# Research on the Combination Optimization Method of Aircraft Climb Parameters Considering the Influence of Pollution Emissions

Authors: Wang Na; Wei Zhiqiang; Han Xiaolan
Journal: Journal of Advanced Transportation
 (2022)

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## Abstract

In order to find the optimum climb parameters for decreasing aircraft pollution emissions, we establish a calculation model of aircraft pollutant emissions to thoroughly examine the economy of aircraft operation and its impact on the environment, based on the International Civil Aviation Organization reference emission data and Boeing Method 2 which can calculate pollutant emissions in different flight phases by correcting the ICAO emission data. Firstly, we propose the concept of integrated flight cost considering the effect of emissions and establish its calculation model. Secondly, we establish the climb speed optimization models based on the traditional flight cost and integrated flight cost. Then, we analyze the effect of climb performance on the integrated flight cost. Next, we establish a combined optimization model of indicated airspeed and thrust using the genetic algorithm. The model simulation and result analysis are carried out using a wide-body aircraft. Finally, we analyze the effect of the optimization parameters and the sensitivity of the pollution index. The results show that the decrease in the flight speed and thrust effectively reduces the cost of pollutant emission during flight. The combination of (Vc, Tr) with the smallest integrated flight cost reduces the integrated flight cost by 1.28% and emission cost by 4.56%.


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## Body

## 1. Introduction

The emissions of aircraft in flight have a significant impact on the environmental degradation and have an increasing concern for the civil aviation departments. To estimate pollutant emissions, Sherry [1] proposed a method to improve the accuracy of airport emission inventory using disparate data sets at thrust reduction setting, based on the International Civil Aviation Organization’s (ICAO) emission model. Wasiuk et al. [2] developed an aircraft performance tool to estimate global and regional commercial aviation fuel burn and emissions. Turgut et al. [3] evaluated NOX emissions in vertical and horizontal directions during takeoff and climbing. Mitchell et al. [4] assessed the environmental effects of flight under the condition of speed limits during departure. Wei et al. [5] estimated pollutant emissions for civil aircraft during cruising, showing that CO2 and NOX are the main pollutants during flight. Han et al. [6] studied the emission characteristics of fine particles during takeoff. The above studies involved only the calculation of the pollutant emissions in flight, without taking into account the differences in the environmental impacts of different pollutants, and the trade-off analysis which considers pollutants and flight efficiency thoroughly.Optimization of performance parameters effectively reduces the effects of flight on the environment during flight. Chen et al. [7] kept the balance of the relationship between the environmental impact and operating costs to decrease the impact of flight on the environment. Koudis et al. [8] used thrust reduction methods as a means to decrease airport pollutant emissions during takeoff. Schumann et al. [9] established a track optimization method for wake clouds and CO2 to reduce the impact of flight on the environment by changing the altitude of the flight. Wei et al. [10] introduced the aircraft emission cost into the flight cost model and optimized the parameters during the cruise.The climb is an important part of performance optimization. Antonakis et al. [11] conducted experiments for optimizing the aircraft climb path using particle swarm optimization, by which the time and fuel consumption will be reduced by 15% and 20%, respectively. Dalmau and Prats [12] evaluated the advantages of continuous climb operation during the cruise phase of flights, which saves fuel and trip time ranging from 0.5% to 2% and 1% to 5%, respectively. Lu and Nan [13] used an adaptive genetic algorithm to optimize the vertical trajectory of the aircraft climb phase taking time and fuel consumption into account. The above-mentioned studies considered only flight time and fuel consumption as optimization targets but not included the effect of flight on the environment. Turgut and Oznur [14] quantified the effect of climb angle on fuel consumption and nitrogen oxide emission which were increased, respectively, by 9–19 and 0.3–0.7 kg per degree of climb angle for the departure climb phase. However, climb parameters, such as speed and thrust, will affect the economy of flight and pollutant emissions as well.Boeing Method 2 (BM2), also known as “Boeing Fuel Flow Method 2” (BFFM2), was proposed by the Boeing Company to predict the emission indexes in a subsonic cruise condition, without the need to get access to any engine proprietary data. It can be used to calculate emissions indices based on fuel flow and ICAO certification data [15]. On the basis of the ICAO’s reference emission data and Boeing method 2 (BM2), the calculation model of pollutant emission cost was established first, and then, the integrated flight cost calculation model that includes emission cost was established. Second, the climb performance optimization method of typical aircraft was established, and the effects of climb speed and thrust on flight cost were analyzed. Third, the optimization model of the combination of indicated airspeed and thrust was established using the genetic algorithm which aimed at minimizing the integrated flight cost. The model simulation and result analysis were carried out based on a wide-body aircraft. Finally, according to the concept of pollutant emission price index, the effect of the integrated flight cost was analyzed.


## 2. Calculation of the Integrated Flight Cost

### 2.1. Calculation of Flight Cost

The formula for flight cost includes time cost and fuel cost, which is written as follows [16]:(1)C=CT+CF,where CT is the total time cost (in ￥) and CF is the total fuel cost (in ￥).The cost index CI is defined as(2)CI=CtCf,where Ct is the time cost (in ￥/min) and Cf is the fuel cost per kg.Substitution of Equation (2) in Equation (1) gives(3)C=Cf⋅CI⋅t+F,where t is the flight time and F is the fuel consumption (kg).The aircraft performance parameters and flight cost calculation process during the climb are shown in the following figure. According to the input parameters, including engine parameters, airspeed, engine thrust percentage, aircraft mass, temperature deviation, and altitude integration step, the performance parameters and flight cost in the phase of the climb can be calculated. The flight cost calculation process during the climb is shown in Figure1.Figure 1
Flow chart of flight cost calculation in the climb.The traditional flight cost calculation model includes only two factors—time and fuel consumption—and it does not involve the cost of environmental impact induced by pollutant emissions in flight.


### 2.2. Calculation of Nitrogen Oxide Emissions

The emission of engine pollutants depends on the type of engine, fuel consumption, and emission index. Table1 presents the basic emission indices at four typical flight stages of a PW4077–2PW061 engine, according to the reference emission model of ICAO.Table 1
Basic emission indices of a PW4077–2PW061 engine.
ModePower setting (%F)Time (min)Fuel flow (kg/h)CO2 index (kg/kg) [17]NOX index (kg/kg)Takeoff1000.710868.43.1550.0398Climb out852.28827.23.1550.0325Approach304.02937.63.1550.0113Idle726.0835.23.1550.0042We can infer from Table1 that the CO2 emission index is independent of the flight stage. For NOX, we use the BM2 method to modify the aircraft emission index using correction and interpolation based on the data in the table [18]. The three steps to correct pollution emission indices are described as follows.


#### 2.2.1. Calculation of Reference Fuel Flow

The data in Table1 were obtained from the engine test under the conditions of International Standard Atmosphere (ISA), sea level, and different work speeds. Therefore, for a specific flight environment, the actual fuel flow of the aircraft should be converted into the reference fuel flow under the above conditions. The equation is written as(4)Wff=Wfδθ3.8e0.2M2,where Wff is the corrected fuel flow in ISA at sea level (kg/h), Wf is the actual fuel flow (kg/h), δ is the ratio of static atmospheric pressure to atmospheric pressure at sea level, θ is the ratio of static temperature to air temperature at sea level, and M is the aircraft flight Mach number.


#### 2.2.2. Calculation of the Reference Emission Index

From Table1, we understand that atmospheric and flight conditions affect the fuel flow, and hence, the reference emission index is calculated by interpolating the reference fuel flow obtained from Equation (4).


#### 2.2.3. Calculation of Actual Emission Index

The value of the NOX emission index is also affected by flight altitude, aircraft speed, atmospheric temperature, and humidity. For the specific flight environment, the reference emission index should be converted into the actual emission index according to the following equation [15]:(5)EI,N=ERIδ0.51θ1.65exp19.00.0063−0.622ΦPvP−ΦPv,where EI,N is the actual emission index of NOX (kg/kg), Φ is the atmospheric relative humidity, P is the local static pressure at a given altitude (Pa), and Pv is saturated vapor pressure (Pa). Then, the emissions of NOX can be calculated from the following equation:(6)SNOX=EI,N⋅F,where SNOX is the emissions of NOX (kg).The total emission costCp is defined, on the basis of traditional flight cost, as(7)Cp=CCO2⋅SCO2+CNOX⋅SNOX,where Cp is the total emission cost (￥/kg), CNOX is the emission cost of NOX per unit mass (￥/kg), and CCO2 is the emission cost of CO2 per kilogram (￥/kg).


### 2.3. Calculation of the Integrated Flight Cost Considering the Influence of Emissions

To measure the impact of aircraft emissions on the external environment, the emission factors were included in the flight cost to obtain the integrated flight cost.(8)C′=CT+CF+Cp.The pollutant emission price indices, including CO2 price index PICO2 and NOX price index PINOX, which indicate the ratio of CO2 price and NOX price to fuel price, respectively, are defined as(9)PICO2=CCO2Cf,(10)PINOX=CNOXCf.Substitution of the above equations in Equation (9) gives(11)C′=CfCI⋅t+F+PICO2⋅SCO2+PINOX⋅SNOX.Based on the Intergovernmental Panel on Climate Change’s estimation of emission index and its impact, EUROCONTROL’s cost-benefit analyses provided an international overview of aircraft emission shadow prices [19], as shown in Table 2.Table 2
Overview of emission costs of different pollutants (￥).
PollutantsMediumLowHighCO2 (per ton)274.091.4456.3NOX (per kg)33.311.655.0


## 2.1. Calculation of Flight Cost

The formula for flight cost includes time cost and fuel cost, which is written as follows [16]:(1)C=CT+CF,where CT is the total time cost (in ￥) and CF is the total fuel cost (in ￥).The cost index CI is defined as(2)CI=CtCf,where Ct is the time cost (in ￥/min) and Cf is the fuel cost per kg.Substitution of Equation (2) in Equation (1) gives(3)C=Cf⋅CI⋅t+F,where t is the flight time and F is the fuel consumption (kg).The aircraft performance parameters and flight cost calculation process during the climb are shown in the following figure. According to the input parameters, including engine parameters, airspeed, engine thrust percentage, aircraft mass, temperature deviation, and altitude integration step, the performance parameters and flight cost in the phase of the climb can be calculated. The flight cost calculation process during the climb is shown in Figure1.Figure 1
Flow chart of flight cost calculation in the climb.The traditional flight cost calculation model includes only two factors—time and fuel consumption—and it does not involve the cost of environmental impact induced by pollutant emissions in flight.


## 2.2. Calculation of Nitrogen Oxide Emissions

The emission of engine pollutants depends on the type of engine, fuel consumption, and emission index. Table1 presents the basic emission indices at four typical flight stages of a PW4077–2PW061 engine, according to the reference emission model of ICAO.Table 1
Basic emission indices of a PW4077–2PW061 engine.
ModePower setting (%F)Time (min)Fuel flow (kg/h)CO2 index (kg/kg) [17]NOX index (kg/kg)Takeoff1000.710868.43.1550.0398Climb out852.28827.23.1550.0325Approach304.02937.63.1550.0113Idle726.0835.23.1550.0042We can infer from Table1 that the CO2 emission index is independent of the flight stage. For NOX, we use the BM2 method to modify the aircraft emission index using correction and interpolation based on the data in the table [18]. The three steps to correct pollution emission indices are described as follows.


### 2.2.1. Calculation of Reference Fuel Flow

The data in Table1 were obtained from the engine test under the conditions of International Standard Atmosphere (ISA), sea level, and different work speeds. Therefore, for a specific flight environment, the actual fuel flow of the aircraft should be converted into the reference fuel flow under the above conditions. The equation is written as(4)Wff=Wfδθ3.8e0.2M2,where Wff is the corrected fuel flow in ISA at sea level (kg/h), Wf is the actual fuel flow (kg/h), δ is the ratio of static atmospheric pressure to atmospheric pressure at sea level, θ is the ratio of static temperature to air temperature at sea level, and M is the aircraft flight Mach number.


### 2.2.2. Calculation of the Reference Emission Index

From Table1, we understand that atmospheric and flight conditions affect the fuel flow, and hence, the reference emission index is calculated by interpolating the reference fuel flow obtained from Equation (4).


### 2.2.3. Calculation of Actual Emission Index

The value of the NOX emission index is also affected by flight altitude, aircraft speed, atmospheric temperature, and humidity. For the specific flight environment, the reference emission index should be converted into the actual emission index according to the following equation [15]:(5)EI,N=ERIδ0.51θ1.65exp19.00.0063−0.622ΦPvP−ΦPv,where EI,N is the actual emission index of NOX (kg/kg), Φ is the atmospheric relative humidity, P is the local static pressure at a given altitude (Pa), and Pv is saturated vapor pressure (Pa). Then, the emissions of NOX can be calculated from the following equation:(6)SNOX=EI,N⋅F,where SNOX is the emissions of NOX (kg).The total emission costCp is defined, on the basis of traditional flight cost, as(7)Cp=CCO2⋅SCO2+CNOX⋅SNOX,where Cp is the total emission cost (￥/kg), CNOX is the emission cost of NOX per unit mass (￥/kg), and CCO2 is the emission cost of CO2 per kilogram (￥/kg).


## 2.2.1. Calculation of Reference Fuel Flow

The data in Table1 were obtained from the engine test under the conditions of International Standard Atmosphere (ISA), sea level, and different work speeds. Therefore, for a specific flight environment, the actual fuel flow of the aircraft should be converted into the reference fuel flow under the above conditions. The equation is written as(4)Wff=Wfδθ3.8e0.2M2,where Wff is the corrected fuel flow in ISA at sea level (kg/h), Wf is the actual fuel flow (kg/h), δ is the ratio of static atmospheric pressure to atmospheric pressure at sea level, θ is the ratio of static temperature to air temperature at sea level, and M is the aircraft flight Mach number.


## 2.2.2. Calculation of the Reference Emission Index

From Table1, we understand that atmospheric and flight conditions affect the fuel flow, and hence, the reference emission index is calculated by interpolating the reference fuel flow obtained from Equation (4).


## 2.2.3. Calculation of Actual Emission Index

The value of the NOX emission index is also affected by flight altitude, aircraft speed, atmospheric temperature, and humidity. For the specific flight environment, the reference emission index should be converted into the actual emission index according to the following equation [15]:(5)EI,N=ERIδ0.51θ1.65exp19.00.0063−0.622ΦPvP−ΦPv,where EI,N is the actual emission index of NOX (kg/kg), Φ is the atmospheric relative humidity, P is the local static pressure at a given altitude (Pa), and Pv is saturated vapor pressure (Pa). Then, the emissions of NOX can be calculated from the following equation:(6)SNOX=EI,N⋅F,where SNOX is the emissions of NOX (kg).The total emission costCp is defined, on the basis of traditional flight cost, as(7)Cp=CCO2⋅SCO2+CNOX⋅SNOX,where Cp is the total emission cost (￥/kg), CNOX is the emission cost of NOX per unit mass (￥/kg), and CCO2 is the emission cost of CO2 per kilogram (￥/kg).


## 2.3. Calculation of the Integrated Flight Cost Considering the Influence of Emissions

To measure the impact of aircraft emissions on the external environment, the emission factors were included in the flight cost to obtain the integrated flight cost.(8)C′=CT+CF+Cp.The pollutant emission price indices, including CO2 price index PICO2 and NOX price index PINOX, which indicate the ratio of CO2 price and NOX price to fuel price, respectively, are defined as(9)PICO2=CCO2Cf,(10)PINOX=CNOXCf.Substitution of the above equations in Equation (9) gives(11)C′=CfCI⋅t+F+PICO2⋅SCO2+PINOX⋅SNOX.Based on the Intergovernmental Panel on Climate Change’s estimation of emission index and its impact, EUROCONTROL’s cost-benefit analyses provided an international overview of aircraft emission shadow prices [19], as shown in Table 2.Table 2
Overview of emission costs of different pollutants (￥).
PollutantsMediumLowHighCO2 (per ton)274.091.4456.3NOX (per kg)33.311.655.0


## 3. Effect of Climb Performance on the Integrated Flight Cost

### 3.1. Effect of Climb Speed

Figure2 shows the results of flight costs, emission costs, and integrated flight costs at different airspeeds, taking a certain aircraft as an example. The cost index was set to 30, for the same aircraft and engine type as a reference. Other calculation conditions are as follows: the climb altitude ranges from 10,000 to 25,000 ft, the fuel price is 6000 ￥/t, the initial climb weight is 200,000 kg, the ambient temperature is 0°C, PICO2=0.094, PINOX=16.46, and the climbing strategy is constant. Under such conditions, the airspeed climbs at a speed range from 200 to 350 kt.Figure 2
Variation of cost withVc.From Figure2, the optimum speeds at different cost indicators are as follows: the optimum speeds of the flight cost Vf are higher than the optimum speeds of the integrated flight cost Vt which is higher than the optimum speeds of emission cost Vp. When the speed is lower, the corresponding fuel flow and the emission index of NOX are smaller. Consequently, the emissions and emission cost of NOX are also lesser. Climbing at a higher speed also indicates that airspeed takes less time, so the flight cost becomes lesser.


### 3.2. Effect of Climb Thrust

Table1 demonstrates that thrust has a significant effect on the emission index of NOX, and hence, it should be taken into account when analyzing the emission cost of pollutants. The flight cost and pollutant emission cost under different thrust conditions were calculated and analyzed, and the results are shown in Figure 3.Figure 3
Variation of cost withTr.The specific conditions used to calculate the flight cost are as follows: the climb speed is 280 kt; the value of the ratio of actual thrust to rated maximum climb thrust (Tr) ranges from 75 to 100%; the climb altitude is 10,000–25,000 ft.Figure3 indicates that the flight cost and integrated flight cost decrease with increasing Tr, whereas the emission cost changes in the opposite way. The fuel flow and the emissions of NOX are in accordance with thrust. Hence, we conclude that when the thrust is lower, the emission cost is lesser. Furthermore, at this moment, the longer the flight time is, the higher the flight cost becomes.


## 3.1. Effect of Climb Speed

Figure2 shows the results of flight costs, emission costs, and integrated flight costs at different airspeeds, taking a certain aircraft as an example. The cost index was set to 30, for the same aircraft and engine type as a reference. Other calculation conditions are as follows: the climb altitude ranges from 10,000 to 25,000 ft, the fuel price is 6000 ￥/t, the initial climb weight is 200,000 kg, the ambient temperature is 0°C, PICO2=0.094, PINOX=16.46, and the climbing strategy is constant. Under such conditions, the airspeed climbs at a speed range from 200 to 350 kt.Figure 2
Variation of cost withVc.From Figure2, the optimum speeds at different cost indicators are as follows: the optimum speeds of the flight cost Vf are higher than the optimum speeds of the integrated flight cost Vt which is higher than the optimum speeds of emission cost Vp. When the speed is lower, the corresponding fuel flow and the emission index of NOX are smaller. Consequently, the emissions and emission cost of NOX are also lesser. Climbing at a higher speed also indicates that airspeed takes less time, so the flight cost becomes lesser.


## 3.2. Effect of Climb Thrust

Table1 demonstrates that thrust has a significant effect on the emission index of NOX, and hence, it should be taken into account when analyzing the emission cost of pollutants. The flight cost and pollutant emission cost under different thrust conditions were calculated and analyzed, and the results are shown in Figure 3.Figure 3
Variation of cost withTr.The specific conditions used to calculate the flight cost are as follows: the climb speed is 280 kt; the value of the ratio of actual thrust to rated maximum climb thrust (Tr) ranges from 75 to 100%; the climb altitude is 10,000–25,000 ft.Figure3 indicates that the flight cost and integrated flight cost decrease with increasing Tr, whereas the emission cost changes in the opposite way. The fuel flow and the emissions of NOX are in accordance with thrust. Hence, we conclude that when the thrust is lower, the emission cost is lesser. Furthermore, at this moment, the longer the flight time is, the higher the flight cost becomes.


## 4. Combination Optimization of Airspeed and Thrust

### 4.1. Combination Optimization Model

From the above analysis in Section2, considering the effect of climb speed and thrust on flight cost, we optimize the performance parameters using the genetic algorithm, obtaining the combination of climb speed and thrust (Vc, Tr) with the minimum integrated flight cost.The actual flight needs to fulfill certain requirements for speed and thrust. The conditions for control variables were set according to the basic performance parameters and actual operation requirements of the aircraft.(12)200≤Vc≤350,0.75≤Tr≤1.The control variables (Vc, Tr) were encoded by the float-point coding method. The length of each control variable was 1.The fitness function was derived from the following equation:(13)f=CfCI⋅t+F+PICO2⋅SCO2+PINOX⋅SNOX.The roulette method was used to select parents for individuals that meet the above constraints, and the probability of selection was 0.5. The probability of parents to cross and mutate was 0.6 and 0.05, respectively.


### 4.2. Realization Flowchart Based on the Genetic Algorithm

The integrated cost under the given temperature deviation, aircraft quality, fuel price, cost index, and emission index was calculated. To obtain the minimum integrated cost, the genetic algorithm was used to optimize the combination of (Vc, Tr). Figure 4 shows the specific optimization process.Figure 4
Flowchart of optimization calculation of comprehensive cost based on the genetic algorithm.First, the given calculation conditions were introduced. The control variables (Vc, Tr) were then coded, and the population was initialized. Second, the fitness function was calculated by calling the integrated cost calculation model. The pollutant emissions can be obtained based on the calculation formulas and models used in this study. The flight cost, emission cost, and integrated flight cost were obtained, as well as thus the fitness value. Third, after selection, crossover, and mutation, the population was updated. Furthermore, the current algebra was evaluated. Finally, by repeating the second and third steps until reaching the setting algebra, the combination of minimum fitness value (Vc, Tr), which was also the best optimization result, can be obtained.


### 4.3. Analysis of Optimization Results

The optimization method was simulated for a wide-body aircraft as an example. The calculation conditions are as follows: the temperature deviation is ISA + 0°C; the initial weight of the aircraft is 200,000 kg; the fuel price is 6000 ￥/t; the cost index is 30; PICO2=0.094; PINOX=16.46; the climb altitude is 10,000–25,000 ft; Vc ranges from 200 to 350; Tr ranges from 75% to 100%. Figure 5 shows the variation of the fitness function value with the algebra, using the genetic algorithm optimization. It illustrates that the optimal fitness is 10,850 ￥, and the function values begin to converge around the 150th generation. The final optimized (Vc, Tr) combination is (278, 86).Figure 5
Variation of the fitness function with the algorithm.The combinations of the lowest integrated flight cost (Vc, Tr) and the lowest flight cost (Vc, Tr) are (278, 86) and (300, 100), respectively. Table 3 compares the climbing parameters in the above two methods.Table 3
Comparison of climbing parameters.
Value of combination parametersIntegrated flight cost (￥)Flight cost (￥)Emission cost (￥)Fuel (kg)(278, 86)1085061244726903(300, 100)1099160394952879Deviation ratio (%)−1.281.40−4.562.73Table3 shows that the combination of the lowest integrated flight cost (Vc, Tr) reduces the emission cost by 4.56% and the integrated flight cost by 1.28%. Although the fuel increases, the emission cost decreases more, and thus, the total cost or the integrated flight cost decreases.


### 4.4. Influencing Factors of Optimization Parameters

When the price of pollutants is constant, the initial climb weight, temperature deviation, and cost index are the main influencing factors of the optimization parameters.The typical calculation conditions are as follows: the ambient temperature is ISA + 0°C; the climb weight is 200,000 kg; the fuel price is 6000 ￥/t; the cost index is 30; PICO2=0.094; PINOX=16.46, the climb height is 10,000–25,000 ft; Vc ranges from 200 to 350 kt; Tr ranges from 75% to 100%.Figure6 shows the effect of the initial climb weight on the optimization parameters, i.e., Vc and Tr. The climb weight ranges from 200,000 to 230,000 kg. Other calculations are the same as typical calculation conditions.Figure 6
Effect of flight weight on optimization parameters.Figure6 illustrates the optimum speed and the optimum thrust as a function of flight weight. It is clearly shown that an increase in aircraft weight increases the optimum speed and the optimum thrust for minimal integrated flight cost and flight cost. This is because when the speed and thrust are constant, the flight cost and emission cost increase with the weight. By increasing the speed and thrust, the flight cost can be reduced.Figure7 shows the effect of external temperature deviation on the optimization parameters. The temperature deviation ranges from −10°C to 30°C, and other calculation conditions are the same as typical calculation conditions.Figure 7
Effect of temperature on optimization parameters.As shown in Figure7, the optimum speed and thrust decrease with the increase of temperature only accounting for flight cost. While when considering the integrated flight cost, the optimum speed and thrust decrease proportionally at lower temperatures, with the temperature ranging from −10°C to 20°C, but begin to increase at higher temperatures. The emission of NOX increases with increasing temperature. Lower airspeed and thrust lead to lower emission cost and integrated flight cost. As the temperature reaches 30°C, under the same conditions, the fuel cost increases sharply. Therefore, high airspeed and thrust are very important to reduce flight cost.Figure8 shows the effect of cost index CI on optimization parameters. CI ranges from 0 to 100, and other calculation conditions are the same as typical calculation conditions.Figure 8
Effect of CI on optimization parameters.As shown in Figure8, for minimal integrated flight cost and flight cost, the optimal thrust and airspeed increase with increasing CI. Since the proportion of time cost to the integrated flight cost increases with increasing CI, increasing the airspeed and thrust to reduce the integrated flight cost is very important.


### 4.5. Impact Analysis of Pollutant Emission Price Index

With the rapid growth of civil aviation transportation, the impact of aircraft emissions on the environment has drawn increasing attention from all walks of life, and the emission price of pollutants has gradually increased. Therefore, it is imperative to analyze the effect of the pollutant emission price index on the optimization parameters. Since the emission index of CO2 is constant and the influencing factors are the same as fuel consumption, it is not analyzed in detail in this work.With other calculation conditions unchanged, the combination ofVc and Tr related to the minimum integrated flight cost under different NOX emission prices was calculated, and the results are shown in Figure 9.Figure 9
Effect ofPINOX on optimization parameters.It can be seen from Figure9 that, for minimal integrated flight cost, the optimal Vc and Tr decrease with increasing PINOX and finally decrease slowly. But the flight cost is not affected by PINOX. With the increase in PINOX, the NOX cost per kilogram increases, and hence, the emission cost also increases accordingly. By reducing the thrust and speed, the fuel flow and NOX emission can be reduced, which in turn reduces the emission cost.


## 4.1. Combination Optimization Model

From the above analysis in Section2, considering the effect of climb speed and thrust on flight cost, we optimize the performance parameters using the genetic algorithm, obtaining the combination of climb speed and thrust (Vc, Tr) with the minimum integrated flight cost.The actual flight needs to fulfill certain requirements for speed and thrust. The conditions for control variables were set according to the basic performance parameters and actual operation requirements of the aircraft.(12)200≤Vc≤350,0.75≤Tr≤1.The control variables (Vc, Tr) were encoded by the float-point coding method. The length of each control variable was 1.The fitness function was derived from the following equation:(13)f=CfCI⋅t+F+PICO2⋅SCO2+PINOX⋅SNOX.The roulette method was used to select parents for individuals that meet the above constraints, and the probability of selection was 0.5. The probability of parents to cross and mutate was 0.6 and 0.05, respectively.


## 4.2. Realization Flowchart Based on the Genetic Algorithm

The integrated cost under the given temperature deviation, aircraft quality, fuel price, cost index, and emission index was calculated. To obtain the minimum integrated cost, the genetic algorithm was used to optimize the combination of (Vc, Tr). Figure 4 shows the specific optimization process.Figure 4
Flowchart of optimization calculation of comprehensive cost based on the genetic algorithm.First, the given calculation conditions were introduced. The control variables (Vc, Tr) were then coded, and the population was initialized. Second, the fitness function was calculated by calling the integrated cost calculation model. The pollutant emissions can be obtained based on the calculation formulas and models used in this study. The flight cost, emission cost, and integrated flight cost were obtained, as well as thus the fitness value. Third, after selection, crossover, and mutation, the population was updated. Furthermore, the current algebra was evaluated. Finally, by repeating the second and third steps until reaching the setting algebra, the combination of minimum fitness value (Vc, Tr), which was also the best optimization result, can be obtained.


## 4.3. Analysis of Optimization Results

The optimization method was simulated for a wide-body aircraft as an example. The calculation conditions are as follows: the temperature deviation is ISA + 0°C; the initial weight of the aircraft is 200,000 kg; the fuel price is 6000 ￥/t; the cost index is 30; PICO2=0.094; PINOX=16.46; the climb altitude is 10,000–25,000 ft; Vc ranges from 200 to 350; Tr ranges from 75% to 100%. Figure 5 shows the variation of the fitness function value with the algebra, using the genetic algorithm optimization. It illustrates that the optimal fitness is 10,850 ￥, and the function values begin to converge around the 150th generation. The final optimized (Vc, Tr) combination is (278, 86).Figure 5
Variation of the fitness function with the algorithm.The combinations of the lowest integrated flight cost (Vc, Tr) and the lowest flight cost (Vc, Tr) are (278, 86) and (300, 100), respectively. Table 3 compares the climbing parameters in the above two methods.Table 3
Comparison of climbing parameters.
Value of combination parametersIntegrated flight cost (￥)Flight cost (￥)Emission cost (￥)Fuel (kg)(278, 86)1085061244726903(300, 100)1099160394952879Deviation ratio (%)−1.281.40−4.562.73Table3 shows that the combination of the lowest integrated flight cost (Vc, Tr) reduces the emission cost by 4.56% and the integrated flight cost by 1.28%. Although the fuel increases, the emission cost decreases more, and thus, the total cost or the integrated flight cost decreases.


## 4.4. Influencing Factors of Optimization Parameters

When the price of pollutants is constant, the initial climb weight, temperature deviation, and cost index are the main influencing factors of the optimization parameters.The typical calculation conditions are as follows: the ambient temperature is ISA + 0°C; the climb weight is 200,000 kg; the fuel price is 6000 ￥/t; the cost index is 30; PICO2=0.094; PINOX=16.46, the climb height is 10,000–25,000 ft; Vc ranges from 200 to 350 kt; Tr ranges from 75% to 100%.Figure6 shows the effect of the initial climb weight on the optimization parameters, i.e., Vc and Tr. The climb weight ranges from 200,000 to 230,000 kg. Other calculations are the same as typical calculation conditions.Figure 6
Effect of flight weight on optimization parameters.Figure6 illustrates the optimum speed and the optimum thrust as a function of flight weight. It is clearly shown that an increase in aircraft weight increases the optimum speed and the optimum thrust for minimal integrated flight cost and flight cost. This is because when the speed and thrust are constant, the flight cost and emission cost increase with the weight. By increasing the speed and thrust, the flight cost can be reduced.Figure7 shows the effect of external temperature deviation on the optimization parameters. The temperature deviation ranges from −10°C to 30°C, and other calculation conditions are the same as typical calculation conditions.Figure 7
Effect of temperature on optimization parameters.As shown in Figure7, the optimum speed and thrust decrease with the increase of temperature only accounting for flight cost. While when considering the integrated flight cost, the optimum speed and thrust decrease proportionally at lower temperatures, with the temperature ranging from −10°C to 20°C, but begin to increase at higher temperatures. The emission of NOX increases with increasing temperature. Lower airspeed and thrust lead to lower emission cost and integrated flight cost. As the temperature reaches 30°C, under the same conditions, the fuel cost increases sharply. Therefore, high airspeed and thrust are very important to reduce flight cost.Figure8 shows the effect of cost index CI on optimization parameters. CI ranges from 0 to 100, and other calculation conditions are the same as typical calculation conditions.Figure 8
Effect of CI on optimization parameters.As shown in Figure8, for minimal integrated flight cost and flight cost, the optimal thrust and airspeed increase with increasing CI. Since the proportion of time cost to the integrated flight cost increases with increasing CI, increasing the airspeed and thrust to reduce the integrated flight cost is very important.


## 4.5. Impact Analysis of Pollutant Emission Price Index

With the rapid growth of civil aviation transportation, the impact of aircraft emissions on the environment has drawn increasing attention from all walks of life, and the emission price of pollutants has gradually increased. Therefore, it is imperative to analyze the effect of the pollutant emission price index on the optimization parameters. Since the emission index of CO2 is constant and the influencing factors are the same as fuel consumption, it is not analyzed in detail in this work.With other calculation conditions unchanged, the combination ofVc and Tr related to the minimum integrated flight cost under different NOX emission prices was calculated, and the results are shown in Figure 9.Figure 9
Effect ofPINOX on optimization parameters.It can be seen from Figure9 that, for minimal integrated flight cost, the optimal Vc and Tr decrease with increasing PINOX and finally decrease slowly. But the flight cost is not affected by PINOX. With the increase in PINOX, the NOX cost per kilogram increases, and hence, the emission cost also increases accordingly. By reducing the thrust and speed, the fuel flow and NOX emission can be reduced, which in turn reduces the emission cost.


## 5. Conclusions

In this study, we established an airspeed–thrust combination optimization model based on the integrated flight cost, according to the ICAO reference emission data and BM2 method. In addition, the concept of emission price index was proposed. The model was evaluated and simulated using a typical aircraft type. We conclude the following:(1)
The combination optimization model of climb parameters based on the integrated flight cost is more scientific and reasonable than the traditional economic climb model. The optimal combination (Vc, Tr) with the minimum integrated flight cost reduces the integrated flight cost by 1.28% and the emission cost by 4.56%.(2)
The climb parameters of aircraft have a significant impact on pollutant emissions. The optimal combinations of (Vc, Tr) are different for considering the integrated flight cost and only considering the flight cost.(3)
The optimal combination (Vc, Tr) with the minimum integrated flight cost is affected by some parameters including weight, temperature, CI, and PINOX. For minimal integrated flight cost, an increase in aircraft weight increases the optimum speed and the optimum thrust; the optimum speed and thrust decrease proportionally at lower temperatures, with the temperature ranging from −10°C to 20°C, but begin to increase at higher temperatures; the optimal thrust and airspeed increase with increasing CI but decrease with increasing PINOX.



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*Source: 3432579-2022-06-16.xml*