modelling Grid (CMG) product (MCD12C1, [51]). The product provides the dominant land cover types on a 0.05\({}^{\circ}\)\(\times\) 0.05\({}^{\circ}\) global grid. It contains three classification schemes that describe the land cover properties derived from observations spanning a year's input of MODIS observations. The primary land cover scheme, which identifies 17 land cover classes defined by the International Geosphere Biosphere Programme (IGBP), was used in this work. We re-gridded the data into 1\({}^{\circ}\)\(\times\) 1\({}^{\circ}\) resolution to match the other observations by calculating the fraction of each land cover class within the 1\({}^{\circ}\)\(\times\) 1\({}^{\circ}\) pixels. We considered a pixel to be dominated by a certain land cover class if the fraction of that type was 50% or larger. In the analysis, we only used pixels dominated by the three most abundant land cover classes in the southeastern USA in 2011: woody savannas, mixed forests and cropland/natural mosaic. The studied domain included in total 82 1\({}^{\circ}\)\(\times\) 1\({}^{\circ}\) pixels. Of those pixels, 22 were classified as woody savannas, 13 as mixed forests and 10 as cropland/natural mosaic.  
The other land cover classes dominated fewer than five \(1^{\circ}\times 1^{\circ}\) pixels, thus they did not provide enough observations for a statistically robust analysis. We also checked how the land cover types had changed during the studied period but no large-scale changes were found.
# An Optimal Power Control Strategy for Grid-Following Inverters in a Synchronous Frame
## 3 Discrete LQR-ORT Controller Design  
The complete scheme of the LQR-ORT controller is presented in Figure 3. The LQR controller \(K_{\rm d}\) is obtained using a modified cost function that weights the tracking error and control input. The use of a modified cost function requires the computation of an auxiliary optimal reference tracking (ORT) matrix \(K_{\rm v}v\) to solve the optimization problem. Finally, since the classic LQR problem does not consider the influence of external disturbances, the superposition principle is used to account for the power contribution of \(V_{\rm dq}\). Superposition can be applied because the grid-following inverter was expressed in (6) as a linear state-space model that not only considers V-I dynamics but also include active and reactive power computations.
## 5 Results and Discussion
### Evaluation Indicators  
The energy consumption intensity and exergy intensity per ton of DRI are adopted as the evaluation indicators of energy utilization of the improved HDRSF process.  
The energy intensity per ton of DRI is calculated according to Equation (39).  
\[E_{DRI\ consumption}=\frac{E_{Input}}{M_{DRI}} \tag{39}\]  
where \(E_{Input}\) represents the net energy input of the process to produce \(M_{DRI}\).  
The exergy intensity per ton of DRI is calculated according to Equation (40).  
\[Ex_{DRI\ consumption}=\frac{Ex_{Input}}{M_{DRI}} \tag{40}\]  
where \(Ex_{Input}\) represents the net exergy input of the process to produce \(M_{DRI}\).  
In terms of overall process flow, the energy intensity of HDRSF production per ton of DRI reduced from 2.75 \(\times\) 10\({}^{7}\) kJ to 1.70 \(\times\) 10\({}^{7}\) kJ, while the exergy intensity decreased from 1.08 \(\times\) 10\({}^{7}\) kJ to 1.05 \(\times\) 10\({}^{7}\) kJ.  
Figure 9: Exergy flow analysis of the HDRSF process with waste heat recovery.
## Appendix A Cooperative Pilot
### Computation of a Reference Space and Lane Coordinate System  
In order to simplify the stages of trajectories computation and their associated risk level, a change of the referential is applied. We move from the XY Cartesian \"world\" perception reference (similar to Lambert coordinate 120 used by Global Navigation Satellite System (GNSS)) to a simpler and mostly linear UW local reference (Figure 11). The UW curvilinear lane coordinate system uses the same origin as the XY coordinate system of the ego-vehicle. In this new linear frame of reference, the U axis is parallel to the middle of each lane and the W axis is perpendicular to U. This UW environment is the easiest environment to use for trajectory calculations involving the ego-vehicle and the surrounding objects. This UW lane coordinate system and the XY ego-vehicle coordinate system are both shown in Figure 11. In the UW lane coordinate system, the lane centres have a constant W coordinate. Thus, the trajectories of the ego-vehicle and objects that target the lane centre can be represented by two sections: a transitional section (with variable W coordinates) and a permanent section (with constant W coordinates).  
Computations with constant W coordinates are much easier and faster than computations in the XY lane geometry, which is usually (but not necessarily) based on a combination of lines, clothoid and circles [78]. After that, all trajectory computations of the ego-vehicle and objects will be performed in UW.
# Delaying the Occurrence of Bar Buckling in RC Columns Confined with SRG Jacketing
## 4 Analysis of the Axial and Lateral Response Due to SRG Confinement
### Criteria for the Effectiveness of SRG Jacketing in Delaying Bar Buckling
The position of the data points relative to the buckling curve denotes (a) when the data points fall below the curve and \(\mu_{es}<1\), then failure occurs due to elastic buckling; (b) when the data points fall under the curve and \(\mu_{es}>1\), then failure occurs due to buckling with steel being into its plateau of zero stiffness--apparently cases (a) and (b) correspond to RC columns with poor reinforcement detailing representative of the old construction practice; (c) when the data points fall on the curve branch denoted by (\(\mathrm{E_{hi}}\)) in Figure 8b, then failure happens owing to buckling with steel being into the hardening region and the concrete core being unable to undertake the load released by the buckled bars [2]; and (d) when the data points are over the curve and \(\mu=\varepsilon_{s,\mathrm{crit}}/\varepsilon_{y}>\varepsilon_{h}/\varepsilon_ {y}\), then buckling is postponed occurring at higher deformation level due to the efficiency of the confined concrete to undertake the load realised by the buckled bars. With reference to Figure 8b, the control specimens (black circles) failed near or below yielding of longitudinal reinforcement except for Group C, which coincided with the buckling curve at strain ductility 2.7. In all other cases, SRG jacketing managed to provide the necessary overstrength to the confined concrete core to successfully undertake the released load by the buckled bars (core and bars act as springs in parallel during compression) and to delay failure due to buckling. For the SRG specimens that are placed above the curve of Figure 8b, the following analysis is carried out. Upon attainment of bar critical strain \(\varepsilon_{s,\mathrm{crit}}>\varepsilon_{y}\) (\(\varepsilon_{s,\mathrm{crit}\,|\,\mathrm{A}}=1.5\varepsilon_{y}\), \(\varepsilon_{s,\mathrm{crit}\,|\,\mathrm{B}}=2\varepsilon_{y}\) and \(\varepsilon_{s,\mathrm{crit}\,|\,\mathrm{C}}=2.7\varepsilon_{y}\), see Figure 8b), the corresponding lateral strain of the SRG jacket due  
Figure 8: (**a**) Bar stress–strain diagram and (**b**) compressive strain ductility, \(\mu_{es}\), versus stirrup spacing \(s/\mathrm{D_{b}}\) for the longitudinal reinforcement used in the tests (dots refer to experimental values of \(\mu=\varepsilon_{\mathrm{ccu}}/\varepsilon_{y}\)). to concrete core dilation \(\epsilon_{j}^{dil,crit}\) is approximated by considering the apparent Poisson \(\
u\) as follows:  
\[\epsilon_{j}^{dil,crit}=\
u\epsilon_{s,crit} \tag{6}\]  
Values for \(\
u\) for low-to-medium confinement range between 0.5 and 1. At the corners, the jacket is stretched locally by a strain amount \(\Delta\epsilon_{j,ch}^{crit}\) to accommodate the lateral deflection w of the buckled bar.
# Proxy-Based Adaptive Transmission of MP-QUIC in Internet-of-Things Environment
## 4 Performance Analysis by Experimentations
### Analysis of Total Transmission Delays  
In this section, we analyze the performance of total transmission delays for three candidate schemes. The total transmission delay is measured by the time that has been elapsed to complete the transmission of all data files from clients to the server.  
Figure 6 shows the total transmission delays of the three candidate schemes for different link delays. The link delays for each path are changed from 10 ms to 50 ms. In the figure, we see that the proposed adaptive MP-QUIC scheme provides smaller response delays than the existing SP-QUIC and MP-QUIC schemes. This is because the proposed scheme can utilize dynamic network information for data transmission adaptively to the current network conditions.  
Figure 7 shows the total transmission delays of the three candidate schemes for different packet loss rates, ranging from 0% to 15%. We see that the proposed adaptive MP-QUIC transmission scheme provides smaller response delays than the existing SP-QUIC and MP-QUIC schemes. This is because the proposed adaptive scheme can transmit the data packets over the best path among the multiple paths by considering the network conditions, including the packet loss rate.  
Figure 5: Comparison of response delays for different sizes of data files transmitted.  
Figure 6: Total transmission delays for different link delays.  
Figure 8 shows the comparison of total transmission delays for different data file sizes, which are ranged from 128 KB to 1024 KB. From the figure, we see that the proposed scheme provides the best performance among the candidate schemes for all test instances. The gaps of transmission delays between the proposed scheme and the existing schemes get larger as the data file size increases.
## 4 Joint Optimization of Altitude and Energy Beamforming for Multiple EHDs
### Optimal Design with Efficient Algorithm
```
1:Input: Parameters of the channel statistics \((a_{1},b_{1},a_{2},b_{2},a_{3},c_{3})\), the minimum and maximum altitude of \(H_{\text{min}}\) and \(H_{\text{max}}\), the sample grid accuracy \(\Delta\), the sum power constraint \(P\), all EHDs' horizontal distance, LoS component, and energy requirements, i.e., \(\big{[}r_{u},a_{u},Q_{u}^{\text{Req}}\big{]}_{u=1}^{U}\)
2:Set:\(H_{\text{min}}\leq h\leq H_{\text{max}}\) with the sample grid \(\Delta\), the iteration number \(Iter=1\)
3:for\(h=H_{\text{min}},H_{\text{min}}+\Delta,...,H_{\text{max}}\)do
4: Compute \(\big{[}\mathbf{G}_{u}(h)\big{]}_{u=1}^{U}\) from (A5)
5: Obtain \(t^{\star}(h)\) from (A2) by solving the dual problem \(\mathbf{D}.\mathbf{P}_{2}^{\mathbf{X}}\)
6:\(Iter\gets Iter+1\)
7:endfor
8: Total number of iterations: \(Iter_{tot}\gets Iter\)
9: Obtain \(h^{\star}=\arg\min\limits_{H_{\text{min}}\leq h\leq H_{\text{max}}}t^{\star}(h)\)
10: Obtain \(X^{\star}\) and \(t^{\star}\) by solving the primal problem \(\mathbf{P}_{2}^{\mathbf{X}}\) with \([\mathbf{G}_{u}(h^{\star})]_{u=1}^{U}\)
11:Output: The optimal altitude \(h^{\star}\) and energy beamforming \(\mathbf{X}^{\star}\)
```  
**Algorithm 3** Proposed method of obtaining an optimal solution of the problem \(\mathbf{P}_{2}\)