Research on multi-objective parameter optimization of split flow dual flash thermodynamic cycle

In addition to considering the intermediate pressure variables, the split flow double flash thermodynamic cycle also needs to consider the operating conditions and split flow coefficient of the cycle. Traditional algorithms require too much computation and have a longer response time. Faced with sudden operating conditions or changes in production conditions, it is difficult to quickly re-optimize the intermediate parameters of the split flow double flash thermal cycle and carry out corresponding system control. This article uses intelligent optimization algorithms instead of global search methods to optimize multi-objective parameters in a split flow double flash thermodynamic cycle. By summarizing and analyzing the characteristics of algorithms such as SIO, EA, and ANN, it is believed that the GWO algorithm in the SIO class is the most suitable for multi-objective parameter optimization of the split flow double flash thermal cycle. It is combined with the SA optimization algorithm to propose the SA-GWO algorithm with faster optimization speed and higher optimization accuracy, taking the initial position of the wolf pack as the optimization objective. Combined with the cycle thermodynamic model, a multi-objective parameter optimization model for the split flow double flash thermal cycle is established.


Introduction
With the development of heating technology, some low-grade energy sources only used for heating and preheating can no longer meet people's application needs.With the gradual increase in requirements for heat production materials, many scholars at home and abroad have begun to study heat pump systems with high condensation temperature characteristics and call them High Temperature Heat Pump (HTHP) technology.
Although many studies have shown that heat pump systems using traditional single-stage compression cycle processes can achieve high-quality thermal energy production under specific conditions, the thermodynamic performance of the thermal cycle heating process is relatively poor.Therefore, more researchers are starting from the perspective of optimizing the thermal cycle process and structure further to improve the thermodynamic performance of high-temperature heat pump systems.
Arpagaus et al. [1] pointed out that most industrial heat pumps are based on steam compression cycles and improve the performance of single stage compression heat pump cycles by adding IHX.Hu et al. [2] designed mechanical steam recompression technology (MVR) and believed that this technology could be used for steam production, and its COP is better than traditional heat pumps under certain conditions.Zhang et al. [3] conducted experimental research on the performance of energy-saving steam injection air heat pumps in cold regions of China.The results show that compared with traditional single stage compression systems, the thermal performance of systems with steam replenishment characteristics has improved by 4% to 6%.Zhang et al. [4] [5] analyzed the reuse of low-temperature heat energy in large-span high-temperature heat pump systems using energy efficiency analysis methods.The results show that compared with traditional heat pumps, the COP of the dual flash heating system has increased by 23.8% to 44.54%.
In addition to considering the pressure variables in the middle, the split double flash thermal cycle also needs to consider the operating conditions and split coefficient parameters.If the global method is still used to optimize the middle multi-objective parameters of the split double flash thermal cycle, the computational complexity is too large, and the response time is long.In the face of sudden operating conditions or changes in production conditions, it is difficult to quickly re-optimize the middle parameters of the split double flash thermal cycle and carry out corresponding system control.To achieve this, it is necessary to combine intelligent algorithms to establish an optimization model that can accurately and quickly optimize the multi-objective parameters of the split flow double flash thermal cycle.
This chapter uses intelligent optimization algorithms instead of global search methods to optimize multi-objective parameters in a split flow dual flash thermodynamic cycle.Adaptability analysis is conducted for multi-objective optimization of split flow double flash thermal cycles based on the principles and characteristics of different types of intelligent optimization algorithms.

System thermodynamic model
Figure 1 Experimental system of double flash heat pump cycle.

Figure 2
The pressure and enthalpy characteristics of the steam system of the dual-flash compound high-temperature heat pump.

ICEEPS-2023
Journal of Physics: Conference Series 2728 (2024) 012008 Figure 1 shows the double flash heat pump system.In Figure 2, the pressure enthalpy diagram of the system is shown.Oilfield wastewater heat is used as the heat source of the system, and the thermodynamic model at the heat source end is the change of sensible heat in the heating process of hot water.If the mass flow rate of the heat source is ms, the temperature change of the inlet and outlet of the hot water in the evaporator is ΔTs.Assuming that the heat supplied by the heat source is relatively stable, the total heat provided by the heat source is represented by Qs as shown in the following formula: In the formula, cp-s is the constant pressure specific heat capacity of the heat source, kJ• (kg• K) -1 .The specific entropy change formula for converting the total heat supply into a hot water supply is shown in the following formula: In addition, the diversion coefficient x also significantly impacts the circulation, and the diversion coefficient x determines the ratio of refrigerant entering Com-2 and Cond-2.Assuming that the mass flow rate of Com-1 passing through the refrigerant is m1, and the mass flow rate of Com-2 passing through the refrigerant is m2, the relationship between m1 and m2 can be expressed as Formula (3).
2 ⋅ (1-) =  1 ⋅  (3) The above a is the proportion of the refrigerant entering the FT-1 and converting it into a gaseous refrigerant, and b is the proportion of the refrigerant entering the FT-2 and converting it into a gaseous refrigerant.According to the law of mass conservation and energy conservation in the stable state of the system, the expressions of a and b are shown in Formula (4): Assuming that there is no heat loss in Eva, combined with Formula (2), Formula (3) and Formula (4), the expression of m1 can be obtained as follows: According to the process flow of the system, the heat released by Cond-1, Sub-C, and Cond-2 in the split double-flash thermodynamic cycle is defined as Q1, Q2, and Q3, which can be expressed by Formula (6).
When the cycle is a closed loop without considering the heat exchange with the outside world, the total power consumption of the compressor can be obtained through the change of the enthalpy value of the compressor suction and discharge refrigerant, where the power consumption of Com-1 and Com-2 is respectively represented by W1 expressed with W2, the calculation formula is shown in the following formula: The COP of the final system can be expressed as Formula (8).
The thermodynamic performance of the system can be preliminarily determined by calculating the COP of the system.

Adaptation analysis of intelligent optimization models
The split flow double flash thermal cycle process is subject to problems such as multiple intermediate parameter variables and complex optimization processes of parameter coupling.This results in some thermal cycles with good heating performance, but complex structures not being favored in the industry.Through the thermodynamic model of the split flow double flash thermodynamic cycle, it can be concluded that the parameters of the closed-loop cycle need to be iteratively calculated through the model, and the control parameters within the cycle have a certain coupling effect with the component parameters [6] .If intelligent algorithms are applied to parameter optimization of split flow double flash thermal cycle, the key problem is establishing a suitable cycle parameter optimization path.
Combining the thermodynamic model characteristics of the split flow double flash thermal cycle, intelligent optimization algorithms are required to have fast convergence speed, high optimization accuracy, and the ability to embed complex models to optimize the multi-objective parameters of the model.For the optimization path problem of multi-objective parameter variables in thermal cycles, it is necessary to clarify the optimization purpose of the thermal cycle.If considering the heating performance of the system itself, the optimization goal can be set to the COP of the cycle.
Figure 3 shows that when the optimization target is selected, a parameter matrix of the variables that need to be optimized for the system is established.Through intelligent optimization algorithms, the optimal target parameters under the requirements of the operating range are obtained under boundary conditions.Then, multiple corresponding system parameters are obtained through numerical feedback.Intelligent optimization algorithms determine the optimization accuracy of system optimization objectives, and selecting appropriate intelligent optimization algorithms plays a crucial role in the multi-objective optimization process of complex systems.
The basic idea of intelligent optimization algorithms is to find an optimal solution in a decision space to achieve the optimal effect of the target in the decision space.Suitable intelligent algorithms for solving multi-objective optimization problems in engineering include Swarm Intelligence Optimization (SIO), Simulated Annealing (SA), Artificial Neural Network (ANN), and Evolutionary Algorithms (EA) [7][9] .Among them, SIO, EA, and ANN algorithms are algorithm clusters.Common SIO algorithms include Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO), while common EA algorithms include Differential Evolution (DE) and Genetic Algorithm (GA) [10][11] .The schematic diagrams of several common optimization algorithms are shown in Figure 4. Figure 4(d) shows that the fitness value represents a small difference the final loop optimal COP, and the optimization results of several optimization algorithms are not significantly different.However, the final optimization difference shows that the GWO algorithm among SIO algorithms is more accurate in optimizing the loop results.From the perspective of optimization speed, the fitness values of the SA algorithm and GWO algorithm reached stability after the 11th iteration.In contrast, the fitness values of the GA algorithm only reached stability after the 60th iteration.Therefore, it can be concluded that SIO algorithms are more suitable for nested multi-objective parameter optimization processes.

A multi-objective parameter optimization model based on the SA-GWO algorithm
The GWO algorithm is a swarm intelligence algorithm proposed by Mirjalili et al. [12] in Australia in 2014, which simulates the social dominance hierarchy of wolf packs and uses this relationship for collaborative predation processes.According to the social relationships of wolf packs, they are divided into α, β, δ, and ω four levels.α wolf represents the head of a pack of wolves, located at the forefront of the hunting process.β wolf blame is defined as the predatory assistant of a wolf, second only to its prey in distance α wolf.δ wolves are the backbone of a pack of wolves, following closely during the siege process α wolf and β wolves surround their prey.ω ranked as the lowest level wolf pack in the pack α under the guidance of the wolf belt, change its position and replace it at any time α.The wolves further rounded up their prey.
In this social relationship, the approximation process of wolf packs is divided into three steps: searching for targets, rounding up and approaching, and launching an attack.In the beginning, when there is no prey target, the wolf pack disperses to search for food to increase the probability of finding the target [13] .When the target is found, the α, β, and δ surround the prey, and the rest ω wolf waits for an opportunity to listen α.The wolf commands to approach the prey and reduce the range of encirclement gradually.The entire process, from discovering the target to capturing prey, is shown in Figures 5 and 6.
When introducing the Grey Wolf algorithm into the split flow double flash thermal cycle, the intermediate variables in the system are converted into the position information of the Grey Wolf, where xi, j, and Z represent the i-th position information of the Z-th wolf in the jth hunting process.Combined with the thermodynamic model of the system, the intermediate variables include intermediate pressure pmid, FT-1 pressure pFT-1, FT-2 pressure pFT-2, and flow coefficient x.Before optimization, we initialize the parameters and after calibrating the range, randomly assign the variable parameters, as shown in Formula ( 9).x By initializing the parameter wolf pack to search for the optimal COP of the system, it is substituted into the thermodynamic model of the split flow double flash thermodynamic cycle, and this process is used as a wolf pack search method.The split flow heat transfer condition is added, and the search results that do not meet the conditions are set to zero, limiting the search direction of the wolf pack.The distance between the head wolf and its prey can be expressed as L. Since the Grey Wolf algorithm requires a target value, L is the distance between the system COP and the target COP calculated by the wolf pack through initialization parameters.The expression is shown in Formula (10).After determining the hunting method and limiting conditions of the wolf pack, the wolf pack begins to approach the prey.It iteratively replaces the distance L between the head wolf and the target.The prey is attacked when the distance between the head wolf and the target is less than or equal to the specified attack range.Mathematical models can represent the process of wolf packs surrounding prey.

( ) ( )
The formula gradually decreases from 2 to 0; r1 and r2 are random vectors.XM(j) represents the position of the prey after j iterations; Xk-wolf(j) represents the position of the k-th wolf after j iterations; L represents the distance between the wolf and its prey and its center point; A. C is the synergy coefficient vector, represented by Formulas ( 13) and ( 14).During the approach process, the wolf pack will pass through α, β, δ command of the wolf, designated ω.The wolf changes position to change the relationship between the wolf pack and further approach the target.We specify ω.The position of the wolf can be expressed as Formula (15).
( ) where Xk-α, Xk-β and Xk-δ represent the position of the j+1 iteration of wolf k relative to wolf α, β and δ, which is consistent with the distance expression of prey, as shown in Formula (16).
( ) After the designated ω wolf is in place, the α, β, δ wolf pack will change the position of the wolf in the wolf pack to form a new round of hunting process until the attack requirements are met, and the optimal value of the system is obtained.
SA algorithm has a simple process and strong robustness.It can control the optimization rate by adjusting the temperature drop rate.It is very suitable for the optimal selection of GWO initial parameters.Its main parameters include initial temperature T0, cooling gradient parameter K, cooling process length L, etc. Through the simulated annealing process, the t0 is continuously reduced to find the optimal solution.On the one hand, combined with the principle of the annealing process, it is considered that the smaller the value of the new solution is, the more stable the target structure is.On the other hand, the local optimization problem is broken through the receiving probability.Therefore, the m criterion (Metropolis criterion) [14] is adopted to accept the new solution of the target.The expression of the M criterion is shown in Formula (17).
In essence, the optimization process of the SA algorithm combined with the GWO algorithm is that the SA algorithm has a faster initial operation speed, so it can quickly select the direction of multiobjective optimization of the GWO algorithm and record it as the SA-GWO optimization algorithm.

Results
Multiple parameters evaluated the thermodynamic performance of the cycle.Combined with the sagwo algorithm and the thermodynamic model of the cycle, the condensation temperature range of the system is set at 110-140℃, and the parameters of the cycle are calculated on the premise of staged heat transfer conditions.Each variable in the optimization algorithm gives the boundary and sets the parameters of the SA-GWO algorithm: the number of wolves nG, the number of iterations NG, the step factor dT, the temperature drop attenuation parameter k, the Markov chain length L, etc., as shown in Table 1.After obtaining the optimal cop of the cycle, the multi-objective parameter optimization process is completed by recording the intermediate parameters of the cycle under the feedback condition.Taking the condensation temperature of 120℃ as an example, in the iterative optimization process of cyclic fitness, the variation of the cyclic intermediate variable with the fitness value is shown in Figure 7.
It can be seen from Figure 8 that the pressure parameter in the cycle fluctuates significantly within 100 iterations, and the subsequent fluctuations are small.However, after 54 steps of iteration, the change amplitude of fitness value in the process of cyclic iteration is relatively small, and the increment amplitude after 10 4 steps is less than 10 -6 , which can be regarded as a cyclic steady state.However, due to the narrow variation range of cycle subcooling and split coefficient, the number of steady-state iterations is almost the same as the fitness value of the cycle.In conclusion, these two variables have a significant impact on the change of fitness value in the optimization process, and the cop of the cycle can reach a steady state after the intermediate pressure variable reaches a specific working condition.

Conclusion
By summarizing and analyzing the characteristics of algorithms such as SIO, EA, and ANN, it is believed that the GWO algorithm in SIO is the most suitable for multi-objective parameter optimization of split flow double flash thermal cycles.
This study combines it with the SA optimization algorithm and proposes the SA-GWO algorithm with faster optimization speed and higher accuracy.Finally, combined with the thermodynamic model of the cycle, a multi-objective parameter optimization model for the split flow double flash thermal cycle is established.
From the optimization results, the cyclic COP optimized by the SA-GWO algorithm has a relatively higher fitness value than the traditional GWO and SA algorithms.Moreover, the optimization time of the SA-GWO algorithm is reduced by nearly 3 hours compared to the global method.From this, it can be concluded that the SA-GWO algorithm is suitable for the multi-objective parameter optimization process of the split flow double flash thermodynamic cycle.

Figure 3 Figure 4
Figure 3 Optimization process of thermal cycle multi-objective parameters.

Figure 5 7 Figure 6
Figure 5The hunting principle of the Grey Wolf algorithm.
Figure 7 Comparison of optimization results under different algorithms.Figure 8 The variation trend of the intermediate variable with a fitness value.