# Multi-objective trade-off optimal control of energy management for hybrid system

## Abstract

Currently, energy management control mainly focuses on single-objective optimization (SOO). Even if multi-objective optimization (MOO) problem is studied, it is often converted into an SOO problem by using the weighted sum method. Obviously, it cannot really reflect the essential strengths of MOO. In this paper, a parallel hybrid electric vehicle is taken as the research object. The fuel economy, emissions, and drivability performance are taken as optimization objectives. The parameters of energy management and driveline system are optimized. Considering the constraint conditions of the dynamic performance and charge balance, the fast non-dominated sorting differential evolution algorithm (NSDEA) is proposed to solve the multi-objective optimization problem. Then multi-group sets of Pareto solutions with good distribution and convergence are obtained. The simulation results of NSDEA show that the fuel economy is increased by 20.26% on average. The emissions evaluation index is optimized by 11.33% on average, and the maximum carbon monoxide (CO) optimization value reaches 21.9%. The average of drivability evaluation index (jerk) is up to 20.84%, and 40.32% for maximum. Obviously, the above obtained results are discrete points. They only represent some optimal solutions. Based on the above sets, the locally weighted scatter plot smoothing method is used to fit continuous curve and surfaces. Then, the multi-objective Pareto trade-off optimal control surface is established to further obtain the optimal solutions. This study can provide more reference for the optimal control strategy and lay a foundation for multi-objective energy management of the actual vehicle.

## Keywords

Hybrid system Energy management Multi-objective Trade-off Optimization## List of symbols

*A*_{i}Fitting coefficient of the external characteristic mathematical model

*A*_{k}Fitting coefficient matrix of the universal characteristic mathematical model

- cl
*_n* Engagement times of clutch

- CR
Crossover probability

- CR
_{max} Maximum value of the crossover probability

- CR
_{min} Minimum value of the crossover probability

- DN
Equation index of components’ action times

*d*_{max–min}Maximum value among the minimum distance between individual vectors

*d*_{min}^{i}Minimum distance between individual

*i*and*j**D*_{i}Crowding distance

*E*Rate of pollutant emissions

*E*_{CO}Emission rate of carbon monoxide (CO)

*E*_{HC}Emission rate of hydrocarbon (HC)

- \(E_{{{\text{NO}}_{x} }}\)
Emission rate of nitrogen oxides (NO

_{ x })*f**m*dimensional target vector*F*Variation constant

*f*_{2}Comprehensive evaluation index of emissions

- fc_
*n* Starting times of engine

*F*(*x*)Target vector

*F*_{min}Minimum value of zoom factor

*F*_{max}Maximum value of zoom factor

*f*(*U*_{i}^{t})Target value of test individuals

*f*(*X*_{i}^{t})Fitness value of target individuals

*f*_{m}^{i}(·)The

*m*th target function of individual*i**f*_{m}^{j}( · )The

*m*th target function of individual*j*- |
*f*_{m}^{i}(*x*) −*f*_{m}^{j}(*x*)| The distance between individual

*i*and*j**G*Current generation number

*G*_{max}Maximum generation number

- gb_
*j* Jerk generated by gearbox

- gb_
*n* Shifting times of transmission

*g*_{e}Specific fuel consumption

*g*_{j}*j*dimensional inequality constraint vectors*h*_{k}*k*dimensional equality constraint vectors*i*Order of engine speed variable for engine torque fitting

*j*Order of engine speed fitting

*k*Fitting order

*k*_{2}Fitting coefficient matrix of emission characteristic for pollutants

*k*(*t*)The gear ratio of transmission

*l*Order of engine speed variable for universal characteristic fitting

*m*The number of target vectors

*n*the nmber of decision variables which constitute

*x*decision space*n*_{e}Engine speed (r/min)

*n*_{m}Motor speed (r/min)

*n*_{w}Wheel speed (r/min)

*N*_{p}Population number

*N*_{obj}Target number

*N*Population size

*P*_{m}Motor power (kW)

*P*_{e}Engine power (kW)

*P*_{w}Vehicle power (kW)

*ρ*_{m}Motor speed ratio

*ρ*(*k*(*t*))Total drive ratio of the corresponding gears

*Q*_{CO}Emission amount of CO pollutant (g/L)

*Q*_{HC}Emission amount of HC pollutant (g/L)

- \(Q_{{{\text{NO}}_{x} }}\)
Emission amount of NO

_{ x }pollutant (g/L)*Q*_{fc}Fuel consumption of the engine (L/100 km)

- rand
_{ij} The random number of corresponding genes

*s*Model order

- sgn
Symbol function

- SOC
State of charge

- SOC
_{max} Maximum value of SOC

- SOC
_{min} Minimum value of SOC

*t*Driving time (s)

*T*Required torque (N m)

*T*_{chg}Preset charging torque (N m)

*T*_{chgs}Additional actual charging torque (N m)

*T*_{cl}Clutch torque (N m)

*T*_{e}Engine torque (N m)

*T*_{m}Motor torque (N m)

*T*_{w}Total of engine and motor torque (N m)

*U*_{i}^{t}Test individual

*u*_{ij}^{t}The

*j*th gene of individual*i*of the*t*th generation*v*_{ij}^{t}Mutated individual gene

*v*_{i}^{t+1}The

*i*th variant individual generated by the*t*th generation*w*_{1}Input power of clutch (kW)

*w*_{2}Output power of clutch (kW)

*W*Weighting coefficient

*x*Decision space

*x*_{min}Minimum value of the optimized vector

*x*_{max}Maximum value of the optimized vector

*x*_{r1}^{t}The target individual 1 of the

*t*th generation*x*_{r2}^{t}The target individual 2 of the

*t*th generation*x*_{r3}^{t}The target individual 3 of the

*t*th generation*x*_{ij}^{t}Target individual gene

*X*_{i}^{t}Target individual

*X*_{i}^{t+1}Selected individual

*y*Objective evaluation index

*η*_{c}Charger efficiency

*η*_{d}Final drive efficiency

*η*_{e}Engine efficiency

*η*_{m}Motor efficiency

*η*_{t}Transmission efficiency

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant no. 51305473); Project Funded by China Postdoctoral Science Foundation (Grant no. 2014M552317); Postdoctoral Science Funded Project of Chongqing (Grant no. xm2014032). Finally, the authors are grateful to the anonymous reviewers for their helpful comments and constructive suggestions.

### Compliance with ethical standards

### Conflict of interest

The authors declare no conflict of interests, including specific financial interests and relationships relevant to the subject of this paper.

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